Anatomy of an Octave

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

All ratios between whole numbers 64 and lower
All ratios between 31-limit numbers up to 96 (31-limit meaning that the numbers contain no prime-number factors larger than 31)
Harmonics up to 255 (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 (or mathematically interesting) 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.

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Ratio:CentsName (if any)
1/10.000tonic
1732/17310.9998superparticular cent approximation
32805/327681.954schisma (3 to the 8th/2 to the 12th x 5/8)
129/12813.473129th harmonic
126/12513.795
121/12014.367
100/9917.399
99/9817.576
96/9518.128
93/9218.716
92/9118.921
91/9019.130
88/8719.786
85/8420.488
81/8021.506syntonic comma
78/7722.339
77/7622.631
76/7522.931
531441/52428823.460Pythagorean comma (3 to the 12th/2 to the 19th)
74/7323.555superparticular Pythagorean comma approximation
70/6924.910
69/6825.274
66/6526.432
65/6426.84165th harmonic
64/6327.264
63/6227.700
62/6128.151
61/6028.616
60/5929.097
59/5829.594
58/5730.109
57/5630.642
56/5531.194Ptolemy's enharmonic
55/5431.767
54/5332.360
53/5232.977
52/5133.617
51/5034.283
50/4934.976
49/4835.697
48/4736.448
95/9336.836
47/4637.232
93/9137.637
46/4538.051inferior quarter-tone (Ptolemy)
45/4438.906
44/4339.800
131/12840.108131st harmonic
87/8540.263
43/4240.737
128/12541.059diminished second (16/15 x 24/25)
42/4141.719
41/4042.749
525/51243.408enharmonic diesis (Avicenna)
40/3943.831
39/3844.970superior quarter-tone (Eratosthenes)
77/7545.561
38/3746.169
37/3647.434
36/3548.770superior quarter-tone (Archytas)
250/24349.166
35/3450.184E.T. 1/4-tone approximation
34/3351.682
33/3253.27333rd harmonic
65/6354.105
32/3154.964inferior quarter-tone (Didymus)
95/9255.552
63/6155.851
125/12156.305
31/3056.767superior quarter-tone (Didymus)
61/5957.713
91/8858.036
30/2958.692
59/5759.704
88/8560.049
29/2860.751
57/5561.836
28/2762.961inferior quarter-tone (Archytas)
55/5364.127
27/2665.337
80/7766.170
133/12866.339133rd harmonic
53/5166.594
26/2567.9001/3-tone (Avicenna)
51/4969.259
126/12170.100
25/2470.672minor 5-limit half-step
49/4772.145
24/2373.681
95/9174.473
47/4575.283
117/11275.612
23/2276.956
91/8777.821
68/6578.114
45/4378.706
67/6479.30767th harmonic
22/2180.537hard 1/2-step (Ptolemy, Avicenna, Safiud)
65/6281.806
43/4182.455
64/6183.115
85/8183.449
21/2084.467septimal semitone
62/5985.864
41/3986.580
61/5887.308
81/7787.676
20/1988.801
256/24390.225Pythagorean half-step
59/5690.346
39/3791.139
58/5591.946
135/12892.179limma ascendant
96/9192.601
19/1893.603
56/5395.321
93/8895.673
37/3596.204
92/8796.742
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
53/50100.877
35/33101.867
52/49102.876
69/65103.388
86/81103.698
17/16104.955overtone half-step, 17th harmonic
50/47107.121
33/31108.237
49/46109.377
81/76110.307
16/15111.731major 5-limit half-step
63/59113.564
47/44114.189
31/29115.458
77/72116.234
46/43116.757
61/57117.417
137/128117.638137th harmonic
91/85118.084
15/14119.443Cowell just half-step
