159edo
| ← 158edo | 159edo | 160edo → |
159 equal divisions of the octave (abbreviated 159edo or 159ed2), also called 159-tone equal temperament (159tet) or 159 equal temperament (159et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 159 equal parts of about 7.55 ¢ each. Each step represents a frequency ratio of 21/159, or the 159th root of 2. The step size of this system is in between the sizes of 243/242 (the rastma) and 225/224 (the marvel comma).
Theory
A salient fact about 159edo is that 159 = 3 × 53, and thus, this system has both 3edo and 53edo as subsets- the former subset being shared with 12edo.
History
Despite being none other than the tripled superset of the famous 53edo, and hence, one would think, fairly easy to find, it is a wonder that the first person to show theoretical interest in it was Ozan Yarman- specifically in terms of extracting a voluminous subset for representing maqams- in late 2005 to early 2006.
Yarman started by dividing the 4/3 perfect fourth into 33 equal parts and continued the resultant 15.1 cent comma until just below the octave, only to then reinstate the final 79th degree with the octave so as to arrive at the "yegah-neva" partition. He then rotated the scale to begin on the Turkish "rast" note which he notated as C according to Sipürde Ahenk (C Ney), and thus the 22.8 cent larger singular comma previously on top now appears just below the Turkish neva note in the midst of his tuning scheme. Not long after proposing said 79-tone system, Yarman visited the now-deceased Ejder Güleç in İzmir who affixed mandals on Yarman's Qanun according to Yarman's instructions.
Gene Ward Smith was the first to point out that Yarman's scheme was a MOS of 159tet and had an 80-pitch twin. Yarman adopted this argument, because his approach and the related MOS subset out of 159tet was, for all intents and purposes, synonymous. Yarman has stated that he thinks he introduced his Qanun to his now-deceased supervisor in Istanbul Technical University Turkish Music State Conservatory sometime during late 2006, and she suggested that Yarman include the double-sharp mandals. At the time this information was added to this article, Yarman remembered that the Qanun in his doctorate defense of 2008 included the double-sharp mandals. The acceptance of his thesis was in June 2008.
Accordingly, it is no coincidence that the first records of 159edo on this Wiki from the days of Wikispaces concern said 79-tone subset related to the yarman temperament which had been proposed by Yarman as a tuning standard for Arabic, Turkish and Persian music. Based on the information given by Ozan Yarman himself, his elder colleague M. Ugur Kececioglu first utilized 159edo in his revamped 2011 release of the Notist score editor and therein allowed the Arel-Ezgi-Uzdilek accidentals to be bent by as little a detail as 1/3rd of a single step of 53edo, while also mapping AEU altogether to a suitable subset of 53edo to allow transpositions throughout.
Mappings and JI approximation quality
This system inherits its approximations of the 3rd, 5th, 13th, and 19th harmonics from 53edo, however, the patent vals differ on the mappings for 7, 11 and 17 – in fact, this edo has a very accurate 11 and an only slightly less accurate 17. Furthermore, 159edo demonstrates 3-to-2 telicity, as despite being contorted in the 5-limit, it is the largest edo to temper out Mercator's comma in which said comma is less than half the size of a single edo step. This means, among other things, that there is a perfect match between the direct approximation and the more complicated traditional mapping for an octave-reduced stack of fifty-three tempered 3/2 perfect fifths – a complete circle of fifths for this edo.
159edo is consistent up to the no-17 29-odd-limit or the no-19 27-odd-limit as {19/17, 34/19} and {29/17, 34/29} exhaust the inconsistently mapped interval pairs in the 29-odd-limit. Thus its full 29-limit interpretation using the patent val is obvious. However, the direct approximation and the val mapping for intervals such as 49/32, 35/32, and 169/128 do not match, and as a result, 159edo can be thought of as having a perfunctory 7-limit that mainly serves to bridge to the 11-limit and divide the nearly just 3/2 into three, as well as a similarly perfunctory 13-limit that mainly serves to bridge to the 17-limit and to absorb complex combinations of 3 and 5.
