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This system shares the same 3rd, 5th and 13th [[harmonic]]s with 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.  Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the ~19/17 mapped to the second closest step. 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 mapping]] 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. However, for intervals such as [[49/32]] and [[128/125]], these two mappings do not match. 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 159e mappings support other members of this temperament family.
This system shares the same 3rd, 5th and 13th [[harmonic]]s with 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.  Although 159edo is [[consistent]] up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the ~19/17 mapped to the second closest step. 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 mapping]] 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. However, for intervals such as [[49/32]] and [[128/125]], these two mappings do not match. 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 159e mappings support other members of this temperament family.


=== Commas ===
=== MOSes and other scales ===
 
No less than 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 [[7L 26s|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 paucitonic, which can be extended to a [[5L 22s|reinatonic]] 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 [[19L 26s|veljentyttonic]] 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 [[22L 5s|antireinatonic]] 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 [[12L 29s|pythamystonic]] MOS.
 
In addition, 159edo has no less than four possible generators for the [[5L 3s|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 "[[11L 2s|Diatonicized Chromatic Scale]]" with large and small scale steps at 13\159 and 8\159 respectively.
 
== Intervals ==
{{see also| 159edo interval names and harmonies}}
 
The following table assumes 17-limit patent val {{val|159 252 369 446 550 588 650}}.