59/55121.540
44/41122.256
29/27123.712
43/40125.204
57/53125.963
14/13128.298
69/64130.22969th harmonic
55/51130.721
41/38131.549
68/63132.220
95/88132.509
27/25133.238alternate Renaissance half-step
121/112133.810
40/37134.970
53/49135.853
92/85137.005
13/12138.5733/4-tone (Avicenna)
64/59140.828
51/47141.404
38/35142.373
139/128142.729139th harmonic
63/58143.159
88/81143.498
25/23144.353
87/80145.218
62/57145.568
37/34146.389
135/124147.143
49/45147.428
61/56148.059
85/78148.786
12/11150.637undecimal "median" 1/2-step
95/87152.295
59/54153.307
47/43153.989
35/32155.14035th harmonic
93/85155.721
58/53156.073
23/21157.493
57/52158.940
34/31159.920
800/729160.897
45/41161.161
56/51161.915
11/10165.004
76/69167.284
141/128167.462141st harmonic
54/49168.213
43/39169.035
75/68169.627
32/29170.423
85/77171.125
53/48171.550
21/19173.268
52/47175.021
31/28176.210
567/512176.646
72/65177.069
41/37177.718
51/46178.636
61/55179.253
71/64179.69771st harmonic
10/9182.404minor whole-tone
69/62185.194
59/53185.667
49/44186.334
39/35187.343
29/26189.050
77/69189.915
125/112190.115
48/43190.437
143/128191.846143rd harmonic
19/17192.558
85/76193.756
160/143194.468
47/42194.726
28/25196.198
121/108196.771
65/58197.264
37/33198.071
46/41199.212
55/49199.980
2 to the 1/6th200.000equal-tempered whole-tone
64/57200.532
91/81201.534
9/8203.910major whole-tone, 9th harmonic
62/55207.404
53/47207.998
44/39208.835
35/31210.104
96/85210.686
61/54211.020
87/77211.388
26/23212.253
95/84213.046
112/99213.598
43/38214.005
60/53214.764
77/68215.188
145/128215.891145th harmonic
17/15216.687
59/52218.644
42/37219.437
92/81220.454
25/22221.309
58/51222.667
91/80223.040
256/225223.463
33/29223.696
41/36225.152
729/640225.416
49/43226.134
57/50226.841
65/57227.373
73/64227.78973rd harmonic
8/7231.174septimal whole-tone
87/76234.019
63/55235.104
55/48235.677
47/41236.444
39/34237.527
225/196238.886
31/27239.171
147/128239.607
54/47240.358
169/147241.449
23/20241.961
61/53243.380
2187/1900243.545
38/33244.240
144/125244.969diminished third (6/5 x 24/25)
53/46245.230
121/105245.541
15/13247.741
52/45250.304
37/32251.34437th harmonic
59/51252.261
81/70252.680
125/108253.076
22/19253.805
51/44255.592
80/69256.084
196/169256.596consonant interval (Avicenna)
29/25256.950
65/56258.015
36/31258.874
43/37260.174
93/80260.677
50/43261.110
57/49261.816
64/55262.368
149/128263.002149th harmonic
7/6266.871septimal minor third
90/77270.080
76/65270.672
62/53271.531
55/47272.125
48/41272.893
41/35273.923
75/64274.582augmented second (9/8 x 25/24)
34/29275.378
95/81276.007
61/52276.357
88/75276.736
27/23277.591
47/40279.193
20/17281.358
53/45283.281
33/28284.447
46/39285.792
151/128286.086151st harmonic
59/50286.544
85/72287.359
13/11289.210
58/49291.925
45/38292.711
77/65293.302
32/27294.135Pythagorean minor third
51/43295.393
19/16297.513overtone minor third, 19th harmonic
63/53299.231
44/37299.974
2 to the 1/4th300.000equal-tempered minor third
69/58300.652
25/21301.847
81/68302.865
56/47303.319
31/26304.508
68/57305.487
105/88305.777
37/31306.308
