Anatomy of an Octave

!"#$%&' %) #" %*$#+, Ratio: 1/1 32805/32768 126/125 121/120 100/99 99/98 81/80 531441/524288 531441/52428 8 65/64 64/63 63/62 58/57 57/56 56/55 55/54 52/51 51/50 50/49 49/48 46/45 45/44 128/125 525/512 40/39 39/38 77/75 36/35 250/243 35/34 34/33 33/32 32/31 125/121 31/30 30/29 29/28 57/55 28/27 80/77 27/26 26/25 51/49 126/121 25/24 24/23 117/112 23/22

Cents 0.000 1.954 13.795 14.367 17.399 17.576 21.506 23.460 26.841 27.264 27.700 30.109 30.642 31.194 31.767 33.617 34.283 34.976 35.697 38.051 38.906 41.059 43.408 43.831 44.970 45.561 48.770 49.166 50.184 51.682 53.273 54.964 56.305 56.767 58.692 60.751 61.836 62.961 66.170 65.337 67.900 69.259 70.100 70.672 73.681 75.612 76.956

Name (if any) tonic schisma (3 to the 8th/2 to the 12th x 5/8)

syntonic comma Pythagorean comma (3 to the 12th/2 to the 19th) 65th harmonic

Ptolemy's enharmonic

inferior quarter-tone (Ptolemy) inferior diminished second (16/15 x 24/25) enharmonic diesis (Avicenna) superior quarter-tone (Eratosthenes) superior superior quarter-tone (Archytas) superior E.T. 1/4-tone approximation E.T. 33rd harmonic inferior quarter-tone (Didymus) inferior superior quarter-tone (Didymus) superior

inferior quarter-tone (Archytas) inferior

1/3-tone (Avicenna)

minor 5-limit half-step

67/64 22/21 21/20 81/77 20/19 256/243 58/55 135/128 96/91 19/18 55/52 128/121 18/17 2 to the 1/12th 89/84 35/33 52/49 86/81 17/16 33/31 49/46 16/15 31/29 77/72 15/14 29/27 14/13 69/64 55/51 27/25 121/112 13/12 64/59 38/35 63/58 88/81 25/23 62/57 135/124 49/45 12/11 59/54 35/32 23/21 57/52 34/31 800/729 56/51 11/10

79.307 80.537 84.467 87.676 88.801 90.225 91.946 92.179 92.601 93.603 97.104 97.364 98.955 100.000 100.099 101.867 102.876 103.698 104.955 108.237 109.377 111.731 115.458 116.234 119.443 123.712 128.298 130.229 130.721 133.238 133.810 138.573 140.828 142.373 143.159 143.498 144.353 145.568 147.143 147.428 150.637 153.307 155.140 157.493 158.940 159.920 160.897 161.915 165.004

67th harmonic hard 1/2-step (Ptolemy, Avicenna, Safiud)

Pythagorean half-step limma ascendant

E.T. half-step approximation equal-tempered half-step ET half-step approximation

overtone half-step

major 5-limit half-step

Cowell just half-step

69th harmonic alternate Renaissance half-step 3/4-tone (Avicenna)

undecimal "median" 1/2-step 35th harmonic

54/49 32/29 21/19 31/28 567/512 51/46 71/64 10/9 49/44 39/35 29/26 125/112 48/43 19/17 160/143 28/25 121/108 55/49 2 to the 1/6th 64/57 9/8 62/55 44/39 35/31 26/23 112/99 17/15 25/22 58/51 256/225 33/29 729/640 57/50 73/64 8/7 63/55 55/48 39/34 225/196 31/27 147/128 169/147 23/20 2187/1900 38/33 144/125 121/105 15/13 52/45

168.213 170.423 173.268 176.210 176.646 178.636 179.697 182.404 186.334 187.343 189.050 190.115 190.437 192.558 194.468 196.198 196.771 199.980 200.000 200.532 203.910 207.404 208.835 210.104 212.253 213.598 216.687 221.309 222.667 223.463 223.696 225.416 226.841 227.789 231.174 235.104 235.677 237.527 238.886 239.171 239.607 241.449 241.961 243.545 244.240 244.969 245.541 247.741 250.304

