Xenharmonic Wiki

159edo: Difference between revisions

← Older edit
VisualWikitext
Aura (talk | contribs)
autopatrolled, emailconfirmed
6,010 edits
No edit summary
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
(343 intermediate revisions by 17 users not shown)
Line 1: Line 1:
'''159edo''' is the '''159 equal division of the octave''' into equal parts of 7.547 [[cent|cents]] each.  
{{interwiki
| de = 159-EDO
| en = 159edo
| es =
| ja =
}}
{{Infobox ET}}
{{ED intro}} The step size of this system is in between the sizes of [[243/242]] (the rastma) and [[225/224]] (the marvel comma).
 
== 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 maqamat—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 [https://en.xen.wiki/index.php?title=159edo&type=revision&diff=5153&oldid=5154 the first records of 159edo on this Wiki from the days of Wikispaces] concern said 79-tone subset related to the [[Turkish maqam music temperaments|yarman]] temperament which had been proposed by Yarman as a tuning standard for [[Arabic, Turkish, Persian music|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 (AEU) accidentals to be bent by as little a detail as 1/3 of a single step of 53edo, while also mapping AEU altogether to a suitable subset of 53edo to allow transpositions throughout.


== Theory ==
== Theory ==
A salient fact about 159edo is that {{nowrap| 159 {{=}} 3 × 53 }}, and thus, this system has both [[3edo]] and [[53edo]] as subsets—the former subset being shared with [[12edo]].
=== Mappings and JI approximation quality ===
This system inherits its approximations of [[3/1|3]], [[5/1|5]], [[13/1|13]], and [[19/1|19]] from 53edo, however, the [[patent val]]s differ on the mappings for [[7/1|7]], [[11/1|11]] and [[17/1|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 edostep. 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's [[29-odd-limit]] or the no-19's [[27-odd-limit]] as [[19/17]], [[29/17]], and their [[octave complement]]s exhaust the inconsistently mapped interval pairs in the 29-odd-limit. Thus its full 29-limit interpretation using the [[patent val]] is obvious, albeit with the catch that it is less than ideal to use the 17-prime at the same time as either the 19-prime or the 29-prime. However, there is more to consider as 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 [[Gamelismic family #Portending|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 [[support]]s 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 ===
{{Harmonics in equal|159|columns=12}}
=== 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 [http://musictheory.zentral.zone/huntsystem2.html#2 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 chord]]s including [[marveltwin chords]], [[minor minthmic 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.
=== Mos and other scales ===
{{See also| List of MOS scales in 159edo }}
Five possible generators for the [[5L 2s|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 ([[7L 26s]]) 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 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 ([[5L 3s]]) 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.
=== Octave stretch ===
159edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[93edf]] or [[252edt]]. This improves the approximated harmonics 5, 7, 11, 13, 19, 23, and 29; only the 17 becomes less accurate as it is tuned sharp already.
== Intervals ==
{{See also| {{PAGENAME}}/Interval names and harmonies }}
The following table assumes 17-limit patent val {{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.
{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 159edo intervals
|-
! Step
! Cents
! 5 limit
! 7 limit
! 11 limit
! 13 limit
! 17 limit
|-
| 0
| 0
| colspan="5"| '''[[1/1]]'''
|-
| 1
| 7.5471698
|
| [[225/224]]
| [[243/242]]
| [[196/195]], [[351/350]]
| [[256/255]]
|-
| 2
| 15.0943396
|
| ?
| [[121/120]], [[100/99]]
| [[144/143]], [[105/104]]
| [[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]]
| [[55/51]], [[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
|
| [[640/567]]
| ?
| [[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
|
| ?
| [[968/729]]
| ?
| [[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
|
| ?
| [[729/484]]
| ?
| [[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
|
| [[567/320]]
| ?
| [[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]]
| [[102/55]], [[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]], [[208/105]]
| [[119/60]]
|-
| 158
| 1192.4528302
|
| [[448/225]]
| [[484/243]]
| [[195/98]], [[700/351]]
| [[255/128]]
|-
| 159
| 1200
| colspan="5"| '''[[2/1]]'''
|-
|}
== Notation ==
{{main| {{PAGENAME}}/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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 1029/1024, 10976/10935, 15625/15552
| {{mapping| 159 252 369 446 }}
| +0.411
| 0.413
| 5.47
|-
| 2.3.5.7.11
| 385/384, 441/440, 4000/3993, 10976/10935
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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 [[152edo|152fg]]. The next equal temperament that does better in the 19-limit is [[161edo|161]], and in the 23-limit, [[183edo|183]].


