音樂中泛音純律之研究－未來人類最完美之律　　　　2001.02.15訂1991傅國權著

四以基音D=Ax2/3=440x2/3 (293.33)HZ 為起音,推算各頻率值.
泛音序列 頻率值 音名
1 440x2/3 x1 =293.33 D
2 440x2/3 x2 =586.66/2 =293.33 D
3 440x2/3 x3 =880.00/2 =440.00 A
4 440x2/3 x4 =1173.3/4 =293.33 D
5 440x2/3 x5 =1466.6/4 =366.66 #F
6 440x2/3 x6 =1760.0/4 =440.00 A
7 440x2/3 x7 =2053.3/4 =513.33
8 440x2/3 x8 =2346.6/8 =293.33 D
9 440x2/3 x9 =2640.0/8 =330.00 E
10 440x2/3 x10=2933.3/8 =366.66 #F
11 440x2/3 x11=3226.6/8 =403.33
12 440x2/3 x12=3520.0/8 =440.00 A
13 440x2/3 x13=3813.3/8 =476.66 #D=311.66=3cent
14 440x2/3 x14=4106.6/16=256.66 349.23-311.66=37.57
15 440x2/3 x15=4400.0/16=275.00 #C 200/37.57=5.323
16 440x2/3 x16=4693.3/16=293.33 D 311.66-311.13=0.53
17 440x2/3 x17=4986.6/16=311.66 #D 0.53x5.323=2.82=3cent
18 440x2/3 x18=5280.0/16=330.00 E
19 440x2/3 x19=5573.3/16=348.33 F G392-F349.23=42.77
20 440x2/3 x20=5866.6/16=366.66 #F 200/42.77=4.67617
21 440x2/3 x21=6160.0/16=385.00 349.23-348.33=0.9
22 440x2/3 x22=6453.3/16=403.33 0.9x4.67617=4.2
23 440x2/3 x23=6746.6/16=421.66 #G F=348.33=-4cent
24 440x2/3 x24=7040.0/16=440.00 A
25 440x2/3 x25=7333.3/16=458.33
26 440x2/3 x26=7626.6/16=476.66
27 440x2/3 x27=7920.0/16=495.00 B
28 440x2/3 x28=8213.3/32=256.66
29 440x2/3 x29=8506.6/32=265.83
30 440x2/3 x30=8800.0/32=275.00
31 440x2/3 x31=9093.3/32=284.16
32 440x2/3 x32=9386.6/32=293.33
33 440x2/3 x33=9680.0/32=302.50
34 440x2/3 x34=9973.3/32=311.66
35 440x2/3 x35=10266 /32=320.83
36 440x2/3 x36=10560 /32=330.00
37 440x2/3 x37=10853 /32=339.16
38 440x2/3 x38=11146 /32=348.33
39 440x2/3 x39=11440 /32=357.50
40 440x2/3 x40=11733 /32=366.66
41 440x2/3 x41=12026 /32=375.83
42 440x2/3 x42=12320 /32=385.00
43 440x2/3 x43=12613 /32=394.16 G
44 440x2/3 x44=12906 /32=403.33
45 440x2/3 x45=13200 /32=412.50 #G
46 440x2/3 x46=13493 /32=421.66
47 440x2/3 x47=13786 /32=430.83
48 440x2/3 x48=14080 /32=440.00
49 440x2/3 x49=14373 /32=449.16
