Introduction
In Just Intonation, a musical interval is specified as a ratio of two frequencies.. When two (or more) pitches are sounded that are in simple proportions to one another, there is a “fusing” quality to the sound which is often described as pleasing; hence the interest in tuning the pitches of musical systems according to such proportions. There is much debate as to what “consonance” means in a musical system, but in Just Intonation, it is generally assumed that lower numbers in frequency ratios lead to greater consonance. In the actual performance of a piece of music, the number of factors involved are enormous, and it is not often helpful to reduce a musical experience to a one-dimensional description of “consonance versus dissonance.” Hence the need for this gallery, to give life to conversation about what an interval means beyond the numerical description: “5/3” or “21/16” or what have you.
What follows is a Gallery of Just Intervals in ascending order from 1/1 to 2/1 and beyond. No such list could possibly be complete (as there are infinite possible ratios).
Gallery of Just Intervals
| frequency ratio |
cents value | some common names |
| 1/1 | 0 | unity, perfect prime, Tonic |
| 32805/32768 | 1.953721 | schisma, |-15, 8, 1> |
| 225/224 | 7.711523 | septimal subcomma, |-5, 2, 2, -1> |
| 100/99 | 17.399484 | Ptolemy’s comma |
| 99/98 | 17.576131 | Mothwellsma |
| 2048/2025 | 19.552569 | |11, -4, -2> |
| 81/80 | 21.506286 | Syntonic comma, Didymus comma |
| 531441/524288 | 23.46001 | Pythagorean comma, Ditonic comma, |-19, 12> |
| 66/65 | 26.431568 | Winmeanma |
| 65/64 | 26.841376 | Wilsorma, 13th-partial chroma |
| 64/63 | 27.264092 | Septimal comma, Archytas’ comma |
| 3125/3072 | 29.613568 | Magic comma, small diesis, |-10, -1, 5> |
| 50/49 | 34.975615 | septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis |
| 49/48 | 35.696812 | large septimal diesis, slendro diesis |
| 45/44 | 38.905773 | undecimal 1/5th tone |
| 128/125 | 41.058858 | Diesis, minor diesis, augmented comma, enharmonic comma, |7, 0, -3> |
| 525/512 | 43.408335 | Avicenna’s enharmonic diesis, |-9, 1, 2, 1> |
| 36/35 | 48.770381 | double comma, septimal quarter tone |
| 250/243 | 49.166137 | Porcupine comma, |1, -5, 3> |
| 59049/57344 | 50.724102 | Harrison’s comma, |-13, 10, 0, -1> |
| 100/97 | 52.732017 | shrutar quarter tone |
| 33/32 | 53.272943 | undecimal quarter tone, undecimal diesis, al-Farabi’s 1/4-tone, octave-reduced 33rd harmonic |
| 648/625 | 62.565148 | diminished comma, major diesis, |8, 4, -4> |
| 28/27 | 62.960904 | septimal chroma, small septimal chromatic semitone, septimal subminor second |
| 25/24 | 70.672427 | chroma, chromatic semitone, Zarlinian semitone |
| 68/65 | 78.114034 | valentine semitone |
| 22/21 | 80.537035 | undecimal minor semitone |
| 64/61 | 83.115195 | harry minor semitone |
| 21/20 | 84.467193 | minor semitone, large septimal chromatic semitone |
| 256/243 | 90.224996 | Pythagorean limma, Pythagorean minor second, |8, -5> |
| 135/128 | 92.178716 | major limma, |-7, 3, 1> |
