tag:blogger.com,1999:blog-80800625246826846322024-03-13T01:13:47.824-07:00СонантометрияМузыка, как математическая дисциплина. Обсудите пожалуйста идеи тональных функций высшего порядка, что здесь развиваются.Mykhaylo Khramovhttp://www.blogger.com/profile/00483331671036933638noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-8080062524682684632.post-71915769581338447622007-06-04T03:46:00.000-07:002007-07-25T08:45:54.936-07:00<A name=0.1_table01></A> <CENTER><FONT face="Times New Roman" size=5><B>Сонантометрия.</B></FONT><FONT face=Verdana size=2> </FONT></CENTER> <CENTER><FONT face="Times New Roman" size=3><B>© 2007 Михаил Храмов.</B></FONT></CENTER> <CENTER><FONT face="Times New Roman" size=3><B>Статья пишется в ходе реализации </B></FONT><A href="http://geocities.com/commator/" target=_blank><FONT face="Times New Roman" color=#800080 size=3><B><U>проекта КОММАТОР.</U></B></FONT></A><FONT face="Times New Roman" size=3> </FONT></CENTER> <CENTER><FONT face="Times New Roman" size=3><B>Последние изменения внесены 25 июля 2007.</B> </FONT></CENTER> <CENTER><A href="http://sonantometry.blogspot.com/" target=_blank><FONT face="Times New Roman" color=#800080 size=3><B><U>Английский перевод</U></B></FONT></A><FONT face="Times New Roman" size=3><B> оригинального текста выполнил </B></FONT><A href="http://geocities.com/kmy180753/" target=_blank><FONT face="Times New Roman" color=#800080 size=3><B><U>Михаил Храмов.</U></B></FONT></A><FONT face="Times New Roman" size=3><B> </B></FONT></CENTER> <CENTER><FONT face="Times New Roman" size=3>В предлагаемых заметках развиваются </FONT><A href="http://www.sibeliusmusic.com/cgi-bin/show_score.pl?scoreid=53047" target=_blank><FONT face="Times New Roman" color=#000080 size=3><U>идеи, касающиеся тональных функций высших порядков</U></FONT></A><FONT face="Times New Roman" size=3>. Говорится </FONT><A href="http://www.sibeliusmusic.com/cgi-bin/show_score.pl?scoreid=96007" target=_blank><FONT face="Times New Roman" color=#000080 size=3><U>о практике применения этих тональных функций, как сонантов</U></FONT></A><FONT face="Times New Roman" size=3> к анализу и синтезу музыки. </FONT></CENTER> <P style="MARGIN: 1ex" align=left><FONT face="Times New Roman" size=3>Сонантометрия, или </FONT><A href="http://sonantometria.blogspot.com/2007_05_01_archive.html" target=_blank><FONT face="Times New Roman" color=#000080 size=3><U>алгебра тональных функций</U></FONT></A><FONT face="Times New Roman" size=3>, уточняет и расширяет возможности функционального анализа и синтеза музыки. Пример моего собственного опыта применения предмета статьи доступен в файле: <A title="Вагнерово начало Тристана. 53РДО аппроксимация предела 7." href="http://commons.wikimedia.org/wiki/Image:Wagner_Tristan_opening_by_Khramov53e7L.jpg" target=_blank>Вагнерово начало Тристана. 53РДО аппроксимация предела 7.</A> Также <A title="MIDI - модель звучания этого фрагмента" href="http://commons.wikimedia.org/wiki/Image:Wagner_Tristan_opening_by_Khramov53e7L.mid" target=_blank>MIDI - модель звучания этого фрагмента</A> даётся. </FONT></P> <P style="MARGIN: 1ex" align=left><FONT face="Times New Roman" size=3>Суть сонантометрии можно пояснить следующей таблицей</FONT><FONT face="Times New Roman" size=2>: </FONT></P> <CENTER style="MARGIN: 1ex"> <TABLE cellSpacing=0 width=0 border=2> <TBODY> <TR vAlign=top> <TD width="36%" colSpan=4 height=1> <P align=center><FONT face="Times New Roman" size=2>Диатоника чистой интонации предела 5</FONT> </P></TD> <TD width="24%" colSpan=2> <P align=center><FONT face="Times New Roman" size=2>Тональная функция</FONT> </P></TD> <TD width="38%" colSpan=4> <P align=center><FONT face="Times New Roman" size=2>Сонант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="11%" colSpan=2 height=1> <P align=center><FONT face="Times New Roman" size=2>Ступень</FONT> </P></TD> <TD width="25%" colSpan=2> <P align=center><FONT face="Times New Roman" size=2>Соотношение частот ступени и 1й ступени</FONT> </P></TD> <TD width="15%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>Имя</FONT> </P></TD> <TD width="9%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>Символ</FONT> </P></TD> <TD width="18%" colSpan=3> <P align=center><FONT face="Times New Roman" size=2>Формулы по порядку близости к Тонанту</FONT> </P></TD> <TD width="20%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>Имя</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>№</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>Символ</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>Дробь</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Факторизация</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>0</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="12%"> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>8</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VIII ~ I</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>2/1 ~ 1</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>2<SUP>1 </SUP>~ 1</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Тоника</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>T</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="12%"> <P align=center><FONT face="Times New Roman" size=2>:Dd</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>Тонант и Дедант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>7</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VII<</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>15/8~15</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>1</SUP>×3<SUP>1</SUP>×2<SUP>-3 </SUP>~ 5<SUP>1</SUP>×3<SUP>1</SUP></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Вводная</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="12%"> <P align=center><FONT face="Times New Roman" size=2>:MD или :DM</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>МеДент или ДеМент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>6</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VI<</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>5/3</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>1</SUP>×3<SUP>-1</SUP>×2<SUP>0 </SUP>~ 5<SUP>1</SUP>×3<SUP>-1</SUP></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Субмедианта</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="12%"> <P align=center><FONT face="Times New Roman" size=2>:Md или :dM</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>Медант или даМент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>5</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>V</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>3/2~3</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>3<SUP>1</SUP>×2<SUP>-1 </SUP>~ 3<SUP>1</SUP></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Доминанта</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>D</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:D</FONT> </P></TD> <TD width="12%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>Доминант или Дент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>4</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>IV</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>4/3~1/3</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>3<SUP>-1</SUP>×2<SUP>2 </SUP>~ 3<SUP>-1</SUP></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Субдоминанта</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>S</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:d</FONT> </P></TD> <TD width="12%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>Субдоминант или дант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>3</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>III<</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>5/4~5</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>1</SUP>×2<SUP>-2 </SUP>~ 5<SUP>1</SUP></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Медианта</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>M</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:M</FONT> </P></TD> <TD width="12%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>Медиант или Мент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>II</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>9/8~9</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>3<SUP>2</SUP>×2<SUP>-3 </SUP>~ 3<SUP>2</SUP></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Двойная дом-та</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>DD or 2D</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="12%"> <P align=center><FONT face="Times New Roman" size=2>:2D или :DD</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>ДваДент или ДеДент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>I ~ VIII</FONT> </P></TD> <TD width="10%"> <P align=center><FONT face="Times New Roman" size=2>1/1=1 ~ 2/1</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>2<SUP>0</SUP>=1 ~ 2<SUP>1</SUP></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Тоника</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>T</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="12%"> <P align=center><FONT face="Times New Roman" size=2>:dD</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>Тонант и даДент</FONT> </P></TD></TR></TBODY></TABLE></CENTER> <DIV style="MARGIN: 1ex" align=left> <P><FONT face="Times New Roman" size=3>Эта таблица показывает связи между хорошо известной мажорной диатоникой чистой интонации предела 5, так же хорошо известными тональными функциями и тем, что названо здесь сонанты и лежит в основе дисциплины, названной здесь сонантометрия. </FONT></P> <P><FONT face="Times New Roman" size=3>Сравнение ступеней I и VIII указывает на сопоставление им одной и той же функции T, или тоники. Другими словами эти ступени функционально эквивалентны, и можно записать VIII ~ I и I ~ VIII. Соотношения частот ступеней при этом показывают, что 2 ~ 1 и все степени простого числа 2 в факторизациях соотношений частот всех ступеней можно заменить единицей и не показывать. Понятно, что тонике соответствует число 1, именно этому числу соответствует первый, вводимый здесь сонант :T (произносится [Te]) наименованный Тонант. Формула сонанта :T имеет нулевой порядок близости к Тонанту, поскольку этот порядок равен сумме абсолютных значений показателей степеней простых нечётных оснований факторизации и для :T равен 0. </FONT></P> <P><FONT face="Times New Roman" size=3>Ступеням V и IV сопоставлены функции D и S, или доминанты и субдоминанты. Легко заметить, что имени ‘доминанта’ соответствует число 3<SUP>1</SUP>, а приставка ‘суб’ указывает лишь на смену знака показателя степени этого числа. В сонантометрии это закреплено тем, что любой сонант :S предполагает наличие своего субсонанта :s. Поэтому функции доминанта соответствует сонант :D (произносится [De]), с именем Доминант или Дент. Его субсонант :d (произносится [da]), с именем субдоминант или дант соответствует функции субдоминанта. Формула :S в сонантометрии обозначает любой возможный сонант со своим субсонантом :s, а функции субдоминанта соответствует субсонант :d сонанта :D. Из факторизации ясно, что Дент и дант имеют порядок близости к Тонанту равный 1. </FONT></P> <P><FONT face="Times New Roman" size=3>В мажорной диатонике чистой интонации предела 5 есть ещё один сонант порядка 1. Он соответствует числу 5<SUP>1</SUP> и функции медианта, что сопоставлена ступени III<. Знак «<» указывает на изгиб высоты ступени пиафагоровой шкалы вниз, и будет описан далее, вместе со знаком изгиба вверх «>». Сонант :M (произносится [Me]), с именем Медиант или Мент, и его субсонант :m (произносится [ma]), с именем субмедиант или мант суть последние из возможных