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Re: Musical notes...

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=============================================================
WARNING!  This is a very long, almost useless list.  WARNING!
  (I dump it here lest it be otherwise lost to posterity.)
=============================================================

Octave, perfect fifth, etc. are handy ratios when discussing music,
but the complexities of tuning mean that various esoteric ratios arise.
For example, the "Pythagorean comma" -- 1.0136432 or 531441/524288 --
is the ratio between C and B-sharp in Pythagoras' tuning.

I chose to sort these ratios in ascending numeric order but do
not otherwise recall details of this list's construction.

1.0000000 1/1 unison, perfect prime
1.0000430 23232/23231 lesser harmonisma
1.0000508 19657/19656 greater harmonisma
1.0000939 10648/10647 harmonisma
1.0001020 9801/9800 kalisma, Gauss' comma
1.0001689 450359962737049600/450283905890997363 monzisma
1.0001880 250047/250000 Landscape comma
1.0002286 4375/4374 ragisma
1.0002442 4096/4095 tridecimal schisma, Sagittal schismina
1.0003306 3025/3024 Lehmerisma
1.0004166 2401/2400 Breedsma
1.0004861 2058/2057 xenisma
1.0004979 7629394531250/7625597484987 ennealimmal comma
1.0005777 1732/1731 approximation to 1 cent
1.0007770 1288/1287 triaphonisma
1.0007999 274877906944/274658203125 semithirds comma
1.0008230 1216/1215 Eratosthenes' comma
1.0011291 32805/32768 schisma
1.0012714 1575/1573 Nicola
1.0013351 750/749 ancient Chinese tempering
1.0013580 65625/65536 horwell comma
1.0013654 2200/2197 Parizek comma
1.0014005 715/714 septendecimal bridge comma
1.0014814 676/675 island comma
1.0016276 19073486328125/19042491875328 '19-tone' comma
1.0017530 4000/3993 undecimal schisma
1.0018552 540/539 Swets' comma
1.0019299 6115295232/6103515625 Vishnu comma
1.0019531 513/512 undevicesimal comma, Boethius' comma
1.0020903 19383245667680019896796723/19342813113834066795298816 Mercator's comma
1.0021997 33554432/33480783 Beta 2, septimal schisma
1.0022727 441/440 Werckmeister's undecimal septenarian schisma
1.0026041 385/384 undecimal kleisma
1.0028490 352/351 minthma
1.0030613 1224440064/1220703125 parakleisma
1.0031020 6144/6125 porwell comma
1.0033313 5120/5103 Beta 5, Garibaldi comma
1.0035200 3136/3125 middle second comma
1.0035607 1600000/1594323 kleisma - schisma
1.0037494 10976/10935 hemimage
1.0038164 118098/117649 stearnsma
1.0039215 256/255 septendecimal kleisma
1.0040459 16875/16807 small BP diesis
1.0041322 243/242 neutral third comma
1.0042265 1188/1183 kestrel comma
1.0042754 390625/388962 dimcomp comma
1.0044642 225/224 septimal kleisma
1.0046939 15625/15552 kleisma, semicomma majeur
1.0047095 640/637 huntma
1.0048605 65536/65219 orgonisma
1.0048828 1029/1024 gamelan residue
1.0050551 2187/2176 septendecimal comma
1.0056116 896/891 undecimal semicomma
1.0057142 176/175 valinorsma
1.0058283 2109375/2097152 semicomma, Fokker's comma
1.0066329 393216/390625 W�rschmidt's comma
1.0069444 145/144 29th-partial chroma
1.0075801 1728/1715 Orwell comma
1.0077696 78732/78125 medium semicomma
1.0078105 4000/3969 septimal semicomma
1.0080000 126/125 small septimal comma
1.0082304 245/243 minor BP diesis
1.0083333 121/120 undecimal seconds comma
1.0084200 50421/50000 Trimyna
1.0093688 33075/32768 mirwomo comma
1.0096021 736/729 vicesimotertial comma
1.0096153 105/104 small tridecimal comma
1.0101010 100/99 Ptolemy's comma
1.0102040 99/98 small undecimal comma
1.0102173 67108864/66430125 Misty comma, diaschisma - schisma
1.0105263 96/95 19th-partial chroma
1.0111111 91/90 medium tridecimal comma
1.0113580 2048/2025 diaschisma
1.0115288 36893488147419103232/36472996377170786403 '41-tone' comma
1.0120783 2430/2401 nuwell comma
1.0123096 3125/3087 major BP diesis
1.0125000 81/80 syntonic comma, Didymus comma
1.0127314 875/864 keema
