大三度純律和12平均律比較 (30個基音) 傅國權 1991著 2007.10.5訂純律和絃產生法 12平均律 1.A4=440 DC>E>G>B 2.F4= A4_F4 =440x 4/5 =352.00 13cent F= F>A>C 349.23 3.C4= F4_C5 =352x 3/2/2=264.00 15cent F= F>A>C 261.63 4.E4= C4_E4 =264x 5/4 =330.00 2cent C= C>E>G 329.63 5.G4= C4_G4 =264x 3/2 =396.00 17cent C= C>E>G 392.00 6.B4= G4_B4 =396x 5/4 =495.00 4cent G= G>B>D 493.89 13B=若#F=366.66則Bx3/2=366.66 B=488.88-16cent Bm=B>D#FF 7.D4= A4_D4 =440x 2/3 =293.33 -2cent D= D>#F>A 293.66 14.D4=G4_D5=396x3/2/2 =297.00 19cent G= G>B>D 8.#C4 #C5_A4 =440x(5/4)/2 =275.00 -14cent A= A>#C E 277.18 15.#c4=#Fx3/2=371.25x3/2 =278.44 7cent #F=#F>#A>#C 9.#F4 B4_#F5 =495x3/2/2 =371.25 6cent B= B>#D #F 369.99 16.#F4= A4_#F4 =440x 5/6 =366.66 -15cent D= D #F#A #C 466.16 #A=#D_#A=309.375x3/2=464.06 18.#A4=#Fx5/4=371.25x5/4 =464.06(-7cent)#F=#F>#A>#C 27.#A4=#D_#A=310x3/2=465 19.#D4= B4_#D4 =495x(5/4)/2 =309.375 -8CENT B= B>#D #F 311.13 10.#G4 E4_ #G4 =330x 5/4 =412.50 -12cent E= E> #G B 415.30 20=#C_#G=278.44x 5/4 =417.66 11.bB4 bB4_D5 =293.33x4/5x2=469.33 11cent bB= bB>D>F 466.16 bB4_F5 =352x2x2/3 =469.33 11cent bB= bB>D>F 466.16 21=bE4_bB4=316.8x3/2 =475.2 34cent bE =bE>G>bB 466.16 G4_bB4 =396x6/5 =475.2 34cent Gm =G>bB.D 466.16 bD4 22=bB_bD5=469.34x(6/5)/2=281.604(25cent)bB=bB>bD>F 277.18 23=bD4_F4 =352 x(4/5) =281.6(25cent) bD=bD>F>bA 277.18 bG4 24=bB4_bG4=469.34x 4/5=375.472(23cent) bG=bG>bB>bD 369.99 bA4 25=bA4_C5 =264x(4/5)x2= 422.4 (30cent) bA=bA>C>bE 415.3 bD4_bA5=281.6x3/2 =422.4 bD=bD>F>bA 415.3 bD_bA=281.6x 3/2=422.4 (30cent) bA=bD>F>bA 415.3 12.bE4 bB4_bE4 =469.33x2/3=312.89(9cent) bEm=bE>bG>bB 311.13 26=C4_bE4=264x(6/5) =316.8 (32cent) Cm= C>bE>G 311.13 C_bE 264x(6/5) =316.8 (32cent) bA=bA>C>bE 311.13 bA-bE 422.4x3/2/2=316.8 (32cent) bA=bA>C>bE 311.13 bE4_G4 =396x4/5 =316.8 (32cent) bE=bE>G>bB 311.13 27=bA bA-bE bAx3/2=bE bAx3/2=312.89 bA=417.1866(7cent) 28=bD bD-bA bDx3/2=bA bDx3/2=417.18 bD=278.1244(5cent) 29=bG bGx3/2=bD bGx3/2=278.1244 bG=370.83(3cent) 30=F=bBx 3/2/2=469.33x 3/2/2=351.99 bB=bB>D>F 純律音階大2度9/8,10/9 小3度6/5 大3度5/4 完全4度5/4 完全5度3/2 大7度15/8 完全8度2/1 C 2 #F 9 293.33 - 264 = 29.33 Hz 412.5 - 371.25 = 40.75 200 /29.33 = 6.8189 CENT 200 / 40.75 =4.907975 264 - 261.63 = 2.37 Hz 371.25-369.99 = 1.26 2.37 x 6.8189 = 16.16 CENT 1.26x4.907975=6.184CENT C264 = 16 CENT #F371.25 = 6 CENT 16cent=264.0972>0.0972 15cent=263.9299<0.0701 C264 = 15 CENT #C=275Hz 8 G 4 305.55 - 275 = 30.55 440 - 396 = 44 200 / 30.55=6.5466CENT 200 / 44 = 4.54 275 - 277.18 =-2.18 396 - 392 = 4 6.5466x -2.18 =- 14.27 CENT 4 x 4.54 = 18.16 CENT #C275 = -14 CENT G396 = 18 CENT -14cent=274.9608<0.0392 18cent=396.1875>0.1875 -13cent=275.1424>0.1424 17cent=395.9365<0.0635 #C275 = -14 CENT G396 = 17 CENT D 7 #G 8 330 - 293.33=36.67 458.33 - 412.5 = 45.83 200/36.67= 5.45 CENT 200 / 45.83 = 4.36 CENT 293.33-293.66=-0.33 412.50 - 415.30 = -2.8 -0.33x5.45 =-1.79 CENT -2.8 x 4.36 = -12.2 CENT D293.33=-2 CENT #G412.50 = -12 CENT -12cent=412.5320>0.0320 -11cent=412.6681<0.1681 #G412.50 = -12 CENT bE 12 A 352 - 312.89 = 39.11 A = 440 200 / 39.11 = 5.11 CENT 312.89 - 311.13 =1.76 5.11 x 1.76 =8.993CENT bE312.89=9 CENT A440 = 0 CENT E 3 bB 11 366.66 - 330 = 36.66 528-469.33 =58.67 200 / 36.66 = 5.45 CENT 200 /58.67=3.409 CENT 330 - 329.63 = 0.37 469.33 - 466.16 = 3.173 0.37 x5.45 = 2.21 CENT 3.409 x 3.173 = 10.816 E330 = 2 CENT bB 469.33= 11 CENT 11cent=469.2083<0.1217 12cent=469.5610>0.2310 bB469.33=11 CENT F 6 B 5 396 - 352 = 44 550 - 495 = 55 200 / 44 = 4.54 CENT 200 / 55 = 3.63 CENT 352 - 349.23 = 2.77 495 - 493.89 = 1.11 2.77 x 4.54 = 12.57 CENT 1.11 x 3.63 = 4.02 CENT 352 = 13 CENT B495 = 4 CENT 13cent=351.9064<0.0936 4cent=495.0380>0.0380 14cent=352.1047>0.1047 3cent=494.8420<0.1580 F352 = 13 CENT B495 = 4 CENT #A4 18 523.23-464.06=59.17 200/59.17=3.38 464.06-466.16=-2.1 -2.1x 3.38=-7.1 #A464.06=-7CENT #C4 8 #G #D4-#C4=311.13-277.18=33.95 200/33.95=5.89 278.44-277.18=1.26 5.894x1.26=7.42 #C4=7cent #D4 19-1 (12平均律音差) 349.23 - 311.13=38.1 200/38.1=5.2493cent 309.375-311.13=-1.775 -1.775x5.2493=-9.21 #D309.375 =-9 CENT --------------------------------------------------------- #D4 19 (純律平均律音差) #A4 17 352 - 309.375 = 42.625 528 - 458.33=69.67 200 / 42.625= 4.672 CENT 200 /69.67=2.87CENT 309.375 - 311.13 =-1.755 458.33-466.16=-7.83 4.672x -1.755=-8.2 -7.83x2.87 = -22.47 #D309.375 =-8 CENT #A458.33 = -23 CENT bD4 22 bA4 25 316.8-281.604=35.196 469.34-422.4=46.94 200 / 35.196 =5.68246 