72 II. 1.,,,,,,,,,,,, 2.,, ( ),,,,,,, (,,,,,,,,...,, 3.,,, ( ),,,,,,, y (, y ), t, y = sin t ( II )

41 2012 71 24 3 31 2011,,,, I.,,,,,, (, 440 Hz,, ), ( 580BC 500BC ) ( ),,,,, (, ), 1636 ( 1588 1648,,, ( ), ) ( ), 17,,,,

73 [2, p13], 440 ( ), 440 Hz, y = sin(440 2π t) ( t : ) 440 Hz,, ( ) 440 Hz (, ), : 880 Hz,, ( ( ), 880 Hz, 1636 = k ( ),, 1 2,, 2,, 1, x y y = 2 x

74 ( II ),, 1., (, y = 2 x ) 2., (, 16.5 cm ), 3., 88, ( ) 5 cm (, 5 5.6 cm ),, 5 cm 2 7 = 640 cm = 6 m 40 cm!!, = k ( ),,,, 1 3, 32 = 9, ( ) 274 cm, 205 cm ( 640 cm 1 3 ), 3 2 = 9,,,,, ( ) (,,,,,,,, ),, Klavins Mod. 370 ( ) 303 cm, 27.5 Hz, Klavins Mod. 370, 1 ( 1909 1910 ) 10 22 30 ( [19, p39] ) 28 ( C1=32.7 Hz ) (, 3 1 ), 50 Hz, Gülsin Onay Klavins Mod.370 1989, (, 1980, 2

ピタゴラス音律 75 満足できず ( 日本の湿度が理由 ), やむなくヤマハのピアノを使用したが, 演奏を行ったのは１公演だけであり, 他の公演は全てキャンセル怒った日本の招聘元がミケランジェリのピアノを差し押さえるという事件が起こった ) の弾くコンサートグランドピアノ ( 1971 年のライブ録音だが録音は極めて優秀 ) である ( 実際に聴き比べを行う少し退屈そうにしていた学生が, 気持ちを集中させる http://www.klavins-pianos.com/index en.htm より引用注釈. 私見では, この２つのピアノは, ピアノという同じ名称を持っているが, 響きはかなり異なり, 同種の楽器とはとても思えない Klavins は, コンサートグランドピアノに比べ, 低い

76,, Klavins,, ( ),, Klavins,, (,,, 16 (, 1565 ), 14 ), Klavins Klavins, ( 4.,,,, ( music temperament ) (, ),,,,,,, ( scale ),,,,,,,, (2212221),,,,,,, (2212221),,,,,,, (2122122) ( 1, 2 ) (2212221), (2122122)

77,, ( church mode ) (, ( gregorian mode ) ),,,,,,, 2212221,,,,,,, 2122212,,,,,,, 1222122,,,,,,, 2221221,,,,,,, 2212212,,,,,,, 2122122,,,,,,, 1221222.,, 1.,, 1 ( : 590 604 ), ( ),,, 2.,, ( ),,, 1, 8,,,,, 9 3., 440 Hz,,,,,,,,,,, ( 6 ) (2212221). ( ) 440 Hz, 1939 5,

78,,,, 17 370 560 Hz, 415 Hz, 422 Hz, 1800 428 Hz, 19,, 1884 432 Hz (,, ),, 442 448 Hz (, ( 1955 1989 ) 446 Hz ) 4.,,,,,,, 1,,,,,,,, 1525 (, ) 5. ( ), ( ) 1600, 19, 6.,,,, ( ), ( ), ( ), ( ),

79,, ( finalis ),,, ),,,,,,,, (, 8, 9 10,, 2 ), (, 5, 6 7 ) 7.,,,,,,,, 8.,, 6,,,, ( ), ( ) 20,,, Kawai ( ),,, (,