1.0131578 77/76 approximation to 53-tone comma
1.0136432 531441/524288 Pythagorean comma, ditonic comma
1.0156250 65/64 13th-partial chroma
1.0158730 64/63 septimal comma, Archytas' comma
1.0161052 20000/19683 minimal diesis
1.0162962 686/675 senga
1.0172526 3125/3072 small diesis
1.0184012 34171875/33554432 Ampersand's comma
1.0195312 261/256 vicesimononal comma
1.0200000 51/50 17th-partial chroma
1.0203667 1594323/1562500 Unicorn comma
1.0204081 50/49 Erlich's decatonic comma, tritonic diesis
1.0205761 248/243 tricesoprimal comma
1.0206414 15625/15309 great BP diesis
1.0208333 49/48 slendro diesis, septimal 1/6-tone
1.0215737 8192/8019 undecimal minor diesis
1.0220275 48828125/47775744 Sycamore comma
1.0222222 46/45 23rd-partial chroma
1.0227272 45/44 1/5-tone
1.0240000 128/125 minor diesis, diesis
1.0251562 6561/6400 Mathieu superdiesis
1.0253906 525/512 Avicenna enharmonic diesis
1.0256410 40/39 tridecimal minor diesis
1.0283203 1053/1024 tridecimal major diesis
1.0285714 36/35 septimal diesis, 1/4-tone
1.0288065 250/243 maximal diesis
1.0292887 246/239 Meshaqah's 1/4-tone
1.0294117 35/34 septendecimal 1/4-tone
1.0297328 59049/57344 Harrison's comma
1.0299682 16875/16384 double augmentation diesis
1.0312500 33/32 undecimal comma, al-Farabi's 1/4-tone
1.0322580 32/31 Greek enharmonic 1/4-tone
1.0333333 31/30 31st-partial chroma
1.0355113 729/704 undecimal major diesis
1.0368000 648/625 major diesis
1.0370370 28/27 Archytas' 1/3-tone
1.0384615 27/26 tridecimal comma
1.0393182 134217728/129140163 Pythagorean double diminished third
1.0400000 26/25 tridecimal 1/3-tone
1.0404917 20480/19683 grave minor second
1.0409948 17496/16807 septimal major diesis
1.0416666 25/24 classic chromatic semitone, minor chroma
1.0434782 24/23 vicesimotertial minor semitone
1.0448979 256/245 septimal minor semitone
1.0476190 22/21 undecimal minor semitone
1.0500000 21/20 minor semitone
1.0526315 20/19 small undevicesimal semitone
1.0534979 256/243 limma, Pythagorean minor second
1.0546875 135/128 major chroma, major limma
1.0555555 19/18 undevicesimal semitone
1.0578512 128/121 undecimal semitone
1.0581580 413343/390625 BP small link
1.0588235 18/17 Arabic lute index finger
1.0594600 52973/50000 Mersenne's quasi-equal semitone
1.0595238 89/84 quasi-equal semitone
1.0625000 17/16 17th harmonic
1.0664062 273/256 Ibn Sina's minor second
1.0666666 16/15 minor diatonic semitone
1.0668672 83349/78125 BP minor link
1.0678710 2187/2048 apotome
1.0711836 6561/6125 BP major link
1.0714285 15/14 major diatonic semitone
1.0769230 14/13 2/3-tone
1.0800000 27/25 large limma, BP small semitone
1.0824403 1162261467/1073741824 Pythagorean-19 comma
1.0833333 13/12 tridecimal 2/3-tone
1.0843696 18225/16807 minimal BP chroma
1.0864197 88/81 2nd undecimal neutral second
1.0872483 162/149 Persian neutral second
1.0880755 4608/4235 Arabic neutral second
1.0888888 49/45 BP minor semitone
1.0904977 241/221 Meshaqah's 3/4-tone
1.0909090 12/11 3/4-tone, undecimal neutral second
1.0922666 2048/1875 double diminished third
1.0932944 375/343 BP major semitone, minor BP chroma
1.0937500 35/32 septimal neutral second
1.0973936 800/729 grave whole tone
1.0986328 1125/1024 double augmented prime
1.1000000 11/10 4/5-tone, Ptolemy's second
1.1020408 54/49 Zalzal's mujannab
1.1022927 625/567 BP great semitone, major BP chroma
1.1034482 32/29 29th subharmonic
1.1098579 65536/59049 Pythagorean diminished third
1.1111111 10/9 minor whole tone
1.1113651 21875/19683 maximal BP chroma
1.1160714 125/112 classic augmented semitone
1.1176470 19/17 quasi-meantone
1.1193415 272/243 Persian whole tone
1.1200000 28/25 middle second
1.1224489 55/49 quasi-equal major second
1.1224637 134217728/119574225 whole tone - 2 schismas, 5-limit approximation to ET whole tone
1.1250000 9/8 major whole tone
1.1363636 25/22 undecimal acute whole tone
1.1377777 256/225 diminished third
