ピッチクラス・セット(pitch-class set)

自由な無調音楽の集合論・分析法の一つで、1990年代に入ってからアメリカで積極的に用いられるようになった。 12等分平均律上の全ての音高を1オクターブ内に還元すると、12の異なった音高類(ピッチクラス)しか無い事になる。 このピッチクラスの3音から9音までの組み合わせをそこに含まれる音程によって整理したものがピッチクラスセットで、主なるものは208種ある。
無調音楽に於ける統一的要素を抽出する分析法だが、作曲技法としても用いられる。


ピッチクラス(pitch-class)

ある所定の音高がゼロまたは整数オクターブ移高されたもの、と云う条件全てを満たすピッチ全ての集合体。例えば全てのDはピッチクラスDに帰属する。
ミルトン・バビット (Milton Babbitt, 1916-2011) によって導入されたこの用語は、それ以前には明確に定義されなかった基本的な音楽概念を示している。最初は十二音技法に関連して用いられたもので、十二音のセットは12のピッチクラスの順序づけされた連なりからなる。

Fortean set-name
ハイフンにより分けられる2種類の数字で表されるPCSの名前。
ハイフンの左側の数字はセットの基数、右側の数字はセットの序数を表す。
この序数は原型のリスト位置。
*
鏡合わせの性質を持つ。逆行型 retrogradeにおいても同じ形を表す。
B
セット名にBをもつセットは、持たないセットの逆行型で同じInterval Vectorを持つ。
(1); (3); (4); (6)
表示の転回型は12種類だが、カッコ数字を持つセットはx種類しかできない。
Prime
Prime form 原型。
移値または転回ののち、移値の操作を行い最初の整数が0になる正規順序のセット。16進法。
Hieroglyphics
可視化しやすくするため、PCS構成音をO、非構成音をXで示す。
interval vector
音程行ベクトル。
PCSの音程内容を表す数値の順序立てられた整列。カッコ内の数字は左から短二度, 長二度, 短三度, 長三度、完全四度, 増四度それぞれの音程関係がx個内包されてるかを表す。
Fortean
set-name
PrimeHieroglyphicsinterval
vector
Fortean
set-name
PrimeHieroglyphicsinterval
vector
0-0(empty)XXXXXXXXXXXX<000000> 12-1*(1)(0,1,2,3,4,5,6,7,8,9,A,B)OOOOOOOOOOOO<C,C,C,C,C,6>
1-1*(0)OXXXXXXXXXXX<000000> 11-1*(0,1,2,3,4,5,6,7,8,9,A)OOOOOOOOOOOX<A,A,A,A,A,5>
2-1*(0,1)OOXXXXXXXXXX<100000> 10-1*(0,1,2,3,4,5,6,7,8,9)OOOOOOOOOOXX<988888>
2-2*(0,2)OXOXXXXXXXXX<010000> 10-2*(0,1,2,3,4,5,6,7,8,A)OOOOOOOOOXOX<898888>
2-3*(0,3)OXXOXXXXXXXX<001000> 10-3*(0,1,2,3,4,5,6,7,9,A)OOOOOOOOXOOX<889888>
2-4*(0,4)OXXXOXXXXXXX<000100> 10-4*(0,1,2,3,4,5,6,8,9,A)OOOOOOOXOOOX<888988>
2-5*(0,5)OXXXXOXXXXXX<000010> 10-5*(0,1,2,3,4,5,7,8,9,A)OOOOOOXOOOOX<888898>
2-6*(6)(0,6)OXXXXXOXXXXX<000001> 10-6*(6)(0,1,2,3,4,6,7,8,9,A)OOOOOXOOOOOX<888889>
3-1*(0,1,2)OOOXXXXXXXXX<210000> 9-1*(0,1,2,3,4,5,6,7,8)OOOOOOOOOXXX<876663>
3-2(0,1,3)OOXOXXXXXXXX<111000> 9-2(0,1,2,3,4,5,6,7,9)OOOOOOOOXOXX<777663>
3-2B(0,2,3)OXOOXXXXXXXX<111000> 9-2B(0,2,3,4,5,6,7,8,9)OXOOOOOOOOXX<777663>
3-3(0,1,4)OOXXOXXXXXXX<101100> 9-3(0,1,2,3,4,5,6,8,9)OOOOOOOXOOXX<767763>
