Anatomy of an Octave

Below, for the reference of tuning enthusiasts, is a table of more than 700 pitches within an octave. The table contains all pitches that meet any one of the following six criteria:

All ratios between whole numbers 32 and lower
All ratios between 31-limit numbers up to 64 (31-limit meaning that the numbers contain no prime-number factors larger than 31)
Harmonics up to 128 (each whole number divided by the closest inferior power of 2)
All ratios between 11-limit numbers up to 128
All ratios between 5-limit numbers up to 1024
Certain historically important ratios such as the schisma and Pythagorean comma

The table is similar to, but much briefer than, that found in Alain Danielou's encyclopedic but long out-of-print Comparative Table of Musical Intervals.

Ratio:CentsName (if any)
1/10.000tonic
32805/327681.954schisma (3 to the 8th/2 to the 12th x 5/8)
126/12513.795
121/12014.367
100/9917.399
99/9817.576
81/80 21.506syntonic comma
531441/52428823.460Pythagorean comma (3 to the 12th/2 to the 19th)
65/6426.84165th harmonic
64/6327.264
63/6227.700
58/5730.109
57/5630.642
56/5531.194Ptolemy's enharmonic
55/5431.767
52/5133.617
51/5034.283
50/4934.976
49/4835.697
46/4538.051inferior quarter-tone (Ptolemy)
45/4438.906
128/12541.059diminished second (16/15 x 24/25)
525/51243.408enharmonic diesis (Avicenna)
40/3943.831
39/3844.970 superior quarter-tone (Eratosthenes)
77/7545.561
36/3548.770superior quarter-tone (Archytas)
250/24349.166
35/3450.184E.T. 1/4-tone approximation
34/3351.682
33/3253.27333rd harmonic
32/3154.964inferior quarter-tone (Didymus)
125/12156.305
31/3056.767superior quarter-tone (Didymus)
30/2958.692
29/2860.751
57/5561.836
28/2762.961inferior quarter-tone (Archytas)
80/7766.170
27/2665.337
26/2567.9001/3-tone (Avicenna)
51/4969.259
126/12170.100
25/2470.672minor 5-limit half-step
24/2373.681
117/11275.612
23/2276.956
67/6479.30767th harmonic
22/2180.537hard 1/2-step (Ptolemy, Avicenna, Safiud)
21/2084.467
81/7787.676
20/1988.801
256/24390.225Pythagorean half-step
58/5591.946
135/12892.179limma ascendant
96/9192.601
19/1893.603
55/5297.104
128/12197.364
18/1798.955E.T. half-step approximation
2 to the 1/12th100.000equal-tempered half-step
89/84100.099ET half-step approximation
35/33101.867
52/49102.876
86/81103.698
17/16104.955overtone half-step
33/31108.237
49/46109.377
16/15111.731major 5-limit half-step
31/29115.458
77/72116.234
15/14119.443 Cowell just half-step
29/27123.712
14/13128.298
69/64130.22969th harmonic
55/51130.721
27/25133.238alternate Renaissance half-step
121/112133.810
13/12138.573 3/4-tone (Avicenna)
64/59140.828
38/35142.373
63/58143.159
88/81143.498
25/23144.353
62/57145.568
135/124147.143
49/45147.428
12/11150.637undecimal "median" 1/2-step
59/54153.307
35/32155.14035th harmonic
23/21157.493
57/52158.940
34/31159.920
800/729160.897
56/51161.915
11/10165.004
54/49168.213
32/29170.423
21/19173.268
31/28176.210
567/512176.646
51/46178.636
71/64179.69771st harmonic
10/9182.404minor whole-tone
49/44186.334
39/35187.343
29/26189.050
125/112190.115
48/43190.437
19/17192.558
160/143194.468
28/25196.198
121/108196.771
55/49199.980
2 to the 1/6th200.000equal-tempered whole-tone
64/57200.532
9/8203.910major whole-tone
62/55207.404
44/39208.835
35/31210.104
26/23212.253
112/99213.598
17/15216.687
25/22221.309
58/51222.667
256/225223.463
33/29223.696
729/640225.416
57/50226.841
73/64227.78973rd harmonic
8/7231.174septimal whole-tone
63/55235.104
55/48235.677
39/34237.527
225/196238.886
31/27239.171
147/128239.607
169/147241.449
23/20241.961
2187/1900243.545
38/33244.240
144/125244.969diminished third (6/5 x 24/25)
121/105245.541
15/13247.741
52/45250.304
37/32251.34437th harmonic
81/70252.680
125/108253.076
22/19253.805
51/44255.592
196/169256.596consonant interval (Avicenna)
29/25256.950
36/31258.874
93/80260.677
57/49261.816
64/55262.368
