159edo: Difference between revisions

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=== Mappings and JI approximation quality ===

This system inherits its approximations of the 3rd, 5th and ~~13th~~ [[harmonic]]s from 53edo, however, the [[patent val]]s differ on the mappings for 7, 11 and 17 – in fact, this ~~EDO~~ has a very accurate 11 and an only slightly less accurate 17. Furthermore, 159edo demonstrates 3-to-2 [[telicity]], as despite being [[contorted]] in the 5-limit, it is the largest ~~EDO~~ to temper out Mercator's comma in which said comma is less than half the size of a single ~~EDO~~ step. This means, among other things, that there is a perfect match between the [[direct 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~~.

This system inherits its approximations of the [[3/1|3rd]], [[5/1|5th]], [[13/1|13th]], and [[19/1|19th]] [[harmonic]]s 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 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, while~~ 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, and this despite 53edo's approximation of~~ [[~~19/16~~]] ~~being inherited from 53edo~~. ~~In addition~~, the [[direct ~~mapping~~]] and the ~~more complicated traditional~~ 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 [[29-odd-limit]] or the no-19 [[27-odd-limit]]. {19/17, 34/19} and {29/17, 34/29} exhaust the inconsistently mapped interval pairs in the 29-odd-limit. Thus its full 29-limit interpretation using the [[patent val]] is obvious. However, the [[direct approximation]] and the val mapping for intervals such as [[49/32]], [[35/32]], and [[169/128]] do not match, and as a result, 159edo can be thought of as having a perfunctory [[7-limit]] that mainly serves to bridge to the [[11-limit]] and divide the nearly just 3/2 into three, as well as a similarly perfunctory [[13-limit]] that mainly serves to bridge to the 17-limit and to absorb complex combinations of 3 and 5.

Notably, 159edo provides the [[optimal patent val]] for 11-limit guiron, 13-limit tritikleismic, the 13-limit rank-3 temperament [[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 22: / Line 22: @@
 === Mappings and JI approximation quality ===
-This system inherits its approximations of the 3rd, 5th and 13th [[harmonic]]s from 53edo, however, the [[patent val]]s differ on the mappings for 7, 11 and 17 – in fact, this EDO has a very accurate 11 and an only slightly less accurate 17.  Furthermore, 159edo demonstrates 3-to-2 [[telicity]], as despite being [[contorted]] in the 5-limit, it is the largest EDO to temper out Mercator's comma in which said comma is less than half the size of a single EDO step. This means, among other things, that there is a perfect match between the [[direct 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.
+This system inherits its approximations of the [[3/1|3rd]], [[5/1|5th]], [[13/1|13th]], and [[19/1|19th]] [[harmonic]]s 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 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, while 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, and this despite 53edo's approximation of [[19/16]] being inherited from 53edo.  In addition, the [[direct mapping]] and the more complicated traditional 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 [[29-odd-limit]] or the no-19 [[27-odd-limit]]. {19/17, 34/19} and {29/17, 34/29} exhaust the inconsistently mapped interval pairs in the 29-odd-limit. Thus its full 29-limit interpretation using the [[patent val]] is obvious. However, the [[direct approximation]] and the val mapping for intervals such as [[49/32]], [[35/32]], and [[169/128]] do not match, and as a result, 159edo can be thought of as having a perfunctory [[7-limit]] that mainly serves to bridge to the [[11-limit]] and divide the nearly just 3/2 into three, as well as a similarly perfunctory [[13-limit]] that mainly serves to bridge to the 17-limit and to absorb complex combinations of 3 and 5.
-Notably, 159edo provides the [[optimal patent val]] for 11-limit guiron, 13-limit tritikleismic, the 13-limit rank-3 temperament [[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 ===