1.1390625 729/640 acute major second
1.1403486 4782969/4194304 Pythagorean double augmented prime
1.1428571 8/7 septimal whole tone
1.1458333 55/48 undecimal semi-augmented whole tone
1.1520000 144/125 classic diminished third
1.1538461 15/13 tridecimal 5/4-tone
1.1550000 231/200 5/4-tone
1.1562500 37/32 37th harmonic
1.1571428 81/70 Al-Hwarizmi's lute middle finger
1.1574074 125/108 semi-augmented whole tone
1.1666666 7/6 septimal minor third
1.1692330 16777216/14348907 Pythagorean double diminished fourth
1.1705532 2560/2187 grave minor third
1.1718750 75/64 classic augmented second
1.1739130 27/23 vicesimotertial minor third
1.1764705 20/17 septendecimal augmented second
1.1785714 33/28 undecimal minor third
1.1818181 13/11 tridecimal minor third
1.1851851 32/27 Pythagorean minor third
1.1865234 1215/1024 wide augmented second
1.1875000 19/16 19th harmonic
1.1904761 25/21 BP second, quasi-tempered minor third
1.1911764 81/68 Persian wosta
1.2000000 6/5 minor third
1.2013549 19683/16384 Pythagorean augmented second
1.2136296 4096/3375 double diminished fourth
1.2142857 17/14 supraminor third
1.2150000 243/200 acute minor third
1.2187500 39/32 39th harmonic, Zalzal wosta of Ibn Sina
1.2190476 128/105 septimal neutral third
1.2203389 72/59 Ibn Sina's neutral third
1.2222222 11/9 undecimal neutral third
1.2240000 153/125 7/4-tone
1.2244897 60/49 smaller approximation to neutral third
1.2250000 49/40 larger approximation to neutral third
1.2272727 27/22 neutral third, Zalzal wosta of al-Farabi
1.2307692 16/13 tridecimal neutral third
1.2345679 100/81 grave major third
1.2352941 21/17 submajor third
1.2359619 10125/8192 double augmented second
1.2485901 8192/6561 Pythagorean diminished fourth
1.2500000 5/4 major third
1.2592592 34/27 septendecimal major third
1.2600000 63/50 quasi-equal major third
1.2631578 24/19 smaller undevicesimal major third
1.2641975 512/405 narrow diminished fourth
1.2656250 81/64 Pythagorean major third
1.2666666 19/15 undevicesimal ditone
1.2692307 33/26 tridecimal major third
1.2698412 80/63 wide major third
1.2727272 14/11 undecimal diminished fourth or major third
1.2777777 23/18 vicesimotertial major third
1.2800000 32/25 classic diminished fourth
1.2814453 6561/5120 acute major third
1.2828922 43046721/33554432 Pythagorean double augmented second
1.2857142 9/7 septimal major third, BP third
1.2962962 35/27 9/4-tone, septimal semi-diminished fourth
1.3000000 13/10 tridecimal semi-diminished fourth
1.3020833 125/96 classic augmented third
1.3061224 64/49 2 septatones or septatonic major third
1.3090909 72/55 undecimal semi-diminished fourth
1.3125000 21/16 narrow fourth
1.3153871 2097152/1594323 Pythagorean double diminished fifth
1.3168724 320/243 grave fourth
1.3183593 675/512 wide augmented third
1.3200000 33/25 2 pentatones
1.3333333 4/3 perfect fourth
1.3348388 10935/8192 fourth + schisma, 5-limit approximation to ET fourth
1.3500000 27/20 acute fourth
1.3515243 177147/131072 Pythagorean augmented third
1.3611111 49/36 Arabic lute acute fourth
1.3636363 15/11 undecimal augmented fourth
1.3653333 512/375 double diminished fifth
1.3714285 48/35 septimal semi-augmented fourth
1.3732910 5625/4096 double augmented third
1.3740000 687/500 11/4-tone
1.3750000 11/8 undecimal semi-augmented fourth
1.3846153 18/13 tridecimal augmented fourth
1.3857576 536870912/387420489 Pythagorean double diminished sixth
1.3888888 25/18 classic augmented fourth
1.3913043 32/23 23rd subharmonic
1.4000000 7/5 septimal or Huygens' tritone, BP fourth
1.4046639 1024/729 Pythagorean diminished fifth
1.4062500 45/32 diatonic tritone
1.4117647 24/17 1st septendecimal tritone
1.4141414 140/99 quasi-equal tritone
1.4142857 99/70 2nd quasi-equal tritone
1.4166666 17/12 2nd septendecimal tritone
1.4222222 64/45 2nd tritone
1.4238281 729/512 Pythagorean tritone
1.4285714 10/7 Euler's tritone
