159edo: Difference between revisions

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=== Relationship to the JND ===

As the step size of 159edo is simultaneously above the average peak melodic [http://musictheory.zentral.zone/huntsystem2.html#2 JND] and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another.

=== Mappings and JI approximation quality ===

<todo>

This system shares the same 3rd, 5th and 13th [[harmonic]]s with 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. 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.

=== Prime harmonics ===

~~=== Mappings ===~~

This system shares the same 3rd, 5th and 13th [[harmonic]]s with 53edo, however, 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.

=== MOSes and other scales ===

@@ Line 23: / Line 23: @@
 === Relationship to the JND ===
 As the step size of 159edo is simultaneously above the average peak melodic [http://musictheory.zentral.zone/huntsystem2.html#2 JND] and small enough to be well within the margin of error between Just 5-limit intervals and their [[12edo]] counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another.
+=== Mappings and JI approximation quality ===
+<todo>
+This system shares the same 3rd, 5th and 13th [[harmonic]]s with 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.  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.
 === Prime harmonics ===
 {{Harmonics in equal|159|columns=12}}
-=== Mappings ===
-This system shares the same 3rd, 5th and 13th [[harmonic]]s with 53edo, however, 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.
 === MOSes and other scales ===