Notably, while 159edo plays host to a number of fun temperaments like [[gamelan]] and [[364/363|gentle]], 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. Assuming the [[val]] {{val|159 252 369 446 550 588 650 }}, 159tet [[tempers out]] the following [[comma]]s.
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="commatable wikitable center-all left-3 right-4 left-6"
{| class="mw-collapsible mw-collapsed wikitable center-1"
|+ style=white-space:nowrap | Table of 159edo intervals
|-
|-
! [[Harmonic limit|Prime <br>limit]]
! Step
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! Cents
! [[Monzo]]
! 5 limit
! [[Cents]]
! 7 limit
! [[Color name]]
! 11 limit
! Name
! 13 limit
! 17 limit
|-
|-
| 3
| 0
| <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>
| 0
| {{monzo| -84 53 }}
| colspan="5"| '''[[1/1]]'''
| 3.6150
| Wa-53
| [[Mercator's comma]]
|-
|-
| 5
| 1
| <abbr title="2109375/2097152">(14 digits)</abbr>
| 7.5471698
| {{monzo| -21 3 7 }}
|  
| 10.0610
| [[225/224]]
| Lasepyo
| [[243/242]]
| [[Semicomma]]
| [[196/195]], [[351/350]]
| 256/255
|-
|-
| 5
| 2
| 15625/15552
| 15.0943396
| {{monzo| -6 -5 6 }}
|  
| 8.1073
| ?
| Tribiyo
| [[121/120]], [[100/99]]
| [[15625/15552|Kleisma]]
| [[144/143]]
| 120/119
|-
|-
| 5
| 3
| <abbr title="1600000/1594323">(14 digits)</abbr>
| 22.6415094
| {{monzo| 9 -13 5 }}
| [[81/80]]
| 6.1536
| ?
| Saquinyo
| ?
| [[Amity comma]]
| [[78/77]]
| 85/84
|-
|-
| 5
| 4
| <abbr title="10485760000/10460353203">(22 digits)</abbr>
| 30.1886792
| {{monzo| 24 -21 4 }}
|  
| 4.1998
| [[64/63]]
| Sasaquadyo
| [[56/55]], [[55/54]]
| [[Vulture comma]]
| ?
| [[52/51]]
|-
|-
| 5
| 5
| 32805/32768
| 37.7358491
| {{monzo| -15 8 1 }}
| 1.9537
| Layo
| [[Schisma]]
|-
| 5
| <abbr title="2954312706550833698643/2951479051793528258560">(44 digits)</abbr>
| {{monzo| -69 45 -1 }}
| 1.6613
|  
|  
| [[Counterschisma]]
| ?
| [[45/44]]
| ?
| [[51/50]]
|-
|-
| 5
| 6
| <abbr title="450359962737049600/450283905890997363">(36 digits)</abbr>
| 45.2830189
| {{monzo| 54 -37 2 }}
| ?
| 0.2924
| ?
|  
| ?
| [[Monzisma]]
| [[40/39]]
| 192/187
|-
|-
| 7
| 7
| <abbr title="117649/116640">(12 digits)</abbr>
| 52.8301887
| {{monzo| -5 -6 -1 6 }}
| 14.9117
|  
|  
| ?
| [[33/32]]
| ?
| [[34/33]]
|-
| 8
| 60.3773585
|  
|  
| [[28/27]]
| ?
| ?
| 88/85
|-
|-
| 7
| 9
| [[1029/1024]]
| 67.9245283
| {{monzo| -10 1 0 3 }}
| [[25/24]]
| 8.4327
| ?
| Latrizo
| ?
| Gamelisma
| [[26/25]], [[27/26]]
| ?
|-
|-
| 7
| 10
| <abbr title="235298/234375">(12 digits)</abbr>
| 75.4716981
| {{monzo| 1 -1 -7 6 }}
| 6.8044
|
|  
|  
| ?
| ?
| ?
| 160/153
|-
|-
| 7
| 11
| 10976/10935
| 83.0188679