| 18/17 | 98.954592 | small septendecimal semitone, Arabic lute index finger |
| 17/16 | 104.95541 | large septendecimal semitone, octave-reduced 17th harmonic |
| 16/15 | 111.731285 | diatonic semitone, classic minor second, octave-reduced 15th subharmonic |
| 2187/2048 | 113.685006 | apotome, |-11, 7> |
| 77/72 | 116.233847 | undecimal secor |
| 15/14 | 119.442808 | septimal diatonic semitone |
| 14/13 | 128.298245 | 2/3-tone, trienthird, tridecimal supraminor second |
| 27/25 | 133.237575 | large limma |
| 13/12 | 138.572661 | tridecimal subtone, tridecimal 2/3-tone |
| 243/224 | 140.949098 | septimal subtone, |-5, 5, 0, -1> |
| 88/81 | 143.497939 | undecimal subtone |
| 49/45 | 147.428097 | swetismic neutral second |
| 12/11 | 150.637059 | small undecimal neutral second, 3/4-tone |
| 35/32 | 155.13962 | septimal neutral second |
| 78/71 | 162.786119 | porcupine neutral second |
| 11/10 | 165.004228 | large undecimal neutral second, 4/5-tone, Ptolemy’s second |
| 54/49 | 168.21319 | Zalzal’s mujannab |
| 10/9 | 182.403712 | minor whole tone |
| 49/44 | 186.333871 | werckismic minor second |
| 19/17 | 192.557607 | quasi-meantone |
| 28/25 | 196.198479 | middle major second |
| 55/49 | 199.979843 | werckismic tone |
| 9/8 | 203.910002 | major whole tone, Pythagorean tone, octave-reduced 9th harmonic |
| 17/15 | 216.686695 | septendecimal whole tone, septendecimal eventone |
| 8/7 | 231.174094 | supermajor second, septimal whole tone, diminished third, octave-reduced 7th subharmonic |
| 63/55 | 235.104252 | werckismic supermajor second |
| 55/48 | 235.676655 | keenanismic supermajor second |
| 15/13 | 247.741053 | semifourth, tridecimal ultramajor second, tridecimal inframinor third |
| 22/19 | 253.804926 | minimal minor third, godzilla third |
| 64/55 | 262.368344 | keenanismic subminor third, octave-reduced 55th subharmonic |
| 7/6 | 266.870906 | subminor third, septimal minor third, augmented second |
| 90/77 | 270.079867 | swetismic subminor third |
| 62/53 | 271.531027 | orwell subminor third |
| 75/64 | 274.582429 | classic augmented second |
| 20/17 | 281.358304 | septendecimal augmented second, septendecimal minor third |
| 13/11 | 289.209179 | tridecimal minor third |
| 32/27 | 294.134997 | Pythagorean minor third, octave-reduced 27th subharmonic |
| 19/16 | 297.513016 | otonal minor third, octave-reduced 19th harmonic |
| 25/21 | 301.84652 | quasi-tempered minor third |
| 61/51 | 309.974395 | myna third |
| 6/5 | 315.641287 | minor third, pental minor third |
| 77/64 | 320.143849 | keenanismic minor third, octave-reduced 77th harmonic |
| 135/112 | 323.35281 | large septimal minor third, marvelous minor third, |-4, 3, 1, -1> |
| 35/29 | 325.562426 | doublewide minor third |
| 17/14 | 336.129503 | septendecimal supraminor third |
| 73/60 | 339.520756 | amity supraminor third |
| 625/512 | 345.254855 | 5-limit neutral third, |-9, 0, 4> |
| 11/9 | 347.407941 | undecimal neutral third |
| 60/49 | 350.616902 | smaller septimal neutral third |
| 49/40 | 351.338099 | larger septimal neutral third |
| 27/22 | 354.54706 | rastmic neutral third |
