＜純律和12平均律比較＞ 傅國權1991著2003.3.3訂
純律音階大3度5/4 小3度6/5 完全5度3/2 完全8度2/1
純律 和絃產生法 12平均律
1.A4=440 D<F< A >C>E>G>B
2.C4= A4_C5 =440x 6/5/2=264.00 15cent Am= A>C>E 261.63
3.E4= C4_E4 =264x 5/4 =330.00 2cent Am= A>C>E 329.63
4.G4= E4_G4 =330x 6/5 =396.00 17cent Em= E>G>B 392.00
5.B4= G4_B4 =396x 5/4 =495.00 4cent Em= E>G>B 493.89
13B=若#F=366.66則B=488.88-16cent Bm=B>D#F
6.F4= A4_F4 =440x 4/5 =352.00 13cent Dm= D<F<A 349.23
7.D4= F4_D4 =352x 5/6 =293.33 -2cent Dm= D<F<A 293.66
14D4= B4_D4 =495x 5/6/2 =297.00
8.#C4 #C5_A4 =440x(5/4)/2 =275.00 -14cent A = A>#C E 277.18
15=#c=#Fx3/2=371.25x3/2=278.44(7cent) #F=#F>#A>#C
9.#F4 B_#F =495x3/2/2 =371.25 6cent B = B>#D #F 369.99
16= A4_#F4 =440x 5/6 =366.66 -15cent D = D #F<A 369.99
#A4 17= #A4_#F =366.66x5/4=458.33(-23cent)#F= #F>#A #C 466.16
#A=#D_#A=309.375x3/2=464.06
18=#A=#Fx5/4=371.25x5/4=464.06(-7cent) #F=#F>#A>#C
#D4 19= B4_#D4 =495x(5/4)/2=309.375 -8CENT B = B>#D #F 311.13
#F4_#D4=366.66x5/6 =305.55 (bEm)#Dm=#D>#F>#A
10.#G4 #G4_E4 =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
C #F
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 G
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 #G
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 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 bB
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 B
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
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 #G
#D-#C=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 #A4
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 bA4
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 bB4
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
bG4 #F4 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 D4
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
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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 bB7和弦 469.33 (11cent)
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)
Am 小調純律 Am 和絃 A=440 C=261.25(-2cent) E=330(2cent)
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
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