159edo: Difference between revisions

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=== Mappings ===

A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] 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 19th harmonic having multiple reasonable mappings. Furthermore, 159edo demonstrates 3-to-2 [[telicity]], as despite being [[contorted]] ~~(or [[enfactored]])~~ 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 don't match. While the [[patent val]] supports [[Mercator family#Cartography|cartography]] temperament, which is 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 [[Overtone series|harmonics]] 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 19th harmonic having multiple reasonable mappings. 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 don't match. While the [[patent val]] supports [[Mercator family#Cartography|cartography]] temperament, which is among the best 13-limit temperaments in the [[Mercator family]], the 159d and 159e mappings support other members of this temperament family.

=== Commas ===

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 === Mappings ===
-A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th [[Overtone series|harmonics]] 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 19th harmonic having multiple reasonable mappings.  Furthermore, 159edo demonstrates 3-to-2 [[telicity]], as despite being [[contorted]] (or [[enfactored]]) 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 don't match.  While the [[patent val]] supports [[Mercator family#Cartography|cartography]] temperament, which is 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 [[Overtone series|harmonics]] 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 19th harmonic having multiple reasonable mappings.  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 don't match.  While the [[patent val]] supports [[Mercator family#Cartography|cartography]] temperament, which is among the best 13-limit temperaments in the [[Mercator family]], the 159d and 159e mappings support other members of this temperament family.
 === Commas ===