сонантов порядка 1 в чистой интонации предела 5. В рассматриваемой диатонике мант отсутствует и функция субмедианты сопоставлена другому сонанту. Однако гармоническая версия мажорной диатоники чистой интонации предела 5 замещает ступень VI< ступенью VIb> (8/5 ~ 5<SUP>-1</SUP>)и соответствие субмедианты манту восстанавливается. </FONT></P> <P><FONT face="Times New Roman" size=3>Существование функции с именем двойная доминанта помогает понять суть сложных сонантов. Это все сонанты порядка 2 и выше. Подобно двойной доминанте, или доминанте от доминанты, любой сонант порядка 2 существует как простой сонант порядка 1 от принятого вместо Тонанта другого сонанта, простого к Тонанту по близости. Следует заметить, что субсонант своего сонанта равен Тонанту, и смена мест простых сонантов в формуле сложного не приводит к другому сонанту. </FONT></P> <P><FONT face="Times New Roman" size=3>В данной диатонике существуют сложные сонанты :2D или :DD, :MD или :DM, :Dd или :dD, :Md или :dM. Их имена и соответствие ступеням отражены в таблице. Понятно, что диатоника охватывает не все возможные сонанты чистой интонации предела 5 и порядка 2. Полный набор этих сонантов отражается в таблице ниже. </FONT></P></DIV> <CENTER style="MARGIN: 1ex"> <TABLE cellSpacing=0 width=0 border=2> <TBODY> <TR vAlign=top> <TD width="49%" colSpan=4 height=1> <P align=center><FONT face="Times New Roman" size=2>Шкала чистой интонации предела 5 и порядка 2</FONT> </P></TD> <TD width="50%" colSpan=4> <P align=center><FONT face="Times New Roman" size=2>Сонант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="13%" colSpan=2 height=1> <P align=center><FONT face="Times New Roman" size=2>Ступень</FONT> </P></TD> <TD width="36%" colSpan=2> <P align=center><FONT face="Times New Roman" size=2>Соотношение частот ступени и 1й ступени</FONT> </P></TD> <TD width="21%" colSpan=3> <P align=center><FONT face="Times New Roman" size=2>Формулы по порядку близости к Тонанту</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Имя</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>№</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>Символ</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>Дробь</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>Факторизация</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>0</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>14</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>VIII ~ I</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>2/1 ~ 1</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>2<SUP>1</SUP>~1</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:Dd и :Mm</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Тонант и Дедант и Мемант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>13</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>VII<</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>15/8 ~ 15</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>1</SUP>×3<SUP>1</SUP>×2<SUP>-3 </SUP>~ 5<SUP>1</SUP>×3<SUP>1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:MD или :DM</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>МеДент или ДеМент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>12</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>VIIb</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>16/9 ~ 1/9</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>3<SUP>-2</SUP>×2<SUP>4 </SUP>~ 3<SUP>-2</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:2d или :dd</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Двадант или дадант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>11</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>VI<</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>5/3</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>1</SUP>×3<SUP>-1</SUP>×2<SUP>0 </SUP>~ 5<SUP>1</SUP>×3<SUP>-1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:Md или :dM</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Медант или даМент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>10</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>VIb></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>8/5 ~ 1/5</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>-1</SUP>×2<SUP>3 </SUP>~ 5<SUP>-1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:m</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Субмедиант или мант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>9</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>V#(</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>25/16 ~ 25</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>2</SUP>×2<SUP>-4 </SUP>~ 5<SUP>2</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:2M или :MM</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>ДваМент или МеМент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>8</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>V</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>3/2 ~ 3</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>3<SUP>1</SUP>×2<SUP>-1 </SUP>~ 3<SUP>1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:D</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Доминант или Дент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>7</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>IV</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>4/3 ~ 1/3</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>3<SUP>-1</SUP>×2<SUP>2 </SUP>~ 3<SUP>-1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:d</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Субдоминант или дант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>6</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>IVb)</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>32/25 ~ 1/25</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>-2</SUP>×2<SUP>5 </SUP>~ 5<SUP>-2</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:2m или :mm</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Двамант или мамант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>5</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>III<</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>5/4 ~ 5</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>1</SUP>×2<SUP>-2 </SUP>~ 5<SUP>1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:M</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Медиант или Мент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>4</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>IIIb></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>6/5</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>-1</SUP>×3<SUP>1</SUP>×2<SUP>1 </SUP>~ 5<SUP>-1</SUP>×3<SUP>1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:mD или :Dm</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>маДент или Демант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>3</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>II</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>9/8 ~ 9</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>3<SUP>2</SUP>×2<SUP>-3 </SUP>~ 3<SUP>2</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:2D или :DD</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>ДваДент или ДеДент</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>IIb></FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>16/15 ~ 1/15</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>5<SUP>-1</SUP>×3<SUP>-1</SUP>×2<SUP>4 </SUP>~ 5<SUP>-1</SUP>×3<SUP>-1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:md или :dm</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>мадант или дамант</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="9%"> <P align=center><FONT face="Times New Roman" size=2>I ~ VIII</FONT> </P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>1/1=1 ~ 2/1</FONT> </P></TD> <TD width="20%"> <P align=center><FONT face="Times New Roman" size=2>2<SUP>0</SUP>=1 ~ 2<SUP>1</SUP></FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="15%"> <P align=center><FONT face="Times New Roman" size=2>:dD и :mM</FONT> </P></TD> <TD width="29%"> <P align=center><FONT face="Times New Roman" size=2>Тонант и даДент и маМент</FONT> </P></TD></TR></TBODY></TABLE></CENTER><FONT face=Verdana size=2> <P style="MARGIN: 1ex"></FONT></P><FONT face="Times New Roman" size=3><A name=0.1_table03></A> <DIV style="MARGIN: 1ex" align=left> <P><FONT face="Times New Roman" size=3>Приступая к выяснению сути знаков «>», «<», «)», «(», вспомним, что шкала чистой интонации предела 5, может быть выражена через шкалу чистой интонации предела 3. Последняя шкала носит имя Пифагора, и по сей день лежит в основе музыкальной нотации и понятия ‘квинтовая спираль’, что через настройку 12 равномерных делений октавы (12 РДО) вырождается в ‘квинтовый круг’. В пифагоровой шкале, как шкале чистой интонации предела 3, мы найдём всеобщий сонант :T, простые к нему сонанты :D и :d, а также все сложные, что можно получить из :T, :D и :d. Другие сонанты в этой шкале невозможны. </FONT></P> <P><FONT face="Times New Roman" size=3>В следующей таблице шкала чистой интонации предела 5 (сонанты :T, :D, :d, :M, :m и их комбинации) выражена через интервальные сонанты (интерсонанты) к ступеням пифагоровой: </FONT></P></DIV> <CENTER style="MARGIN: 1ex"> <TABLE cellSpacing=0 width=0 border=2> <TBODY> <TR vAlign=top> <TD width="39%" colSpan=10 height=1> <P align=center><FONT face="Times New Roman" size=2>Пифагорова шкала чистой интонации предела 3 и порядка 8</FONT> </P></TD> <TD width="60%" colSpan=7> <P align=center><FONT face="Times New Roman" size=2>Шкала чистой интонации предела 5 и порядка 2.</FONT> </P></TD></TR> <TR vAlign=top> <TD width="10%" colSpan=2 height=1> <P align=center><FONT face="Times New Roman" size=2>Ступень</FONT> </P></TD> <TD width="28%" colSpan=8> <P align=center><FONT face="Times New Roman" size=2>Сонанты</FONT> </P> <P align=center><FONT face="Times New Roman" size=2>по порядку близости к Тонанту</FONT> </P></TD> <TD width="12%" colSpan=2> <P align=center><FONT face="Times New Roman" size=2>Ступень</FONT> </P></TD> <TD width="14%" colSpan=3> <P align=center><FONT face="Times New Roman" size=2>Сонанты по порядку близости к Тонанту</FONT> </P></TD> <TD width="33%" colSpan=2> <P align=center><FONT face="Times New Roman" size=2>Изгиб высоты в интерсонантах по порядку близости к ступеням пифагоровой шкалы.