Notably, 159edo provides the optimal patent val for 11-limit guiron, 13-limit tritikleismic, the 13-limit rank-3 temperament portending, as well as the 17-limit rank-6 temperament tempering out 273/272. In addition to this, it also supports both forms of the yarman temperament, with a generator of 2\159 which can be taken as an approximate 105/104. Both have a mos of 79 or 80 notes to the octave, and have their optimal patent vals supplied by 159edo in 7-limit, 11-limit, 13-limit, 17-limit and even 19-limit forms. While the patent val supports both cartography and iodine temperaments, which are among the best 13-limit temperaments in the Mercator family, the 159d and 159f mappings support other members of this temperament family.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.07 | -1.41 | -2.79 | -0.37 | -2.79 | +0.70 | -3.17 | -1.86 | -3.16 | +2.13 | -2.29 |
| Relative (%) | +0.0 | -0.9 | -18.7 | -36.9 | -5.0 | -37.0 | +9.3 | -42.0 | -24.6 | -41.9 | +28.3 | -30.3 | |
| Steps (reduced) |
159 (0) |
252 (93) |
369 (51) |
446 (128) |
550 (73) |
588 (111) |
650 (14) |
675 (39) |
719 (83) |
772 (136) |
788 (152) |
828 (33) | |
Additional Properties of 159-tone equal temperament
The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic JND, and similarly this system can approximate the sounds of other systems such as 10edo, 13edo, 22edo and 31edo. Furthermore, the step size of 159edo is simultaneously above the average peak melodic JND and small enough to be well within the margin of error between Just 5-limit intervals and their 12edo counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another, even as it also allows one to also control the bandwidth of certain sounds. As a result of tempering out some of the commas, it allows essentially tempered chords including marveltwin chords, gentle chords, keenanismic chords, werckismic chords, and sinbadmic chords in the 13-odd-limit, in addition to island chords and nicolic chords in the 15-odd-limit.
MOSes and other scales
No less than five possible generators for the diatonic MOS Scale are supported by 159edo, though these different diatonic scales have different descendants. The 91\159 generator results in large and small scale steps at 23\159 and 22\159 respectively, making for a quasi-equalized diatonic scale, which can be extended to a siskontyttonic MOS, while the 95\159 results in large and small scale steps at 31\159 and 2\159 respectively, making for a version approaching collapsed, which can be extended to a reinatonic (5L 22s) MOS. The 92\159 generator results in large and small scale steps at 25\159 and 17\159 respectively, and this makes for a very meantone-like diatonic scale perfect for xenharmonic pieces that follow in the classical tradition, though this can be extended to a veljentyttonic (19L 26s) MOS. Conversely, the 94\159 generator results in results in large and small scale steps at 29\159 and 7\159 respectively, and this makes for a superpyth diatonic scale that is slightly harder and better than that of 22edo, which can be extended to an antireinatonic (22L 5s) MOS. Finally, the patent 93\159 generator results in the same diatonic MOS scale found in 53edo, which, despite now having competition from other possible generators, is still the go-to for those looking for something more akin to the classic Pythagorean tuning, as well as for those looking to deal with good approximations of related 5-limit scales, or even just the basic form of the pythamystonic (12L 29s) MOS.
In addition, 159edo has no less than four possible generators for the oneirotonic MOS scale, and of these, two of them are also supported by 53edo. The 60\159 generator results in large and small scale steps at 21\159 and 18\159 respectively, making for a distinctly ultra-soft scale, while the 63\159 generator results in large and small scale steps at 30\159 and 3\159 respectively, making for a distinctly ultra-hard scale. As for the remaining two generators, the 61\159 generator results in large and small scale steps at 24\159 and 13\159 respectively and comes the closest to any sort of basic form of this scale, however, the 62\159 generator is also a solid choice, and is also useful for at least one related non-MOS scale due to 62\159 approximating 21/16.
Furthermore, this EDO supports Wyschnegradsky's "Diatonicized Chromatic Scale" (11L 2s) with large and small scale steps at 13\159 and 8\159 respectively.
Intervals
The following table assumes 17-limit patent val ⟨159 252 369 446 550 588 650].