3-3B(0,3,4)OXXOOXXXXXXX<101100> 9-3B(0,1,3,4,5,6,7,8,9)OOXOOOOOOOXX<767763>
3-4(0,1,5)OOXXXOXXXXXX<100110> 9-4(0,1,2,3,4,5,7,8,9)OOOOOOXOOOXX<766773>
3-4B(0,4,5)OXXXOOXXXXXX<100110> 9-4B(0,1,2,4,5,6,7,8,9)OOOXOOOOOOXX<766773>
3-5(0,1,6)OOXXXXOXXXXX<100011> 9-5(0,1,2,3,4,6,7,8,9)OOOOOXOOOOXX<766674>
3-5B(0,5,6)OXXXXOOXXXXX<100011> 9-5B(0,1,2,3,5,6,7,8,9)OOOOXOOOOOXX<766674>
3-6*(0,2,4)OXOXOXXXXXXX<020100> 9-6*(0,1,2,3,4,5,6,8,A)OOOOOOOXOXOX<686763>
3-7(0,2,5)OXOXXOXXXXXX<011010> 9-7(0,1,2,3,4,5,7,8,A)OOOOOOXOOXOX<677673>
3-7B(0,3,5)OXXOXOXXXXXX<011010> 9-7B(0,1,2,3,4,5,7,9,A)OOOOOOXOXOOX<677673>
3-8(0,2,6)OXOXXXOXXXXX<010101> 9-8(0,1,2,3,4,6,7,8,A)OOOOOXOOOXOX<676764>
3-8B(0,4,6)OXXXOXOXXXXX<010101> 9-8B(0,1,2,3,4,6,8,9,A)OOOOOXOXOOOX<676764>
3-9*(0,2,7)OXOXXXXOXXXX<010020> 9-9*(0,1,2,3,5,6,7,8,A)OOOOXOOOOXOX<676683>
3-10*(0,3,6)OXXOXXOXXXXX<002001> 9-10*(0,1,2,3,4,6,7,9,A)OOOOOXOOXOOX<668664>
3-11(0,3,7)OXXOXXXOXXXX<001110> 9-11(0,1,2,3,5,6,7,9,A)OOOOXOOOXOOX<667773>
3-11B(0,4,7)OXXXOXXOXXXX<001110> 9-11B(0,1,2,3,5,6,8,9,A)OOOOXOOXOOOX<667773>
3-12*(4)(0,4,8)OXXXOXXXOXXX<000300> 9-12*(4)(0,1,2,4,5,6,8,9,A)OOOXOOOXOOOX<666963>
4-1*(0,1,2,3)OOOOXXXXXXXX<321000> 8-1*(0,1,2,3,4,5,6,7)OOOOOOOOXXXX<765442>
4-2(0,1,2,4)OOOXOXXXXXXX<221100> 8-2(0,1,2,3,4,5,6,8)OOOOOOOXOXXX<665542>
4-2B(0,2,3,4)OXOOOXXXXXXX<221100> 8-2B(0,2,3,4,5,6,7,8)OXOOOOOOOXXX<665542>
4-3*(0,1,3,4)OOXOOXXXXXXX<212100> 8-3*(0,1,2,3,4,5,6,9)OOOOOOOXXOXX<656542>
4-4(0,1,2,5)OOOXXOXXXXXX<211110> 8-4(0,1,2,3,4,5,7,8)OOOOOOXOOXXX<655552>
4-4B(0,3,4,5)OXXOOOXXXXXX<211110> 8-4B(0,1,3,4,5,6,7,8)OOXOOOOOOXXX<655552>
4-5(0,1,2,6)OOOXXXOXXXXX<210111> 8-5(0,1,2,3,4,6,7,8)OOOOOXOOOXXX<654553>
4-5B(0,4,5,6)OXXXOOOXXXXX<210111> 8-5B(0,1,2,4,5,6,7,8)OOOXOOOOOXXX<654553>
4-6*(0,1,2,7)OOOXXXXOXXXX<210021> 8-6*(0,1,2,3,5,6,7,8)OOOOXOOOOXXX<654463>
4-7*(0,1,4,5)OOXXOOXXXXXX<201210> 8-7*(0,1,2,3,4,5,8,9)OOOOOOXXOOXX<645652>
4-8*(0,1,5,6)OOXXXOOXXXXX<200121> 8-8*(0,1,2,3,4,7,8,9)OOOOOXXOOOXX<644563>