Compared to [[94edo]], 159edo offers both potential advantages and potential disadvantages.  On one hand is the potential disadvantage of 159edo being [[consistent]] only up to the 17 odd-limit- with it proving to be inconsistent in the 19-limit.  On the other hand, the step size of 159edo itself, due to being slightly above the average peak JND of human pitch perception, allows for a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having so many steps as to have individual steps blend completely into one another.  Furthermore, the septimal kleisma, [[225/224]], maps to a single step in 159edo- a third the size of the tempered version of [[81/80]]- which not only allows for the septimal kleisma to be easily accounted for in notation systems, but also for easy distinctions between certain fairly important intervals such as [[25/16]] and [[14/9]] that are otherwise tempered out in 94edo.
Notably, while 159edo plays host to a number of fun temperaments like [[portent]] and [[364/363|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.


A salient fact about 159edo is that 159 = 3*53, so that it shares the same 5-limit thirds and fifths with [[53edo]]. However, compared to 53edo, the patent vals differ on the mapping for 7. In the 7-limit it tempers out 1029/1024 and 10976/10935 in addition to the 5-limit commas [[32805/32768]] and [[15625/15552]]. This makes it among other things an excellent tuning for [[Gamelismic_clan #Guiron|guiron]] and [[Gamelismic_clan #Tritikleismic|tritikleismic]] temperaments. It has a very accurate 11, and in the 11-limit tempers out not only [[385/384]], 441/440, and 4000/3993, but - in a first for EDOs that are multiples of 53 - 117440512/117406179 as well. In the 13-limit it tempers out 325/324, 364/363, and 10985/10976.  It also has an accurate 17, and in the 17-limit tempers out 273/272 and 375/374.  In the 19-limit it tempers out 343/342 and 361/360. It also provides the [[optimal patent val]] for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament [[Gamelismic_family #Portending|portending]].
=== Commas ===
Assuming the patent val {{val| 159 252 369 446 550 588 650 675 719 }}, 159tet [[tempering out|tempers out]] the following [[comma]]s in the 23-limit.