Intervals highlighted in bold are prime harmonics or subharmonics. In addition, intervals that differ from assigned steps by more than 50%, multiples of such intervals, and intervals of odd limit higher than 1024, are not shown. Furthermore, when multiple well-known intervals for a given prime-limit share a step size, they may share a cell in the chart; conversely, a "?" in the chart means that no known interval meets the criteria for inclusion. Note that no 5-limit intervals can be represented by degrees other than multiples of 3, so those entries are left blank.
| Step | Cents | 5 limit | 7 limit | 11 limit | 13 limit | 17 limit |
|---|---|---|---|---|---|---|
| 0 | 0 | 1/1 | ||||
| 1 | 7.5471698 | 225/224 | 243/242 | 196/195, 351/350 | 256/255 | |
| 2 | 15.0943396 | ? | 121/120, 100/99 | 144/143 | 120/119 | |
| 3 | 22.6415094 | 81/80 | ? | ? | 78/77 | 85/84 |
| 4 | 30.1886792 | 64/63 | 56/55, 55/54 | ? | 52/51 | |
| 5 | 37.7358491 | ? | 45/44 | ? | 51/50 | |
| 6 | 45.2830189 | ? | ? | ? | 40/39 | 192/187 |
| 7 | 52.8301887 | ? | 33/32 | ? | 34/33 | |
| 8 | 60.3773585 | 28/27 | ? | ? | 88/85 | |
| 9 | 67.9245283 | 25/24 | ? | ? | 26/25, 27/26 | ? |
| 10 | 75.4716981 | ? | ? | ? | 160/153 | |
| 11 | 83.0188679 | 21/20 | 22/21 | ? | ? | |
| 12 | 90.5660377 | 256/243, 135/128 | ? | ? | ? | ? |
| 13 | 98.1132075 | ? | 128/121 | 55/52 | 18/17 | |
| 14 | 105.6603774 | ? | ? | ? | 17/16 | |
| 15 | 113.2075472 | 16/15 | ? | ? | ? | ? |
| 16 | 120.7547170 | 15/14 | 275/256 | ? | ? | |
| 17 | 128.3018868 | ? | ? | 14/13 | 128/119 | |
| 18 | 135.8490566 | 27/25 | ? | ? | 13/12 | ? |
| 19 | 143.3962264 | ? | 88/81 | ? | ? | |
| 20 | 150.9433962 | ? | 12/11 | ? | ? | |
| 21 | 158.4905660 | ? | ? | ? | 128/117 | 561/512, 1024/935 |
| 22 | 166.0377358 | ? | 11/10 | ? | ? | |
| 23 | 173.5849057 | 567/512 | 243/220 | ? | 425/384 | |
| 24 | 181.1320755 | 10/9 | ? | 256/231 | ? | ? |
| 25 | 188.6792458 | ? | ? | 143/128 | 512/459 | |
| 26 | 196.2264151 | 28/25 | 121/108 | ? | ? | |
| 27 | 203.7735849 | 9/8 | ? | ? | ? | ? |
| 28 | 211.3207547 | ? | ? | 44/39 | 289/256 | |
| 29 | 218.8679245 | ? | 25/22 | ? | 17/15 | |
| 30 | 226.4150943 | 256/225 | ? | 154/135 | ? | ? |
| 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 | ? | ? | 117/100 | ? |
| 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 | ? | ? | 26/21 | 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 | ? | 34/27 | |
| 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 | ? | ? | 50/39 | ? |
| 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 | 256/189 | 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 | 189/128 | 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 | ? | ? | 39/25 | ? |
| 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 | ? | 27/17 | |
| 107 | 807.5471698 | ? | ? | ? | 51/32 | |
| 108 | 815.0943396 | 8/5 | ? | 77/48 | ? | ? |
| 109 | 822.6415094 | 45/28 | 825/512 | ? | ? | |
| 110 | 830.1886792 | ? | ? | 21/13 | 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 | ? | ? | 200/117 | ? |
| 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 | ? | 44/25 | ? | 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 | ||||
Notation
Because of the complexity of 159edo, notation requires systems that make use of multiple extra pairs of accidentals. This is because at high EDOs, systems with only a single extra accidental pair become unwieldy due to the sheer number of such accidentals required for notating some pitches, which in turn results in high amounts of clutter on scores. So far, several notation systems addressing this problem have been proposed.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 1029/1024, 10976/10935, 15625/15552 | [⟨159 252 369 446]] | +0.411 | 0.413 | 5.47 |
| 2.3.5.7.11 | 385/384, 441/440, 4000/3993, 10976/10935 | [⟨159 252 369 446 550]] | +0.350 | 0.389 | 5.15 |
| 2.3.5.7.11.13 | 325/324, 364/363, 385/384, 625/624, 10976/10935 | [⟨159 252 369 446 550 588]] | +0.418 | 0.385 | 5.11 |
| 2.3.5.7.11.13.17 | 273/272, 325/324, 364/363, 375/374, 385/384, 3773/3757 | [⟨159 252 369 446 550 588 650]] | +0.333 | 0.412 | 5.46 |
| 2.3.5.7.11.13.17.19 | 273/272, 325/324, 343/342, 361/360, 364/363, 375/374, 385/384 | [⟨159 252 369 446 550 588 650 675]] | +0.385 | 0.409 | 5.42 |
| 2.3.5.7.11.13.17.19.23 | 273/272, 325/324, 343/342, 361/360, 364/363, 375/374, 385/384, 392/391 | [⟨159 252 369 446 550 588 650 675 719]] | +0.388 | 0.386 | 5.11 |
- 159et has lower absolute errors in the 19- and 23-limit than any previous equal temperaments, past 152fg. The next equal temperament that does better in the 19-limit is 161, and in the 23-limit, 183.