43/36307.608
92/77308.130
49/41308.589
153/128308.865153rd harmonic
55/46309.357
61/51309.974
91/76311.841
6/5315.6415-limit minor third
77/64320.14477th harmonic
65/54320.976
59/49321.520
53/44322.187
47/39323.024
41/34324.107
76/63324.777
35/29325.562
64/53326.495
93/77326.847
29/24327.622
52/43329.010
75/62329.547
98/81329.832
121/100330.008
23/19330.761
155/128331.349155th harmonic
63/52332.208
40/33333.041
57/47333.961
91/75334.771
17/14336.130
243/200337.148
62/51338.125
45/37338.880
28/23340.552
95/78341.344
39/32342.48339th harmonic
128/105342.905
8000/6561343.301
50/41343.565
61/50344.257
11/9347.408undecimal "median" third
93/76349.478
60/49350.617
49/40351.338
38/31352.477
157/128353.545157th harmonic
92/75353.692
27/22354.547
43/35356.378
59/48357.217
16/13359.472
85/69361.040
69/56361.403
53/43361.987
37/30363.075
95/77363.683
58/47364.071
79/64364.53779th harmonic
100/81364.807
121/98364.984
21/17365.825
68/55367.324
47/38367.994
99/80368.914
26/21369.747
57/46371.194
31/25372.408
36/29374.333
159/128375.460159th harmonic
41/33375.789
87/70376.393
46/37376.930
51/41377.848
56/45378.602
61/49379.233
81/65380.979
96/77381.811
8192/6561384.360Pythagorean "schismatic" third
5/4386.3145-limit major third, 5th harmonic
69/55392.598
64/51393.090
59/47393.665
54/43394.347
49/39395.169
44/35396.178
161/128397.100161st harmonic
39/31397.447
34/27399.090
2 to the 1/3rd400.000equal-tempered major third
63/50400.108
121/96400.681
29/23401.303
53/42402.724
125/99403.713
24/19404.442
91/72405.444
512/405405.866
43/34406.562
62/49407.384
81/64407.820Pythagorean major third
19/15409.244
52/41411.465
33/26412.745
80/63413.578
47/37414.163
61/48414.930
14/11417.508
163/128418.474163rd harmonic
65/51419.931
51/40420.597
88/69421.089
125/98421.289
37/29421.767
60/47422.762
23/18424.364
55/43426.114
87/68426.577
32/25427.373diminished fourth
41/32429.06241st harmonic
50/39430.145
59/46430.897
77/60431.875
9/7435.084septimal major third
85/66437.996
58/45439.353
165/128439.587165th harmonic
49/38440.139
40/31441.278
31/24443.081
1323/1024443.517
84/65443.940
53/41444.442
128/99444.772
22/17446.363
57/44448.150
162/125448.879
35/27449.275
83/64450.04783rd harmonic
48/37450.611
61/47451.378
100/77452.484
13/10454.214
125/96456.986augmented third (5/4 x 25/24)
56/43457.308
43/33458.245
30/23459.994
167/128460.445167th harmonic
47/36461.597
64/49462.348
98/75463.069
17/13464.428
89/68465.925
72/55466.278
55/42466.851
38/29467.936
59/45468.948
21/16470.781septimal fourth
46/35473.135
25/19475.114
320/243476.539
54/41476.803
29/22478.259
675/512478.492
91/69479.124
62/47479.529
95/72479.917
33/25480.646
169/128481.055169th harmonic
37/28482.518
41/31484.027
45/34485.286
49/37486.308
53/40487.191
57/43487.950
61/46488.610
65/49489.190
69/52489.702
85/64491.26985th harmonic
93/70491.851
4/3498.045perfect fourth
2 to the 5/12ths500.000equal-tempered perfect fourth
171/128501.423171st harmonic
91/68504.398
87/65504.691
75/56505.757
63/47507.229
59/44507.854
55/41508.569
51/38509.397
47/35510.367
43/32511.51843rd harmonic
121/90512.412