| 16/13 | 359.472338 | tridecimal neutral third |
| 21/17 | 365.825498 | septendecimal submajor third |
| 56/45 | 378.602191 | narrow perde segah, marvelous major third |
| 51/41 | 377.848005 | maja third |
| 71/57 | 380.228526 | witchcraft major third |
| 76/61 | 380.628211 | magic major third |
| 96/77 | 381.811152 | undecimal perde segah, keenanismic major third |
| 5/4 | 386.313714 | major third, octave-reduced 5th harmonic, pental major third |
| 81/64 | 407.820003 | Pythagorean major third, octave-reduced 81st harmonic |
| 80/63 | 413.577806 | werckismic sharp major third |
| 14/11 | 417.507964 | undecimal major third, undecimal diminished fourth |
| 32/25 | 427.372572 | classic diminished fourth |
| 77/60 | 431.875134 | swetismic supermajor third |
| 9/7 | 435.084095 | supermajor third, septimal major third, septimal diminished fourth |
| 31/24 | 443.080572 | sensi supermajor third |
| 22/17 | 446.362533 | septendecimal supermajor third |
| 35/27 | 449.274618 | semi-diminished fourth |
| 13/10 | 454.213948 | Barbados third, tridecimal 9/4 tone, tridecimal semidiminished fourth, tridecimal ultramajor third |
| 64/49 | 462.348187 | septatonic major third |
| 17/13 | 464.427748 | septendecimal sub-fourth |
| 21/16 | 470.780907 | sub-fourth, narrow fourth, augmented third, 8ve-reduced 21st harmonic |
| 33/25 | 480.645516 | "5-EDO"-esque fourth |
| 117/88 | 493.119721 | tridecimal gentle fourth, |-3, 2, 0, 0, -1, 1> |
| 4/3 | 498.044999 | just perfect fourth, octave-reduced 3rd subharmonic, diatessaron |
| 75/56 | 505.756522 | marvelous fourth |
| 27/20 | 519.551289 | acute fourth |
| 19/14 | 528.68711 | 19-limit wide fourth |
| 49/36 | 533.741811 | Arabic lute acute fourth |
| 15/11 | 536.950772 | undecimal augmented fourth, subaugmented fourth |
| 48/35 | 546.815381 | septimal super-fourth |
| 11/8 | 551.317942 | super-fourth, undecimal semi-augmented fourth, octave-reduced 11th harmonic or harmonic 11th, Alphorn-Fa |
| 18/13 | 563.38234 | tridecimal augmented fourth |
| 25/18 | 568.717426 | classic augmented fourth, pental augmented fourth |
| 88/63 | 578.582034 | werckismic augmented fourth |
| 7/5 | 582.512193 | augmented fourth, septimal tritone, Huygen’s tritone |
| 45/32 | 590.223716 | |
| 108/77 | 585.721154 | swetismic augmented fourth, |2, 3, 0, -1, -1> |
| 24/17 | 596.999591 | smaller septendecimal tritone |
| 17/12 | 603.000409 | larger septendecimal tritone |
| 64/45 | 609.776284 | |
| 10/7 | 617.487807 | diminished fifth, Euler’s tritone, superaugmented fourth |
| 23/16 | 628.274347 | 23-limit superaugmented fourth, octave-reduced 23rd harmonic |
| 36/25 | 631.282574 | pental diminished fifth, classic diminshed fifth |
| 13/9 | 636.61766 | tridecimal diminished fifth |
| 16/11 | 648.682058 | sub-fifth, octave-reduced 11th subharmonic |
| 35/24 | 653.184619 | septimal sub-fifth |
| 22/15 | 663.049228 | undecimal diminished fifth, semidiminished fifth |
| 72/49 | 666.258889 | septimal catafifth |
| 81/55 | 670.188347 | undecimal catafifth |
| 28/19 | 671.31289 | 19-limit narrow fifth |
| 40/27 | 680.448711 | grave fifth |