7/6266.871septimal minor third
90/77270.080
75/64274.582augmented second (9/8 x 25/24)
34/29275.378
88/75276.736
27/23277.591
20/17281.358
33/28284.447
46/39285.792
13/11289.210
58/49291.925
45/38292.711
32/27294.135Pythagorean minor third
19/16297.513overtone minor third
2 to the 1/4th300.000equal-tempered minor third
25/21301.847
31/26304.508
105/88305.777
55/46309.357
6/5315.6415-limit minor third
77/64320.14477th harmonic
35/29325.562
29/24327.622
75/62329.547
98/81329.832
121/100330.008
23/19330.761
63/52332.208
40/33333.041
17/14336.130
243/200337.148
62/51338.125
28/23340.552
39/32342.48339th harmonic
128/105342.905
8000/6561343.301
11/9347.408undecimal "median" third
60/49350.617
49/40351.338
38/31352.477
27/22354.547
16/13359.472
79/64364.53779th harmonic
100/81364.807
121/98364.984
21/17365.825
99/80368.914
26/21369.747
57/46371.194
31/25372.408
36/29374.333
56/45378.602
96/77381.811
8192/6561384.360Pythagorean "schismatic" third
5/4386.3145-limit major third
64/51393.090
49/39395.169
44/35396.178
39/31397.447
34/27399.090
2 to the 1/3rd400.000equal-tempered major third
63/50400.108
121/96400.681
29/23401.303
125/99403.713
24/19404.442
512/405405.866
62/49407.384
81/64407.820Pythagorean major third
19/15409.244
33/26412.745
80/63413.578
14/11417.508
51/40 420.597
125/98421.289
23/18424.364
32/25427.373diminished fourth
41/32429.06241st harmonic
50/39430.145
77/60431.875
9/7435.084septimal major third
58/45439.353
49/38440.139
40/31441.278
31/24443.081
1323/1024443.517
128/99444.772
22/17446.363
57/44448.150
162/125448.879
35/27449.275
83/64450.04783rd harmonic
100/77452.484
13/10454.214
125/96456.986augmented third (5/4 x 25/24)
30/23459.994
64/49462.348
98/75463.069
17/13464.428
72/55466.278
55/42466.851
38/29467.936
21/16470.781septimal fourth
46/35473.135
25/19475.114
320/243476.539
29/22478.259
675/512478.492
33/25480.646
45/34485.286
85/64491.26985th harmonic
4/3498.045perfect fourth
2 to the 5/12ths500.000equal-tempered perfect fourth
75/56505.757
51/38509.397
43/32511.51843rd harmonic
121/90512.412
39/29512.905
35/26514.612
66/49515.621
31/23516.761
27/20519.551
23/17523.319
42/31525.745
19/14528.687
110/81529.812
87/64531.53287th harmonic
34/25532.328
49/36533.742
15/11536.951
512/375539.104
26/19543.015
63/46544.462
48/35546.815
1000/729547.211
11/8551.318undecimal tritone (11th harmonic)
62/45554.812
40/29556.737
29/21558.796
112/81561.006
18/13563.382
25/18568.717augmented fourth (4/3 x 25/24)
89/64570.88089th harmonic
32/23571.726
39/28 573.657
46/33575.001
88/63 578.582
7/5 582.512septimal tritone
108/77 585.721
1024/729588.270low Pythagorean tritone
45/32590.224high 5-limit tritone
38/27591.648
31/22593.718
55/39595.149
24/17597.000
Square root of 2600.000equal-tempered tritone
99/70600.088
17/12603.000
44/31606.282
125/88607.623
27/19608.352
91/64609.35491st harmonic
64/45609.776low 5-limit tritone
729/512611.730high Pythagorean tritone
57/40613.154
77/54614.279
10/7617.488septimal tritone
63/44621.418
33/23624.999
56/39626.343
23/16628.27423rd harmonic
36/25631.283diminished fifth (3/2 x 24/25)
121/84631.855
49/34632.696
13/9636.618
81/56638.994
55/38640.119
42/29641.204
29/20643.263
45/31645.188
93/64646.99193rd harmonic
16/11648.682
51/35651.771
729/500652.789
35/24653.185
19/13656.985
375/256660.896
22/15663.049
47/32665.50747th harmonic
72/49 666.258
25/17667.672
81/55670.188
28/19671.313
31/21674.255
189/128674.691
34/23676.681
40/27680.449dissonant "wolf" 5-limit fifth
46/31683.239
95/64683.82795th harmonic
49/33684.379
52/35685.388
58/39687.095
125/84688.160
112/75694.243
121/81694.816
2 to the 7/12ths700.000equal-tempered perfect fifth
3/2701.955perfect fifth
121/80716.322
50/33719.354
97/64719.89597th harmonic
1024/675721.508
44/29721.741
243/160723.461
38/25724.886
35/23726.865
32/21729.219
29/19732.064
84/55733.149
55/36733.722
26/17735.572
75/49736.931