Another and notable temperament supported by 159 is [[Turkish_maqam_music_temperaments|yarman temperament]], with a generator of 2\159 which can be taken as an approximate 105/104. 159 supplies the optimal patent val for 7, 11, 13, 17 and 19-limit yarman, so they are very closely associated. Curiously, the temperament does not temper out 1029/1024, however.
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color name]]
! Name
|-
| 3
| <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>
| {{monzo| -84 53 }}
| 3.6150
| Wa-53
| [[Mercator's comma]]
|-
| 5
| <abbr title="2109375/2097152">(14 digits)</abbr>
| {{monzo| -21 3 7 }}
| 10.0610
| Lasepyo
| [[Semicomma]]
|-
| 5
| 15625/15552
| {{monzo| -6 -5 6 }}
| 8.1073
| Tribiyo
| [[15625/15552|Kleisma]]
|-
| 5
| <abbr title="1600000/1594323">(14 digits)</abbr>
| {{monzo| 9 -13 5 }}
| 6.1536
| Saquinyo
| [[Amity comma]]
|-
| 5
| <abbr title="10485760000/10460353203">(22 digits)</abbr>
| {{monzo| 24 -21 4 }}
| 4.1998
| Sasaquadyo
| [[Vulture comma]]
|-
| 5
| <abbr title="68719476736000/68630377364883">(28 digits)</abbr>
| {{monzo| 39 -29 3 }}
| 2.2461
|
| [[Alphatricot comma]]
|-
| 5
| 32805/32768
| {{monzo| -15 8 1 }}
| 1.9537
| Layo
| [[Schisma]]
|-
| 5
| <abbr title="2954312706550833698643/2951479051793528258560">(44 digits)</abbr>
| {{monzo| -69 45 -1 }}
| 1.6613
|
| [[Counterschisma]]
|-
| 5
| <abbr title="450359962737049600/450283905890997363">(36 digits)</abbr>
| {{monzo| 54 -37 2 }}
| 0.2924
|
| [[Monzisma]]
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.4327
| Latrizo
| Gamelisma
|-
| 7
| <abbr title="235298/234375">(12 digits)</abbr>
| {{monzo| 1 -1 -7 6 }}
| 6.8044
|
| [[Triwellisma]]
|-
| 7
| 10976/10935
| {{monzo| 5 -7 -1 3 }}
| 6.4790
| Satrizo-agu
| [[Hemimage comma]]
|-
| 7
| <abbr title="7710244864/7688671875">(20 digits)</abbr>
| {{monzo| 16 -9 -8 6 }}
| 4.8507
|
|
|-
| 7
| <abbr title="1640558367/1638400000">(20 digits)</abbr>
| {{monzo| -19 14 -5 3 }}
| 2.2792
|
| [[Forge comma]]
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.6283
|
| [[Meter]]
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{monzo| -4 6 -6 3 }}
| 0.3254
|
| [[Landscape comma]]
|-
| 7
| <abbr title="37252902984619140625000000000/37252879910233655318543787489">(58 digits)</abbr>
| {{monzo| 9 -28 37 -18 }}
| 0.0011
| Satritribiru-athiseyo
| [[Termite comma]]
|-
| 11
| 4375/4356
| {{monzo| -2 -2 4 1 -2 }}
| 7.5349
|
|
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.5026
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.9302
| Luzozogu
| Werckisma
|-
| 11
| [[6250/6237]]
| {{monzo| 1 -4 5 -1 -1 }}
| 3.6047
|
| Liganellus comma
|-
| 11
| <abbr title="252710532568634085/252201579132747776">(36 digits)</abbr>
| {{monzo| -55 11 1 -1 11 }}
| 3.4902
|
| [[Tritonoquartisma]]
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.0323
| Triluyo
| Wizardharry comma
|-
| 11
| 19712/19683
| {{monzo| 8 -9 0 1 1 }}
| 2.5488
| Salozo
| [[Symbiotic comma]]
|-
| 11
| <abbr title="1771561/1769472">(14 digits)</abbr>
| {{Monzo| 16 -3 0 0 6 }}
| 2.0427
| Tribilo
| [[Nexus comma]]
|-
| 11
| <abbr title="171885556953/171798691840">(24 digits)</abbr>
| {{monzo| -35 17 -1 0 3 }}
| 0.8751
| Trila-trilo-agu
| [[Triagnoshenisma]]
|-
| 11
| <abbr title="22876792454961/22866405883904">(28 digits)</abbr>
| {{monzo| -34 28 0 0 -3 }}
| 0.7862
| Quadla-trilu