| 112/75 | 694.243478 | marvelous fifth |
| 3/2 | 701.955001 | just perfect fifth, octave-reduced 3rd harmonic, diapente |
| 182/121 | 706.717684 | tridecimal gentle fifth, |1, 0, 0, 1, -2, 1 > |
| 176/117 | 706.880279 | tridecimal gentle fifth, |4, -2, 0, 0, 1, -1> |
| 50/33 | 719.354484 | "5-EDO"-esque fifth |
| 32/21 | 729.219093 | super-fifth, wide fifth, diminished sixth, octave-reduced 21st subharmonic |
| 26/17 | 735.572252 | septendecimal super-fifth |
| 49/32 | 737.651813 | superduper fifth, octave-reduced 49th harmonic |
| 20/13 | 745.786052 | Barbados sixth, ratwolf wolf fifth, tridecimal semi-augmented fifth, tridecimal ultraminor sixth |
| 17/11 | 753.637467 | septendecimal subminor sixth |
| 14/9 | 764.915905 | subminor sixth, septimal minor sixth, augmented fifth |
| 25/16 | 772.627428 | pental augmented fifth, classic augmented fifth, otonal minor sixth, octave-reduced 25th harmonic |
| 11/7 | 782.492036 | undecimal subminor sixth, undecimal augmented fifth |
| 63/40 | 786.422194 | |
| 128/81 | 792.179997 | |
| 8/5 | 813.686286 | minor sixth, octave-reduced 5th subharmonic |
| 413/256 | 827.997565 | octave-reduced 413th harmonic, homestuck sixth (2-8 * 7 * 59) |
| 13/8 | 840.527662 | tridecimal neutral sixth, octave-reduced 13th harmonic |
| 80/49 | 848.661901 | |
| 49/30 | 849.383198 | |
| 18/11 | 852.592059 | undecimal neutral sixth |
| 28/17 | 863.870497 | septendecimal submajor sixth |
| 5/3 | 884.358713 | major sixth |
| 42/25 | 898.15348 | |
| 27/16 | 905.865003 | Pythagorean major sixth, octave-reduced 27th harmonic |
| 22/13 | 910.790821 | tridecimal major sixth |
| 17/10 | 918.641696 | septendecimal diminished seventh, septendecimal major sixth |
| 12/7 | 933.129094 | supermajor sixth, septimal major sixth, diminished seventh |
| 26/15 | 952.258947 | semitwelfth, tridecimal inframinor seventh, tridecimal ultramajor sixth |
| 7/4 | 968.825906 | subminor seventh, harmonic seventh, augmented sixth, octave-reduced 7th harmonic |
| 225/128 | 976.537429 | marvel five-limit harmonic seventh, octave-reduced 225th harmonic, |-7, 2, 2> |
| 30/17 | 983.313305 | septendecimal minor seventh |
| 16/9 | 996.089998 | Pythagorean minor seventh, small minor seventh, octave-reduced 9th subharmonic |
| 25/14 | 1003.801521 | Middle minor seventh |
| 34/19 | 1007.442393 | Quasi-meantone minor seventh |
| 9/5 | 1017.596288 | minor seventh, large minor seventh |
| 29/16 | 1029.577194 | 29-limit large minor seventh, octave-reduced 29th harmonic |
| 20/11 | 1034.995772 | undecimal minor seventh, small undecimal neutral seventh |
| 64/35 | 1044.86038 | |
| 11/6 | 1049.362941 | undecimal neutral seventh, 21/4-tone |
| 24/13 | 1061.427339 | tridecimal neutral seventh |
| 13/7 | 1071.701755 | 16/3-tone, tridecimal submajor seventh |
| 28/15 | 1080.557192 | grave major seventh, octave minus a reddish aug unison |
| 15/8 | 1088.268715 | major seventh, just major seventh, octave-reduced 15th harmonic |
| 32/17 | 1095.04459 | small septendecimal major seventh, 8ve-reduced 17th subharmonic |
| 17/9 | 1101.045408 | large septendecimal major seventh |