50 440x2/3 x50=14666 /32=458.33
51 440x2/3 x51=14960 /32=467.50 #A
52 440x2/3 x52=15253 /32=476.66
53 440x2/3 x53=15546 /32=485.83
54 440x2/3 x54=15840 /32=495.00
55 440x2/3 x55=16133 /64=252.07
56 440x2/3 x56=16426 /64=256.65
57 440x2/3 x57=16720 /64=261.25 C
58 440x2/3 x58=17013 /64=265.82
59 440x2/3 x59=17306 /64=270.41
60 440x2/3 x60=17600 /64=275.00
61 440x2/3 x61=17893 /64=279.58
62 440x2/3 x62=18186 /64=284.16
63 440x2/3 x63=18480 /64=288.75
64 440x2/3 x64=18773 /64=293.33 D
65 440x2/3 x65=19066 /64=297.91
66 440x2/3 x66=19360 /64=302.50
67 440x2/3 x67=19653 /64=307.08
68 440x2/3 x68=19946 /64=311.66
69 440x2/3 x69=20240 /64=316.25
70 440x2/3 x70=20533 /64=320.83
71 440x2/3 x71=20826 /64=325.41
72 440x2/3 x72=21120 /64=330.00 E
73 440x2/3 x73=21413 /64=334.58
74 440x2/3 x74=21706 /64=339.16
75 440x2/3 x75=22000 /64=343.75
76 440x2/3 x76=22293 /64=348.33
77 440x2/3 x77=22586 /64=352.91
78 440x2/3 x78=22880 /64=357.50
79 440x2/3 x79=23173 /64=362.07
80 440x2/3 x80=23466 /64=366.66
81 440x2/3 x81=23760 /64=371.25
82 440x2/3 x82=24053 /64=375.83
83 440x2/3 x83=24346 /64=380.41
84 440x2/3 x84=24640 /64=385.00
85 440x2/3 x85=24933 /64=389.58
86 440x2/3 x86=25226 /64=394.16 c=261.25 -2 cent 57泛音
87 440x2/3 x87=25520 /64=398.75 #c=275 -14cent 15泛音
88 440x2/3 x88=25813 /64=403.33 d=293.33 -2 13泛音
89 440x2/3 x89=26106 /64=407.91 #d=311.66 3 17泛音
90 440x2/3 x90=26400 /64=412.50 e=330 2 9泛音
91 440x2/3 x91=26693 /64=417.08 f=348.33 -4 19泛音
92 440x2/3 x92=26986 /64=421.66 #f=366.66 -16 5泛音
93 440x2/3 x93=27280 /64=426.25 g=394.16 -9 43泛音
94 440x2/3 x94=27573 /64=430.83 #g=412.5 -12 45泛音
95 440x2/3 x95=27866 /64=435.41 a=440 0 3泛音
96 440x2/3 x96=28160 /64=440.00 #a=467.5 5 51泛音
97 440x2/3 x97=28453 /64=444.58 b=495 4 27泛音
98 440x2/3 x98=28746 /64=449.16
99 440x2/3 x99=29040 /64=453.75
100 440x2/3x100=29333 /64=458.33
五.以基音F=Ax4/5=440x4/5 (352.00)HZ 為起音,推算各頻率值.