39/29512.905
35/26514.612
66/49515.621
31/23516.761
58/43518.059
85/63518.533
27/20519.551
50/37521.283
173/128521.554173rd harmonic
23/17523.319
88/65524.477
65/48524.886
42/31525.745
61/45526.661
19/14528.687
110/81529.812
53/39531.022
87/64531.53287th harmonic
34/25532.328
49/36533.742
64/47534.493
15/11536.951
512/375539.104
56/41539.764
41/30540.794
175/128541.453175th harmonic
93/68542.035
26/19543.015
63/46544.462
37/27545.479
85/62546.234
48/35546.815
1000/729547.211
59/43547.654
11/8551.318undecimal tritone, 11th harmonic
95/69553.597
62/45554.812
51/37555.556
91/66556.081
40/29556.737
69/50557.602
29/21558.796
76/55559.881
47/34560.551
112/81561.006
177/128561.127177th harmonic
18/13563.382
61/44565.567
43/31566.482
68/49567.304
25/18568.717augmented fourth (4/3 x 25/24)
57/41570.406
89/64570.88089th harmonic
32/23571.726
39/28573.657
46/33575.001
53/38575.992
60/43576.751
88/63578.582
95/68578.871
179/128580.579179th harmonic
7/5 582.512septimal tritone
108/77585.721
87/62586.497
1024/729588.270low Pythagorean tritone
59/42588.391
52/37589.184
45/32590.224high 5-limit tritone
38/27591.648
69/49592.578
31/22593.718
55/39595.149
24/17597.000
65/46598.567
41/29599.485
181/128599.815181st harmonic
Square root of 2600.000equal-tempered tritone
99/70600.088
58/41600.515
92/65601.433
17/12603.000
78/55604.851
61/43605.367
44/31606.282
125/88607.623
27/19608.352
91/64609.35491st harmonic
64/45609.776low 5-limit tritone
37/26610.816
729/512611.730high Pythagorean tritone
47/33612.234
57/40613.154
77/54614.279
10/7617.488septimal tritone
183/128618.840183rd harmonic
93/65620.149
63/44621.418
53/37622.161
43/30623.249
33/23624.999
56/39626.343
23/16628.27423rd harmonic
59/41630.109
95/66630.554
36/25631.283diminished fifth (3/2 x 24/25)
121/84631.855
49/34632.696
62/43633.518
13/9636.618
185/128637.658185th harmonic
81/56638.994
55/38640.119
42/29641.204
29/20643.263
45/31645.188
61/42646.104
93/64646.99193rd harmonic
16/11648.682
51/35651.771
729/500652.789
35/24653.185
54/37654.521
92/63655.538
187/128656.273187th harmonic
19/13656.985
60/41659.206
41/28660.236
375/256660.896
63/43661.218
85/58661.692
22/15663.049
91/62664.318
47/32665.50747th harmonic
72/49666.258
25/17667.672
53/36669.595
81/55670.188
28/19671.313
59/40672.858
31/21674.255
189/128674.691
96/65675.114
65/44675.523
34/23676.681
37/25678.717
40/27680.449dissonant "wolf" 5-limit fifth
43/29681.941
46/31683.239
95/64683.82795th harmonic
49/33684.379
52/35685.388
55/37686.288
58/39687.095
61/41687.822
125/84688.160
64/43688.482
76/51690.603
85/57691.801
191/128692.915191st harmonic
112/75694.243
121/81694.816
2 to the 7/12ths700.000equal-tempered perfect fifth
3/2701.955perfect fifth, 3rd harmonic
193/128710.948193rd harmonic
95/63711.091
68/45714.732
62/41715.973
121/80716.322
59/39716.689
56/37717.482
53/35718.365
50/33719.354
97/64719.89597th harmonic
47/31720.471
91/60721.085
1024/675721.508
44/29721.741
85/56722.443
41/27723.197
243/160723.461
38/25724.886
35/23726.865
195/128728.796195th harmonic
32/21729.219
61/40730.571
29/19732.064