200 / 46.94 =4.26 281.604-277.18=4.424 422.4-415.3 =7.1 4.424 x5.68246=25.1 7.1 x 4.26 =30.24 bD281.604=25 CENT bA422.4=30 CENT bE4 26 bB4 21 352-316.8=35.2 528-475.2=52.8 200 /35.2=5.681818 200/52.8=3.78787 316.8-311.13=5.67 475.2-466.16=9.04 5.67x5.681818=32021 9.04x3.78787=34.2423 bE316.8=32CENT bB475.2=34CENT #F4 16 bG4 24 412.5 - 366.66 = 45.84 422.4-375.472=46.928 200 / 4 .84 = 4.36 CENT 200 / 46.928 =4.2618 366.66 369.99 =-3.33 375.472-369.99=5.482 -3.33x .36 =-14.5 CENT 5.482x4.2618=23.363 #F366.6 =-15 CENT bG4 375.472 =23 CENT -16cent=366.6675>0.0009 -17cent=366.4531<0.2135 #F366.66 = -16 CENT D4 14-1 D4 14 329.63-293.66=35.97 330-297=33 200/35.97=5.56 200/33=6.06 297-293.66=3.34 297-293.66=3.34 3.34x5.56=18.57 3.34x6.06=20.24 D297=19cent D297=20cent G和弦B和弦不適用前著可用G7和弦代替 D有2個音 G和弦5 7 2 D=297 不採用 Dm 和弦2 4 6 D=293.33 近A泛音純律293.25 #F有2個音 B和弦7 #2 #4 #F=371.25 同A泛音純律與12平均律音距較近有2個音 D和弦2 #4 A #F=366.66不採用.只用在G大調 ------------------------------------------------------------------------ 純律純5度(3/2) +大3度(4/5) F和弦 _____440_______ F - A C 352 528 bB和弦 C和弦 __________ ________ bB - D F C - E G 469.33 330 396 D和弦 A和弦 __________ ________ D #F - A -#C E 293.33 366.66 440 275 330 bE和弦 E和弦 B和弦 ________ ___________ ___________ bE-G bB E - #G B B - #D- #F 312.89 469.33 412.5 495 495 371.25 (bE=bE4_G4 =396x4/5 =316.8 (32cent) bE =bE>G>bB 311.13 ) #A bB 黑鍵有6個音 457.8525 458.325 462 467.5 469.33 475.2 1. C261.63第7泛音bB=457.8525 2. #F366.33大3度#A=458.325 3. C264泛音純律第7音bB=462 -13cent C7=1 3 5 b7 4. A440 第17泛音bB=467.5 4.86cent 5. bB=bE4_bB4=316.8x3/2 =475.2 34cent bE =bE>G>bB bE=(bE4_G4 =396x4/5 =316.8 32cent bE =bE>G>bB 311.13 6. bB4_D5 =293.33x4/5x2 =469.33 11cent bB= bB>D>F 466.16 F352 第21泛音bB=462 ( 同3) -13cent ( D293.33第51泛音bB=467.5 同4) 1. A=440 2. F=440x4/5=352 F和弦 3. C=352x3/2=528/2=264 F和弦 4. E=264x5/4=330 C和弦 5. G=264x3/2=396 C和弦 6. bB=352x2/3x2=469.33 bB和弦 7. #C=440x5/4/2=275 A和弦 8. D=440x2/3=293.33 D和弦 9. #F=293.33x5/4=366.66 D和弦 10.#G=330x5/4=412.5 E和弦 11.B =330x3/2=495 E和弦 12.bE=469.33x2/3=312.89 bE和弦 1. 