159edo: Difference between revisions

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=== Mappings and JI approximation quality ===

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 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~~.

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 EDO step. 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, 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.

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 === Mappings and JI approximation quality ===
-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 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.
+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 EDO step. 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, 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.