1.4375000 23/16 23rd harmonic
1.4400000 36/25 classic diminished fifth
1.4432537 387420489/268435456 Pythagorean double augmented third
1.4444444 13/9 tridecimal diminished fifth
1.4545454 16/11 undecimal semi-diminished fifth
1.4555555 131/90 13/4-tone
1.4563555 8192/5625 double diminished sixth
1.4583333 35/24 septimal semi-diminished fifth
1.4648437 375/256 double augmented fourth
1.4666666 22/15 undecimal diminished fifth
1.4693877 72/49 Arabic lute grave fifth
1.4798105 262144/177147 Pythagorean diminished sixth
1.4814814 40/27 grave fifth
1.4927113 512/343 3 septatones or septatonic fifth
1.4980000 749/500 ancient Chinese quasi-equal fifth
1.4983081 16384/10935 fifth - schisma, 5-limit approximation to ET fifth
1.5000000 3/2 perfect fifth
1.5151515 50/33 3 pentatones
1.5170370 1024/675 narrow diminished sixth
1.5187500 243/160 acute fifth
1.5204648 1594323/1048576 Pythagorean double augmented fourth
1.5238095 32/21 wide fifth
1.5277777 55/36 undecimal semi-augmented fifth
1.5306122 75/49 BP fifth
1.5360000 192/125 classic diminished sixth
1.5384615 20/13 tridecimal semi-augmented fifth
1.5423728 91/59 15/4-tone
1.5428571 54/35 septimal semi-augmented fifth
1.5555555 14/9 septimal minor sixth
1.5589773 67108864/43046721 Pythagorean double diminished seventh
1.5607376 10240/6561 grave minor sixth
1.5625000 25/16 classic augmented fifth
1.5714285 11/7 undecimal augmented fifth
1.5750000 63/40 narrow minor sixth
1.5757575 52/33 tridecimal minor sixth
1.5789473 30/19 smaller undevicesimal minor sixth
1.5802469 128/81 Pythagorean minor sixth
1.5820312 405/256 wide augmented fifth
1.5833333 19/12 undevicesimal minor sixth
1.5873015 100/63 quasi-equal minor sixth
1.5882352 27/17 septendecimal minor sixth
1.6000000 8/5 minor sixth
1.6018066 6561/4096 Pythagorean augmented fifth
1.6181728 16384/10125 double diminished seventh
1.6190476 34/21 supraminor sixth
1.6200000 81/50 acute minor sixth
1.6250000 13/8 tridecimal neutral sixth
1.6296296 44/27 neutral sixth
1.6326530 80/49 smaller approximation to neutral sixth
1.6333333 49/30 larger approximation to neutral sixth
1.6339869 250/153 17/4-tone
1.6363636 18/11 undecimal neutral sixth
1.6406250 105/64 septimal neutral sixth
1.6410256 64/39 39th subharmonic
1.6460905 400/243 grave major sixth
1.6470588 28/17 submajor sixth
1.6479492 3375/2048 double augmented fifth
1.6647868 32768/19683 Pythagorean diminished seventh
1.6666666 5/3 major sixth, BP sixth
1.6770186 270/161 Kirnberger's sixth
1.6800000 42/25 quasi-tempered major sixth
1.6842105 32/19 19th subharmonic
1.6855967 2048/1215 narrow diminished seventh
1.6875000 27/16 Pythagorean major sixth
1.6923076 22/13 tridecimal major sixth
1.7000000 17/10 septendecimal diminished seventh
1.7059558 4096/2401 4 septatones or septatonic major sixth
1.7066666 128/75 diminished seventh
1.7085937 2187/1280 acute major sixth
1.7105230 14348907/8388608 Pythagorean double augmented fifth
1.7142857 12/7 septimal major sixth
1.7280000 216/125 semi-augmented sixth
1.7297297 64/37 37th subharmonic
1.7311827 161/93 19/4-tone
1.7333333 26/15 tridecimal semi-augmented sixth
1.7361111 125/72 classic augmented sixth
1.7500000 7/4 harmonic seventh
1.7538495 8388608/4782969 Pythagorean double diminished octave
1.7558299 1280/729 grave minor seventh
1.7578125 225/128 augmented sixth
1.7600000 44/25 undecimal grave minor seventh
1.7777777 16/9 Pythagorean minor seventh
1.7818181 98/55 quasi-equal minor seventh
1.7857142 25/14 middle minor seventh
1.8000000 9/5 just minor seventh, BP seventh
1.8020324 59049/32768 Pythagorean augmented sixth
1.8125000 29/16 29th harmonic
1.8181818 20/11 large minor seventh
1.8204444 2048/1125 double diminished octave
1.8225000 729/400 acute minor seventh
1.8285714 64/35 septimal neutral seventh
1.8310546 1875/1024 double augmented sixth
1.8333333 11/6 21/4-tone, undecimal neutral seventh