| {{monzo| 5 -7 -1 3 }}
| 6.4790
| Satrizo-agu
| [[Hemimage comma]]
|-
| 7
| <abbr title="1640558367/1638400000">(20 digits)</abbr>
| {{monzo| -19 14 -5 3 }}
| 2.2792
|  
|  
| [[Forge comma]]
| [[21/20]]
| [[22/21]]
| ?
| ?
|-
|-
| 7
| 12
| <abbr title="703125/702464">(12 digits)</abbr>
| 90.5660377
| {{monzo| -11 2 7 -3 }}
| [[256/243]], [[135/128]]
| 1.6283
| ?
|  
| ?
| [[Meter]]
| ?
| ?
|-
|-
| 7
| 13
| <abbr title="250047/250000">(12 digits)</abbr>
| 98.1132075
| {{monzo| -4 6 -6 3 }}
| 0.3254
|  
|  
| [[Landscape comma]]
| ?
| [[128/121]]
| [[55/52]]
| [[18/17]]
|-
|-
| 7
| 14
| <abbr title="37252902984619140625000000000/37252879910233655318543787489">(58 digits)</abbr>
| 105.6603774
| {{monzo| 9 -28 37 -18 }}
|  
| 0.0011
| ?
| Satritribiru-athiseyo
| ?
| [[Termite comma]]
| ?
| '''[[17/16]]'''
|-
|-
| 11
| 15
| 4375/4356
| 113.2075472
| {{monzo| -2 -2 4 1 -2 }}
| [[16/15]]
| 7.5349
| ?
| ?
| ?
| ?
|-
| 16
| 120.7547170
|  
|  
| [[15/14]]
| 275/256
| ?
| ?
|-
| 17
| 128.3018868
|  
|  
| ?
| ?
| [[14/13]]
| 128/119
|-
|-
| 11
| 18
| [[385/384]]
| 135.8490566
| {{monzo| -7 -1 1 1 1 }}
| [[27/25]]
| 4.5026
| ?
| Lozoyo
| ?
| Keenanisma
| [[13/12]]
| ?
|-
|-
| 11
| 19
| [[441/440]]
| 143.3962264
| {{monzo| -3 2 -1 2 -1 }}
| 3.9302
| Luzozogu
| Werckisma
|-
| 11
| [[6250/6237]]
| {{monzo| 1 -4 5 -1 -1 }}
| 3.6047
|  
|  
| Liganellus comma
| ?
| [[88/81]]
| ?
| ?
|-
|-
| 11
| 20
| [[4000/3993]]
| 150.9433962
| {{monzo| 5 -1 3 0 -3 }}
|
| 3.0323
| ?
| Triluyo
| [[12/11]]
| Wizardharry comma
| ?
| ?
|-
|-
| 11
| 21
| 19712/19683
| 158.4905660
| {{monzo| 8 -9 0 1 1 }}
| ?
| 2.5488
| ?
| Salozo
| ?
| [[Symbiotic comma]]
| [[128/117]]
| 561/512, 1024/935
|-
|-
| 11
| 22
| <abbr title="1771561/1769472">(14 digits)</abbr>
| 166.0377358
| {{Monzo| 16 -3 0 0 6 }}
|  
| 2.0427
| ?
| Tribilo
| [[11/10]]
| [[Nexus comma]]
| ?
| ?
|-
|-
| 11
| 23
| <abbr title="22876792454961/22866405883904">(28 digits)</abbr>
| 173.5849057
| {{monzo| -34 28 0 0 -3 }}
|  
| 0.7862
| 567/512
| Quadla-trilu
| 243/220
| [[Frameshift comma]]
| ?
| 425/384
|-
|-
| 11
| 24
| [[3025/3024]]
| 181.1320755
| {{monzo| -4 -3 2 -1 2 }}
| [[10/9]]
| 0.5724
| ?
| Loloruyoyo
| 256/231
| Lehmerisma
| ?
| ?
|-
|-
| 11
| 25
| <abbr title="117440512/117406179">(18 digits)</abbr>
| 188.6792458
| {{monzo| 24 -6 0 1 -5 }}
|  
| 0.5062
| ?
| Saquinlu-azo
| ?
| [[Quartisma]]
| [[143/128]]
| 512/459
|-
|-
| 11
| 26
| <abbr title="1771561/1771470">(14 digits)</abbr>
| 196.2264151
| {{monzo| -1 -11 -1 0 6 }}
|
| 0.0889
| [[28/25]]
| Satribilo-agu
| [[121/108]]
| [[Parimo]]
| ?
| ?
|-
| 27
| 203.7735849
| [[9/8]]
| ?
| ?
| ?
| ?
|-
|-
|}
| 28
 