</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>№</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>Символ</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>0</FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>3</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>4</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>5</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>…</FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>8</FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>№</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>Символ</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>0</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman" size=2>5</FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Times New Roman" size=2>10</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>14</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VIII ~ I</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>14</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>VIII ~ I</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>13</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VII</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:5D</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>:δ[5D = :M4d[5D</FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>13</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>VII<</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:MD</FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>12</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VIIb</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:2d</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>12</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>VIIb</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:2d</FONT> </P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>11</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VI</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:3D</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>:δ[3D = :M4d[3D</FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>11</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>VI<</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:Md</FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>10</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>V#</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>:8D</FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Times New Roman" size=2></FONT></P></TD> <TD width="17%" rowSpan=4> <P align=center><FONT face="Times New Roman" size=2>:2δ[8d = :2M8d[8D</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>10</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>VIb></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:m</FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>:Δ[4D = :m4D[4d</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>9</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>VIb</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:4d</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>9</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>V#(</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:2M</FONT> </P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>8</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>V</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>:D</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>8</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>V</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:D</FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>7</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>IV</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>:d</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>7</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>IV</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:d</FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>6</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>IVb)</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:2m</FONT> </P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%" rowSpan=4> <P align=center><FONT face="Times New Roman" size=2>:2Δ[8d = :2m8D[8d</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>6</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>III</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:4D</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>:δ[4d = :M4d[4D</FONT> </P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>5</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>III<</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:M</FONT> </P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>5</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>IVb</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>:8d</FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>4</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>IIIb></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:mD</FONT> </P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>:Δ[3d = :m4D[3d</FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>4</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>IIIb</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:3d</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>3</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>II</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:2D</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>3</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>II</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:2D</FONT> </P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="7%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>IIb></FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Times New Roman" size=2>:md</FONT> </P></TD> <TD width="16%" rowSpan=2> <P align=center><FONT face="Times New Roman" size=2>:Δ[5d = :m4D[5d</FONT> </P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>2</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>IIb</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:5d</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="8%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD></TR> <TR vAlign=top> <TD width="3%" height=1> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="7%"> <P align=center><FONT face="Times New Roman" size=2>I ~ VIII</FONT> </P></TD> <TD width="2%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="2%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="3%"> <P align=center><FONT face="Times New Roman" size=2>1</FONT> </P></TD> <TD width="8%"> <P align=center><FONT face="Times New Roman" size=2>I ~ VIII</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Times New Roman" size=2>:T</FONT> </P></TD> <TD width="4%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="6%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="16%"> <P align=center><FONT face="Arial Unicode MS" size=3></FONT></P></TD> <TD width="17%"><FONT face="Arial Unicode MS" size=3> </FONT> </TD></TR></TBODY></TABLE></CENTER></FONT><FONT face=Verdana size=3> <P><FONT face="Times New Roman" size=3>Как можно заметить, шкалы в точности совпадают по ступеням I, II, IV, V, VIIb. По другим ступеням находим расхождение пятого порядка на маЧетыреДент или обратный ему МеЧетыредант. Ступени IVb и V# расходятся на ДвамаВосемьДент и обратный ему ДваМеВосемьдант, оба десятого порядка. </FONT></P> <P><FONT face="Times New Roman" size=3>Здесь даётся простое правило нахождения интерсонанта между любой парой сонантов. Достаточно написать первый сонант, дописать к нему элементы второго, элементы обратного второму, и вынести за скобку элементы второго, как точки отсчёта. Перед скобкой останется искомый интерсонант. </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:md = :md 5d 5D = :md5D[5d = :m4D[5d; </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:mD = :mD 3d 3D = :mD3D[3d = :m4D[3d; </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:M = :M 4D 4d = :M4d[4D; </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:2m = :2m 8d 8D = 2m8D[8d; </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:2M = :2M 8D 8d = 2M8d[8D; </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:m = :m 4d 4D = :m4D[4d; </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:Md = :Md 3D 3d = :Md3d[3D = :M4d[3D; </FONT></P> <P align=center><FONT face="Times New Roman" size=3>:MD = :MD 5D 5d = :MD5d[5D = :M4d[5D. </FONT></P> <P><FONT face="Times New Roman" size=3>Восстановление числовых значений интерсонанта :m4D по формуле 5<SUP>-1</SUP>×3<SUP>4</SUP>×2<SUP><B>Z</B></SUP>, <B>Z</B> ={-n, …, 0, … n}, n </FONT><FONT face=Symbol size=3><SPAN lang=EN-US style="FONT-SIZE: 10pt; COLOR: black; FONT-FAMILY: Symbol">Î </SPAN></FONT><FONT face="Times New Roman" size=3><B>N</B>,<B> N </B>= {1, 2, 3, …, n} даёт множество октавных эквивалентов дроби 81/80 и изгибает высоту пифагоровой ступени (далее ступени) на дидимову комму вверх. Обратный сонант :M4d (множество октавных эквивалентов дроби 80/81) изгибает высоту ступени на дидимову комму вниз. Поэтому :m4D можно назвать Дидимент, и обозначить :Δ. При этом :M4d назовём дидимант, и обозначим :δ. </FONT></P> <P><FONT face="Times New Roman" size=3>Таблица показывает, что «>» изгибает высоту ступени вверх на Дидимент, а «<» вниз на дидимант. Знак «)» предписывает ДваДидимент вверх, и «(» означает Двадидимант вниз.</FONT><FONT face=Verdana size=3> </FONT></P> <P><FONT face="Times New Roman" size=3>Заметим, если в сонанте ноты присутствует:</FONT> </P> <P align=center><FONT face="Times New Roman" size=3>M, возникает "<" и ступень понижается интерсонантом :δ;</FONT> </P> <P align=center><FONT face="Times New Roman" size=3>2M возникает "(" и ступень понижается интерсонантом :2δ;</FONT> </P> <P align=center><FONT face="Times New Roman" size=3>m, возникает ">" и ступень повышается интерсонантом :Δ;</FONT> </P> <P align=center><FONT face="Times New Roman" size=3>2 m, возникает ")" и ступень повышается интерсонантом :2Δ.</FONT> </P> <P><FONT face="Times New Roman" size=3>Другими словами, в чистой интонации предела 5, изгиб высоты ступени возникает только у нот, содержащих M, или m. Величина изгиба составляет, соответственно, ровно столько интерсонантов :δ или :Δ, сколько M, или m в составе ноты.</FONT> </P> <P><FONT face="Times New Roman" size=3>Это наблюдение даёт следующее, состоящее из двух фаз, правило расстановки дополнительных знаков альтерации для записи нот чистой интонации предела 5 в партитурах пифагорова строя:</FONT> </P> <OL type=1> <LI><FONT face="Times New Roman" size=3>опираясь на контекст партитуры, присвоить каждой нотной головке, соответствующий ей сонант;</FONT> <LI><FONT face="Times New Roman" size=3>если в сонанте присутствуют M, 2M, 3M, соответственно изогнуть высоту ноты на :δ, :2δ, :3δ, и обозначить эти понижения соответственно дополнительными знаками альтерации "<", "(", "{". Присутствие m, 2m, 3m требует соответствующих изгибов на :Δ, :2Δ, :3Δ, и обозначений повышения соответствующими дополнительными знаками альтерации ">", ")" и "}".</FONT> </LI></OL></FONT>Mykhaylo Khramovhttp://www.blogger.com/profile/00483331671036933638noreply@blogger.com15tag:blogger.com,1999:blog-8080062524682684632.post-61614777217425903432007-05-30T06:59:00.000-07:002007-06-05T05:18:26.188-07:00<P class=western> </P> <P class=western align=center><B><FONT size=5>Алгебра<SPAN lang=ru-RU> </SPAN>Тональных<SPAN lang=ru-RU> </SPAN>Функций<SPAN lang=ru-RU>.</SPAN></FONT></B> </P> <P class=western align=center><BR></P> <P class=western align=center><B><SPAN lang=ru-RU>© 2005 </SPAN>Геннадий<SPAN lang=ru-RU> </SPAN>Воль</B> </P> <P class=western align=center> </P> <P class=western align=center> <B>Статья<SPAN lang=ru-RU> </SPAN>написана<SPAN lang=ru-RU> </SPAN>для<SPAN lang=ru-RU> </SPAN></B><FONT color=#000080><U><A href="http://geocities.com/commator/"><B>проекта<SPAN lang=ru-RU> </SPAN>КОММАТОР<SPAN lang=ru-RU>.</SPAN></B></A></U></FONT> </P> <P class=western align=center> </P> <P class=western align=center> <FONT color=#000080><U><A title="Английский перевод" href="http://sonantometry.blogspot.com/2007_05_01_archive.html"><B>Английский<SPAN lang=ru-RU> </SPAN>перевод</B></A></U></FONT><B><SPAN lang=ru-RU> </SPAN>оригинального<SPAN lang=ru-RU> </SPAN>текста<SPAN lang=ru-RU> </SPAN>выполнил<SPAN lang=ru-RU> </SPAN></B><FONT color=#000080><U><A href="http://geocities.com/kmy180753/"><B>Михаил<SPAN lang=ru-RU> </SPAN>Храмов<SPAN lang=ru-RU>.</SPAN></B></A></U></FONT> </P> <P class=western align=center> </P> <P class=western align=center><B>Переводчик<SPAN lang=ru-RU> </SPAN>благодарит<SPAN lang=ru-RU> </SPAN>Геннадия<SPAN lang=ru-RU> </SPAN>Воля<SPAN lang=ru-RU>, </SPAN>который<SPAN lang=ru-RU> </SPAN>любезно<SPAN lang=ru-RU> </SPAN>согласился<SPAN lang=ru-RU> </SPAN>изложить<SPAN lang=ru-RU> </SPAN>математические<SPAN lang=ru-RU> </SPAN>аспекты<SPAN lang=ru-RU> </SPAN>предложенной<SPAN lang=ru-RU> </SPAN>ему<SPAN lang=ru-RU> </SPAN>темы<SPAN lang=ru-RU>.