159edo: Difference between revisions

Line 12:

=== History ===

~~<todo>~~

Due to being a relative of the famous [[53edo]], this system is fairly easy to find, so there are lingering questions as to just who found this system first. However, [https://en.xen.wiki/index.php?title=159edo&type=revision&diff=5153&oldid=5154 the first records of it on this site] concern a 79-tone subset of this system related to [[Turkish maqam music temperaments|yarman]] temperament which had been proposed by [[Ozan Yarman]] as a tuning standard for [[Arabic, Turkish, Persian|Arabic, Turkish and Persian]] music.

=== Prime harmonics ===

Line 18:

=== Mappings ===

A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[harmonic]]s with [[53edo]]. However, compared to 53edo, 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.

A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[harmonic]]s with 53edo. However, compared to 53edo, 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 ===

Line 36:

* 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.

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 ~~[[Turkish maqam music temperaments|~~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 ~~– either way, a form of yarman temperament has been proposed by [[Ozan Yarman]] as a tuning standard for [[Arabic, Turkish, Persian|Arabic, Turkish and Persian]] music~~.

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.

@@ Line 12: / Line 12: @@
 === History ===
-<todo>
+Due to being a relative of the famous [[53edo]], this system is fairly easy to find, so there are lingering questions as to just who found this system first.  However, [https://en.xen.wiki/index.php?title=159edo&type=revision&diff=5153&oldid=5154 the first records of it on this site] concern a 79-tone subset of this system related to [[Turkish maqam music temperaments|yarman]] temperament which had been proposed by [[Ozan Yarman]] as a tuning standard for [[Arabic, Turkish, Persian|Arabic, Turkish and Persian]] music.
 === Prime harmonics ===
@@ Line 18: / Line 18: @@
 === Mappings ===
-A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[harmonic]]s with [[53edo]]. However, compared to 53edo, 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.
+A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[harmonic]]s with 53edo. However, compared to 53edo, 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 ===
@@ Line 36: / Line 36: @@
 * 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.
-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 [[Turkish maqam music temperaments|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 – either way, a form of yarman temperament has been proposed by [[Ozan Yarman]] as a tuning standard for [[Arabic, Turkish, Persian|Arabic, Turkish and Persian]] music.
+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.