4-9*(6)(0,1,6,7)OOXXXXOOXXXX<200022> 8-9*(6)(0,1,2,3,6,7,8,9)OOOOXXOOOOXX<644464>
4-10*(0,2,3,5)OXOOXOXXXXXX<122010> 8-10*(0,2,3,4,5,6,7,9)OXOOOOOOXOXX<566452>
4-11(0,1,3,5)OOXOXOXXXXXX<121110> 8-11(0,1,2,3,4,5,7,9)OOOOOOXOXOXX<565552>
4-11B(0,2,4,5)OXOXOOXXXXXX<121110> 8-11B(0,2.4.5.6.7.8.9)OXOXOOOOOOXX<565552>
4-12<(0,2,3,6)OXOOXXOXXXXX<112101> 8-12<(0,1,3,4,5,6,7,9)OOXOOOOOXOXX<556543>
4-12B<(0,3,4,6)OXXOOXOXXXXX<112101> 8-12B<(0,2,3,4,5,6,8,9)OXOOOOOXOOXX<556543>
4-13(0,1,3,6)OOXOXXOXXXXX<112011> 8-13(0,1,2,3,4,6,7,9)OOOOOXOOXOXX<556453>
4-13B(0,3,5,6)OOXOXXOXXXXX<112011> 8-13B(0,2,3,5,6,7,8,9)OXOOXOOOOOXX<556453>
4-14<(0,2,3,7)OXOOXXXOXXXX<111120> 8-14<(0,1,2,4,5,6,7,9)OOOXOOOOXOXX<555562>
4-14B<(0,4,5,7)OXXXOOXOXXXX<111120> 8-14B<(0,2,3,4,5,7,8,9)OXOOOOXOOOXX<555562>
4-Z15..29(0,1,4,6)OOXXOXOXXXXX<111111> 8-Z15..29(0,1,2,3,4,6,8,9)OOOOOXOXOOXX<555553>
4-Z15B..29(0,2,5,6)OXOXXOOXXXXX<111111> 8-Z15B..29(0,1,3,5,6,7,8,9)OOXOXOOOOOXX<555553>
4-16(0,1,5,7)OOXXXOXOXXXX<110121> 8-16(0,1,2,3,5,7,8,9)OOOOXOXOOOXX<554563>
4-16B(0,2,6,7)OXOXXXOOXXXX<110121> 8-16B(0,1,2,4,6,7,8,9)OOOXOXOOOOXX<554563>
4-17*(0,3,4,7)OXXOOXXOXXXX<102210> 8-17*(0,1,3,4,5,6,8,9)OOXOOOOXOOXX<546652>
4-18(0,1,4,7)OOXXOXXOXXXX<102111> 8-18(0,1,2,3,5,6,8,9)OOOOXOOXOOXX<546553>
4-18B(0,3,6,7)OXXOXXOOXXXX<102111> 8-18B(0,1,3,4,6,7,8,9)OOXOOXOOOOXX<546553>
4-19(0,1,4,8)OOXXOXXXOXXX<101310> 8-19(0,1,2,4,5,6,8,9)OOOXOOOXOOXX<545752>
4-19B(0,3,4,8)OXXOOXXXOXXX<101310> 8-19B(0,1,3,4,5,7,8,9)OOXOOOXOOOXX<545752>
4-20*(0,1,5,8)OOXXXOXXOXXX<101220> 8-20*(0,1,2,4,5,7,8,9)OOOXOOXOOOXX<545662>
4-21*(0,2,4,6)OXOXOXOXXXXX<030201> 8-21*(0,1,2,3,4,6,8,A)OOOOOXOXOXOX<474643>
4-22(0,2,4,7)OXOXOXXOXXXX<021120> 8-22(0,1,2,3,5,6,8,A)OOOOXOOXOXOX<465562>
4-22B(0,3,5,7)OXXOXOXOXXXX<021120> 8-22B(0,1,2,3,5,7,9,A)OOOOXOXOXOOX<465562>
4-23*(0,2,5,7)OXOXXOXOXXXX<021030> 8-23*(0,1,2,3,5,7,8,A)OOOOXOXOOXOX<465472>
4-24*(0,2,4,8)OXOXOXXXOXXX<020301> 8-24*(0,1,2,4,5,6,8,A)OOOXOOOXOXOX<464743>