| [[Frameshift comma]]
|-
| 11
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
| 0.5724
| Loloruyoyo
| Lehmerisma
|-
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{monzo| 24 -6 0 1 -5 }}
| 0.5062
| Saquinlu-azo
| [[Quartisma]]
|-
| 11
| <abbr title="1771561/1771470">(14 digits)</abbr>
| {{monzo| -1 -11 -1 0 6 }}
| 0.0889
| Satribilo-agu
| [[Parimo]]
|-
| 13
| [[2197/2187]]
| {{monzo| 0 -7 0 0 0 3 }}
| 7.8980
| Satritho
| Threedie
|-
| 13
| [[325/324]]
| {{monzo| -2 -4 2 0 0 1 }}
| 5.3351
| Thoyoyo
| Marveltwin comma
|-
| 13
| [[364/363]]
| {{monzo| 2 -1 0 1 -2 1 }}
| 4.7627
| Tholuluzo
| Minor minthma
|-
| 13
| 13720/13689
| {{monzo| 3 -4 1 3 0 -2 }}
| 3.9161
|
|
|-
| 13
| [[625/624]]
| {{monzo| -4 -1 4 0 0 -1 }}
| 2.7722
| Thuquadyo
| Tunbarsma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.5629
| Bithogu
| Island comma
|-
| 13
| 85293/85184
| {{monzo| -6 8 0 0 -3 1 }}
| 2.2138
| Lathotrilu
| [[Sinarabian comma]]
|-
| 13
| [[1575/1573]]
| {{monzo| 0 2 2 1 -2 -1 }}
| 2.1998
|
| Nicola
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.7304
|
| Fairytale comma
|-
| 13
| 10985/10976
| {{monzo| -5 0 1 -3 0 3 }}
| 1.4190
|
| [[Cantonisma]]
|-
| 13
| 43904/43875
| {{monzo| 7 -3 -3 3 0 1 }}
| 1.1439
|
| [[Punctisma]]
|-
| 13
| [[2080/2079]]
| {{monzo| 5 -3 1 -1 -1 1 }}
| 0.8325
| Tholuruyo
| Ibnsinma
|-
| 13
| <abbr title="256000/255879">(12 digits)</abbr>
| {{monzo| 11 -9 3 0 0 -1 }}
| 0.8185
| Sathutriyo
| [[Phaotic comma]]
|-
| 13
| [[6656/6655]]
| {{monzo| 9 0 -1 0 -3 1 }}
| 0.2601
| Thotrilu-agu
| Jacobin comma
|-
| 13
| <abbr title="140625/140608">(12 digits)</abbr>
| {{monzo| -6 2 6 0 0 -3 }}
| 0.2093
|
| [[Catasma]]
|-
| 13
| <abbr title="123201/123200">(12 digits)</abbr>
| {{monzo| -6 6 -2 -1 -1 2 }}
| 0.0141
| Lathotholurugugu
| [[Chalmersia]]
|-
| 17
| 15379/15300
| {{monzo| -2 -2 -2 1 0 3 -1 }}
| 8.9161
|
|
|-
| 17
| [[273/272]]
| {{monzo| -4 1 0 1 0 1 -1 }}
| 6.3532
| Suthozo
| Tannisma
|-
| 17
| [[375/374]]
| {{monzo| -1 1 3 0 -1 0 -1 }}
| 4.6228
|
| Ursulisma
|-
| 17
| [[595/594]]
| {{monzo| -1 -3 1 1 -1 0 1 }}
| 2.9121
|
| Dakotisma
|-
| 17
| [[715/714]]
| {{monzo| -1 -1 1 -1 1 1 -1 }}
| 2.4230
|
| September comma
|-
| 17
| [[833/832]]
| {{monzo| -6 0 0 2 0 -1 1 }}
| 2.0796
| Sothuzozo
| Horizon comma
|-
| 17
| [[936/935]]
| {{monzo| 3 2 -1 0 -1 1 -1 }}
| 1.8506
|
| Ainos comma
|-
| 17
| [[2025/2023]]
| {{monzo| 0 4 2 -1 0 0 -2 }}
| 1.7107
|
| Fidesma
|-
| 17
| [[1089/1088]]
| {{monzo| -6 2 0 0 2 0 -1 }}
| 1.5905
|
| Twosquare comma
|-
| 17
| [[1701/1700]]
| {{monzo| -2 5 -2 1 0 0 -1 }}
| 1.0181
|
| Palingenetic comma
|-
| 17
| 24576/24565
| {{monzo| 13 1 -1 0 0 0 -3 }}
| 0.7751
|
| [[24576/24565|Archagallisma]]
|-
| 17
| [[2431/2430]]
| {{monzo| -1 -5 -1 0 1 1 1 }}
| 0.7123
|
| Heptacircle comma
|-
| 17
| <abbr title="248897/248832">(12 digits)</abbr>
| {{monzo| -10 -5 0 0 4 0 1 }}
| 0.44522
|
|
|-
| 17
| 12376/12375
| {{monzo| 3 -2 -3 1 -1 1 1 }}
| 0.1399
|
| [[Flashma]]
|-
| 17
| 14400/14399
| {{monzo| 6 2 2 -1 -2 0 -1 }}
| 0.1202
| Sululuruyoyo
| [[Sparkisma]]
|-
| 19
| [[343/342]]
| {{monzo| -1 -2 0 3 0 0 0 -1 }}
| 5.0547
|
|
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.8023
|
|
|-
| 19
| [[513/512]]
| {{monzo| -9 3 0 0 0 0 0 1 }}
| 3.3780
|
|
|-
| 19
| [[1216/1215]]
| {{monzo| 6 -5 -1 0 0 0 0 1 }}
| 1.4243
|
|
|-
|}
<references group="note" />


