159edo: Difference between revisions

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{{ED intro}} The step size of this system is in between the sizes of [[243/242]] (the rastma) and [[225/224]] (the marvel comma).

~~== Theory ==~~

== History ==

~~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]].~~

=== 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.

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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 ==

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 [[29-odd-limit]] or the no-19 [[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.

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.

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 ===

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| ?

| [[121/120]], [[100/99]]

| [[144/143]]

| [[144/143]], [[105/104]]

| [[120/119]]

|-

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| 211.3207547

|

| ?

| [[640/567]]

| ?

| [[44/39]]

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| 988.6792458

|

| ?

| [[567/320]]

| ?

| [[39/22]]

Line 1,320:

| ?

| [[240/121]], [[99/50]]

| [[143/72]]

| [[143/72]], [[208/105]]

| [[119/60]]

|-

Line 1,452:

| 2.2461

|

| [[~~Tricot~~ comma]]

| [[Alphatricot comma]]

|-

| 5

Line 1,868:

|-

|}

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.

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.

=== Rank-2 temperaments ===

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| [[Iodine]]

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<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 ==

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== 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]]

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* [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]]

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* [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 ==

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* [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)]

~~== Notes ==~~

~~<references group="note" />~~

== References ==

@@ Line 8: / Line 8: @@
 {{ED intro}} The step size of this system is in between the sizes of [[243/242]] (the rastma) and [[225/224]] (the marvel comma).
-== Theory ==
+== History ==
-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]].
-=== 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.
@@ Line 19: / Line 16: @@
 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 ==
+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.
-edo is [[consistent]] up to the no-17 [[29-odd-limit]] or the no-19 [[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.
+edo 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.
+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 ===
@@ Line 80: / Line 80: @@
 | ?
 | [[121/120]], [[100/99]]
-| [[144/143]]
+| [[144/143]], [[105/104]]
 | [[120/119]]
 |-
@@ Line 286: / Line 286: @@
 | 211.3207547
 |
-| ?
+| [[640/567]]
 | ?
 | [[44/39]]
@@ Line 1,110: / Line 1,110: @@
 | 988.6792458
 |
-| ?
+| [[567/320]]
 | ?
 | [[39/22]]
@@ Line 1,320: / Line 1,320: @@
 | ?
 | [[240/121]], [[99/50]]
-| [[143/72]]
+| [[143/72]], [[208/105]]
 | [[119/60]]
 |-
@@ Line 1,452: / Line 1,452: @@
 | 2.2461
 |
-| [[Tricot comma]]
+| [[Alphatricot comma]]
 |-
 | 5
@@ Line 1,868: / Line 1,868: @@
 |-
 |}
+<references group="note" />
-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.
+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.
 === Rank-2 temperaments ===
@@ Line 1,997: / Line 1,998: @@
 | [[Iodine]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<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 ==
@@ Line 2,004: / Line 2,005: @@
 == 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]]
@@ Line 2,013: / Line 2,017: @@
 * [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]]
@@ Line 2,019: / Line 2,026: @@
 * [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 ==
@@ Line 2,027: / Line 2,036: @@
 * [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)]
-== Notes ==
-<references group="note" />
 == References ==