1.8409090 81/44 2nd undecimal neutral seventh
1.8461538 24/13 tridecimal neutral seventh
1.8518518 50/27 grave major seventh
1.8571428 13/7 16/3-tone
1.8666666 28/15 grave major seventh
1.8728852 4096/2187 Pythagorean diminished octave
1.8750000 15/8 classic major seventh
1.8823529 32/17 17th subharmonic
1.8876404 168/89 quasi-equal major seventh
1.8888888 17/9 septendecimal major seventh
1.8947368 36/19 smaller undevicesimal major seventh
1.8962962 256/135 octave - major chroma
1.8984375 243/128 Pythagorean major seventh
1.9000000 19/10 undevicesimal major seventh
1.9047619 40/21 acute major seventh
1.9090909 21/11 undecimal major seventh
1.9166666 23/12 vicesimotertial major seventh
1.9200000 48/25 classic diminished octave
1.9221679 19683/10240 acute major seventh
1.9243383 129140163/67108864 Pythagorean double augmented sixth
1.9285714 27/14 septimal major seventh
1.9290123 625/324 octave - major diesis
1.9375000 31/16 31st harmonic
1.9393939 64/33 33rd subharmonic
1.9418074 32768/16875 octave - double augmentation diesis
1.9428571 68/35 23/4-tone
1.9440000 243/125 octave - maximal diesis
1.9444444 35/18 septimal semi-diminished octave
1.9496638 32768/16807 5 septatones or septatonic diminished octave
1.9531250 125/64 classic augmented seventh, octave - minor diesis
1.9600000 49/25 BP eighth
1.9660800 6144/3125 octave - small diesis
1.9683000 19683/10000 octave - minimal diesis
1.9687500 63/32 octave - septimal comma
1.9730807 1048576/531441 Pythagorean diminished ninth
1.9753086 160/81 octave - syntonic comma
1.9775390 2025/1024 2 tritones
1.9868214 390625/196608 octave - W�rschmidt's comma
1.9977442 65536/32805 octave - schisma
2.0000000 2/1 octave
2.0272865 531441/262144 Pythagorean augmented seventh
2.0833333 25/12 classic augmented octave
2.1250000 17/8 septendecimal minor ninth
2.1333333 32/15 minor ninth
2.1428571 15/7 septimal minor ninth, BP ninth
2.1648806 1162261467/536870912 Pythagorean double augmented seventh
2.2000000 11/5 neutral ninth
2.2222222 20/9 small ninth
2.2500000 9/4 major ninth
2.2857142 16/7 septimal major ninth
2.3333333 7/3 minimal tenth, BP tenth
2.5200000 63/25 quasi-equal major tenth, BP eleventh
2.7777777 25/9 classic augmented eleventh, BP twelfth

-----
James

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Thread

Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 08:20 +0000
  Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 12:58 +0000
    Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 18:06 +0000
      Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-28 10:15 +0000
  Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-27 16:25 +0000
    Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 17:10 +0000
  Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-28 12:30 +0000
    Re: Musical notes... HenHanna@NewsGrouper <user4055@newsgrouper.org.invalid> - 2026-05-28 18:21 +0000
    Re: Musical notes... msb@vex.net (Mark Brader) - 2026-05-28 18:50 +0000
    Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-29 17:19 +0000
    Re: Musical notes... Charlie Roberts <croberts@gmail.com> - 2026-05-29 15:48 -0400
      Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-29 20:59 +0000
        Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:07 +0000
          Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:25 +0000
            Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:33 +0000
            Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-30 12:56 +0000
    Re: Musical notes... Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-05-29 22:38 +0100
      Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-30 06:38 +0000
    Re: Musical notes... Phil Carmody <pc+usenet@asdf.org> - 2026-06-05 16:41 +0300
      Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-06-05 14:54 +0000

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