71st harmonic minor whole-tone

equal-tempered whole-tone major whole-tone

73rd harmonic septimal whole-tone

diminished third (6/5 x 24/25)

37/32 81/70 125/108 22/19 51/44 196/169 29/25 36/31 93/80 57/49 64/55 7/6 90/77 75/64 34/29 88/75 27/23 20/17 33/28 46/39 13/11 58/49 45/38 32/27 19/16 2 to the 1/4th 25/21 31/26 105/88 55/46 6/5 77/64 35/29 29/24 75/62 98/81 121/100 23/19 63/52 40/33 17/14 243/200 62/51 28/23 39/32 128/105 8000/6561 11/9 60/49

251.344 252.680 253.076 253.805 255.592 256.596 256.950 258.874 260.677 261.816 262.368 266.871 270.080 274.582 275.378 276.736 277.591 281.358 284.447 285.792 289.210 291.925 292.711 294.135 297.513 300.000 301.847 304.508 305.777 309.357 315.641 320.144 325.562 327.622 329.547 329.832 330.008 330.761 332.208 333.041 336.130 337.148 338.125 340.552 342.483 342.905 343.301 347.408 350.617

37th harmonic

consonant interval (Avicenna)

septimal minor third augmented second (9/8 x 25/24)

Pythagorean minor third overtone minor third equal-tempered minor third

5-limit minor third 77th harmonic

39th harmonic

undecimal "median" third

49/40 38/31 27/22 16/13 79/64 100/81 121/98 21/17 99/80 26/21 57/46 31/25 36/29 56/45 96/77 8192/6561 5/4 64/51 49/39 44/35 39/31 34/27 2 to the 1/3rd 63/50 121/96 29/23 125/99 24/19 512/405 62/49 81/64 19/15 33/26 80/63 14/11 51/40 125/98 23/18 32/25 41/32 50/39 77/60 9/7 58/45 49/38 40/31 31/24 1323/1024 128/99

351.338 352.477 354.547 359.472 364.537 364.807 364.984 365.825 368.914 369.747 371.194 372.408 374.333 378.602 381.811 384.360 386.314 393.090 395.169 396.178 397.447 399.090 400.000 400.108 400.681 401.303 403.713 404.442 405.866 407.384 407.820 409.244 412.745 413.578 417.508 420.597 421.289 424.364 427.373 429.062 430.145 431.875 435.084 439.353 440.139 441.278 443.081 443.517 444.772

79th harmonic

Pythagorean "schismatic" third 5-limit major third

equal-tempered major third

Pythagorean major third

diminished fourth 41st harmonic

septimal major third

22/17 57/44 162/125 35/27 83/64 100/77 13/10 125/96 30/23 64/49 98/75 17/13 72/55 55/42 38/29 21/16 46/35 25/19 320/243 29/22 675/512 33/25 45/34 85/64 4/3 2 to the 5/12ths 75/56 51/38 43/32 121/90 39/29 35/26 66/49 31/23 27/20 23/17 42/31 19/14 110/81 87/64 34/25 49/36 15/11 512/375 26/19 63/46 48/35 1000/729 11/8

446.363 448.150 448.879 449.275 450.047 452.484 454.214 456.986 459.994 462.348 463.069 464.428 466.278 466.851 467.936 470.781 473.135 475.114 476.539 478.259 478.492 480.646 485.286 491.269 498.045 500.000 505.757 509.397 511.518 512.412 512.905 514.612 515.621 516.761 519.551 523.319 525.745 528.687 529.812 531.532 532.328 533.742 536.951 539.104 543.015 544.462 546.815 547.211 551.318