| 243/128 | 1109.775004 | Pythagorean major seventh, octave-reduced 243rd harmonic |
| 40/21 | 1115.532907 | acute major seventh |
| 61/32 | 1116.884905 | octave-reduced 61st harmonic |
| 48/25 | 1129.327573 | octave minus a deep yellow augmented unison |
| 27/14 | 1137.039096 | |
| 31/16 | 1145.035572 | 31-limit ultramajor seventh, octave-reduced 31st harmonic |
| 64/33 | 1146.727057 | octave-reduced 33rd subharmonic |
| 35/18 | 1151.239619 | octave minus a greenish comma |
| 96/49 | 1164.303188 | |
| 49/25 | 1165.024385 | |
| 160/81 | 1178.493814 | octave minus syntonic comma |
| 2/1 | 1200 | octave, diapason |
Intervals larger than 2/1
| 13/6 | 1338.573 |
| 11/5 | 1365.004 |
| 16/7 | 1431.174 |
| 5/2 | 1586.314 |
| 8/3 | 1698.045 |
| 11/4 | 1751.318 |
| 16/5 | 2013.686 |
| 13/4 | 2040.528 |
| 10/3 | 2084.359 |
| 7/2 | 2168.826 |
| 11/3 | 2249.363 |
| 15/4 | 2288.269 |
| 4/1 | 2400 |
| 13/3 | 2538.573 |
| 9/2 | 2603.91 |
| 14/3 | 2666.871 |
| 5/1 | 2786.314 |
| 16/3 | 2898.045 |
| 11/2 | 2951.318 |
| 6/1 | 3101.955 |
| 13/2 | 3240.528 |
| 7/1 | 3368.826 |
| 15/2 | 3488.269 |
| 8/1 | 3600 |
| 9/1 | 3803.91 |
| 10/1 | 3986.314 |
| 11/1 | 4151.318 |
| 12/1 | 4301.955 |
| 13/1 | 4440.528 |
| 14/1 | 4568.826 |
| 15/1 | 4688.269 |
| 16/1 | 4800 |
List of root-3rd-P5 triads in JI
The basic structure of major and minor triads — two stacked thirds which total to a perfect fifth — can be generalized to produce an infinity of chords with their own distinct qualities. What follows is a list of all such chords that are possible in 47-prime-limit Just Intonation, assuming a 3/2 perfect fifth. Wiki authors can feel free to extend this list beyond the 47-limit or leave it at that, but of course, it should be noted that a complete list would be infinite. The narrowest “third” is 27/25, which is decidedly not a third; and the widest “third” is 50/27, which ditto. Thus, the entire conceptual category of a third and then some is covered, and composers can decide for themselves what counts as a “third” and what doesn’t.
List of root-3rd-P5 triads in JI
| chord | first interval | second interval | prime | odd | comments | ||
| ratio | cents | ratio | cents | limit | limit | ||
| 50:54:75 | 27/25 | 133.238 | 25/18 | 568.717 | 5 | 75 | |
| 12:13:18 | 13/12 | 138.573 | 18/13 | 563.382 | 13 | 13 | |
| 46:50:69 | 25/23 | 144.353 | 69/50 | 557.602 | 23 | 69 | |
| 22:24:33 | 12/11 | 150.637 | 11/8 | 551.318 | 11 | 33 | |
| 42:46:63 | 23/21 | 157.493 | 63/46 | 544.462 | 23 | 63 | |
| 10:11:15 | 11/10 | 165.004 | 15/11 | 536.951 | 11 | 15 | |
| 38:42:57 | 21/19 | 173.268 | 19/14 | 528.687 | 19 | 57 | |
| 18:20:27 | 10/9 | 182.404 | 27/20 | 519.551 | 5 | 27 | |
| 34:38:51 | 19/17 | 192.558 | 51/38 | 509.397 | 19 | 51 | Quasi-meantone Suspended 2nd |
| 8:9:12 | 9/8 | 203.910 | 4/3 | 498.045 | 3 | 9 | Suspended 2nd |
| 30:34:45 | 17/15 | 216.687 | 45/34 | 485.268 | 17 | 45 | |
| 22:25:33 | 25/22 | 221.309 | 33/25 | 480.646 | 11 | 33 | |
| 36:41:54 | 41/36 | 225.152 | 54/41 | 476.803 | 41 | 41 | |
| 14:16:21 | 8/7 | 231.174 | 21/16 | 470.781 | 7 | 21 | |