純律應用實例少女的祈禱the maidens prayer (t.badarzewska) 本曲bE調和弦 bA bB7 bE Fm bA=bA>C>bE bA=bA4_C5 =264x(4/5)x2=422.4 (30cent) bE=C_bE =264x(6/5) =316.8 (32cent) bE=bA-bE 422.4 x3/2/2 =316.8 (32cent) 若 C=264則 bA=422.4(30cent) bE=316.8 (32cent) bE =bE>G>bB bE=bE4_G4 =396x4/5 =316.8 (32cent) 若 G=396則 bB=bE4_bB4 316.8x3/2 =475.2 (34cent) bB7=bB>D>F>bA bB=bB4_D5 293.33x4/5x2=469.33(11cent) 本首曲子 bB有2個音 bB= bB7和弦 469.33 (11cent) bB= bE 和弦 475.2 (34cent) 本首曲子主調bE大調結束故和弦採用 475.2 (34cent) 2. 純律應用實例大本詩歌283首羨慕-與主交通本曲G 調和弦有 G G7 D7 G=396(17cent) B=495( 4cent) D=297 #F=371.25(6cent) 3. 泛音純律樂曲應用實例給愛麗絲 (貝多芬) for elise (Beethoven) 本曲和弦 C7 F Gm7 C G7 A9 G Cm E7 Am #D7 A9 = A #C E G bB bB=467.5Hz (A=440之17泛音5cent) G391.88 57泛音-0.5cent C7 = C E G bB bB=462Hz (C=264之7泛音-12.6cent) G396 3泛音 17 cent C7和弦之bB=462 bB有4個音 12平均律bB=466.16 528-462=66 bB和弦469.33=11cent 200/66=3.0303 bE和弦475.2 =34cent 462-466.16=-4.16 C7和弦462=-12.6cent -4.16x3.0303=-12.6 A9和弦467.5 = 5cent 若是3度純律 bB有2個音 bB7和弦 469.33(11cent) bE和弦 475.2(34cent) * 在此上述2音皆不和諧採用泛音純律A9和弦bB=467.5Hz (A=440之17泛音5cent) 也較近12平均律傳統聽覺習慣. G大調純律D=297 4. F大調大本詩歌167首有C7和絃 F調當C=264 C7=C E G bB C264泛音純律第7音bB=462 (-13cent) 5. 大本詩歌405首 F大調純律bB以C7和弦為主=C E G bB 當C=264 E-330 G=396 bB=462 (-13cent) 附記:若D大調Dm和絃 D=293.33為基音時其19泛音的 F=348.33( -4CENT) D 和絃 D=293.33為基音時其 5泛音的#F=366.66(-15CENT) 這時音階分布上有點異樣感.因為G=396(18CENT) #F-G純律音階和12平均律音階有33CENT之差在音程上有大距離故要特別留意旋律進行時特殊現象. C大調純律 C 和絃 C=264(16cent) E=330(2cent) G=396(18cent) A小調純律 Am 和絃 A=440 C=261.25(-2cent) E=330(2cent) D小調和絃 Dm 和絃 D=293.33其19泛音 F=348.33(-4cent) A=440 結論:最好的調音應該以曲子作和聲學的和絃頻率分析再來訂律製音階. 和絃純律PURE CHORD TEMPERMENT 大提琴定弦 C2 G2 D3 A3 12平均律 65.406 98.000 146.83 220.00 五度相生律65.185 97.775 146.66 220.00 -5cent -4cent -2cent 0cent 大提琴C大調五度定弦之純律音階 12平均律 27 C2=Ax2/3 x2/3 x2/3=65.185(c4=260.74) 比值 -6cent C 261.63 #C4=275 比值 -14cent #C277.18 D3=A3x2/3=146.66(D4=293.33) 比值 9/8 -2cent D293.66 28 #D=Bx5/4=122.22x5/4=152.78(#D4=305.56) 比值 -25cent #D311.13 29 E2=65.185x5/4 =81.49 (E4=325.96) 比值 -20cent E329.63 F=第19分音440x2/3 x19=5573.3/16=348.33 比值 76/57 -4cent F349.23 #F3=183.33(#F4=366.66 ) 比值 -16cent #F369.99 30 G2=Ax2/3x2/3=97.775 (G4=394.16) 比值 3/2 -9cent G392.00 #G=97.775 =412.50 -12cent #G415.30 A3=220.00(A4=440) 比值 96/57 0 cent A440.00 31 #A=#Fx5/4=366.66x5/4=458.33 比值 -23cent #A466.16 32 B2=G2x5/4=97.775x5/4=122.22(B4=488.89=391.11x5/4)比值 -18cent #B493.89