159edo: Difference between revisions

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== Theory ==

~~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~~.

A salient fact about 159edo is that 159 = 3 × 53, and thus, this system has both [[3edo]] and [[53edo]] as subsets.

=== History ===

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 Wiki from the days of Wikispaces] 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.

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 Wiki from the days of Wikispaces] 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.

=== 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.

=== Prime harmonics ===

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=== 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.

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.

=== Commas ===

@@ Line 9: / Line 9: @@
 == Theory ==
-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.
+A salient fact about 159edo is that 159 = 3 × 53, and thus, this system has both [[3edo]] and [[53edo]] as subsets.
 === History ===
-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 Wiki from the days of Wikispaces] 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.
+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 Wiki from the days of Wikispaces] 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.
+=== 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.
 === Prime harmonics ===
@@ Line 18: / Line 21: @@
 === 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.
+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.
 === Commas ===