</SPAN></B> </P> <P lang=ru-RU style="BORDER-RIGHT: medium none; PADDING-RIGHT: 0in; BORDER-TOP: medium none; PADDING-LEFT: 0in; PADDING-BOTTOM: 0in; BORDER-LEFT: medium none; PADDING-TOP: 0in; BORDER-BOTTOM: medium none" align=center><B>Переводчик<SPAN lang=ru-RU> </SPAN>также<SPAN lang=ru-RU> </SPAN>благодарит<SPAN lang=ru-RU> </SPAN>Вадима<SPAN lang=ru-RU> </SPAN>Бондаря<SPAN lang=ru-RU> </SPAN>за<SPAN lang=ru-RU> </SPAN>помощь<SPAN lang=ru-RU> </SPAN>и<SPAN lang=ru-RU> </SPAN>будет<SPAN lang=ru-RU> </SPAN>рад<SPAN lang=ru-RU> </SPAN>любой<SPAN lang=ru-RU> </SPAN>критике<SPAN lang=ru-RU>, </SPAN>улучшающей<SPAN lang=ru-RU> </SPAN>качество<SPAN lang=ru-RU> </SPAN>перевода<SPAN lang=ru-RU>.</SPAN></B> </P> <P class=western align=center><BR>В<SPAN lang=ru-RU> </SPAN>предлагаемых<SPAN lang=ru-RU> </SPAN>заметках<SPAN lang=ru-RU> </SPAN>кратко<SPAN lang=ru-RU> </SPAN>излагаются<SPAN lang=ru-RU> </SPAN><FONT color=#000080><U><A title="идеи М. Ю. Храмова, касающиеся тональных функций высших порядков" href="http://sonantometria.blogspot.com/2007/06/blog-post.html" target=_blank>идеи<SPAN lang=ru-RU> </SPAN>М<SPAN lang=ru-RU>. </SPAN>Ю<SPAN lang=ru-RU>. </SPAN>Храмова<SPAN lang=ru-RU>, </SPAN>касающиеся<SPAN lang=ru-RU> </SPAN>тональных<SPAN lang=ru-RU> </SPAN>функций<SPAN lang=ru-RU> </SPAN>высших<SPAN lang=ru-RU> </SPAN>порядков</A></U></FONT><SPAN lang=ru-RU>. </SPAN>Приводятся<SPAN lang=ru-RU> </SPAN>элементарные<SPAN lang=ru-RU> </SPAN>математические<SPAN lang=ru-RU> </SPAN>формулы<SPAN lang=ru-RU>, </SPAN>относящиеся<SPAN lang=ru-RU> </SPAN>к<SPAN lang=ru-RU> </SPAN>формальной<SPAN lang=ru-RU> </SPAN>части<SPAN lang=ru-RU> </SPAN>проблемы<SPAN lang=ru-RU> </SPAN>соответствия<SPAN lang=ru-RU> </SPAN>тон<SPAN lang=ru-RU> - </SPAN>функция<SPAN lang=ru-RU>, </SPAN>а<SPAN lang=ru-RU> </SPAN>также<SPAN lang=ru-RU> </SPAN>формулы<SPAN lang=ru-RU> </SPAN>для<SPAN lang=ru-RU> </SPAN>подсчета<SPAN lang=ru-RU> </SPAN>числа<SPAN lang=ru-RU> </SPAN>тональных<SPAN lang=ru-RU> </SPAN>функций<SPAN lang=ru-RU>.</SPAN> </P> <P class=western lang=ru-RU> </P> <P class=western align=left><SPAN style="FONT-SIZE: 12pt"><STRONG>1.</STRONG><SPAN> </SPAN></SPAN><B><SPAN style="FONT-SIZE: 12pt">Неформальное описание тональных функций.</SPAN></B></P> <P class=western align=left><B><SPAN style="FONT-SIZE: 12pt"></SPAN></B> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">В некоторой тональности пусть </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt"> означает тонику, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SPAN style="FONT-SIZE: 12pt"> - доминанту, <I>М</I> - медианту, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><SPAN style="FONT-SIZE: 12pt"> - субдоминанту, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN lang=UK style="FONT-SIZE: 12pt"> - </SPAN><SPAN style="FONT-SIZE: 12pt">субмедианту. (В обычном звукоряде функциям </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN style="FONT-SIZE: 12pt"> соответствуют </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">I</SPAN><SPAN style="FONT-SIZE: 12pt">, </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">III</SPAN><SPAN style="FONT-SIZE: 12pt">, </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">IV</SPAN><SPAN style="FONT-SIZE: 12pt">, </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">V</SPAN><SPAN style="FONT-SIZE: 12pt">, </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">VI</SPAN><SPAN style="FONT-SIZE: 12pt"> ступени). Если частоту </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt"> условно принять за 1, то частоты </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN style="FONT-SIZE: 12pt"> равны соответственно 3/2, 4/3, 5/4, 8/5 (в пределах одной октавы). При перенесении тона на октаву вверх или вниз его функция не меняется; следовательно, частоты, соответствующие </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN style="FONT-SIZE: 12pt"> могут быть умножены или разделены на 2. Пренебрегая множителем, равным 2, получим, что </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN style="FONT-SIZE: 12pt"> соответствуют числа 1, 3, 1/3, 5, 1/5. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Пусть </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt"> - некоторая функция; предположим, что тон, которому соответствует </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">, принят за тонику; в новой тональности функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><I><SPAN style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">соответствует некоторый тон, которому по отношению к исходной тональности приписываем функцию </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">. Очевидно, что </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">. Частота, соответствующая функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">, получается перемножением частот функций </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1<I> </I></SPAN></SUB><SPAN style="FONT-SIZE: 12pt">и<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><I><SPAN style="FONT-SIZE: 12pt">.</SPAN></I><SPAN style="FONT-SIZE: 12pt"> Например, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">DD</SPAN></I><SPAN style="FONT-SIZE: 12pt"> соответствует частота 3·3=9; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">MMD</SPAN></I><SPAN style="FONT-SIZE: 12pt"> - частота 5·5·3=75 и т. д. Вместо </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">DD</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">MMD</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">будем писать 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SPAN style="FONT-SIZE: 12pt">,<I> </I>2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">MD</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">и т. д. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Заметим, что </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">Dd</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">=</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">Mm</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">=</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt">; этому соответствуют числовые равенства 3·(1/3)=1 и 5·(1/5)=1. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"><B><SPAN style="FONT-SIZE: 12pt">2.<SPAN> </SPAN></SPAN></B><B><SPAN style="FONT-SIZE: 12pt">Формальное описание тональных функций. </SPAN></B> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">Рассмотрим свободную абелеву группу </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>F</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt"> с двумя образующими </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SPAN style="FONT-SIZE: 12pt"> и </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SPAN style="FONT-SIZE: 12pt">; единицу группы обозначим </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt">. Пусть </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">=</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">=</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">. Элемент </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>Î</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>F</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt"> назовем тональной функцией; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt"> однозначно записывается в виде </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">, где </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt"> и </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt"> целые числа. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">В мультипликативной группе </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">Q</SPAN></I><I><SUP><SPAN style="FONT-SIZE: 12pt">+</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt"> положительных рациональных чисел рассмотрим подгруппу </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">G</SPAN></I><SPAN style="FONT-SIZE: 12pt"> порожденную числами 3 и 5. Определим изоморфизм </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>F</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>@</SPAN></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">G</SPAN></I><SPAN style="FONT-SIZE: 12pt">, задавая его на образующих: </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>j</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=3; </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>j</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=5; т. о., </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>j</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">)=3</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">5</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">. Для каждой функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt"> определим множество частот </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">)={2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></SUP></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>j</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">)</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">}, где </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">пробегает<I> </I>целые числа. (Частота </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">, т. о. соответствует<SPAN> </SPAN></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt">). Кроме того, определим для каждой функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt"> порядок </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=|</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">|+|</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">|. Существует единственная функция порядка </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">=0, а именно </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt">; при </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">>0 существует 4</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt"> функций порядка </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">Примеры: </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">=1; функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN style="FONT-SIZE: 12pt">. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">=2; функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">DM</SPAN></I><SPAN style="FONT-SIZE: 12pt">,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">DM</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SPAN style="FONT-SIZE: 12pt">; в обозначениях п.1: 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SPAN style="FONT-SIZE: 12pt">, 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">MD</SPAN></I><SPAN style="FONT-SIZE: 12pt">, 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><SPAN style="FONT-SIZE: 12pt">, 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">md</SPAN></I><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">mD</SPAN></I><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">Md</SPAN></I><SPAN style="FONT-SIZE: 12pt">. </SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt"></SPAN><SPAN style="FONT-SIZE: 12pt">При </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">=3, 4, 5 имеется соответственно 12, 16 и 20 функций. </SPAN> </P> <P class=MsoNormal> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"><B><SPAN style="FONT-SIZE: 12pt">3.