Notably, while 159edo plays host to a number of fun temperaments like portent and minor minthmic, a number of microtemperaments have also been found hiding within its structure. This means that 159edo is well-balanced in terms of the possibilities for interesting temperament usage.
Commas
Assuming the patent val ⟨159 252 369 446 550 588 650 675 719], 159tet tempers out the following commas in the 23-limit.
| Prime limit |
Ratio[1] | Monzo | Cents | Color name | Name |
|---|---|---|---|---|---|
| 3 | (52 digits) | [-84 53⟩ | 3.6150 | Wa-53 | Mercator's comma |
| 5 | (14 digits) | [-21 3 7⟩ | 10.0610 | Lasepyo | Semicomma |
| 5 | 15625/15552 | [-6 -5 6⟩ | 8.1073 | Tribiyo | Kleisma |
| 5 | (14 digits) | [9 -13 5⟩ | 6.1536 | Saquinyo | Amity comma |
| 5 | (22 digits) | [24 -21 4⟩ | 4.1998 | Sasaquadyo | Vulture comma |
| 5 | (28 digits) | [39 -29 3⟩ | 2.2461 | Tricot comma | |
| 5 | 32805/32768 | [-15 8 1⟩ | 1.9537 | Layo | Schisma |
| 5 | (44 digits) | [-69 45 -1⟩ | 1.6613 | Counterschisma | |
| 5 | (36 digits) | [54 -37 2⟩ | 0.2924 | Monzisma | |
| 7 | 1029/1024 | [-10 1 0 3⟩ | 8.4327 | Latrizo | Gamelisma |
| 7 | (12 digits) | [1 -1 -7 6⟩ | 6.8044 | Triwellisma | |
| 7 | 10976/10935 | [5 -7 -1 3⟩ | 6.4790 | Satrizo-agu | Hemimage comma |
| 7 | (20 digits) | [16 -9 -8 6⟩ | 4.8507 | ||
| 7 | (20 digits) | [-19 14 -5 3⟩ | 2.2792 | Forge comma | |
| 7 | (12 digits) | [-11 2 7 -3⟩ | 1.6283 | Meter | |
| 7 | (12 digits) | [-4 6 -6 3⟩ | 0.3254 | Landscape comma | |
| 7 | (58 digits) | [9 -28 37 -18⟩ | 0.0011 | Satritribiru-athiseyo | Termite comma |
| 11 | 4375/4356 | [-2 -2 4 1 -2⟩ | 7.5349 | ||
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.5026 | Lozoyo | Keenanisma |
| 11 | 441/440 | [-3 2 -1 2 -1⟩ | 3.9302 | Luzozogu | Werckisma |
| 11 | 6250/6237 | [1 -4 5 -1 -1⟩ | 3.6047 | Liganellus comma | |
| 11 | (36 digits) | [-55 11 1 -1 11⟩ | 3.4902 | Tritonoquartisma | |
| 11 | 4000/3993 | [5 -1 3 0 -3⟩ | 3.0323 | Triluyo | Wizardharry comma |
| 11 | 19712/19683 | [8 -9 0 1 1⟩ | 2.5488 | Salozo | Symbiotic comma |
| 11 | (14 digits) | [16 -3 0 0 6⟩ | 2.0427 | Tribilo | Nexus comma |
| 11 | (24 digits) | [-35 17 -1 0 3⟩ | 0.8751 | Trila-trilo-agu | Triagnoshenisma |
| 11 | (28 digits) | [-34 28 0 0 -3⟩ | 0.7862 | Quadla-trilu | Frameshift comma |
| 11 | 3025/3024 | [-4 -3 2 -1 2⟩ | 0.5724 | Loloruyoyo | Lehmerisma |
| 11 | (18 digits) | [24 -6 0 1 -5⟩ | 0.5062 | Saquinlu-azo | Quartisma |