Yarman temperament has [[MOS]] of 79 and 80 notes to the octave, and the 79-note MOS has been proposed by Ozan Yarman as a tuning standard for [[Arabic,_Turkish,_Persian|arabic/turkish/persian]] music.  
In the 23-limit, with the 19-prime 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.


=== Articles ===
=== Rank-2 temperaments ===
* [http://www.ozanyarman.com/files/doctorate_thesis.pdf ''79-Tone Tuning & Theory for Turkish Maqam Music''] - Ozan Yarman's dissertation
Note: 5-limit temperaments supported by 53et are not included.  
* [http://www.ozanyarman.com/files/34ten79a.pdf ''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.
* [http://www.bestii.com/~mschulter/17-MOS-tunings_Letter-to-Ozan.txt Letter to Ozan Yarman] by Margo Schulter [http://www.webcitation.org/5xepptaan (permalink)]


== Just approximation ==
{| class="wikitable center-all left-5"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! colspan="2" |
|-
! prime 2
! Periods<br />per 8ve
! prime 3
! Generator*
! prime 5
! Cents*
! prime 7
! Associated<br />ratio*
! prime 11
! Temperament
! prime 13
|-
! prime 17
| 1
! prime 19
| 2\159
! prime 23
| 15.09
! prime 29
| 121/120
! prime 31
| [[Yarman I]] / [[yarman II]]
|-
| 1
| 7\159
| 52.83
| 33/32
| [[Quartkeenlig]]
|-
| 1
| 11\159
| 83.02
| 21/20
| [[Sextilifourths]]
|-
| 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]]
|-
|-
! rowspan="2" |Error
| 3
! absolute (¢)
| 4\159
| 0.00
| 30.19
| -0.07
| 55/54
| -1.41
| [[Hemichromat]]
| -2.79
| -0.37
| -2.79
| +0.70
| -3.17
| -1.86
| -3.16
| +2.13
|-
|-
![[Relative error|relative]] (%)
| 3
| 0.0
| 8\159
| -0.9
| 60.38
| -18.7
| 28/27
| -36.9
| [[Chromat]]
| -5.0
|-
| -37.0
| 3
| +9.3
| 22\159
| -42.0
| 166.04
| -24.6
| 11/10
| -41.9
| [[Tritricot]]
| +28.3
|-
| 3
| 33\159<br>(20\159)
| 249.06<br>(150.94)
| 15/13<br>(12/11)
| [[Altinex]] / [[hemiterm]]
|-
| 3
| 42\159<br>(11\159)
| 316.981<br>(83.02)
| 6/5<br>(21/20)
| [[Tritikleismic]]
|-
| 3
| 66\159<br>(13\159)
| 498.11<br>(98.11)
| 4/3<br>(35/33)
| [[Term]] / [[terminal]]
|-
| 53
| 31\159<br>(1\159)
| 233.96<br>(7.55)
| 8/7<br>(225/224)
| [[Schismerc]] / [[cartography]]
|-
| 53
| 121\159<br>(1\159)
| 913.21<br>(7.55)
| 441/260<br>(196/195)
| [[Iodine]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
== Instruments ==
Currently, there is an instrument under development by [[Erik Natanael]] called the "Neod"<ref>[https://www.youtube.com/watch?v=HwTCF0zeo4o Erik Natanael - neod.1: Explaining my 53edo musical instrument]</ref>, 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.
== Music ==