<SPAN> </SPAN></SPAN></B><B><SPAN style="FONT-SIZE: 12pt">Равномерная 53-ступенная темперация. </SPAN></B> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><B><SPAN style="FONT-SIZE: 12pt"></SPAN></B></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Пусть </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt"> - частота тоники; разделим октаву (т. е. звуковой диапазон между </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt"> и 2</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">) на 53 равные части. Пронумеруем ступени данного строя от 0 (соответствует </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">) до 53 (соответствует 2</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">). Частота ступени с номером </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt"> равна </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">·2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></SUP></I><SUP><SPAN style="FONT-SIZE: 12pt">/53</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Выбор числа 53 связан с тем, что (3/2)<SUP>53</SUP>/2<SUP>31</SUP>=1,002…, т. е. 53 чистые квинты дают "почти замкнутый" строй (заметим, что для 12-ступенной темперации (3/2)<SUP>12</SUP>/2<SUP>7</SUP>=1,01…). Кроме того, 53 ступени дают возможность воспроизводить довольно точно многие другие применяемые строи. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Заметим, что числа 12 и 53 имеют более глубокий математический смысл: это знаменатели подходящих дробей при разложении в цепную дробь числа </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3; действительно, </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3=[1;1,1,2,2,3,1,5, …]; легко проверить, что 5-ой и 7-ой подходящей дробью соответственно являются 19/12 и 84/53. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"> </P> <P class=MsoNormal><SPAN lang=EN-US style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"><B><SPAN style="FONT-SIZE: 12pt">4.<SPAN> </SPAN></SPAN></B><B><SPAN style="FONT-SIZE: 12pt">Сопоставление ступеням 53-ступенной темперации тональных функций (мотивировки). </SPAN></B> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"> </P> <P class=MsoNormal><B><SPAN style="FONT-SIZE: 12pt"><SPAN></SPAN></SPAN></B></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">При интерпретации музыкальных произведений<SPAN> </SPAN>обычной нотации в 53-ступенный темперированный строй мы сталкиваемся с неоднозначностью выбора каких-либо из 53 ступеней, соответствующих данной ноте. Однако эта неоднозначность исчезает, если тону, обозначенному нотой, приписать некоторую функцию. Например, если тону соответствует медианта </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SPAN style="FONT-SIZE: 12pt">, то ее частота 5/4=1.25 ближе всего к частоте 17-ой ступени 2<SUP>17./53</SUP>=1.24898…, и мы сопоставляем именно 17-ю ступень 53-ступенного строя данному тону. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Неформальная часть задачи (и, видимо очень сложная) заключается в разработке правил, по которым из конкретного музыкального контекста мы приписываем ту или иную функцию рассматриваемому тону. Так, если тону соответствует функция 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">MD</SPAN></I><SPAN style="FONT-SIZE: 12pt"> с частотой 5<SUP>2</SUP>·3=75 или, в пределах одной октавы, 75/64=1.171875, то это ближе всего к частоте 12-го тона 53-ступенной темперации. 2<SUP>12/53</SUP>=1.169924…; однако трудность именно в разработке <U>музыкальной</U> мотивировки, которая позволит приписать данному тону функцию 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">MD</SPAN></I><SPAN style="FONT-SIZE: 12pt">.<B> </B></SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><B><SPAN style="FONT-SIZE: 12pt"><SPAN></SPAN></SPAN></B></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><B><SPAN style="FONT-SIZE: 12pt"></SPAN></B></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"><B><SPAN style="FONT-SIZE: 12pt">5.<SPAN> </SPAN></SPAN></B><B><SPAN style="FONT-SIZE: 12pt">Сопоставление ступеням 53-ступенной темперации тональных функций (формальная процедура). </SPAN></B> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><B><SPAN style="FONT-SIZE: 12pt"></SPAN></B></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Пусть </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>¹</SPAN></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt">. Функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt"> соответствует набор частот </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">)={2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt">3</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">5</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">}, где </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">пробегает<I> </I>все целые числа. (</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt"> полагаем условно равной 1). Неравенство 1<2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt">3</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">5</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt"><2 (т. е. выбор частоты в пределах октавы) однозначно определяет </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></I><SPAN style="FONT-SIZE: 12pt">. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Номер </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">ступени 53-ступенной темперации, соответствующей функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">, определяем из условия наибольшей близости числа 2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></SUP></I><SUP><SPAN style="FONT-SIZE: 12pt">/53</SPAN></SUP><SPAN style="FONT-SIZE: 12pt"> к 2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt">3</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">5</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">. Поскольку </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt">3</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">5</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">)=</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3+</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">5+</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>Î</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(0,1), то </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt">3</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">5</SPAN><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">)={</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3+</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">5}, где {</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">x</SPAN></I><SPAN style="FONT-SIZE: 12pt">} обозначает дробную часть числа </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">x</SPAN></I><SPAN style="FONT-SIZE: 12pt">. </SPAN><SPAN lang=UK style="FONT-SIZE: 12pt">Т. о.</SPAN><SPAN style="FONT-SIZE: 12pt">, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt">/53 должно быть близко к {</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3+</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">5} или, что то же самое, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt"> должно быть близко к числу 53{</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3+</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">5}. Другими словами, номер ступени </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt"><SPAN> </SPAN>53-ступенной темперации, соответствующей функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt">, есть ближайшее целое число к числу 53{</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3+</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">5}. Обозначая ближайшее целое к </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">x</SPAN></I><SPAN style="FONT-SIZE: 12pt"> символом [</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">x</SPAN></I><SPAN style="FONT-SIZE: 12pt">]<SUB>1</SUB>, получим: </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt">=[53{</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">3+</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt">log</SPAN><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">5}]<SUB>1</SUB> </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Пример: </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">=2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">MD</SPAN></I><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">; </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">=1; </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">=2; </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN lang=EN-US style="FONT-SIZE: 12pt">log<SUB>2</SUB>3+2log<SUB>2</SUB>5=2.2288186…; </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN lang=EN-US style="FONT-SIZE: 12pt">53·{log<SUB>2</SUB>3+2log<SUB>2</SUB>5}=53·0.2288186=12.12739…; </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt">=12. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">Приведем таблицу функций порядка </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>£</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">5 </SPAN><SPAN lang=UK style="FONT-SIZE: 12pt">и </SPAN><SPAN style="FONT-SIZE: 12pt">соответствующих им ступеней</SPAN><SPAN lang=UK style="FONT-SIZE: 12pt">.</SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"> </P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=541 border=1> <COLGROUP> <COL width=29> <COL width=30> <COL width=30> <COL width=30> <COL width=52></COLGROUP> <COLGROUP> <COL width=30> <COL width=30> <COL width=30> <COL width=30> <COL width=52></COLGROUP> <COLGROUP> <COL width=29> <COL width=29> <COL width=29> <COL width=29> <COL width=52></COLGROUP> <TBODY> <TR> <TD width=29 height=15> <P align=center><EM>h</EM> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a</FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">b</FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=52> <P class=western align=center><I>f</I> </P></TD> <TD width=30> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a</FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">b</FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=52> <P class=western align=center><I>f</I> </P></TD> <TD width=29> <P class=western align=center><I>h</I> </P></TD> <TD width=29> <P class=western align=center><FONT face="Symbol, serif">a</FONT> </P></TD> <TD width=29> <P class=western align=center><FONT face="Symbol, serif">b</FONT> </P></TD> <TD width=29> <P class=western align=center><I>k</I> </P></TD> <TD width=52> <P class=western align=center><I>f</I> </P></TD></TR></TBODY> <TBODY> <TR> <TD width=29 height=16> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=52> <P class=western align=center><I>T</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>12 </P></TD> <TD width=52> <P class=western align=center>2<I>MD</I> </P></TD> <TD width=29> <P class=western align=center>5 </P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>18 </P></TD> <TD width=52> <P class=western align=center><I>m</I>4<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>22 </P></TD> <TD width=52> <P class=western align=center><I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>45 </P></TD> <TD width=52> <P class=western align=center><I>m</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>52 </P></TD> <TD width=52> <P class=western align=center><I>M</I>4<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>36 </P></TD> <TD width=52> <P class=western align=center><I>m</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>26 </P></TD> <TD width=52> <P class=western align=center><I>M</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>32 </P></TD> <TD width=52> <P class=western align=center><I>2m</I>3<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>17 </P></TD> <TD width=52> <P class=western align=center><I>M</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>40 </P></TD> <TD width=52> <P class=western align=center>3<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>47 </P></TD> <TD width=52> <P class=western align=center><I>2M</I>3<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>31 </P></TD> <TD width=52> <P class=western align=center><I>D</I> </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=30> <P class=western align=center>-4 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>35 </P></TD> <TD width=52> <P class=western align=center>4<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>46 </P></TD> <TD width=52> <P class=western align=center>3<I>m</I>2<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>44 </P></TD> <TD width=52> <P class=western align=center>2<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>49 </P></TD> <TD width=52> <P class=western align=center><I>m</I>3<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>42 </P></TD> <TD width=52> <P class=western align=center>3<I>M</I>2<I>d</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>5 </P></TD> <TD width=52> <P class=western align=center><I>md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>30 </P></TD> <TD width=52> <P class=western align=center><I>M</I>3<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>7 </P></TD> <TD width=52> <P class=western align=center>4<I>md</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>39 </P></TD> <TD width=52> <P class=western align=center><I>Md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>10 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I>2<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>37 </P></TD> <TD width=52> <P class=western align=center>4<I>Md</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>19 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>25 </P></TD> <TD width=52> <P class=western align=center>2<I>M</I>2<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>0 </P></TD> <TD width=29> <P class=western align=center>-5 </P></TD> <TD width=29> <P class=western align=center>21 </P></TD> <TD width=52> <P class=western align=center>5<I>m</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>34 </P></TD> <TD width=52> <P class=western align=center>2<I>M</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>24 </P></TD> <TD width=52> <P class=western align=center>3<I>md</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>0 </P></TD> <TD width=29> <P class=western align=center>5 </P></TD> <TD width=29> <P class=western align=center>32 </P></TD> <TD width=52> <P class=western align=center>5<I>M</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>14 </P></TD> <TD width=52> <P class=western align=center><I>mD</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>20 </P></TD> <TD width=52> <P class=western align=center>3<I>Md</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>-4 </P></TD> <TD width=29> <P class=western align=center>16 </P></TD> <TD width=52> <P class=western align=center>4<I>mD</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>48 </P></TD> <TD width=52> <P class=western align=center><I>MD</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-4 </P></TD> <TD width=30> <P class=western align=center>38 </P></TD> <TD width=52> <P class=western align=center>4<I>m</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>46 </P></TD> <TD width=52> <P class=western align=center>4<I>MD</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>9 </P></TD> <TD width=52> <P class=western align=center>2<I>D</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=30> <P class=western align=center>15 </P></TD> <TD width=52> <P class=western align=center>4<I>M</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>-3 </P></TD> <TD width=29> <P class=western align=center>11 </P></TD> <TD width=52> <P class=western align=center>3<I>m</I>2<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>13 </P></TD> <TD width=52> <P class=western align=center>3<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>33 </P></TD> <TD width=52> <P class=western align=center>3<I>mD</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>7 </P></TD> <TD width=52> <P class=western align=center>3<I>M</I>2<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>27 </P></TD> <TD width=52> <P class=western align=center><I>m</I>2<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>29 </P></TD> <TD width=52> <P class=western align=center>3<I>MD</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>-2 </P></TD> <TD width=29> <P class=western align=center>6 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I>3<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>8 </P></TD> <TD width=52> <P class=western align=center><I>M</I>2<I>d</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>28 </P></TD> <TD width=52> <P class=western align=center>2<I>m</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>3 </P></TD> <TD width=29> <P class=western align=center>2 </P></TD> <TD width=29> <P class=western align=center>21 </P></TD> <TD width=52> <P class=western align=center>2<I>M3D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>41 </P></TD> <TD width=52> <P class=western align=center>2<I>md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>43 </P></TD> <TD width=52> <P class=western align=center>2<I>M</I>2<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>-1 </P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=52> <P class=western align=center><I>m</I>4<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=52> <P class=western align=center>2<I>Md</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>23 </P></TD> <TD width=52> <P class=western align=center><I>m</I>3<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>4 </P></TD> <TD width=29> <P class=western align=center>1 </P></TD> <TD width=29> <P class=western align=center>35 </P></TD> <TD width=52> <P class=western align=center><I>M</I>4<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=52> <P class=western align=center><I>3m</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=52> <P class=western align=center><I>M</I>3<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center>5 </P></TD> <TD width=29> <P class=western align=center>0 </P></TD> <TD width=29> <P class=western align=center>49 </P></TD> <TD width=52> <P class=western align=center>5<I>D</I> </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>51 </P></TD> <TD width=52> <P class=western align=center>3<I>M</I> </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>18 </P></TD> <TD width=52> <P class=western align=center>4<I>D</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=52> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=29 height=15> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>50 </P></TD> <TD width=52> <P class=western align=center>2<I>mD</I> </P></TD> <TD width=30> <P class=western align=center>5 </P></TD> <TD width=30> <P class=western align=center>-5 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD> <TD width=52> <P class=western align=center>5<I>d</I> </P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=29> <P class=western align=center><BR></P></TD> <TD width=52> <P class=western align=center><BR></P></TD></TR></TBODY></TABLE></CENTER> <P class=western><BR></P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN lang=EN-US style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">Заметим, что если </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt"> соответствует ступени </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt">, то </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-</SPAN></SUP><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-</SPAN></SUP><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>b</SPAN></SPAN></SUP><SPAN style="FONT-SIZE: 12pt"> соответствует ступени 53-</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">.</SPAN></I><SPAN style="FONT-SIZE: 12pt"> </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">Пример: 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">m</SPAN></I><SPAN style="FONT-SIZE: 12pt">2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">d</SPAN></I><SPAN style="FONT-SIZE: 12pt"> соответствует 10; 2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><SPAN style="FONT-SIZE: 12pt">2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><SPAN style="FONT-SIZE: 12pt"> соответствует 43. Общее число функций с </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>£</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">5 равно 61; при этом всем ступеням от 0 до 52 соответствует, по крайней мере, одна функция. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"><B><SPAN style="FONT-SIZE: 12pt">6.<SPAN> </SPAN></SPAN></B><B><SPAN style="FONT-SIZE: 12pt">Обобщение. </SPAN></B> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-INDENT: -18pt"> </P> <P class=MsoNormal><SPAN style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Предположим, что имеется </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">"исходных" тональных функций </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN style="FONT-SIZE: 12pt">,…, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">, которым соответствуют частоты </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN style="FONT-SIZE: 12pt">,…, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">, где </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN style="FONT-SIZE: 12pt">,…, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt"> - нечетные простые числа (изложенное выше соответствует случаю </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN style="FONT-SIZE: 12pt">=2,<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">D</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">M</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">,<SPAN> </SPAN></SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=3<I>, </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=5). Множество </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>F</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt"> тональных функций тогда является свободной абелевой группой с </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN style="FONT-SIZE: 12pt"><SPAN> </SPAN>образующими </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN style="FONT-SIZE: 12pt">,…, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">. Любой элемент </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt"> из </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>F</SPAN></SPAN><I><SPAN style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">однозначно представляется в виде </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SUP><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUP><I><SPAN style="FONT-SIZE: 12pt">…</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f<SUB>n</SUB></SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt">.