4-25*(6)(0,2,6,8)OXOXXXOXOXXX<020202> 8-25*(6)(0,1,2,4,6,7,8,A)OOOXOXOOOXOX<464644>
4-26*(0,3,5,8)OXXOXOXXOXXX<012120> 8-26*(0,1,2,4,5,7,9,A)OOOXOOXOXOOX<456562>
4-27(0,2,5,8)OXOXXOXXOXXX<012111> 8-27(0,1,2,4,5,7,8,A)OOOXOOXOOXOX<456553>
4-27B(0,3,6,8)OXXOXXOXOXXX<012111> 8-27B(0,1,2,4,6,7,9,A)OOOXOXOOXOOX<456553>
4-28*(3)(0,3,6,9)OXXOXXOXXOXX<004002> 8-28*(3)(0,1,3,4,6,7,9,A)OOXOOXOOXOOX<448444>
8-Z29..15(0,1,3,7)OOXOXXXOXXXX<111111> 8-Z29..15(0,1,2,3,5,6,7,9)OOOOXOOOXOXX<555553>
8-Z29B..15(0,4,6,7)OXXXOXOOXXXX<111111> 8-Z29B..15(0,2,3,4,6,7,8,9)OXOOOXOOOOXX<555553>
5-1*(0,1,2,3,4)OOOOOXXXXXXX<432100> 7-1*(0,1,2,3,4,5,6)OOOOOOOXXXXX<654321>
5-2(0,1,2,3,5)OOOOXOXXXXXX<332110> 7-2(0,1,2,3,4,5,7)OOOOOOXOXXXX<554331>
5-2B(0,2,3,4,5)OXOOOOXXXXXX<332110> 7-2B(0,2,3,4,5,6,7)OXOOOOOOXXXX<554331>
5-3(0,1,2,4,5)OOOXOOXXXXXX<322210> 7-3(0,1,2,3,4,5,8)OOOOOOXXOXXX<544431>
5-3B(0,1,3,4,5)OOXOOOXXXXXX<322210> 7-3B(0,3,4,5,6,7,8)OXXOOOOOOXXX<544431>
5-4(0,1,2,3,6)OOOOXXOXXXXX<322111> 7-4(0,1,2,3,4,6,7)OOOOOXOOXXXX<544332>
5-4B(0,3,4,5,6)OXXOOOOXXXXX<322111> 7-4B(0,1,3,4,5,6,7)OOXOOOOOXXXX<544332>
5-5(0,1,2,3,7)OOOOXXXOXXXX<321121> 7-5(0,1,2,3,5,6,7)OOOOXOOOXXXX<543342>
5-5B(0,4,5,6,7)OXXXOOOOXXXX<321121> 7-5B(0,1,2,4,5,6,7)OOOXOOOOXXXX<543342>
5-6(0,1,2,5,6)OOOXXOOXXXXX<311221> 7-6(0,1,2,3,4,7,8)OOOOOXXOOXXX<533442>
5-6B(0,1,4,5,6)OOXXOOOXXXXX<311221> 7-6B(0,1,4,5,6,7,8)OOXXOOOOOXXX<533442>
5-7(0,1,2,6,7)OOOXXXOOXXXX<310132> 7-7(0,1,2,3,6,7,8)OOOOXXOOOXXX<532353>
5-7B(0,1,5,6,7)OOXXXOOOXXXX<310132> 7-7B(0,1,2,5,6,7,8)OOOXXOOOOXXX<532353>
5-8*(0,2,3,4,6)OXOOOXOXXXXX<232201> 7-8*(0,2,3,4,5,6,8)OXOOOOOXOXXX<454422>
5-9(0,1,2,4,6)OOOXOXOXXXXX<231211> 7-9(0,1,2,3,4,6,8)OOOOOXOXOXXX<453432>
5-9B(0,2,4,5,6)OXOXOOOXXXXX<231211> 7-9B(0,2,4,5,6,7,8)OXOXOOOOOXXX<453432>
5-10(0,1,3,4,6)OOXOOXOXXXXX<223111> 7-10(0,1,2,3,4,6,9)OOOOOXOXXOXX<445332>
5-10B(0,2,3,5,6)OXOOXOOXXXXX<223111> 7-10B(0,2,3,4,5,6,9)OXOOOOOXXOXX<445332>
5-11(0,2,3,4,7)OXOOOXXOXXXX<222220> 7-11(0,1,3,4,5,6,8)OOXOOOOXOXXX<444441>