| 20:23:30 | 23/20 | 241.961 | 30/23 | 459.994 | 23 | 23 | |
| 26:30:39 | 15/13 | 247.741 | 13/10 | 454.214 | 13 | 15 | Inverse "barbados" triad |
| 32:37:48 | 37/32 | 251.344 | 48/37 | 450.611 | 37 | 37 | Rooted inframinor triad |
| 6:7:9 | 7/6 | 266.871 | 9/7 | 435.084 | 7 | 9 | Septimal subminor |
| 40:47:60 | 47/40 | 279.193 | 60/47 | 422.762 | 47 | 47 | |
| 28:33:42 | 33/28 | 284.447 | 14/11 | 417.508 | 11 | 33 | |
| 22:26:33 | 13/11 | 289.210 | 33/26 | 412.745 | 13 | 33 | Neo-Gothic minor triad |
| 16:19:24 | 19/16 | 297.513 | 24/19 | 404.442 | 19 | 19 | Rooted minor triad |
| 26:31:39 | 31/26 | 304.508 | 39/31 | 397.447 | 31 | 39 | |
| 36:43:54 | 43/36 | 307.608 | 54/43 | 394.347 | 43 | 43 | |
| 10:12:15 | 6/5 | 315.641 | 5/4 | 386.314 | 5 | 5 | 5-limit minor |
| 24:29:36 | 29/24 | 327.622 | 36/29 | 374.333 | 29 | 29 | |
| 14:17:21 | 17/14 | 336.130 | 21/17 | 365.825 | 17 | 21 | 17-limit supraminor |
| 32:39:48 | 39/32 | 342.483 | 16/13 | 359.472 | 13 | 39 | Rooted neutral triad |
| 18:22:27 | 11/9 | 347.408 | 27/22 | 354.547 | 11 | 27 | Neutral |
| 22:27:33 | 27/22 | 354.547 | 11/9 | 347.408 | 11 | 33 | Neutral |
| 26:32:39 | 16/13 | 359.472 | 39/32 | 342.483 | 13 | 39 | |
| 30:37:45 | 37/30 | 363.075 | 45/37 | 338.880 | 37 | 45 | |
| 4:5:6 | 5/4 | 386.314 | 6/5 | 315.641 | 5 | 5 | 5-limit major |
| 30:38:45 | 19/15 | 409.244 | 45/38 | 292.711 | 19 | 45 | |
| 26:33:39 | 33/26 | 412.745 | 13/11 | 289.210 | 13 | 33 | |
| 22:28:33 | 14/11 | 417.508 | 33/28 | 284.447 | 11 | 33 | Neo-Gothic major triad |
| 94:120:141 | 60/47 | 422.762 | 47/40 | 279.193 | 47 | 141 | |
| 18:23:27 | 23/18 | 424.364 | 27/23 | 277.591 | 23 | 27 | |
| 32:41:48 | 41/32 | 429.062 | 48/41 | 272.893 | 41 | 41 | Rooted supermajor triad |
| 14:18:21 | 9/7 | 435.084 | 7/6 | 266.871 | 7 | 9 | Septimal supermajor |
| 24:31:36 | 31/24 | 443.081 | 36/31 | 258.874 | 31 | 31 | |
| 74:96:111 | 48/37 | 450.611 | 37/32 | 251.344 | 37 | 37 | Rooted ultramajor triad |
| 10:13:15 | 13/10 | 454.214 | 15/13 | 247.741 | 13 | 15 | "Barbados" triad |
| 36:47:54 | 47/36 | 461.597 | 54/47 | 240.358 | 47 | 47 | |
| 26:34:39 | 17/13 | 464.428 | 39/34 | 237.527 | 17 | 39 | |
| 16:21:24 | 21/16 | 470.781 | 8/7 | 231.174 | 7 | 21 | |
| 22:29:33 | 29/22 | 478.259 | 33/29 | 223.696 | 29 | 29 | |
| 28:37:42 | 37/28 | 482.518 | 42/37 | 219.437 | 37 | 37 | |
| 34:45:51 | 45/34 | 485.268 | 17/15 | 216.687 | 17 | 51 | |
| 6:8:9 | 4/3 | 498.045 | 9/8 | 203.910 | 3 | 9 | Suspended 4th |
| 38:51:57 | 51/38 | 509.397 | 19/17 | 192.558 | 19 | 57 | Quasi-meantone Suspended 4th |
| 20:27:30 | 27/20 | 519.551 | 10/9 | 182.404 | 5 | 27 | |
| 14:19:21 | 19/14 | 529.687 | 21/19 | 173.268 | 19 | 21 | |
| 22:30:33 | 15/11 | 536.951 | 11/10 | 165.004 | 11 | 33 | |
| 46:63:69 | 63/46 | 544.462 | 23/21 | 157.493 | 23 | 69 | |
| 8:11:12 | 11/8 | 551.318 | 12/11 | 150.637 | 11 | 11 | |
| 50:69:75 | 69/50 | 557.602 | 25/23 | 144.353 | 23 | 75 | |
| 26:36:39 | 18/13 | 563.382 | 13/12 | 138.573 | 13 | 39 | |
| 18:25:27 | 25/18 | 568.717 | 27/25 | 133.238 | 5 | 27 | |
Taken from the xenharmonic wiki to preserve some knowledge