159edo/Interval names and harmonies
159edo contains all the intervals of 53edo, however, as some of the interpretations differ due 159edo having different mappings for certain primes, those differences show up in how harmonies are constructed. It should be noted that since 159edo does a better job of representing the 2.3.11 subgroup than 24edo, some of the chords listed on the page for 24edo interval names and harmonies carry over to this page, even though the exact sets of enharmonics differ between the two systems.
| Step | Cents | 5 limit | 7 limit | 11 limit | 13 limit | 17 limit | Interval Names | Notes | ||
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 1/1 | P1 | Perfect Unison | D | The root of any chord, as well as the base representation of the Tonic. | ||||
| 1 | 7.5471698 | 225/224 | 243/242 | 196/195, 351/350 | 256/255 | R1 | Wide Prime | D/ | As the approximation of both the rastma and the marvel comma, this interval is useful for slight dissonances that convey something less than satisfactory, however it can also add to the bandwidth of a sound. | |
| 2 | 15.0943396 | ? | 121/120, 100/99 | 144/143 | 120/119 | rK1 | Narrow Superprime | D↑\ | As the approximation of the ptolemisma and the biyatisma, this interval is useful for slight dissonances that create more noticeable tension. | |
| 3 | 22.6415094 | 81/80 | ? | ? | 78/77 | 85/84 | K1 | Lesser Superprime | D↑ | As the approximation of the syntonic comma and the Pythagorean comma, this interval is useful for appoggiaturas, acciaccaturas, and quick passing tones. |
| 4 | 30.1886792 | 64/63 | 56/55, 55/54 | ? | 52/51 | S1, kU1 | Greater Superprime, Narrow Inframinor Second | Edb<, Dt<↓ | As the approximation of septimal comma and the telepathma, this interval is useful for various types of subchromatic gestures, as well as for appoggiaturas, acciaccaturas, and quick passing tones. | |
| 5 | 37.7358491 | ? | 45/44 | ? | 51/50 | um2, RkU1 | Inframinor Second, Wide Superprime | Edb>, Dt>↓ | By default, this interval is a type of paradiatonic quartertone and is used in much the same way as 24edo's own Inframinor Second. | |
| 6 | 45.2830189 | ? | ? | ? | 40/39 | 192/187 | kkm2, Rum2, rU1 | Wide Inframinor Second, Narrow Ultraprime, Semilimma | Eb↓↓, Dt<\ | This interval is particularly likely to be used as a cross between an Ultraprime and an Inframinor Second; furthermore, as the name "semilimma" suggests, this interval is one half of a Pythagorean Minor Second. |
| 7 | 52.8301887 | ? | 33/32 | ? | 34/33 | U1, rKum2 | Ultraprime, Narrow Subminor Second | Dt<, Edb<↑ | By default, this interval is a type of parachromatic quartertone and is thus used in much the same way as 24edo's own Ultraprime. | |
| 8 | 60.3773585 | 28/27 | ? | ? | 88/85 | sm2, Kum2, uA1 | Lesser Subminor Second, Wide Ultraprime, Infra-Augmented Prime | Dt>, Eb↓\ | Although this interval can act as a leading tone, it can also act as a trienstone- that is, a third of a tone- since it's one third of the Ptolemaic Major Second. | |
| 9 | 67.9245283 | 25/24 | ? | ? | 26/25, 27/26 | ? | km2, rKU1, kkA1 | Greater Subminor Second, Diptolemaic Augmented Prime | Eb↓, Dt<↑\, D#↓↓ | Although this interval frequently acts as the Diptolemaic Chroma, it can also act as a trienstone- that is, a third of a tone- since it's one third of the Pythagorean Major Second. |