83rd harmonic

augmented third (5/4 x 25/24)

septimal fourth

85th harmonic perfect fourth equal-tempered perfect fourth

43rd harmonic

87th harmonic

undecimal tritone (11th harmonic)

62/45 40/29 29/21 112/81 18/13 25/18 89/64 32/23 39/28 46/33 88/63 7/5 108/77 1024/729 45/32 38/27 31/22 55/39 24/17 Square root of 2 99/70 17/12 44/31 125/88 27/19 91/64 64/45 729/512 57/40 77/54 10/7 63/44 33/23 56/39 23/16 36/25 121/84 49/34 13/9 81/56 55/38 42/29 29/20 45/31 93/64 16/11 51/35 729/500 35/24

554.812 556.737 558.796 561.006 563.382 568.717 570.880 571.726 573.657 575.001 578.582 582.512 585.721 588.270 590.224 591.648 593.718 595.149 597.000 600.000 600.088 603.000 606.282 607.623 608.352 609.354 609.776 611.730 613.154 614.279 617.488 621.418 624.999 626.343 628.274 631.283 631.855 632.696 636.618 638.994 640.119 641.204 643.263 645.188 646.991 648.682 651.771 652.789 653.185

augmented fourth (4/3 x 25/24) 89th harmonic

septimal tritone low Pythagorean tritone high 5-limit tritone

equal-tempered tritone

91st harmonic low 5-limit tritone high Pythagorean tritone

septimal tritone

23rd harmonic diminished fifth (3/2 x 24/25)

93rd harmonic

19/13 375/256 22/15 47/32 72/49 25/17 81/55 28/19 31/21 189/128 34/23 40/27 46/31 95/64 49/33 52/35 58/39 125/84 112/75 121/81 2 to the 7/12ths 3/2 121/80 50/33 97/64 1024/675 44/29 243/160 38/25 35/23 32/21 29/19 84/55 55/36 26/17 75/49 49/32 23/15 192/125 20/13 77/50 54/35 125/81 17/11 99/64 48/31 31/20 45/29 14/9

656.985 660.896 663.049 665.507 666.258 667.672 670.188 671.313 674.255 674.691 676.681 680.449 683.239 683.827 684.379 685.388 687.095 688.160 694.243 694.816 700.000 701.955 716.322 719.354 719.895 721.508 721.741 723.461 724.886 726.865 729.219 732.064 733.149 733.722 735.572 736.931 737.652 740.006 743.014 745.786 747.516 750.725 751.121 753.637 755.228 756.919 758.722 760.674 764.916

47th harmonic

dissonant "wolf" 5-limit fifth 95th harmonic

equal-tempered perfect fifth perfect fifth

97th harmonic

49th harmonic diminished sixth (8/5 x 24/25)

99th harmonic

septimal minor sixth

120/77 39/25 25/16 36/23 11/7 63/40 52/33 101/64 30/19 128/81 49/31 405/256 19/12 46/29 100/63 2 to the 2/3rds 27/17 62/39 35/22 51/32 8/5 6561/4096 77/48 45/28 103/64 29/18 50/31 121/75 21/13 55/34 34/21 81/50 125/77 13/8 57/35 44/27 31/19 80/49 49/30 18/11 105/64 64/39 23/14 51/31 400/243 28/17 33/20 38/23 81/49

768.125 769.855 772.627 775.636 782.492 786.422 787.255 789.854 790.756 792.180 792.616 794.134 795.558 798.697 799.892 800.000 800.910 802.553 803.822 806.910 813.686 815.640 818.189 821.398 823.801 825.667 827.592 828.053 830.253 832.676 834.175 835.193 838.797 840.528 844.328 845.453 847.523 848.662 849.383 852.592 857.095 857.517 859.448 861.875 862.852 863.870 866.959 869.239 870.168

augmented fifth undecimal minor sixth

101st harmonic Pythagorean minor sixth

equal-tempered minor sixth

51st harmonic 5-limit minor sixth Pythagorean "schismatic" sixth

103rd harmonic

overtone sixth

undecimal "median" sixth 105th harmonic

48/29 53/32 58/35 63/38 128/77 107/64 5/3 57/34 52/31 42/25 121/72 2 to the 3/4ths 32/19 27/16 49/29 22/13 39/23 56/33 17/10 109/64 46/27 75/44 29/17 128/75 77/45 12/7 55/32 31/18 441/256 50/29 19/11 216/125 121/70 45/26 26/15 111/64 125/72 33/19 40/23 54/31 96/55 110/63 7/4 58/33 225/128 51/29 44/25 30/17 113/64