84/55733.149
55/36733.722
26/17735.572
75/49736.931
49/32737.65249th harmonic
95/62738.791
87/55739.901
23/15740.006
43/28742.692
192/125743.014diminished sixth (8/5 x 24/25)
63/41743.674
20/13745.786
197/128746.462197th harmonic
77/50747.516
57/37748.124
37/24749.389
54/35750.725
125/81751.121
17/11753.637
99/64755.22899th harmonic
65/42756.060
48/31756.919
31/20758.722
76/49759.861
45/29760.674
59/38761.659
87/56762.706
199/128763.950199th harmonic
14/9764.916septimal minor sixth
120/77768.125
53/34768.549
39/25769.855
64/41770.938
25/16772.627augmented fifth
61/39774.402
36/23775.636
47/30777.238
58/37778.233
69/44778.911
91/58779.776
201/128781.262201st harmonic
11/7782.492undecimal minor sixth
85/54785.404
63/40786.422
52/33787.255
41/26788.535
101/64789.854101st harmonic
30/19790.756
128/81792.180Pythagorean minor sixth
49/31792.616
405/256794.134
19/12795.558
203/128798.403203rd harmonic
46/29798.697
100/63799.892
2 to the 2/3rds800.000equal-tempered minor sixth
27/17800.910
62/39802.553
35/22803.822
78/49804.831
43/27805.653
51/32806.91051st harmonic
59/37807.828
91/57809.886
8/5813.6865-limit minor sixth
205/128815.376205th harmonic
6561/4096815.640Pythagorean "schismatic" sixth
93/58817.413
77/48818.189
61/38819.372
53/33820.232
45/28821.398
37/23823.070
103/64823.801103rd harmonic
29/18825.667
50/31827.592
121/75828.053
92/57828.806
21/13830.253
207/128832.184207th harmonic
55/34832.676
34/21834.175
81/50835.193
47/29835.929
60/37836.925
125/77838.797
13/8840.528overtone sixth, 13th harmonic
57/35844.328
44/27845.453
31/19847.523
80/49848.662
209/128848.831209th harmonic
49/30849.383
85/52850.741
18/11852.592undecimal "median" sixth
95/58854.250
59/36855.262
41/25856.435
105/64857.095105th harmonic
64/39857.517
23/14859.448
51/31861.875
400/243862.852
28/17863.870
211/128865.319211th harmonic
61/37865.541
33/20866.959
38/23869.239
81/49870.168
43/26870.990
91/55871.722
48/29872.378
53/32873.50553rd harmonic
58/35874.438
63/38875.223
128/77879.856
213/128881.652213th harmonic
5/3884.3595-limit major sixth
107/64889.760107th harmonic
92/55890.643
87/52891.005
62/37893.692
57/34894.513
52/31895.492
47/28896.681
215/128897.314215th harmonic
42/25898.153
121/72898.726
2 to the 3/4ths900.000equal-tempered major sixth
37/22900.026
32/19902.487
91/54903.489
59/35904.032
27/16905.865Pythagorean major sixth
76/45907.289
49/29908.075
22/13910.790
61/36912.975
217/128913.861217th harmonic
39/23914.208
95/56915.001
56/33915.553
17/10918.642
63/37921.392
109/64921.821109th harmonic
46/27922.409
75/44923.264
29/17924.622
128/75925.418diminished seventh (16/9 x 24/25)
41/24927.107
53/31928.469
65/38929.328
219/128929.744219th harmonic
77/45929.920
12/7933.129septimal major sixth
55/32937.63255th harmonic
43/25938.890
31/18941.126
441/256941.562
50/29943.050
69/40943.916
221/128945.483221st harmonic
19/11946.195
216/125946.924
121/70947.496
64/37948.656
45/26949.696
26/15952.259
111/64953.299111th harmonic
85/49953.617
59/34954.216
125/72955.031augmented sixth (5/3 x 25/24)
33/19955.760
40/23958.039
87/50958.905