5-11B(0,3,4,5,7)OXXOOOXOXXXX<222220> 7-11B(0,2,3,4,5,7,8)OXOOOOXOOXXX<444441>
5-Z12*..36(0,1,3,5,6)OOXOXOOXXXXX<222121> 7-Z12*..36(0,1,2,3,4,7,9)OOOOOXXOXOXX<444342>
5-13(0,1,2,4,8)OOOXOXXXOXXX<221311> 7-13(0,1,2,4,5,6,8)OOOXOOOXOXXX<443532>
5-13B(0,2,3,4,8)OXOOOXXXOXXX<221311> 7-13B(0,2,3,4,6,7,8)OXOOOXOOOXXX<443532>
5-14(0,1,2,5,7)OOOXXOXOXXXX<221131> 7-14(0,1,2,3,5,7,8)OOOOXOXOOXXX<443352>
5-14B(0,2,5,6,7)OXOXXOOOXXXX<221131> 7-14B(0,1,3,5,6,7,8)OOXOXOOOOXXX<443352>
5-15*(0,1,2,6,8)OOOXXXOXOXXX<220222> 7-15*(0,1,2,4,6,7,8)OOOXOXOOOXXX<442443>
5-16(0,1,3,4,7)OOXOOXXOXXXX<213211> 7-16(0,1,2,3,5,6,9)OOOOXOOXXOXX<435432>
5-16B(0,3,4,6,7)OXXOOXOOXXXX<213211> 7-16B(0,1,3,4,5,6,9)OOXOOOOXXOXX<435432>
5-Z17*..37(0,1,3,4,8)OOXOOXXXOXXX<212320> 7-Z17*..37(0,1,2,4,5,6,9)OOOXOOOXXOXX<434541>
5-Z18<..38(0,1,4,5,7)OOXXOOXOXXXX<212221> 7-Z18<..38(0,1,2,3,5,8,9)OOOOXOXXOOXX<434442>
5-Z18B<..38(0,2,3,6,7)OXOOXXOOXXXX<212221> 7-Z18B<..38(0,1,4,6,7,8,9)OOXXOXOOOOXX<434442>
5-19(0,1,3,6,7)OOXOXXOOXXXX<212122> 7-19(0,1,2,3,6,7,9)OOOOXXOOXOXX<434343>
5-19B(0,1,4,6,7)OOXXOXOOXXXX<212122> 7-19B(0,1,2,3,6,8,9)OOOOXXOXOOXX<434343>
5-20(0,1,3,7,8)OOXOXXXOOXXX<211231> 7-20(0,1,2,4,7,8,9)OOOXOXXOOOXX<433452>
5-20B(0,1,5,7,8)OOXXXOXOOXXX<211231> 7-20B(0,1,2,5,7,8,9)OOOXXOXOOOXX<433452>
5-21(0,1,4,5,8)OOXXOOXXOXXX<202420> 7-21(0,1,2,4,5,8,9)OOOXOOXXOOXX<424641>
5-21B(0,3,4,7,8)OXXOOXXOOXXX<202420> 7-21B(0,1,3,4,5,8,9)OOXOOOXXOOXX<424641>
5-22*(0,1,4,7,8)OOXXOXXOOXXX<202321> 7-22*(0,1,2,5,6,8,9)OOOXXOOXOOXX<424542>
5-23(0,2,3,5,7)OXOOXOXOXXXX<132130> 7-23(0,2,3,4,5,7,9)OXOOOOXOXOXX<354351>
5-23B(0,2,4,5,7)OXOXOOXOXXXX<132130> 7-23B(0,2,4,5,6,7,9)OXOXOOOOXOXX<354351>
5-24(0,1,3,5,7)OOXOXOXOXXXX<131221> 7-24(0,1,2,3,5,7,9)OOOOXOXOXOXX<353442>
5-24B(0,2,4,6,7)OXOXOXOOXXXX<131221> 7-24B(0,2,4,6,7,8,9)OXOXOXOOOOXX<353442>
5-25(0,2,3,5,8)OXOOXOXXOXXX<123121> 7-25(0,2,3,4,6,7,9)OXOOOXOOXOXX<345342>
5-25B(0,3,5,6,8)OXXOXOOXOXXX<123121> 7-25B(0,2,3,5,6,7,9)OXOOXOOOXOXX<345342>