| 10 | 75.4716981 | ? | ? | ? | 160/153 | Rkm2, rKuA1 | Wide Subminor Second, Lesser Sub-Augmented Prime | Eb↓/, Dt<↑ | This interval acts as a type of semitone, however, whether it's a diatonic or chromatic semitone depends on the situation. | |
| 11 | 83.0188679 | 21/20 | 22/21 | ? | ? | rm2, KuA1 | Narrow Minor Second, Greater Sub-Augmented Prime | Eb\, Dt>↑ | Not only does this interval serve as a type of leading tone due to it being the approximation of the Septimal Minor Semitone, but it should be noted that six of these add up to a Perfect Fourth. | |
| 12 | 90.5660377 | 256/243, 135/128 | ? | ? | ? | ? | m2, kA1 | Pythagorean Minor Second, Ptolemaic Augmented Prime | Eb, D#↓ | As the approximation of both the Pythagorean Minor Second and the Ptolemaic Augmented Prime, this interval is used accordingly. |
| 13 | 98.1132075 | ? | 128/121 | 55/52 | 18/17 | Rm2, RkA1 | Artomean Minor Second, Artomean Augmented Prime | Eb/, D#↓/ | This interval is one of two in this system that are essential in executing the frameshift cadence; it is also the closest approximation of the 12edo semitone found in this system. | |
| 14 | 105.6603774 | ? | ? | ? | 17/16 | rKm2, rA1 | Tendomean Minor Second, Tendomean Augmented Prime | D#\, Eb↑\ | As the approximation of both the seventeenth harmonic and the interval formed from stacking two Ultraprimes, this interval is used accordingly. | |
| 15 | 113.2075472 | 16/15 | ? | ? | ? | ? | Km2, A1 | Ptolemaic Minor Second, Pythagorean Augmented Prime | D#, Eb↑ | As the approximation of both the Pythagorean Augmented Prime and the Ptolemaic Minor Second, this interval is used accordingly; it is also one of two in this system that are essential in executing the frameshift cadence. |
| 16 | 120.7547170 | 15/14 | 275/256 | ? | ? | RKm2, kn2, RA1 | Wide Minor Second, Artoretromean Augmented Prime | Ed<↓, Eb↑/, D#/ | In addition to being the approximation of the Septimal Major Semitone, this interval is also one third of a Lesser Submajor Third in this system, and is thus used accordingly. | |
| 17 | 128.3018868 | ? | ? | 14/13 | 128/119 | kN2, rKA1 | Lesser Supraminor Second, Tendoretromean Augmented Prime | Ed>↓, D#↑\ | In addition to its properties as a type of Supraminor Second, this interval is also one third of a Ptolemaic Major Third in this system and is thus used accordingly. | |
| 18 | 135.8490566 | 27/25 | ? | ? | 13/12 | ? | KKm2, rn2, KA1 | Greater Supraminor Second, Diptolemaic Limma, Retroptolemaic Augmented Prime | Ed<\, Eb↑↑, D#↑ | This interval is not only both two thirds of Pythagorean Major Second and the approximation of the Large Limma or Diptolemaic Limma in this system, but also a type of Supraminor Second, and is thus used accordingly. |
| 19 | 143.3962264 | ? | 88/81 | ? | ? | n2, SA1, kUA1 | Artoneutral Second, Lesser Super-Augmented Prime | Ed<, Dt#<↓ | As one of two Neutral Seconds in this system, this interval is notable for being half of the Neo-Gothic Minor Third, though it is also sometimes used in much the same way as 24edo's own Neutral Second. | |
| 20 | 150.9433962 | ? | 12/11 | ? | ? | N2, RkUA1 | Tendoneutral Second, Greater Super-Augmented Prime | Ed>, Dt#>↓ | As one of two Neutral Seconds in this system, this interval is the one that most closely resembles the low-complexity JI Neutral Second, and thus, it is frequently used in much the same way as 24edo's own Neutral Second. | |