| 11 | (14 digits) | [-1 -11 -1 0 6⟩ | 0.0889 | Satribilo-agu | Parimo |
| 13 | 2197/2187 | [0 -7 0 0 0 3⟩ | 7.8980 | Satritho | Threedie |
| 13 | 325/324 | [-2 -4 2 0 0 1⟩ | 5.3351 | Thoyoyo | Marveltwin comma |
| 13 | 364/363 | [2 -1 0 1 -2 1⟩ | 4.7627 | Tholuluzo | Minor minthma |
| 13 | 13720/13689 | [3 -4 1 3 0 -2⟩ | 3.9161 | ||
| 13 | 625/624 | [-4 -1 4 0 0 -1⟩ | 2.7722 | Thuquadyo | Tunbarsma |
| 13 | 676/675 | [2 -3 -2 0 0 2⟩ | 2.5629 | Bithogu | Island comma |
| 13 | 1575/1573 | [0 2 2 1 -2 -1⟩ | 2.1998 | Nicola | |
| 13 | 1001/1000 | [-3 0 -3 1 1 1⟩ | 1.7304 | Fairytale comma | |
| 13 | 10985/10976 | [-5 0 1 -3 0 3⟩ | 1.4190 | Cantonisma | |
| 13 | 43904/43875 | [7 -3 -3 3 0 1⟩ | 1.1439 | Punctisma | |
| 13 | 2080/2079 | [5 -3 1 -1 -1 1⟩ | 0.8325 | Tholuruyo | Ibnsinma |
| 13 | (12 digits) | [11 -9 3 0 0 -1⟩ | 0.8185 | Sathutriyo | Phaotic comma |
| 13 | 6656/6655 | [9 0 -1 0 -3 1⟩ | 0.2601 | Thotrilu-agu | Jacobin comma |
| 13 | (12 digits) | [-6 2 6 0 0 -3⟩ | 0.2093 | Catasma | |
| 13 | (12 digits) | [-6 6 -2 -1 -1 2⟩ | 0.0141 | Lathotholurugugu | Chalmersia |
| 17 | 15379/15300 | [-2 -2 -2 1 0 3 -1⟩ | 8.9161 | ||
| 17 | 273/272 | [-4 1 0 1 0 1 -1⟩ | 6.3532 | Suthozo | Tannisma |
| 17 | 375/374 | [-1 1 3 0 -1 0 -1⟩ | 4.6228 | Ursulisma | |
| 17 | 595/594 | [-1 -3 1 1 -1 0 1⟩ | 2.9121 | Dakotisma | |
| 17 | 715/714 | [-1 -1 1 -1 1 1 -1⟩ | 2.4230 | September comma | |
| 17 | 833/832 | [-6 0 0 2 0 -1 1⟩ | 2.0796 | Sothuzozo | Horizon comma |
| 17 | 936/935 | [3 2 -1 0 -1 1 -1⟩ | 1.8506 | Ainos comma | |
| 17 | 2025/2023 | [0 4 2 -1 0 0 -2⟩ | 1.7107 | Fidesma | |
| 17 | 1089/1088 | [-6 2 0 0 2 0 -1⟩ | 1.5905 | Twosquare comma | |
| 17 | 1701/1700 | [-2 5 -2 1 0 0 -1⟩ | 1.0181 | Palingenetic comma | |
| 17 | 24576/24565 | [13 1 -1 0 0 0 -3⟩ | 0.7751 | Archagallisma | |
| 17 | 2431/2430 | [-1 -5 -1 0 1 1 1⟩ | 0.7123 | Heptacircle comma | |
| 17 | (12 digits) | [-10 -5 0 0 4 0 1⟩ | 0.44522 | ||
| 17 | 12376/12375 | [3 -2 -3 1 -1 1 1⟩ | 0.1399 | Flashma | |
| 17 | 14400/14399 | [6 2 2 -1 -2 0 -1⟩ | 0.1202 | Sululuruyoyo | Sparkisma |
| 19 | 343/342 | [-1 -2 0 3 0 0 0 -1⟩ | 5.0547 | ||
| 19 | 361/360 | [-3 -2 -1 0 0 0 0 2⟩ | 4.8023 | ||
| 19 | 513/512 | [-9 3 0 0 0 0 0 1⟩ | 3.3780 |
In the 23-limit, with the 19-limit skipped, this system is known to temper out 392/391, 460/459, 507/506, 529/528, 897/896, 1105/1104, 1288/1287, 2024/2023, 2025/2024, and 2646/2645 among others.