| 211.3207547
 
|
* In the 5-limit, it tempers out the same commas as 53edo, including the [[kleisma]], the [[schisma]], the [[amity comma]], the [[semicomma]], and the [[vulture comma]].
| ?
 
| ?
* In the 7-limit, it tempers out [[1029/1024]], [[10976/10935]], 117649/116640, 235298/234375, [[Landscape comma|250047/250000]] and [[Meter comma|703125/702464]]; this makes it among other things an excellent tuning for [[Schismatic family #Guiron|guiron]] and [[Gamelismic clan #Tritikleismic|tritikleismic]] temperaments, as well as a possible tuning for the metric temperament. In addition, it also tempers out the [[termite comma]] and the [[forge comma]].
| [[44/39]]
 
| 289/256
* In the 11-limit, it tempers out not only [[385/384]], [[441/440]], [[3025/3024]], [[4000/3993]], [[4375/4356]], and [[6250/6237]], but both the [[nexus comma]] and the [[quartisma]], which in turn means that the [[symbiotic comma]] is tempered out as well. In addition, it also tempers out the [[frameshift comma]] – a comma known to be associated with higher-accuracy systems.
|-
 
| 29
* In the 13-limit, it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[6656/6655]], [[cantonisma|10985/10976]], 13720/13689 and [[43904/43875]]. In addition, it also tempers out the [[chalmersia]] and the [[phaotic comma]].
| 218.8679245
 
|
* In the 17-limit, it tempers out [[273/272]], [[375/374]], [[595/594]], [[715/714]], [[833/832]], [[936/935]], [[1089/1088]], [[1701/1700]], [[2025/2023]], 2028/2023, [[2431/2430]], 8624/8619, 11271/11264, 15379/15300, [[24576/24565]], 57375/57344 and [[248897/248832]]. In addition, it also tempers out the [[flashma]] and the [[sparkisma]], the latter of which is also tempered out by 53edo despite 53edo having a different mapping for 17.
| ?
 
| [[25/22]]
* In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas.
| ?
 
| [[17/15]]
* 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.
 
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.
 