| 21 | 158.4905660 | ? | ? | ? | 128/117 | 561/512, 1024/935 | kkM2, RN2, rUA1 | Lesser Submajor Second, Diretroptolemaic Augmented Prime | Ed>/, E↓↓, Dt#>↓/, D#↑↑, Fb↓ | In addition to its properties as a type of Submajor Second, this interval is also one half of a Ptolemaic Minor Third in this system and is thus used accordingly. |
| 22 | 166.0377358 | ? | 11/10 | ? | ? | Kn2, UA1 | Greater Submajor Second, Ultra-Augmented Prime | Ed<↑, Dt#<, Fb↓/ | In addition to its properties as the interval that most closely resembles the low-complexity JI Submajor Second, this interval serves as both the Ultra-Augmented Prime and as one third of a Perfect Fourth, and is used accordingly. | |
| 23 | 173.5849057 | 567/512 | 243/220 | ? | 425/384 | rkM2, KN2 | Narrow Major Second | Ed>↑, E↓\, Dt#>, Fb\ | While this interval is large enough to act as a type of whole tone, it is worth noting that two of these add up to the approximation of the low-complexity JI Neutral Third in this system. | |
| 24 | 181.1320755 | 10/9 | ? | 256/231 | ? | ? | kM2 | Ptolemaic Major Second | E↓, Fb | As the approximation of the Ptolemaic Major Second, this interval is used accordingly, however, it is worth noting that in this system, two of these add up to the approximation of the thirteenth subharmonic; furthermore, it is also one the intervals in this system that are essential in executing any sort of variation on Jacob Collier's "Four Magical chords" from his rendition of "In the Bleak Midwinter". |
| 25 | 188.6792458 | ? | ? | 143/128 | 512/459 | RkM2 | Artomean Major Second | E↓/, Fb/ | This interval has surprising utility in modulating to keys that are not found on the same circle of fifths owing to both its size and its ease of access through octave-reducing stacks of approximated low-complexity JI intervals. | |
| 26 | 196.2264151 | 28/25 | 121/108 | ? | ? | rM2 | Tendomean Major Second | E\, Fb↑\ | This interval is created from stacking two of this system's closet approximation of the 12edo semitone, and thus, it is one of two intervals that come the closest to approximating the 12edo whole tone found in this system. | |
| 27 | 203.7735849 | 9/8 | ? | ? | ? | ? | M2 | Pythagorean Major Second | E, Fb↑ | This interval is the standard-issue whole tone in this system as it is one of two intervals that come the closest to approximating the 12edo whole tone, and the only one of the two that actually approximates the Pythagorean Major Second; furthermore, it is the whole tone that is used as a reference interval in diatonic-and-chromatic-style interval logic in this system as it pertains to both semitones and quartertones. |
| 28 | 211.3207547 | ? | ? | 44/39 | 289/256 | RM2 | Wide Major Second | E/, Fd<↓ | This interval is interesting on the basis that it is formed by stacking two instances of the approximated seventeenth harmonic. | |
| 29 | 218.8679245 | ? | ? | ? | 17/15 | rKM2 | Narrow Supermajor Second | E↑\, Fd>↓ | This interval is of note because it is utilized in approximations of the 17-odd-limit; what's more, it is also the whole tone in this system's superpyth diatonic scale. | |
| 30 | 226.4150943 | 256/225 | ? | 154/135 | ? | ? | KM2 | Lesser Supermajor Second | E↑, Dx | This interval can be interpreted as a type of second on the basis of it approximating the sum of the syntonic comma and the Pythagorean Major Second; it also appears in approximations of 5-limit Neapolitan scales as the interval formed from stacking two Ptolemaic Minor Seconds, making it double as a type of diminished third. |