872.378 873.505 874.438 875.223 879.856 889.760 884.359 894.513 895.492 898.153 898.726 900.000 902.487 905.865 908.075 910.790 914.208 915.553 918.642 921.821 922.409 923.264 924.622 925.418 929.920 933.129 937.632 941.126 941.562 943.050 946.195 946.924 947.496 949.696 952.259 953.299 955.031 955.760 958.039 960.829 964.323 964.896 968.826 976.304 976.537 977.333 978.691 983.313 984.215

53rd harmonic

107th harmonic 5-limit major sixth

equal-tempered major sixth Pythagorean major sixth

109th harmonic

diminished seventh (16/9 x 24/25) septimal major sixth 55th harmonic

111th harmonic augmented sixth (5/3 x 25/24)

septimal minor seventh

113th harmonic

99/56 23/13 62/35 39/22 55/31 16/9 57/32 2 to the 5/6ths 98/55 25/14 34/19 52/29 88/49 115/64 9/5 56/31 38/21 29/16 49/27 20/11 51/28 729/400 31/17 42/23 117/64 64/35 4000/2187 11/6 90/49 57/31 46/25 81/44 35/19 59/32 24/13 50/27 63/34 13/7 119/64 54/29 28/15 58/31 15/8 62/33 32/17 49/26 66/35 2 to the 11/12ths 17/9

986.402 987.747 989.896 991.165 992.596 996.090 999.468 1000.000 1000.020 1003.802 1007.442 1010.950 1013.666 1014.588 1017.596 1023.790 1026.732 1029.577 1031.787 1034.996 1038.085 1039.103 1040.080 1042.507 1044.438 1044.860 1045.256 1049.363 1052.572 1054.432 1055.647 1056.502 1057.627 1059.172 1061.427 1066.762 1067.780 1071.702 1073.781 1076.288 1080.557 1084.542 1088.269 1091.763 1095.045 1097.124 1098.133 1100.000 1101.045

Pythagorean small min. seventh 57th harmonic equal-tempered minor seventh

115th harmonic 5-limit large minor seventh

29th harmonic

117th harmonic

undecimal "median" seventh

59th harmonic

119th harmonic

5-limit major seventh

equal-tempered major seventh

121/64 125/66 36/19 256/135 55/29 243/128 19/10 40/21 61/32 21/11 44/23 23/12 48/25 121/63 123/64 25/13 77/40 52/27 27/14 56/29 29/15 60/31 31/16 64/33 33/17 243/125 35/18 39/20 125/64 88/45 45/23 96/49 49/25 51/26 108/55 55/28 57/29 63/32 160/81 99/50 125/63 127/64 2/1

1102.636 1105.668 1106.397 1107.821 1108.054 1109.775 1111.199 1115.533 1116.885 1119.463 1123.044 1126.319 1129.328 1129.900 1131.017 1132.100 1133.830 1134.663 1137.039 1139.249 1141.308 1143.233 1145.036 1146.727 1148.318 1150.834 1151.230 1156.169 1158.941 1161.094 1161.949 1164.303 1165.024 1166.383 1168.233 1168.806 1169.891 1172.736 1178.494 1182.601 1186.205 1186.422 1200.000

121st harmonic

Pythagorean major seventh

61st harmonic

123rd harmonic

septimal major seventh

31st harmonic

augmented seventh (15/8 x 25/24)

63rd harmonic

127th harmonic octave