As a result of tempering out some of the commas, it allows [[essentially tempered chord]]s 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 [[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 [[7L 26s|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 paucitonic, which can be extended to a [[5L 22s|reinatonic]] 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 [[19L 26s|veljentyttonic]] 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 [[22L 5s|antireinatonic]] 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 [[12L 29s|pythamystonic]] MOS.
 
In addition, 159edo has no less than four possible generators for the [[5L 3s|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 "[[11L 2s|Diatonicized Chromatic Scale]]" with large and small scale steps at 13\159 and 8\159 respectively.
 
== Intervals ==
{{see also| 159edo 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=white-space:nowrap | Table of 159edo intervals
|-
|-
! Step
| 30
! Cents
| 226.4150943
! 5 limit
| [[256/225]]
! 7 limit
| ?
! 11 limit
| [[154/135]]
! 13 limit
| ?
! 17 limit
| ?
|-
|-
| 0
| 31
| 0
| 233.9622642
| colspan="5"| '''[[1/1]]'''
|-
| 1
| 7.5471698
|  
|  
| [[225/224]]
| '''[[8/7]]'''
| [[243/242]]
| [[55/48]]
| [[196/195]], [[351/350]]
| ?
| 256/255
| ?
|-
|-
| 2
| 32
| 15.0943396
| 241.5094340
|  
|  
| ?
| ?
| [[121/120]], [[100/99]]
| 1024/891
| [[144/143]]
| ?
| 120/119
| ?
|-
|-
| 3
| 33
| 22.6415094
| 249.0566038
| [[81/80]]
| ?
| ?
| ?
| ?
| [[15/13]]
| ?
| ?
| [[78/77]]
| 85/84
|-
|-
| 4
| 34
| 30.1886792
| 256.6037736
|  
|  
| [[64/63]]
| [[56/55]], [[55/54]]
| ?
| ?
| [[52/51]]
| 297/256
|-
| 5
| 37.7358491
|
| ?
| ?
| [[45/44]]
| ?
| ?
| [[51/50]]
|-
|-
| 6
| 35
| 45.2830189
| 264.1509434
|
| [[7/6]]
| [[64/55]]
| ?
| ?
| ?
| ?
|-
| 36
| 271.6981132
| [[75/64]]
| ?
| ?
| [[40/39]]
| 192/187
|-
| 7
| 52.8301887
|
| ?
| ?
| [[33/32]]
| 117/100
| ?
| ?
| [[34/33]]
|-
|-
| 8
| 37
| 60.3773585
| 279.2452830
|  
|  
| [[28/27]]
| ?
| ?
| ?
| ?
| 88/85
| ?
| [[20/17]]
|-
| 38
| 286.7924528
|
| ?
| [[33/28]]
| [[13/11]]
| 85/72
|-
|-
| 9
| 39
| 67.9245283
| 294.3396226
| [[25/24]]
| [[32/27]]
| ?
| ?
| ?
| ?
| ?
| [[26/25]], [[27/26]]
| ?
| ?
|-
|-  
| 10
| 40
| 75.4716981
| 301.8867925
|  
|  
| [[25/21]]
| 144/121
| ?
| ?
| ?
| ?
| ?
| 160/153
|-
|-
| 11
| 41
| 83.0188679
| 309.4339622
|  
|  
| [[21/20]]
| [[22/21]]
| ?
| ?
| ?
| ?
| 512/429
| 153/128
|-
|-
| 12
| 42
| 90.5660377
| 316.9811321
| [[256/243]], [[135/128]]
| [[6/5]]
| ?
| ?
| [[77/64]]
| ?
| ?
| ?
| ?
|-
| 43
| 324.5283019
|
| [[135/112]]
| ?
| ?
| ?
| 512/425
|-
|-
| 13
| 44
| 98.1132075
| 332.0754717
|  
|  
| ?
| ?
| [[128/121]]
| [[40/33]], [[121/100]]
| [[55/52]]
| ?
| [[18/17]]
| 144/119, 165/136
|-
|-
| 14
| 45
| 105.6603774
| 339.6226415
|
| ?
| ?
| ?
| ?
| ?
| ?
| '''[[17/16]]'''
| [[39/32]]
| [[17/14]]
|-
|-
| 15
| 46
| 113.2075472
| 347.1698113
| [[16/15]]
|
| ?
| [[11/9]]
| ?
| ?
| ?
|-
| 47
| 354.7169811
|
| ?
| ?
| [[27/22]]
| ?
| ?
| ?
| ?
|-
|-
| 16
| 48
| 120.7547170
| 362.2641509
|  
| ?
| [[15/14]]
| 275/256
| ?
| ?
| ?
| ?
| '''[[16/13]]'''
| [[21/17]]
|-
|-
| 17
| 49
| 128.3018868
| 369.8113208
|  
|  
| ?
| ?
| ?
| ?
| [[14/13]]
| [[26/21]]
| 128/119
| 68/55
|-
|-
| 18
| 50
| 135.8490566
| 377.3584906
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Line 499: Line 564:
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Line 787: Line 828:
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| 82
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| 89
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| 92
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|-
| 96
| 132
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Line 1,051: Line 1,116:
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|-
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| 1049.0566038
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| [[80/51]]
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|-
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| 1056.6037736
|  
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| 105
| 792.4528302
| [[128/81]]
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| 800
| 1071.6981132
|  
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|-
| 107
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| 1079.2452830
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| 144
| 1086.7924528
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| 108
| 815.0943396
| '''[[8/5]]'''
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| '''[[32/17]]'''
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|-
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| 1101.8867925
|  
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| 147
| 1109.4339622
| [[243/128]], [[256/135]]
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| '''[[13/8]]'''
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| 148
| 845.2830189
| 1116.9811321
|  
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| 1124.5283019
|  
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| ?
| ?
| ?
| 153/80
|-
|-
| 114
| 150
| 860.3773585
| 1132.0754717
| [[48/25]]
| ?
| ?
| ?
| ?
| [[25/13]], [[52/27]]
| ?
| ?
| [[64/39]]
| [[28/17]]
|-
|-
| 115
| 151
| 867.9245283
| 1139.6226415
|  
|  
| [[27/14]]
| ?
| ?
| [[33/20]], [[200/121]]
| ?
| ?  
| 85/44
| 119/72, 272/165
|-
|-
| 116
| 152
| 875.4716981
| 1147.1698113
|  
|  
| [[224/135]]
| ?
| ?
| ?  
| [[64/33]]
| 425/256
| ?
| [[33/17]]
|-
|-
| 117
| 153
| 883.0188679
| 1154.7169811
| [[5/3]]
| ?
| ?
| [[128/77]]
| ?
| ?
| ?
| ?
| [[39/20]]
| 187/96
|-
|-
| 118
| 154
| 890.5660377
| 1162.2641509
|  
|  
| ?
| ?
| [[88/45]]
| ?
| ?
| 429/256
| [[100/51]]
| [[256/153]]
|-
|-
| 119
| 155
| 898.1132075
| 1169.8113208
|  
|  
| [[42/25]]
| [[63/32]]
| 121/72
| [[55/28]], [[108/55]]
| ?
| ?
| ?
| [[51/26]]
|-
|-
| 120
| 156
| 905.6603774
| 1177.3584906
| [[27/16]]
| [[160/81]]
| ?
| ?
| ?
| ?
| ?
| ?
| [[77/39]]
| 168/85
|-
|-
| 121
| 157
| 913.2075472
| 1184.9056604
|  
|  
| ?
| ?
| [[56/33]]
| [[240/121]], [[99/50]]
| [[22/13]]
| [[143/72]]
| 144/85
| 119/60
|-
|-
| 122
| 158
| 920.7547170
| 1192.4528302
|  
|  
| ?
| [[448/225]]
| ?
| [[484/243]]
| ?
| [[195/98]], [[700/351]]
| [[17/10]]
| 255/128
|-
| 159
| 1200
| colspan="5"| '''[[2/1]]'''
|-
|-
| 123
|}
| 928.3018868
 
| [[128/75]]
== Notation ==
| ?
{{main| 159edo notation }}
| ?
 