49/32737.65249th harmonic
23/15740.006
192/125743.014diminished sixth (8/5 x 24/25)
20/13745.786
77/50747.516
54/35750.725
125/81751.121
17/11753.637
99/64755.22899th harmonic
48/31756.919
31/20758.722
45/29760.674
14/9764.916septimal minor sixth
120/77768.125
39/25769.855
25/16772.627augmented fifth
36/23775.636
11/7782.492undecimal minor sixth
63/40786.422
52/33787.255
101/64789.854101st harmonic
30/19790.756
128/81792.180Pythagorean minor sixth
49/31792.616
405/256794.134
19/12795.558
46/29798.697
100/63799.892
2 to the 2/3rds800.000equal-tempered minor sixth
27/17800.910
62/39802.553
35/22803.822
51/32806.91051st harmonic
8/5813.6865-limit minor sixth
6561/4096815.640Pythagorean "schismatic" sixth
77/48818.189
45/28821.398
103/64823.801103rd harmonic
29/18825.667
50/31827.592
121/75828.053
21/13830.253
55/34832.676
34/21834.175
81/50835.193
125/77838.797
13/8840.528overtone sixth
57/35844.328
44/27845.453
31/19847.523
80/49848.662
49/30849.383
18/11852.592undecimal "median" sixth
105/64857.095105th harmonic
64/39857.517
23/14859.448
51/31861.875
400/243862.852
28/17863.870
33/20866.959
38/23869.239
81/49870.168
48/29872.378
53/32873.50553rd harmonic
58/35 874.438
63/38875.223
128/77879.856
107/64889.760107th harmonic
5/3884.3595-limit major sixth
57/34 894.513
52/31895.492
42/25898.153
121/72 898.726
2 to the 3/4ths900.000equal-tempered major sixth
32/19902.487
27/16905.865Pythagorean major sixth
49/29908.075
22/13910.790
39/23914.208
56/33915.553
17/10918.642
109/64921.821109th harmonic
46/27922.409
75/44923.264
29/17924.622
128/75925.418diminished seventh (16/9 x 24/25)
77/45929.920
12/7933.129septimal major sixth
55/32937.632 55th harmonic
31/18941.126
441/256941.562
50/29943.050
19/11946.195
216/125946.924
121/70947.496
45/26949.696
26/15952.259
111/64953.299111th harmonic
125/72955.031augmented sixth (5/3 x 25/24)
33/19 955.760
40/23958.039
54/31960.829
96/55964.323
110/63964.896
7/4968.826septimal minor seventh
58/33976.304
225/128976.537
51/29977.333
44/25978.691
30/17983.313
113/64984.215113th harmonic
99/56986.402
23/13987.747
62/35989.896
39/22991.165
55/31992.596
16/9996.090Pythagorean small min. seventh
57/32999.46857th harmonic
2 to the 5/6ths1000.000equal-tempered minor seventh
98/551000.020
25/141003.802
34/191007.442
52/291010.950
88/491013.666
115/641014.588115th harmonic
9/51017.5965-limit large minor seventh
56/311023.790
38/211026.732
29/161029.57729th harmonic
49/271031.787
20/111034.996
51/281038.085
729/4001039.103
31/171040.080
42/231042.507
117/641044.438117th harmonic
64/351044.860
4000/21871045.256
11/61049.363undecimal "median" seventh
90/491052.572
57/311054.432
46/251055.647
81/441056.502
35/191057.627
59/321059.17259th harmonic
24/131061.427
50/271066.762
63/341067.780
13/71071.702
119/641073.781119th harmonic
54/291076.288
28/151080.557
58/311084.542
15/81088.2695-limit major seventh
62/331091.763
32/171095.045
49/261097.124
66/351098.133
2 to the 11/12ths1100.000equal-tempered major seventh
17/91101.045
121/641102.636121st harmonic
125/661105.668
36/191106.397
256/1351107.821
55/291108.054
243/1281109.775Pythagorean major seventh
19/101111.199
40/211115.533
61/321116.88561st harmonic
21/111119.463
44/231123.044
23/121126.319
48/251129.328
121/631129.900
123/641131.017123rd harmonic
25/131132.100
77/401133.830
52/271134.663
27/141137.039septimal major seventh
56/291139.249
29/151141.308
60/311143.233
31/161145.03631st harmonic
64/331146.727
33/171148.318
243/1251150.834
35/181151.230
39/201156.169
125/641158.941augmented seventh (15/8 x 25/24)
88/451161.094
45/231161.949
96/491164.303
49/251165.024
51/261166.383
108/551168.233
55/281168.806
57/291169.891
63/321172.73663rd harmonic
160/811178.494
99/501182.601
125/631186.205
127/641186.422127th harmonic
2/11200.000octave


Copyright, Kyle Gann, 1998
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