Rank-2 temperaments
Note: 5-limit temperaments supported by 53et are not included.
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 2\159 | 15.09 | 121/120 | Yarman I / yarman II |
| 1 | 7\159 | 52.83 | 33/32 | Quartkeenlig |
| 1 | 11\159 | 83.02 | 21/20 | Sextilififths |
| 1 | 22\159 | 166.04 | 11/10 | Tertiaschis |
| 1 | 31\159 | 233.96 | 8/7 | Guiron |
| 1 | 38\159 | 286.79 | 13/11 | Gamity |
| 1 | 41\159 | 309.43 | 448/375 | Triwell |
| 1 | 64\159 | 483.02 | 160/121 | Quarterframe |
| 1 | 67\159 | 505.66 | 75/56 | Marfifths |
| 1 | 68\159 | 513.21 | 121/90 | Trinity |
| 1 | 74\159 | 558.49 | 112/81 | Condor |
| 3 | 4\159 | 30.19 | 55/54 | Hemichromat |
| 3 | 8\159 | 60.38 | 28/27 | Chromat |
| 3 | 22\159 | 166.04 | 11/10 | Tritricot |
| 3 | 33\159 (20\159) |
249.06 (150.94) |
15/13 (12/11) |
Altinex / hemiterm |
| 3 | 42\159 (11\159) |
316.981 (83.02) |
6/5 (21/20) |
Tritikleismic |
| 3 | 66\159 (13\159) |
498.11 (98.11) |
4/3 (35/33) |
Term / terminal |
| 53 | 31\159 (1\159) |
233.96 (7.55) |
8/7 (225/224) |
Schismerc / cartography |
| 53 | 121\159 (1\159) |
913.21 (7.55) |
441/260 (196/195) |
Iodine |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
The songs below are written in 159edo, or, in approximations that differ from the actual 159edo by only fractions of a cent.
- Space Tour (2020)
- Welcome to Dystopia (2021)
- The Forest of Loss (2021)
- Life in a Brave New World (2024)
- Where the Heart Is (2024)
- Foes and Forgiveness (2024)
- ENTIRE ORCHESTRAL 79-tone "Pesendide Fugue" (III. Selim & Ozan Yarman) - Ozan Yarman's (5-voice & 2-subject) Choral Fugue of Sultan III. Selim's Pesendide Ağır Semai in Ağır Aksak usul] (re-scored in accordance with Yarman's 79-tone tuning)
- Whitecap Visualizer assisted (CHOIR + Digitized Ottoman Singing VocalWriter + STRINGS only) - flow of the ENTIRE "Pesendide Fugue" in the 79-tone Qanun tuning
- 79'lu sistemde MISIRLI Udi İbrahim Efendi'nin çoksesli Acemaşiran Sazsemaisi -- Ozan Yarman - Adjemashiran Sazsemai of "Egyptian" Ud-player Ibrahim Efendi in 79 MOS 159-tET as polyphonalized by Ozan Yarman (2005-2022)
- 79-ton RAST KAR-I NATIK (Ozan Yarman -- thicc vokal ver.) - RAST KAR-I NATIK by Ozan Yarman in his 79-tone Qanun tuning
Instruments
Currently, there is an instrument under development by Erik Natanael called the "Neod"[2], which utilizes this system. Although 53edo is the basis for most of the keys on this instrument, there are additional buttons which modify the pitch by a single step of 159edo.
Articles
- 79-Tone Tuning & Theory for Turkish Maqam Music – Ozan Yarman's dissertation
- Search For A Theoretical Model Conforming To Turkish Maqam Music Practice: A Selection Of Fixed-Pitch Settings From 34-tone Equal Temperament To The 79-tone Tuning – also by Ozan Yarman, gives a summary
- Letter to Ozan Yarman by Margo Schulter (permalink)
References
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
- ↑ Erik Natanael - neod.1: Explaining my 53edo musical instrument