80,,, ( 1983 ),,,,,,,,

81, ( pentatonic scale ), 1 5,,,,,, (22323) ( 4 7 ), (, ),, ( ),,,,, 5 5 (1) (2),,,,, (23223), ) (,,,,,, (14214), (3),,,,, (23232), (4),,,,, (32232),, ( 2 6,,,,, (32232) ) (5),,,,, (41241),, ( 2 6 ) 5.,, (, ), 500,, 12,,,,, ( ), 1. ( ),

82 2.,, 3., 4. 16, 5., (,,,,,,,,,... 12,,, ( ), 2, 3, { } x : 1 2 < x < 1 ( ),,, 2 3,,, 2 3, 12, 2 (, 1 ), 1 (,,,, 2 ( ),, 1 2 ( ),,, 1 ( ), 2 (, 2 (, 1 2, 1 ), ( 1 ) (, 1 ), 2 2 3

83 ( 1, 2 1., 2., 2 3,, 20 3.,, 2 3, 3 2, 3 : 2, 1 2, 2 2, 2, 2 2 3, 2 3 2 3 = 22 3 2 = 4 9,, 3, ( 1 ) ( 1 2, 4 9,,,, 3 4 9 2 ( = 23 3 2 ) = 8 9 ( > 1 2 ) ( 3, 1 2 ( 7 ), 3,, 1, 2 5,, ( 2 12, ),,,,,, 3, 3, 3 2 3, 2 3 3 2 2 3 = 24 3 3 = 16 27 ( > 1 2 )

84,, 4 ( n + 1 n, n, n 2 3, 1 2, n + 1,,, n 2 3 2 n + 1,,, 1,, 1 1 1 1 2 1 2 3 = 2 3 0.667 3 2 = 1.5 3 4 5 6 7 8 9 10 11 12 13 2 3 2 3 2 = 23 2 3 3 2 2 3 1 = 24 2 4 3 3 2 3 2 = 26 2 6 3 4 2 3 1 = 27 2 7 3 5 2 3 2 = 29 3 2 0.889 3 2 3 3 0.593 3 3 3 4 0.790 3 4 3 5 0.527 3 5 3 6 0.702 3 6 2 9 3 6 2 3 2 = 211 3 7 0.936 2 11 3 7 2 3 1 = 212 3 8 0.624 2 12 3 8 2 3 2 = 214 3 9 0.832 2 14 3 9 2 3 1 = 215 2 15 3 10 2 3 2 = 217 2 17 3 11 2 3 2 = 219 3 10 0.555 3 10 3 11 0.740 3 11 3 12 0.987 3 12 1.13 23 1.69 24 1.27 26 1.90 27 1.42 29 3 7 1.07 211 3 8 1.60 212 3 9 1.20 214 1.80 215 1.35 217 1.01 219

85, 13, 1 0.987, 13 13, 1 12,., 1, 2 m 3 n ( m, n: ) 1 (, m, n 2 m = 3 n ), 2 3,,,, 25 0.973, 37 0.960 1, 13 0.987 1, 12, (, ), 1 : 2 3 = 3 : 2 (, 3 ), 2 : 3 (, 2 ), (, ), 12 215 3 10 2 3 2 = 217 3 11 0.740 3 11 1.35 217,, 2 17 3 11 : 1 2 3 : 2.03, 3 : 2, 3 11 : 2 2 : 2.96 217, 2 : 3, ( 1 % 1/4 ) (,,, 9 10 )

86, 1 7,,,,,,,, (2221221), ( 440 Hz ), (2221221),,,,,,,, (2221221),, 1 1. 1,,, 1, 2, 3,, 7,,,,,,,, (2221221) 1 1 1 2 1 2 3 = 2 3 0.667 1.5 3 4 5 6 7 2 3 2 3 2 = 23 0.889 1.13 32 2 3 3 2 2 3 1 = 24 0.593 1.69 33 2 4 3 3 2 3 2 = 26 0.790 1.27 34 2 6 3 4 2 3 1 = 27 0.527 1.90 35 2 7 3 5 2 3 2 = 29 0.702 1.42 36 2. 1 7 7, 1 5,,,,, (22323) 5.,,,,, ( ) ( ) ( ),, ( cent ) ( )