5-26<(0,2,4,5,8)OXOXOOXXOXXX<122311> 7-26<(0,1,3,4,5,7,9)OOXOOOXOXOXX<344532>
5-26B<(0,3,4,6,8)OXXOOXOXOXXX<122311> 7-26B<(02,4,5,6,8,9)OXOXOOOXOOXX<344532>
5-27(0,1,3,5,8)OOXOXOXXOXXX<122230> 7-27(0,1,2,4,5,7,9)OOOXOOXOXOXX<344451>
5-27B(0,3,5,7,8)OXXOXOXOOXXX<122230> 7-27B(0,2,4,5,7,8,9)OXOXOOXOOOXX<344451>
5-28<(0,2,3,6,8)OXOOXXOXOXXX<122212> 7-28<(0,1,3,5,6,7,9)OOXOXOOOXOXX<344433>
5-28B<(0,2,5,6,8)OXOXXOOXOXXX<122212> 7-28B<(0,2,3,4,6,8,9)OXOOOXOXOOXX<344433>
5-29(0,1,3,6,8)OOXOXXOXOXXX<122131> 7-29(0,1,2,4,6,7,9)OOOXOXOOXOXX<344352>
5-29B(0,2,5,7,8)OXOXXOXOOXXX<122131> 7-29B(0,2,3,5,7,8,9)OXOOXOXOOOXX<344352>
5-30(0,1,4,6,8)OOXXOXOXOXXX<121321> 7-30(0,1,2,4,6,8,9)OOOXOXOXOOXX<343542>
5-30B(0,2,4,7,8)OXOXOXXOOXXX<121321> 7-30B(0,1,3,5,7,8,9)OOXOXOXOOOXX<343542>
5-31(0,1,3,6,9)OOXOXXOXXOXX<114112> 7-31(0,1,3,4,6,7,9)OOXOOXOOXOXX<336333>
5-31B(0,2,3,6,9)OXOOXXOXXOXX<114112> 7-31B(0,2,3,5,6,8,9)OXOOXOOXOOXX<336333>
5-32(0,1,4,6,9)OOXXOXOXXOXX<113221> 7-32(0,1,3,4,6,8,9)OOXOOXOXOOXX<335442>
5-32B(0,1,4,7,9)OOXXOXXOXOXX<113221> 7-32B(0,1,3,5,6,8,9)OOXOXOOXOOXX<335442>
5-33*(0,2,4,6,8)OXOXOXOXOXXX<040402> 7-33(0,1,2,4,6,8,A)OOOXOXOXOXOX<262623>
5-34*(0,2,4,6,9)OXOXOXOXXOXX<032221> 7-34(0,1,3,4,6,8,A)OOXOOXOXOXOX<254442>
5-35*(0,2,4,7,9)OXOXOXXOXOXX<032140> 7-35(0,1,3,5,6,8,A)OOXOXOOXOXOX<254361>
5-Z36..12(0,1,2,4,7)OOOXOXXOXXXX<222121> 7-Z36..12(0,1,2,3,5,6,8)OOOOXOOXOXXX<444342>
5-Z36B..12(0,3,5,6,7)OXXOXOOOXXXX<222121> 7-Z36B..12(0,2,3,5,6,7,8)OXOOXOOOOXXX<444342>
5-Z37*..17(0,3,4,5,8)OXXOOOXXOXXX<212320> 7-Z37*..17(0,1,3,4,5,7,8)OOXOOOXOOXXX<434541>
5-Z38(0,1,2,5,8)OOOXXOXXOXXX<212221> 7-Z38(0,1,2,4,5,7,8)OOOXOOXOOXXX<434442>
5-Z38B(0,3,6,7,8)OXXOXXOOOXXX<212221> 7-Z38B(0,1,3,4,6,7,8)OOXOOXOOOXXX<434442>
6-1*(0,1,2,3,4,5)OOOOOOXXXXXX<543210> 6-Z25..47B(0,1,3,5,6,8)OOXOXOOXOXXX<233241>
6-2(0,1,2,3,4,6)OOOOOXOXXXXX<443211> 6-Z25B..47(0,2,3,5,7,8)OXOOXOXOOXXX<233241>
6-2B(0,2,3,4,5,6)OXOOOOOXXXXX<443211> 6-Z26*..48(0,1,3,5,7,8)OOXOXOXOOXXX<232341>