| 31 | 233.9622642 | 8/7 | 55/48 | ? | ? | |||||
| 32 | 241.5094340 | ? | 1024/891 | ? | ? | |||||
| 33 | 249.0566038 | ? | ? | ? | 15/13 | ? | ||||
| 34 | 256.6037736 | ? | 297/256 | ? | ? | |||||
| 35 | 264.1509434 | 7/6 | 64/55 | ? | ? | |||||
| 36 | 271.6981132 | 75/64 | ? | ? | ? | ? | ||||
| 37 | 279.2452830 | ? | ? | ? | 20/17 | |||||
| 38 | 286.7924528 | ? | 33/28 | 13/11 | 85/72 | |||||
| 39 | 294.3396226 | 32/27 | ? | ? | ? | ? | ||||
| 40 | 301.8867925 | 25/21 | 144/121 | ? | ? | |||||
| 41 | 309.4339622 | ? | ? | 512/429 | 153/128 | |||||
| 42 | 316.9811321 | 6/5 | ? | 77/64 | ? | ? | ||||
| 43 | 324.5283019 | 135/112 | ? | ? | 512/425 | |||||
| 44 | 332.0754717 | ? | 40/33, 121/100 | ? | 144/119, 165/136 | |||||
| 45 | 339.6226415 | ? | ? | ? | 39/32 | 17/14 | ||||
| 46 | 347.1698113 | ? | 11/9 | ? | ? | |||||
| 47 | 354.7169811 | ? | 27/22 | ? | ? | |||||
| 48 | 362.2641509 | ? | ? | ? | 16/13 | 21/17 | ||||
| 49 | 369.8113208 | ? | ? | ? | 68/55 | |||||
| 50 | 377.3584906 | 56/45 | 1024/825 | ? | ? | |||||
| 51 | 384.9056604 | 5/4 | ? | 96/77 | ? | ? | ||||
| 52 | 392.4528302 | ? | ? | ? | 64/51 | |||||
| 53 | 400 | 63/50 | 121/96 | ? | ? | |||||
| 54 | 407.5471698 | 81/64 | ? | ? | ? | ? | ||||
| 55 | 415.0943396 | ? | 14/11 | 33/26 | 108/85 | |||||
| 56 | 422.6415094 | ? | ? | 143/112 | 51/40 | |||||
| 57 | 430.1886792 | 32/25 | ? | ? | ? | ? | ||||
| 58 | 437.7358491 | 9/7 | 165/128 | ? | ? | |||||
| 59 | 445.2830189 | ? | 128/99 | ? | 22/17 | |||||
| 60 | 452.8301887 | ? | ? | ? | 13/10 | ? | ||||
| 61 | 460.3773585 | ? | 176/135 | ? | ? | |||||
| 62 | 467.9245283 | 21/16 | 55/42, 72/55 | ? | 17/13 | |||||
| 63 | 475.4716981 | 320/243, 675/512 | ? | ? | ? | ? | ||||
| 64 | 483.0188679 | ? | 33/25 | ? | 45/34 | |||||
| 65 | 490.5660377 | ? | ? | ? | 85/64 | |||||
| 66 | 498.1132075 | 4/3 | ? | ? | ? | ? | ||||
| 67 | 505.6603774 | 75/56 | 162/121 | ? | ? | |||||
| 68 | 513.2075472 | ? | 121/90 | ? | ? | |||||
| 69 | 520.7547170 | 27/20 | ? | ? | 104/77 | ? | ||||
| 70 | 528.3018868 | ? | 110/81 | ? | ? | |||||
| 71 | 535.8490566 | ? | 15/11 | ? | ? | |||||
| 72 | 543.3962264 | ? | ? | ? | 160/117 | 256/187 | ||||
| 73 | 550.9433962 | ? | 11/8 | ? | ? | |||||
| 74 | 558.4905660 | 112/81 | ? | ? | ? | |||||
| 75 | 566.0377358 | 25/18 | ? | ? | 18/13 | ? | ||||
| 76 | 573.5849057 | ? | ? | ? | 357/256 | |||||
| 77 | 581.1320755 | 7/5 | ? | ? | ? | |||||
| 78 | 588.6792458 | 1024/729, 45/32 | ? | ? | ? | ? | ||||
| 79 | 596.2264151 | ? | ? | ? | 24/17 | |||||
| 80 | 603.7735849 | ? | ? | ? | 17/12 | |||||
| 81 | 611.3207547 | 729/512, 64/45 | ? | ? | ? | ? | ||||
| 82 | 618.8679245 | 10/7 | ? | ? | ? | |||||
| 83 | 626.4150943 | ? | ? | ? | 512/357 | |||||
| 84 | 633.9622642 | 36/25 | ? | ? | 13/9 | ? | ||||
| 85 | 641.5094340 | 81/56 | ? | ? | ? | |||||
| 86 | 649.0566038 | ? | 16/11 | ? | ? | |||||
| 87 | 656.6037736 | ? | ? | ? | 117/80 | 187/128 | ||||
| 88 | 664.1509434 | ? | 22/15 | ? | ? | |||||
| 89 | 671.6981132 | ? | 81/55 | ? | ? | |||||
| 90 | 679.2452830 | 40/27 | ? | ? | 77/52 | ? | ||||
| 91 | 686.7924528 | ? | 180/121 | ? | ? | |||||
| 92 | 694.3396226 | 112/75 | 121/81 | ? | ? | |||||
| 93 | 701.8867925 | 3/2 | ? | ? | ? | ? | ||||
| 94 | 709.4339622 | ? | ? | ? | 128/85 | |||||
| 95 | 716.9811321 | ? | 50/33 | ? | 68/45 | |||||
| 96 | 724.5283019 | 243/160, 1024/675 | ? | ? | ? | ? | ||||
| 97 | 732.0754717 | 32/21 | 84/55, 55/36 | ? | 26/17 | |||||