| 200/117
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
|-
|-
| 124
! [[TE error|Absolute]] (¢)
| 935.8490566
! [[TE simple badness|Relative]] (%)
|
| [[12/7]]
| [[55/32]]
| ?
| ?
|-
|-
| 125
| 2.3.5.7
| 943.3962264
| 1029/1024, 10976/10935, 15625/15552
|  
| [{{val| 159 252 369 446 }}]
| ?
| +0.411
| 512/297
| 0.413
| ?
| 5.47
| ?
|-
|-
| 126
| 2.3.5.7.11
| 950.9433962
| 385/384, 441/440, 4000/3993, 10976/10935
| ?
| [{{val| 159 252 369 446 550 }}]
| ?
| +0.350
| ?
| 0.389
| [[26/15]]
| 5.15
| ?
|-
|-
| 127
| 2.3.5.7.11.13
| 958.4905660
| 325/324, 364/363, 385/384, 625/624, 10976/10935
|  
| [{{val| 159 252 369 446 550 588 }}]
| ?
| +0.418
| 891/512
| 0.385
| ?
| 5.11
| ?
|-
|-
| 128
| 2.3.5.7.11.13.17
| 966.0377358
| 273/272, 325/324, 364/363, 375/374, 385/384, 3773/3757
|  
| [{{val| 159 252 369 446 550 588 650 }}]
| '''[[7/4]]'''
| +0.333
| [[96/55]]
| 0.412
| ?
| 5.46
| ?
|}
|-
Notably, while 159edo plays host to a number of fun temperaments like [[gamelan]] and [[364/363|gentle]], 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.
| 129
 
| 973.5849057
=== Commas ===
| [[225/128]]
Assuming the [[val]] {{val|159 252 369 446 550 588 650 }}, 159tet [[tempers out]] the following [[comma]]s.
| ?
 
