159edo: Difference between revisions

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Accordingly, it is no coincidence that [https://en.xen.wiki/index.php?title=159edo&type=revision&diff=5153&oldid=5154 the first records of 159edo on this Wiki from the days of Wikispaces] concern said 79-tone subset related to the [[Turkish maqam music temperaments|yarman]] temperament which had been proposed by Yarman as a tuning standard for [[Arabic, Turkish, Persian|Arabic, Turkish and Persian music]]. Based on the information given by Ozan Yarman himself, his elder colleague [[M. Ugur Kececioglu]] first utilized 159edo in his revamped 2011 release of the [[Notist]] score editor and therein allowed the Arel-Ezgi-Uzdilek accidentals to be bent by as little a detail as 1/3rd of a single step of 53edo, while also mapping AEU altogether to a suitable subset of 53edo to allow transpositions throughout.

=== Mappings, JI approximation quality ~~and the JND~~ ===

=== 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.

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.

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.

With all that said, the approximations of everything in the 17-odd-limit and even the approximation of 19/16 still fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and as a result, this system has additional properties that are worth noting. Specifically, the step size of 159edo is simultaneously above the average peak melodic 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, even as it also allows one to also control the bandwidth of certain sounds.

=== Prime harmonics ===

=== Additional Properties of 159-tone equal temperament ===

The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and similarly this system can approximate the sounds of other systems such as [[10edo]], [[13edo]], [[22edo]] and [[31edo]]. Furthermore, the step size of 159edo is simultaneously above the average peak melodic 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, even as it also allows one to also control the bandwidth of certain sounds.

=== MOSes and other scales ===

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 Accordingly, it is no coincidence that [https://en.xen.wiki/index.php?title=159edo&type=revision&diff=5153&oldid=5154 the first records of 159edo on this Wiki from the days of Wikispaces] concern said 79-tone subset related to the [[Turkish maqam music temperaments|yarman]] temperament which had been proposed by Yarman as a tuning standard for [[Arabic, Turkish, Persian|Arabic, Turkish and Persian music]]. Based on the information given by Ozan Yarman himself, his elder colleague [[M. Ugur Kececioglu]] first utilized 159edo in his revamped 2011 release of the [[Notist]] score editor and therein allowed the Arel-Ezgi-Uzdilek accidentals to be bent by as little a detail as 1/3rd of a single step of 53edo, while also mapping AEU altogether to a suitable subset of 53edo to allow transpositions throughout.
-=== Mappings, JI approximation quality and the JND ===
+=== 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 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.
+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.
-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.
-With all that said, the approximations of everything in the 17-odd-limit and even the approximation of 19/16 still fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and as a result, this system has additional properties that are worth noting.  Specifically, the step size of 159edo is simultaneously above the average peak melodic 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, even as it also allows one to also control the bandwidth of certain sounds.
 === Prime harmonics ===
 {{Harmonics in equal|159|columns=12}}
+=== Additional Properties of 159-tone equal temperament ===
+The approximations of everything in the 17-odd-limit and even the approximations of 19/16, 29/16 and 31/16 all fall within the boundaries of the harmonic [http://musictheory.zentral.zone/huntsystem2.html#2 JND], and similarly this system can approximate the sounds of other systems such as [[10edo]], [[13edo]], [[22edo]] and [[31edo]].  Furthermore, the step size of 159edo is simultaneously above the average peak melodic 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, even as it also allows one to also control the bandwidth of certain sounds.
 === MOSes and other scales ===