| 98 | 739.6226415 | ? | 135/88 | ? | ? | |||||
| 99 | 747.1698113 | ? | ? | ? | 20/13 | ? | ||||
| 100 | 754.7169811 | ? | 99/64 | ? | 17/11 | |||||
| 101 | 762.2641509 | 14/9 | 256/165 | ? | ? | |||||
| 102 | 769.8113208 | 25/16 | ? | ? | ? | ? | ||||
| 103 | 777.3584906 | ? | ? | 224/143 | 80/51 | |||||
| 104 | 784.9056604 | ? | 11/7 | 52/33 | 85/54 | |||||
| 105 | 792.4528302 | 128/81 | ? | ? | ? | ? | ||||
| 106 | 800 | 100/63 | 192/121 | ? | ? | |||||
| 107 | 807.5471698 | ? | ? | ? | 51/32 | |||||
| 108 | 815.0943396 | 8/5 | ? | 77/48 | ? | ? | ||||
| 109 | 822.6415094 | 45/28 | 825/512 | ? | ? | |||||
| 110 | 830.1886792 | ? | ? | ? | 55/34 | |||||
| 111 | 837.7358491 | ? | ? | ? | 13/8 | 34/21 | ||||
| 112 | 845.2830189 | ? | 44/27 | ? | ? | |||||
| 113 | 852.8301887 | ? | 18/11 | ? | ? | |||||
| 114 | 860.3773585 | ? | ? | ? | 64/39 | 28/17 | ||||
| 115 | 867.9245283 | ? | 33/20, 200/121 | ? | 119/72, 272/165 | |||||
| 116 | 875.4716981 | 224/135 | ? | ? | 425/256 | |||||
| 117 | 883.0188679 | 5/3 | ? | 128/77 | ? | ? | ||||
| 118 | 890.5660377 | ? | ? | 429/256 | 256/153 | |||||
| 119 | 898.1132075 | 42/25 | 121/72 | ? | ? | |||||
| 120 | 905.6603774 | 27/16 | ? | ? | ? | ? | ||||
| 121 | 913.2075472 | ? | 56/33 | 22/13 | 144/85 | |||||
| 122 | 920.7547170 | ? | ? | ? | 17/10 | |||||
| 123 | 928.3018868 | 128/75 | ? | ? | ? | ? | ||||
| 124 | 935.8490566 | 12/7 | 55/32 | ? | ? | |||||
| 125 | 943.3962264 | ? | 512/297 | ? | ? | |||||
| 126 | 950.9433962 | ? | ? | ? | 26/15 | ? | ||||
| 127 | 958.4905660 | ? | 891/512 | ? | ? | |||||
| 128 | 966.0377358 | 7/4 | 96/55 | ? | ? | |||||
| 129 | 973.5849057 | 225/128 | ? | 135/77 | ? | ? | ||||
| 130 | 981.1320755 | ? | ? | ? | 30/17 | |||||
| 131 | 988.6792458 | ? | ? | 39/22 | 512/289 | |||||
| 132 | 996.2264151 | 16/9 | ? | ? | ? | ? | ||||
| 133 | 1003.7735849 | 25/14 | 216/121 | ? | ? | |||||
| 134 | 1011.3207547 | ? | ? | 256/143 | 459/256 | |||||
| 135 | 1018.8679245 | 9/5 | ? | 231/128 | ? | ? | ||||
| 136 | 1026.4150943 | 1024/567 | 440/243 | ? | 768/425 | |||||
| 137 | 1033.9622642 | ? | 20/11 | ? | ? | |||||
| 138 | 1041.5094340 | ? | ? | ? | 117/64 | 1024/561, 935/512 | ||||
| 139 | 1049.0566038 | ? | 11/6 | ? | ? | |||||
| 140 | 1056.6037736 | ? | 81/44 | ? | ? | |||||
| 141 | 1064.1509434 | 50/27 | ? | ? | 24/13 | ? | ||||
| 142 | 1071.6981132 | ? | ? | 13/7 | 119/64 | |||||
| 143 | 1079.2452830 | 28/15 | 512/275 | ? | ? | |||||
| 144 | 1086.7924528 | 15/8 | ? | ? | ? | ? | ||||
| 145 | 1094.3396226 | ? | ? | ? | 32/17 | |||||
| 146 | 1101.8867925 | ? | 121/64 | 104/55 | 17/9 | |||||
| 147 | 1109.4339622 | 243/128, 256/135 | ? | ? | ? | ? | ||||
| 148 | 1116.9811321 | 40/21 | 21/11 | ? | ? | |||||
| 149 | 1124.5283019 | ? | ? | ? | 153/80 | |||||
| 150 | 1132.0754717 | 48/25 | ? | ? | 25/13, 52/27 | ? | ||||
| 151 | 1139.6226415 | 27/14 | ? | ? | 85/44 | |||||
| 152 | 1147.1698113 | ? | 64/33 | ? | 33/17 | |||||
| 153 | 1154.7169811 | ? | ? | ? | 39/20 | 187/96 | ||||
| 154 | 1162.2641509 | ? | 88/45 | ? | 100/51 | |||||
| 155 | 1169.8113208 | 63/32 | 55/28, 108/55 | ? | 51/26 | |||||
| 156 | 1177.3584906 | 160/81 | ? | ? | 77/39 | 168/85 | ||||
| 157 | 1184.9056604 | ? | 240/121, 99/50 | 143/72 | 119/60 | |||||
| 158 | 1192.4528302 | 448/225 | 484/243 | 195/98, 700/351 | 255/128 | |||||
| 159 | 1200 | 2/1 | P8 | Perfect Octave | D | Reduplication of the root or Tonic. | ||||