The songs below are written in 159edo, or, in approximations that differ from the actual 159edo by only fractions of a [[cent]].
; [[Back Of The Class]] & [[Dawson Berry]]
* [https://soundcloud.com/backoftheclassss/embrace ''Embrace''] (2025)
; [[Dawson Berry]]
* [[:File:Space Tour.mp3|''Space Tour'']] (2020)
* [[:File:Welcome to Dystopia.mp3|''Welcome to Dystopia'']] (2021)
* [[:File:The Forest of Loss.mp3|''The Forest of Loss'']] (2021)
* [https://www.youtube.com/watch?v=8IFWh_srH-I ''Life in a Brave New World''] (2024)
* [https://www.youtube.com/watch?v=16b0ReKhi98 ''Where the Heart Is''] (2024)
* [https://www.youtube.com/watch?v=niSgSld0lUA ''Foes and Forgiveness''] (2024)
* [https://www.youtube.com/watch?v=a2YLIxQZZPk ''Youthful Fun''] (2025)
; [[Eufalesio]]
* [https://soundcloud.com/eufalesio/transhypertonal-ai-megalomania ''TRANSHYPERTONAL AI MEGALOMANIA''] (2024)
; [[Ozan Yarman]]
* [https://www.youtube.com/watch?v=O5Dn6v3UOtM ''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)
* [https://www.youtube.com/watch?v=xoCR1rDXhWI ''Whitecap Visualizer assisted (CHOIR + Digitized Ottoman Singing VocalWriter + STRINGS only)''] – flow of the ENTIRE "Pesendide Fugue" in the 79-tone Qanun tuning
* [https://www.youtube.com/watch?v=tAcmvGgERC4 ''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)
* [https://www.youtube.com/watch?v=dT9RssZ1950 ''79-ton RAST KAR-I NATIK (Ozan Yarman -- thicc vokal ver.)''] – RAST KAR-I NATIK by Ozan Yarman in his 79-tone Qanun tuning
* [https://www.youtube.com/watch?v=gRRnKzjshHg ''Ushshaq Improvisation on the 79-tone Qanun''] (2025)
* [https://www.youtube.com/watch?v=zYvIaVyWrS8 ''Urmo-Yarmanian BUSELİK Improvisation on the 79-tone Qanun''] (2025)
== See also ==
* [[User:Aura/Aura's introduction to 159edo|Aura's approach]]
== Additional Links ==
* [http://www.ozanyarman.com/files/doctorate_thesis.pdf ''79-Tone Tuning & Theory for Turkish Maqam Music''] – Ozan Yarman's dissertation
* [http://www.ozanyarman.com/files/34ten79a.pdf ''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
* [http://www.bestii.com/~mschulter/17-MOS-tunings_Letter-to-Ozan.txt Letter to Ozan Yarman] by Margo Schulter [https://www.webcitation.org/5xepptaan (permalink)]
== References ==
<references />


[[Category:Edo]]
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:53edo]]
[[Category:159edo]]
[[Category:159edo]]
[[Category:Guiron]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Maqam]]
[[Category:Maqam]]
[[Category:Nexus]]
[[Category:Quartismic]]
[[Category:Tritikleismic]]
[[Category:Tritikleismic]]
[[Category:Yarman]]
[[Category:Yarman]]
Retrieved from "https://en.xen.wiki/w/159edo"