6-Z3..36B(0,1,2,3,5,6)OOOOXOOXXXXX<433221> 6-27(0,1,3,4,6,9)OOXOOXOXXOXX<225222>
6-Z3B..36(0,1,3,4,5,6)OOXOOOOXXXXX<433221> 6-27B(0,2,3,5,6,9)OXOOXOOXXOXX<225222>
6-Z4*..37(0,1,2,4,5,6)OOOXOOOXXXXX<432321> 6-Z28*..49(0,1,3,5,6,9)OOXOXOOXXOXX<224322>
6-5(0,1,2,3,6,7)OOOOXXOOXXXX<422232> 6-Z29*..50(0,1,3,6,8,9)OOXOXXOXOOXX<224232>
6-5B(0,1,4,5,6,7)OOXXOOOOXXXX<422232> 6-30 (6)(0,1,3,6,7,9)OOXOXXOOXOXX<224223>
6-Z6*..38(0,1,2,5,6,7)OOOXXOOOXXXX<421242> 6-30B (6)(0,2,3,6,8,9)OXOOXXOXOOXX<224223>
6-7* (6)(0,1,2,6,7,8)OOOXXXOOOXXX<420243> 6-31(0,1,3,5,8,9)OOXOXOXXOOXX<223431>
6-8*(0,2,3,4,5,7)OXOOOOXOXXXX<343230> 6-31B(0,1,4,6,8,9)OOXXOXOXOOXX<223431>
6-9(0,1,2,3,5,7)OOOOXOXOXXXX<342231> 6-32*(0,2,4,5,7,9)OXOXOOXOXOXX<143250>
6-9B(0,2,4,5,6,7)OXOXOOOOXXXX<342231> 6-33(0,2,3,5,7,9)OXOOXOXOXOXX<143241>
6-Z10..39(0,1,3,4,5,7)OOXOOOXOXXXX<333321> 6-33B(0,2,4,6,7,9)OXOXOXOOXOXX<143241>
6-Z10B..39B(0,2,3,4,6,7)OXOOOXOOXXXX<333321> 6-34(0,1,3,5,7,9)OOXOXOXOXOXX<142422>
6-Z11..40B(0,1,2,4,5,7)OOOXOOXOXXXX<333231> 6-34B(0,2,4,6,8,9)OXOXOXOXOOXX<142422>
6-Z11B..40(0,2,3,5,6,7)OXOOXOOOXXXX<333231> 6-35*(2)(0,2,4,6,8,A)OXOXOXOXOXOX<060603>
6-Z12..41B(0,1,2,4,6,7)OOOXOXOOXXXX<332232> 6-Z36..3B(0,1,2,3,4,7)OOOOOXXOXXXX<433221>
6-Z12B..41(0,1,3,5,6,7)OOXOXOOOXXXX<332232> 6-Z36B..3(0,3,4,5,6,7)OXXOOOOOXXXX<433221>
6-Z13*..42(0,1,3,4,6,7)OOXOOXOOXXXX<324222> 6-Z37*..4(0,1,2,3,4,8)OOOOOXXXOXXX<432321>
6-14..14(0,1,3,4,5,8)OOXOOOXXOXXX<323430> 6-Z38*..6(0,1,2,3,7,8)OOOOXXXOOXXX<421242>
6-14B..14B(0,3,4,5,7,8)OXXOOOXOOXXX<323430> 6-Z39..10(0,2,3,4,5,8)OXOOOOXXOXXX<333321>
6-15(0,1,2,4,5,8)OOOXOOXXOXXX<323421> 6-Z39B..10B(0,3,4,5,6,8)OXXOOOOXOXXX<333321>
6-15B(0,3,4,6,7,8)OXXOOXOOOXXX<323421> 6-Z40..11B(0,1,2,3,5,8)OOOOXOXXOXXX<333231>
6-16(0,1,4,5,6,8)OOXXOOOXOXXX<322431> 6-Z40B..11(0,3,5,6,7,8)OXXOXOOOOXXX<333231>
6-16B(0,2,3,4,7,8)OXOOOXXOOXXX<322431> 6-Z41..12B(0,1,2,3,6,8)OOOOXXOXOXXX<332232>
6-Z17..43B(0,1,2,4,7,8)OOOXOXXOOXXX<322332> 6-Z41B..12(0,2,5,6,7,8)OXOXXOOOOXXX<332232>