| [[135/77]]
{| class="commatable wikitable center-all left-3 right-4 left-6"
| ?
| ?
|-
|-
| 130
! [[Harmonic limit|Prime <br>limit]]
| 981.1320755
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
|
! [[Monzo]]
| ?
! [[Cents]]
| [[44/25]]
! [[Color name]]
| ?
! Name
| [[30/17]]
|-
|-
| 131
| 3
| 988.6792458
| <abbr title="19383245667680019896796723/19342813113834066795298816">(52 digits)</abbr>
|  
| {{monzo| -84 53 }}
| ?
| 3.6150
| ?
| Wa-53
| [[39/22]]
| [[Mercator's comma]]
| 512/289
|-
|-
| 132
| 5
| 996.2264151
| <abbr title="2109375/2097152">(14 digits)</abbr>
| [[16/9]]
| {{monzo| -21 3 7 }}
| ?
| 10.0610
| ?
| Lasepyo
| ?
| [[Semicomma]]
| ?
|-
|-
| 133
| 5
| 1003.7735849
| 15625/15552
|  
| {{monzo| -6 -5 6 }}
| [[25/14]]
| 8.1073
| [[216/121]]
| Tribiyo
| ?
| [[15625/15552|Kleisma]]
| ?
|-
|-
| 134
| 5
| 1011.3207547
| <abbr title="1600000/1594323">(14 digits)</abbr>
|  
| {{monzo| 9 -13 5 }}
| ?
| 6.1536
| ?
| Saquinyo
| [[256/143]]
| [[Amity comma]]
| 459/256
|-
|-
| 135
| 5
| 1018.8679245
| <abbr title="10485760000/10460353203">(22 digits)</abbr>
| [[9/5]]
| {{monzo| 24 -21 4 }}
| ?
| 4.1998
| 231/128
| Sasaquadyo
| ?
| [[Vulture comma]]
| ?
|-
| 5
| 32805/32768
| {{monzo| -15 8 1 }}
| 1.9537
| Layo
| [[Schisma]]
|-
|-
| 136
| 5
| 1026.4150943
| <abbr title="2954312706550833698643/2951479051793528258560">(44 digits)</abbr>
| {{monzo| -69 45 -1 }}
| 1.6613
|  
|  
| 1024/567
| [[Counterschisma]]
| 440/243
| ?
| 768/425
|-
|-
| 137
| 5
| 1033.9622642
| <abbr title="450359962737049600/450283905890997363">(36 digits)</abbr>
| {{monzo| 54 -37 2 }}
| 0.2924
|  
|  
| ?
| [[Monzisma]]
| [[20/11]]
| ?
| ?
|-
|-
| 138
| 7
| 1041.5094340
| <abbr title="117649/116640">(12 digits)</abbr>
| ?
| {{monzo| -5 -6 -1 6 }}
| ?
| 14.9117
| ?
|  
| [[117/64]]
| 1024/561, 935/512
|-
| 139
| 1049.0566038
|  
|  
| ?
| [[11/6]]
| ?
| ?
|-
|-
| 140
| 7
| 1056.6037736
| [[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
|
|  
|  
| ?
| [[81/44]]
| ?
| ?
|-
|-
| 141
| 7
| 1064.1509434
| 10976/10935
| [[50/27]]
| {{monzo| 5 -7 -1 3 }}
| ?
| 6.4790
| ?
| Satrizo-agu
| [[24/13]]
| [[Hemimage comma]]
| ?
|-
|-
| 142
| 7
| 1071.6981132
| <abbr title="1640558367/1638400000">(20 digits)</abbr>
| {{monzo| -19 14 -5 3 }}
| 2.2792
|  
|  
| ?
| [[Forge comma]]
| ?
| [[13/7]]
| 119/64
|-
|-
| 143
| 7
| 1079.2452830
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.6283
|  
|  
| [[28/15]]
| [[Meter]]
| 512/275
| ?
| ?
|-
|-
| 144
| 7
| 1086.7924528
| <abbr title="250047/250000">(12 digits)</abbr>
| [[15/8]]
| {{monzo| -4 6 -6 3 }}
| ?
| 0.3254
| ?
| ?
| ?
|-
| 145
| 1094.3396226
|  
|  
| ?
| [[Landscape comma]]
| ?
| ?
| '''[[32/17]]'''
|-
|-
| 146
| 7
| 1101.8867925
| <abbr title="37252902984619140625000000000/37252879910233655318543787489">(58 digits)</abbr>
|  
| {{monzo| 9 -28 37 -18 }}
| ?
| 0.0011
| [[121/64]]
| Satritribiru-athiseyo
| [[104/55]]
| [[Termite comma]]
| [[17/9]]
|-
|-
| 147
| 11
| 1109.4339622
| 4375/4356
| [[243/128]], [[256/135]]
| {{monzo| -2 -2 4 1 -2 }}
| ?
| 7.5349
| ?
|  
| ?
| ?
|-
| 148
| 1116.9811321
|  
|  
| [[40/21]]
| [[21/11]]
| ?
| ?
|-
|-
| 149
| 11
| 1124.5283019
| [[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
| ?
| ?
| 153/80
|-
|-
| 150
| 11
| 1132.0754717
| [[4000/3993]]
| [[48/25]]
| {{monzo| 5 -1 3 0 -3 }}
| ?
| 3.0323
| ?
| Triluyo
| [[25/13]], [[52/27]]
| Wizardharry comma
| ?
|-
|-
| 151
| 11
| 1139.6226415
| 19712/19683
|  
| {{monzo| 8 -9 0 1 1 }}
| [[27/14]]
| 2.5488
| ?
| Salozo
| ?
| [[Symbiotic comma]]
| 85/44
|-
|-
| 152
| 11
| 1147.1698113
| <abbr title="1771561/1769472">(14 digits)</abbr>
|  
| {{Monzo| 16 -3 0 0 6 }}
| ?
| 2.0427
| [[64/33]]
| Tribilo
| ?
| [[Nexus comma]]
| [[33/17]]
|-
|-
| 153
| 11
| 1154.7169811
| <abbr title="22876792454961/22866405883904">(28 digits)</abbr>
| ?
| {{monzo| -34 28 0 0 -3 }}
| ?
| 0.7862
| ?
| Quadla-trilu
| [[39/20]]
| [[Frameshift comma]]
| 187/96
|-
|-
| 154
| 11
| 1162.2641509
| [[3025/3024]]
|  
| {{monzo| -4 -3 2 -1 2 }}
| ?
| 0.5724
| [[88/45]]
| Loloruyoyo
| ?
| Lehmerisma
| [[100/51]]
|-
|-
| 155
| 11
| 1169.8113208
| <abbr title="117440512/117406179">(18 digits)</abbr>
|  
| {{monzo| 24 -6 0 1 -5 }}
| [[63/32]]
| 0.5062
| [[55/28]], [[108/55]]
| Saquinlu-azo
| ?
| [[Quartisma]]
| [[51/26]]
|-
|-
| 156
| 11
| 1177.3584906
| <abbr title="1771561/1771470">(14 digits)</abbr>
| [[160/81]]
| {{monzo| -1 -11 -1 0 6 }}
| ?
| 0.0889
| ?
| Satribilo-agu
| [[77/39]]
| [[Parimo]]
| 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
| colspan="5"| '''[[2/1]]'''
|-
|-
|}
|}