87, 1 1200, X, Y x ( x 1, X Y ),, 2 ( II ) 1200 log 2 x, 1 : 2 1 12, 1200 log 2 2 1 12 = 1200 1 12 = 100 ( ), 1 12, (, y = 2 x, 1 x 1, 12 y,, ),,,,,, (,,,, 3 ) ( ) 1 0 1 0 1 0 3 7 2 11 = 2187 2048 1.07 113.69 2 1 12 1.06 100 3 2 = 9 2 3 8 1.13 203.91 2 12 2 3 2 1.12 200 2 5 3 3 = 32 27 1.19 294.13 2 3 12 1.19 300 2 3 = 9 8 3 4 2 6 = 81 64 1.27 407.82 2 4 12 1.26 400 5 2 2 = 5 4 2 2 3 = 4 3 1.33 498.04 2 5 12 1.33 500 22 3 1 = 4 3 3 6 2 9 = 729 512 1.42 611.73 2 6 12 1.41 600 3 2 = 1.5 701.96 2 12 7 3 1.50 700 2 3 8 2 12 = 6561 4096 1.60 815.64 2 8 12 1.59 800 3 3 = 27 2 4 16 1.69 905.87 2 12 9 5 1.68 900 3 2 4 = 16 3 2 9 3 5 = 243 2 7 128 1.78 996.09 2 10 12 1.78 1, 000 1.90 1, 109.78 2 11 12 1.89 1, 100 3 5 2 3 = 15 8 1.13 203.91 = 1.25 386.31 1.33 498.04 = 1.5 701.96 1.67 884.36 1.88 1, 088.27 2 1, 200 2 1, 200 2 1, 200

88,, ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 13.69 3.91 0 3.91 5.87 7.82 21.51 13.69 1.96 0 1.96 11.73 1.96 0 1.96 15.64 5.87 21.51 15.64 3.91 9.78 21.51 11.73 0 0 0,, (,, 5 ( 2 3 ), ),,,, 13.69,,,,,, [10] (, ),,, 2,,,,,, 5 4, 2 3 ( 1 : 2 ),, 5 701.96, 5 700,,,,,,

89,,, 7,,,, 4 ( ),,,,,,, 5.,, ( harmonia : ),,,,,,,, ( musica : ),, 6, 800,,,, 7 ( Seven Liberal Arts ) 1. 2. 3. 4. ( ) 5. ( ) 6. ( ) 7. ( )

90 12,,, 7 15,,,., ( ) ( liberal arts ), 7,,,,,, 9, ( ), 12, 16,,, ( 1600 1650 ),,,,, III., (,,,,,

91,,,,, sin,, 12,,,,,,,,,,,,,,,,,,

92,,,,,,,,,,,,,,,,,, 2500,,,,,,,,,,

93,,,,,,,,,,,,,,., ( ),, Klavins Piano,, [1],, 1999, [2],, 2009, [3],, 2007, [4],, 2010, [5],, 1999, [6],, 1979, [7],, 1964,

94 [8],,!!, 2009, [9],, 1999, [10],, 2003 9 2006 3 31, [11],, 2004, [12],, 2008, [13] ;, 17, 2008, [14],, 1999, [15],, 1970, [16],, 1977, [17] ; V. ( ), ;, ( ), /, 1998, [18], 1977, [19] C. Debussy, Complete preludes Books 1 and 2, 1989, Dover publications, Inc. [20] The Beatles complete scores, 1993, Hal Leonard Publishing Corportion and Wise Publications [21] Gülsin Onay live on Klavins Mod. 370, Klavins KM007 (CD) [22] Arturo Benedetti Michelangeli plays Debussy, Deutsche Grammophon 449 438-2 (CD) [23] Gregorian Chant Gregorianischer Choral, Choralschola der Wiener Hofburgkapelle, P. Hubertt Dopf S.J., Philips 32CD-36 (CD)