<I> </I>Если </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">G</SPAN></I><SPAN style="FONT-SIZE: 12pt"> подгруппа в </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">Q</SPAN></I><I><SUP><SPAN style="FONT-SIZE: 12pt">+</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt">, порожденная числами </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><I><SPAN style="FONT-SIZE: 12pt">,…, </SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">, то имеет место изоморфизм </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>F</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>@</SPAN></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">G</SPAN></I><SPAN style="FONT-SIZE: 12pt">, заданный на образующих равенства </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>j</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f<SUB>i</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">)=<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p<SUB>i</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">. Для любого </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt"> определено множество частот {2</SPAN><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>j</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">)</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">0</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">}, где </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">S</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">пробегает целые числа; </SPAN><I><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>n</SPAN></SPAN></I><I><SUB><SPAN style="FONT-SIZE: 12pt">0 </SPAN></SUB></I><SPAN style="FONT-SIZE: 12pt">соответствует тонике </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">T</SPAN></I><SPAN style="FONT-SIZE: 12pt"> - единице группы. Порядок функции<I> </I></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><I><SPAN style="FONT-SIZE: 12pt">=</SPAN></I><SPAN style="FONT-SIZE: 12pt"> </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><SUP><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUP><I><SPAN style="FONT-SIZE: 12pt">…</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f<SUB>n</SUB></SPAN></I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN></SUP><I><SUP><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUP></I><SPAN style="FONT-SIZE: 12pt"> определяется равенством </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=|</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">|+<I>…</I>+|</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><I><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUB></I><SPAN style="FONT-SIZE: 12pt">|. Номер </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt"> ступени 53-ступенного темперированного строя, соответствующий функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt">, вычисляется по формуле: </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-ALIGN: center" align=center><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt">=[53{</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">log<SUB>2</SUB><I> p</I><SUB>1</SUB>+…+</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>a</SPAN></SPAN><I><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUB></I><SPAN lang=EN-US style="FONT-SIZE: 12pt">log<SUB>2</SUB><I> p<SUB>n</SUB></I>}]<SUB>1</SUB></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">По-прежнему, если функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SPAN style="FONT-SIZE: 12pt"> соответствует </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt">, то<SPAN> </SPAN>функции </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">f</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">-1</SPAN></SUP><SPAN style="FONT-SIZE: 12pt"> соответствует 53-</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">k</SPAN></I><SPAN style="FONT-SIZE: 12pt"> . В качестве примера рассмотрим случай </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN style="FONT-SIZE: 12pt">=3; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=3; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=5; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">p</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">3</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">=7. Имеет место следующая таблица:</SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"> </P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=480 border=1> <COLGROUP> <COL width=29> <COL width=30> <COL width=30> <COL width=30> <COL width=30></COLGROUP> <COLGROUP> <COL width=30> <COL width=30> <COL width=30> <COL width=30> <COL width=30></COLGROUP> <COLGROUP> <COL width=30> <COL width=30> <COL width=30> <COL width=30> <COL width=30></COLGROUP> <TBODY> <TR> <TD width=29 height=17> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=30> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD> <TD width=30> <P class=western align=center><I>h</I> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=30> <P class=western align=center><I>k</I> </P></TD></TR></TBODY> <TBODY> <TR> <TD width=29 height=16> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>21 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>22 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>22 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>48 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>37 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>36 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>9 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>50 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>10 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>13 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>44 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>43 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>27 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>24 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>17 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>51 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>31 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>34 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>50 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>44 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>8 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>24 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>5 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>41 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>4 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>32 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>15 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>51 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>12 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>48 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>11 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>39 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>42 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>5 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>19 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>38 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>46 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>49 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>12 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>26 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>29 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>45 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>20 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>19 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>33 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>52 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>27 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>29 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>26 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>7 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>9 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>40 </P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>34 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>-2 </P></TD> <TD width=30> <P class=western align=center>3 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=29 height=16> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>14 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>2 </P></TD> <TD width=30> <P class=western align=center>16 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=29 height=15> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>1 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-1 </P></TD> <TD width=30> <P class=western align=center>41 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>0 </P></TD> <TD width=30> <P class=western align=center>-3 </P></TD> <TD width=30> <P class=western align=center>31 </P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD> <TD width=30> <P class=western align=center><BR></P></TD></TR></TBODY></TABLE></CENTER> <P class=western lang=ru-RU><BR></P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">При </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN style="FONT-SIZE: 12pt">=3 и </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">>0 существует 4</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">+2 функций порядка </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">. Общее число функций порядка </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>£</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">3 равно 63; тем не менее ступеням 6, 18, 23, 30, 35, 47 не соответствует ни одной функции порядка </SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>£</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">3; этим ступеням соответствуют функции порядка 4: </SPAN> </P><B><SPAN style="FONT-SIZE: 12pt"> <P class=western><BR></P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=542 border=1> <COLGROUP> <COL width=33> <COL width=34> <COL width=34> <COL width=34> <COL width=34></COLGROUP> <COLGROUP> <COL width=34> <COL width=34> <COL width=34> <COL width=34> <COL width=34></COLGROUP> <COLGROUP> <COL width=34> <COL width=34> <COL width=34> <COL width=34> <COL width=34></COLGROUP> <TBODY> <TR> <TD width=33 height=17> <P class=western align=center><I>h</I> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><I>k</I> </P></TD> <TD width=34> <P class=western align=center><I>h</I> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><I>k</I> </P></TD> <TD width=34> <P class=western align=center><I>h</I> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>1</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>2</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><FONT face="Symbol, serif">a<SUB>3</SUB></FONT> </P></TD> <TD width=34> <P class=western align=center><I>k</I> </P></TD></TR></TBODY> <TBODY> <TR> <TD width=33 height=16> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>-2 </P></TD> <TD width=34> <P class=western align=center>6 </P></TD> <TD width=34> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>3 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>23 </P></TD> <TD width=34> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>2 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>35 </P></TD></TR> <TR> <TD width=33 height=16> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-2 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>18 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-3 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>30 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>2 </P></TD> <TD width=34> <P class=western align=center>47 </P></TD></TR> <TR> <TD width=33 height=16> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>4 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>18 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>3 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>1 </P></TD> <TD width=34> <P class=western align=center>30 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD></TR> <TR> <TD width=33 height=15> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-3 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>-1 </P></TD> <TD width=34> <P class=western align=center>23 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center>-4 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>0 </P></TD> <TD width=34> <P class=western align=center>35 </P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD> <TD width=34> <P class=western align=center><BR></P></TD></TR></TBODY></TABLE></CENTER> <P class=western><BR></P> <P class=MsoNormal>7.