6-Z17B..43(0,1,4,6,7,8)OOXXOXOOOXXX<322332> 6-Z42*..13(0,1,2,3,6,9)OOOOXXOXXOXX<324222>
6-18(0,1,2,5,7,8)OOOXXOXOOXXX<322242> 6-Z43..17B(0,1,2,5,6,8)OOOXXOOXOXXX<322332>
6-18B(0,1,3,6,7,8)OOXOXXOOOXXX<322242> 6-Z43B..17(0,2,3,6,7,8)OXOOXXOOOXXX<322332>
6-Z19..44B(0,1,3,4,7,8)OOXOOXXOOXXX<313431> 6-Z44..19B(0,1,2,5,6,9)OOOXXOOXXOXX<313431>
6-Z19B..44(0,1,4,5,7,8)OOXXOOXOOXXX<313431> 6-Z44B..19(0,1,2,5,8,9)OOOXXOXXOOXX<313431>
6-20*(4)(0,1,4,5,8,9)OOXXOOXXOOXX<303630> 6-Z45*..23(0,2,3,4,6,9)OXOOOXOXXOXX<234222>
6-21(0,2,3,4,6,8)OXOOOXOXOXXX<242412> 6-Z46..24B(0,1,2,4,6,9)OOOXOXOXXOXX<233331>
6-21B(0,2,4,5,6,8)OXOXOOOXOXXX<242412> 6-Z46B..24(0,2,4,5,6,9)OXOXOOOXXOXX<233331>
6-22(0,1,2,4,6,8)OOOXOXOXOXXX<241422> 6-Z47..25B(0,1,2,4,7,9)OOOXOXXOXOXX<233241>
6-22B(0,2,4,6,7,8)OXOXOXOOOXXX<241422> 6-Z47B..25(0,2,3,4,7,9)OXOOOXXOXOXX<233241>
6-Z23*..45(0,2,3,5,6,8)OXOOXOOXOXXX<234222> 6-Z48*..26(0,1,2,5,7,9)OOOXXOXOXOXX<232341>
6-Z24..46B(0,1,3,4,6,8)OOXOOXOXOXXX<233331> 6-Z49*..28(0,1,3,4,7,9)OOXOOXXOXOXX<224322>
6-Z24B..46(0,2,4,5,7,8)OXOXOOXOOXXX<233331> 6-Z50*..29(0,1,4,6,7,9)OOXXOXOOXOXX<224232>
Combination possibilities in our twelve-tone system:
Duotonic: (12.11/2!-6)/12 + 1= 6 (= the six intervall-classes)
Tritonic: (12.11.10/3! -6)/12 +1 = 19 (=19 triads)
Tetratonic: (12.11.10.9 / 4! -15)/12 +3 = 43 (=43 tetrachords)
Pentatonic: (12.11.10.9.8 /5!)/12 = 66 (=66 pentatonic combinations)
Hexatonic: (12.11.10.9.8.7 /6! - 24)/12 +5 = 80
Heptatonic: (12.11.10.9.8.7.6 /7!) /12 = 66 (inversion of pentatonisism)
Octotonic: ......................=43 (inversion of tetratonisism)
Enneatonic: .....................=19 (inversion of tritonisism)
Decatonic: ......................=6 (inversion of Duotonisism)
参考文献:
小畑郁男「楽器の音色を視野に入れた音高構成理論の研究 : 感覚的協和理論の音楽への応用」九州芸術工科大学大学院博士論文 2001年
アレン フォート(著), 森あかね(翻訳)「無調音楽の構造: ピッチクラス・セットの基本的な概念とその考察」音楽之友社 2011年
Set Theory Primer
Table of Pitch Class Sets (Set Classes)
Set Theory Glossary