47/27959.642
54/31960.829
223/128961.080223rd harmonic
61/35961.745
68/39962.473
96/55964.323
110/63964.896
7/4968.826septimal minor seventh, 7th harmonic
58/33976.304
225/128976.537225th harmonic
51/29977.333
95/54977.962
44/25978.691
37/21980.563
30/17983.313
113/64984.215113th harmonic
53/30985.236
99/56986.402
23/13987.747
85/48989.314
62/35989.896
39/22991.165
227/128991.858227th harmonic
55/31992.596
87/49993.880
16/9996.090Pythagorean small min. seventh
57/32999.46857th harmonic
2 to the 5/6ths1000.000equal-tempered minor seventh
98/551000.020
41/231000.788
91/511002.443
25/141003.802
59/331005.899
229/1281007.045229th harmonic
34/191007.442
43/241009.563
52/291010.950
61/341011.929
88/491013.666
115/641014.588115th harmonic
9/51017.5965-limit large minor seventh
92/511021.364
231/1281022.099231st harmonic
65/361022.931
56/311023.790
47/261024.979
38/211026.732
29/161029.57729th harmonic
49/271031.787
69/381032.716
20/111034.996
91/501036.726
233/1281037.023233rd harmonic
51/281038.085
729/4001039.103
31/171040.080
42/231042.507
95/521043.299
53/291043.927
117/641044.438117th harmonic
64/351044.860
4000/21871045.256
11/61049.363undecimal "median" seventh
235/1281051.820235th harmonic
90/491052.572
57/311054.432
46/251055.647
81/441056.502
35/191057.627
59/321059.17259th harmonic
24/131061.427
85/461062.995
61/331063.612
37/201065.030
237/1281066.492237th harmonic
50/271066.762
63/341067.780
13/71071.702
119/641073.781119th harmonic
54/291076.288
95/511076.916
41/221077.744
28/151080.557
239/1281081.040239th harmonic
43/231083.243
58/311084.542
15/81088.2695-limit major seventh, 15th harmonic
92/491090.623
62/331091.763
47/251092.879
32/171095.045
241/1281095.467241st harmonic
49/261097.124
66/351098.133
2 to the 11/12ths1100.000equal-tempered major seventh
17/91101.045
121/641102.636121st harmonic
87/461103.258
53/281104.679
125/661105.668
36/191106.397
91/481107.399
256/1351107.821
55/291108.054
243/1281109.775Pythagorean major seventh, 243rd harmonic
19/101111.199
59/311114.136
40/211115.533
61/321116.88561st harmonic
21/111119.463
65/341121.886
44/231123.044
245/1281123.966245th harmonic
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
247/1281138.041247th harmonic
56/291139.249
85/441139.951
29/151141.308
60/311143.233
31/161145.03631st harmonic
95/491146.175
64/331146.727
33/171148.318
68/351149.816
243/1251150.834
35/181151.230
249/1281152.002249th harmonic
37/191153.831
76/391155.030
39/201156.169
41/211158.281
125/641158.941augmented seventh (15/8 x 25/24)
43/221160.200
88/451161.094
45/231161.949
47/241163.552
96/491164.303
49/251165.024
251/1281165.852251st harmonic
51/261166.383
53/271167.640
108/551168.233
55/281168.806
57/291169.891
59/301170.903
61/311171.849
63/321172.73663rd harmonic
65/331173.568
69/351175.090
160/811178.494
253/1281179.592253rd harmonic
87/441180.214
91/461181.079
95/481181.872
99/501182.601
125/631186.205
127/641186.422127th harmonic
255/1281193.224255th harmonic
2/11200.000octave


Copyright, Kyle Gann, 1998
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