泛音序列 頻率值 音名
1 352x1 =352 F
2 352x2 =704 /2=352 F
3 352x3 =1056 /4=264 C
4 352x4 =1408 /4=352 F
5 352x5 =1760 /4=440 A
6 352x6 =2112 /8=264 C
7 352x7 =2464 /8=308
8 352x8 =2816 /8=352 F
9 352x9 =3168 /8=396 G
10 352x10=3520 /8=440 A
11 352x11=3872 /8=484
12 352x12=4224/16=264 C
13 352x13=4576/16=286
14 352x14=4928/16=308 bE
15 352x15=5280/16=330 E
16 352x16=5632/16=352 F
17 352x17=5984/16=374 #F
18 352x18=6336/16=396 G
19 352x19=6688/16=418
20 352x20=7040/16=440 A
21 352x21=7392/16=462 bB
22 352x22=7744/16=484
23 352x23=6746/16=421 bA
24 352x24=7040/16=440 A
25 352x25=8800/32=275 #C
26 352x26=9152/32=286
27 352x27=9504/32=297 D 不採用
28 352x28=9856/32=308
29 352x29=10208/32=319
30 352x30=10560/32=330 E
31 352x31=10912/32=341
32 352x32=11264/32=352 F
33 352x33=11616/32=363
34 352x34=11968/32=374
35 352x35=12320/32=385
36 352x36=12672/32=396 G
37 352x37=13024/32=407
38 352x38=13376/32=410
39 352x39=13728/32=429
40 352x40=14080/32=440
41 352x41=14432/32=451
42 352x42=14784/32=462
43 352x43=15136/32=473
44 352x44=15488/32=484
45 352x45=15840/32=495 B
46 352x46=16192/64=253
47 352x47=16544/64=258.5
48 352x48=16896/64=264 C
49 352x49=17248/64=269.5
50 352x50=17600/64=275 #C
51 352x51=17952/64=280.5
52 352x52=18304/64=286
53 352x53=18656/64=291.5
54 352x54=19008/64=297
55 352x55=19360/64=302.5
56 352x56=19712/64=308
57 352x57=20064/64=313.5 #D
58 352x58=20416/64=319
59 352x59=20768/64=324.5
60 352x60=21120/64=330
61 352x61=21472/64=335.5
62 352x62=21824/64=341
63 352x63=22176/64=346.5
64 352x64=22528/64=352
65 352x65=22880/64=357.5
66 352x66=23232/64=363
67 352x67=23584/64=368.5
68 352x68=21120/64=330
69 352x69=24288/64=379.5
70 352x70=24640/64=385
71 352x71=24992/64=390.5
72 352x72=25344/64=396
73 352x73=25696/64=401.5
74 352x74=26048/64=407
75 352x75=26400/64=412.5 #G
76 352x76=26752/64=418
77 352x77=27104/64=423.5
78 352x78=27456/64=429
79 352x79=27808/64=434.5
80 352x80=28160/64=440
81 352x81=28512/64=445.5
82 352x82=28864/64=451
83 352x83=29216/64=456.5
84 352x84=29568/64=462
85 352x85=29920/64=467.5
86 352x86=30272/64=473
87 352x87=30624/64=478.5
88 352x88=30976/64=484
89 352x89=31328/64=489.5
90 352x90=31680/64=495
91 352x91=32032/64=500.5
92 352x92=32384/64=506
93 352x93=32736/64=511.5
94 352x94=33088/64=517
95 352x95=33440/64=522.5
96 352x96=33792/128=264 C
97 352x97=34144/128=266.75
98 352x98=34496/128=269.5
99 352x99=34848/128=272.25
100 352x100=35200/128=275 #C
由上表推論可知: 比數 比值 12平均律
C= F 第 3 分音 = 264 1.00 1/1 16 cent 261.63
#C= F 第 25 分音 = 275 1.04 25/24 -14 cent 277.18
D= 440x2/3 A之下屬音= 293.33 1.11 10/9 -2 cent 293.66
bE= F 第 14 分音 = 308 1.16 14/12 -14 cent 311.13
E= F 第 15 分音 = 330 1.25 15/12 2 cent 329.63
F= F 第 1 分音 = 352 1.33 4/3 13 cent 349.23
#F= F 第 17 分音 = 374 1.42 17/12 18 cent 349.23
G= F 第 9 分音 = 396 1.50 3/2 18 cent 392.00
#G= F 第 75 分音 = 412.5 1.59 25/16 -12 cent 415.30
A= F 第 5 分音 = 440 1.66 5/3 0 cent 440.00
bB= F 第 21 分音 = 462 1.75 21/12 -13 cent 466.16
B= F 第 45 分音 = 495 1.87 45/12 4 cent 493.89
bE 308 =-14cent bB 462 =-13cent #F374=18cent
352-308=44 528-462=66 415.3-369.99=45.31
200/44=4.55 200/66=3.03 200/45.31=4.41
(308-311.13)x4.55=-14.24 (462-466.16)x3.03=-12.6 (374-369.99)x4.41=17.68