<SPAN> </SPAN></SPAN></B><B><SPAN style="FONT-SIZE: 12pt">Число тональных функций. </SPAN></B> </P> <P class=MsoNormal><SPAN style="FONT-SIZE: 12pt"> </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Обозначим </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">) число тональных функций порядка </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">при числе образующих </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>³</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">0, </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>³</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">1). С математической точки зрения </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">) есть число решений уравнения |</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">x</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">|+|</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">x</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">|+<I>…</I>+|</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">x<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">|=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt"> в целых числах. Очевидно, что </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(0)=1 и при </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>³</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">1 </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=2. Имеет место рекуррентная формула: </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-ALIGN: center" align=center><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"></SPAN></I></P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-ALIGN: center" align=center><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt">(<I>h</I>)=<I> N<SUB>n</SUB></I>(<I>h-</I>1)+<I>N<SUB>n-</SUB></I><SUB>1</SUB>(<I>h</I>)+<I>N<SUB>n-</SUB></I><SUB>1</SUB>(<I>h-</I>1), </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt; TEXT-ALIGN: center" align=center><SPAN lang=EN-US style="FONT-SIZE: 12pt"></SPAN></P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt"><SPAN style="FONT-SIZE: 12pt">позволяющая вычислять значения </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">). При заполнении приводимой ниже таблицы чисел </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">) удобно пользоваться рекуррентной формулой в виде следующего правила: каждое "внутреннее" число равно сумме чисел, находящихся от него слева, слева вверху и вверху. </SPAN> </P><SPAN style="FONT-SIZE: 12pt"> <P class=western><BR></P> <CENTER> <TABLE borderColor=#000000 cellSpacing=0 cellPadding=1 rules=groups width=434 border=1> <COLGROUP> <COL width=19> <COL width=20></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=54></COLGROUP> <COLGROUP> <COL width=53></COLGROUP> <TBODY> <TR> <TD width=19 height=15> <P class=western align=center><BR></P></TD> <TD width=20> <P class=western align=center>h </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>0 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>1 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>2 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>3 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>4 </P></TD> <TD width=54 rowSpan=2> <P class=western align=center>5 </P></TD> <TD width=53 rowSpan=2> <P class=western align=center>6 </P></TD></TR> <TR> <TD width=19 height=16> <P class=western align=center>n </P></TD> <TD width=20> <P class=western align=center><BR></P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>2 </P></TD> <TD width=53> <P class=western align=center>2 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>2 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>4 </P></TD> <TD width=54> <P class=western align=center>8 </P></TD> <TD width=54> <P class=western align=center>12 </P></TD> <TD width=54> <P class=western align=center>16 </P></TD> <TD width=54> <P class=western align=center>20 </P></TD> <TD width=53> <P class=western align=center>24 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>3 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>6 </P></TD> <TD width=54> <P class=western align=center>18 </P></TD> <TD width=54> <P class=western align=center>38 </P></TD> <TD width=54> <P class=western align=center>66 </P></TD> <TD width=54> <P class=western align=center>102 </P></TD> <TD width=53> <P class=western align=center>146 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>4 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>8 </P></TD> <TD width=54> <P class=western align=center>32 </P></TD> <TD width=54> <P class=western align=center>88 </P></TD> <TD width=54> <P class=western align=center>192 </P></TD> <TD width=54> <P class=western align=center>360 </P></TD> <TD width=53> <P class=western align=center>608 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=16> <P class=western align=center>5 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>10 </P></TD> <TD width=54> <P class=western align=center>50 </P></TD> <TD width=54> <P class=western align=center>170 </P></TD> <TD width=54> <P class=western align=center>450 </P></TD> <TD width=54> <P class=western align=center>1002 </P></TD> <TD width=53> <P class=western align=center>1970 </P></TD></TR></TBODY> <TBODY> <TR> <TD width=42 colSpan=2 height=15> <P class=western align=center>6 </P></TD> <TD width=54> <P class=western align=center>1 </P></TD> <TD width=54> <P class=western align=center>12 </P></TD> <TD width=54> <P class=western align=center>72 </P></TD> <TD width=54> <P class=western align=center>292 </P></TD> <TD width=54> <P class=western align=center>912 </P></TD> <TD width=54> <P class=western align=center>2364 </P></TD> <TD width=53> <P class=western align=center>5336 </P></TD></TR></TBODY></TABLE></CENTER> <P class=western><BR></P> <P class=MsoNormal style="MARGIN-LEFT: 17.85pt">При фиксированном </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN style="FONT-SIZE: 12pt"> и </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>³</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">1 имеют место формулы </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=4</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">3</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=4</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">+2; (эти формулы приводились в п.2 и п.6) </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">4</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=(8</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">+2))/3; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">5</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=(4</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">+5))/3; </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">6</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=(4</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">(2</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">4</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">+20</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SUP><SPAN style="FONT-SIZE: 12pt">2</SPAN></SUP><SPAN style="FONT-SIZE: 12pt">+23))/15. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Обозначим <I>Ñ</I></SPAN><I><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUB></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">) число тональных функций порядка </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>£</SPAN></SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><I><SPAN lang=EN-US style="FONT-SIZE: 12pt"> </SPAN></I><SPAN style="FONT-SIZE: 12pt">при числе образующих </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN style="FONT-SIZE: 12pt">; т. о. <I>Ñ</I></SPAN><I><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUB></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(0)+</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(1)+<I>…</I>+</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">). Имеет место формула:<I> Ñ</I></SPAN><I><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUB></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)=(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><I><SUB><SPAN style="FONT-SIZE: 12pt">+</SPAN></SUB></I><SUB><SPAN style="FONT-SIZE: 12pt">1</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">)+</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N<SUB>n</SUB></SPAN></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">))/2, позволяющая легко вычислять значения <I>Ñ</I></SPAN><I><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></SUB></I><SPAN style="FONT-SIZE: 12pt">(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">h</SPAN></I><SPAN style="FONT-SIZE: 12pt">). Например при количестве образующих </SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">n</SPAN></I><SPAN style="FONT-SIZE: 12pt">=5 число тональных функций порядка </SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>£</SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">5 равно <I>Ñ</I><SUB>5</SUB>(5)=(</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">6</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(5)+</SPAN><I><SPAN lang=EN-US style="FONT-SIZE: 12pt">N</SPAN></I><SUB><SPAN style="FONT-SIZE: 12pt">5</SPAN></SUB><SPAN style="FONT-SIZE: 12pt">(5))/2=(2364+1002)/2=1683. </SPAN> </P> <P class=MsoNormal style="MARGIN-LEFT: 18pt"><SPAN style="FONT-SIZE: 12pt">Приведем также общую формулу: </SPAN> </P><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'"><SPAN> <DIV align=left> <TABLE cellSpacing=0 cellPadding=3 width=300 border=0> <TBODY> <TR> <TD width="50%"> <P align=right><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'"><EM><SUB>n</SUB></EM></SPAN></P></TD> <TD width="50%"></TD></TR> <TR> <TD width="50%"> <P align=right><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'"><EM>N</EM></SPAN><SUB><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'">n</SPAN></SUB><I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'">(h)</SPAN></I><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'">=</SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: Symbol"><SPAN>S</SPAN></SPAN></P></TD> <TD width="50%"><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'"><EM>(<SUB><SPAN>i</SPAN></SUB><SUP><SPAN>n</SPAN></SUP>)(<SUB><SPAN>i</SPAN></SUB><SUP><SPAN>h</SPAN></SUP><SUB><SPAN>-</SPAN></SUB><SUP><SPAN>-</SPAN></SUP><SUB><SPAN>1</SPAN></SUB><SUP><SPAN>1</SPAN></SUP>)</EM></SPAN><SPAN lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'">2<I><SUP>i</SUP></I></SPAN></TD></TR> <TR> <TD width="50%"> <P align=right><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'"><EM><SUP>i=1</SUP></EM></SPAN></P></TD> <TD width="50%"></TD></TR></TBODY></TABLE></SPAN></SPAN><SPAN style="FONT-SIZE: 12pt; FONT-FAMILY: 'Courier New'"><SPAN> </SPAN></SPAN><SPAN style="FONT-SIZE: 12pt">16 июня 2005 </SPAN> </DIV> <P class=western><BR></P>Mykhaylo Khramovhttp://www.blogger.com/profile/00483331671036933638noreply@blogger.com0