== Notation ==
{{main| 159edo 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.
* In the 5-limit, it tempers out the same commas as 53edo, including the [[kleisma]], the [[schisma]], the [[amity comma]], the [[semicomma]], and the [[vulture comma]].
 
* In the 7-limit, it tempers out [[1029/1024]], [[10976/10935]], 117649/116640, 235298/234375, [[Landscape comma|250047/250000]] and [[Meter comma|703125/702464]]; this makes it among other things an excellent tuning for [[Schismatic family #Guiron|guiron]] and [[Gamelismic clan #Tritikleismic|tritikleismic]] temperaments, as well as a possible tuning for the metric temperament. In addition, it also tempers out the [[termite comma]] and the [[forge comma]].
 
* In the 11-limit, it tempers out not only [[385/384]], [[441/440]], [[3025/3024]], [[4000/3993]], [[4375/4356]], and [[6250/6237]], but both the [[nexus comma]] and the [[quartisma]], which in turn means that the [[symbiotic comma]] is tempered out as well. In addition, it also tempers out the [[frameshift comma]] – a comma known to be associated with higher-accuracy systems.
 
* In the 13-limit, it tempers out [[325/324]], [[364/363]], [[625/624]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[6656/6655]], [[cantonisma|10985/10976]], 13720/13689 and [[43904/43875]]. In addition, it also tempers out the [[chalmersia]] and the [[phaotic comma]].
 
* In the 17-limit, it tempers out [[273/272]], [[375/374]], [[595/594]], [[715/714]], [[833/832]], [[936/935]], [[1089/1088]], [[1701/1700]], [[2025/2023]], 2028/2023, [[2431/2430]], 8624/8619, 11271/11264, 15379/15300, [[24576/24565]], 57375/57344 and [[248897/248832]]. In addition, it also tempers out the [[flashma]] and the [[sparkisma]], the latter of which is also tempered out by 53edo despite 53edo having a different mapping for 17.  


== Regular temperament properties ==
* In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas.
{| class="wikitable center-4 center-5 center-6"
 
! rowspan="2" | Subgroup
* 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.
! rowspan="2" | [[Comma list]]
 
! rowspan="2" | [[Mapping]]
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.
! rowspan="2" | Optimal<br>8ve stretch (¢)
 
! colspan="2" | Tuning error
As a result of tempering out some of the commas, it allows [[essentially tempered chord]]s 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.
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 1029/1024, 10976/10935, 15625/15552
| [{{val| 159 252 369 446 }}]
| +0.411
| 0.413
| 5.47
|-
| 2.3.5.7.11
| 385/384, 441/440, 4000/3993, 10976/10935
| [{{val| 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
| [{{val| 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
| [{{val| 159 252 369 446 550 588 650 }}]
| +0.333
| 0.412
| 5.46
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Retrieved from "https://en.xen.wiki/w/159edo"