Kite's Genchain mode numbering: Difference between revisions

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== Temperaments with split octaves ==

If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel~~, using~~ ups and downs. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice must be complete. The number in the brackets becomes two numbers, and the Ls pattern is ~~broken up into periods with~~ hyphens.

If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel. Using ups and downs notation, each genchain has its own height. There is a plain one, an up one, perhaps a down one, etc. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice generated by the 5th and the up must be complete. The number in the brackets becomes two numbers, and the Ls pattern as written here is grouped by period, using hyphens.

[[Srutal]] aka Diaschismatic aka Sagugu has a half-8ve period of ~45/32. All five Srutal[2x5] modes. Every other scale note has a down.

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Using ~25/24 as the generator yields the same scales and mode numbers. 1st Diminished[4x2] would have genchains C – ^^C#, vEb – ^E, ^^F# – G and ^A – vBb, just like the scale.

[[Blackwood|'''Blackwood''']] '''aka ~~5edo~~+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period.

[[Blackwood|'''Blackwood''']] '''aka Sawa+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period.

{| class="wikitable"

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The modes of the sister MOS 3L 5s are the same, just exchange L and s:

* 1st ~~3s 5L~~ = ssLssLsL

* 1st 3L 5s = ssLssLsL

* 2nd ~~3s 5L~~ = ssLsLssL

* 2nd 3L 5s = ssLsLssL

* 3rd ~~3s 5L~~ = sLssLssL

* 3rd 3L 5s = sLssLssL

* etc.

In this context, when no specific temperament is named, the choice of generator follows slightly different rules. Prioritizing 3/2 over 4/3 doesn't make any sense, because the generator doesn't have an actual ratio. Thus while Meantone[7]'s generator is 3/2, i.e. a 5th, 5L 2s's generator is in the 480-514¢ range, i.e. a 4th.

In this context, when no specific temperament is named, the choice of generator follows slightly different rules. Prioritizing 3/2 over 4/3 doesn't make any sense, because the generator doesn't have an actual ratio. So the generator is always chosen to be as small as possible, i.e. less than half of a period. Thus while Meantone[7]'s generator is 3/2, i.e. a 5th, 5L 2s's generator is in the 480-514¢ range, i.e. a 4th.

== Rationale ==

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 == Temperaments with split octaves ==
-If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel, using ups and downs. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice must be complete. The number in the brackets becomes two numbers, and the Ls pattern is broken up into periods with hyphens.
+If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel. Using ups and downs notation, each genchain has its own height. There is a plain one, an up one, perhaps a down one, etc. In order to be a MOS scale, the parallel genchains must not only be the right length, and without any gaps, but also must line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the lattice generated by the 5th and the up must be complete. The number in the brackets becomes two numbers, and the Ls pattern as written here is grouped by period, using hyphens.
 [[Srutal]] aka Diaschismatic aka Sagugu has a half-8ve period of ~45/32. All five Srutal[2x5] modes. Every other scale note has a down.
@@ Line 625: / Line 625: @@
 Using ~25/24 as the generator yields the same scales and mode numbers. 1st Diminished[4x2] would have genchains  C – ^^C#, vEb – ^E, ^^F# – G and ^A – vBb, just like the scale.
-[[Blackwood|'''Blackwood''']] '''aka 5edo+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period.
+[[Blackwood|'''Blackwood''']] '''aka Sawa+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period.
 {| class="wikitable"
@@ Line 854: / Line 854: @@
 The modes of the sister MOS 3L 5s are the same, just exchange L and s:
-* 1st 3s 5L = ssLssLsL
+* 1st 3L 5s = ssLssLsL
-* 2nd 3s 5L = ssLsLssL
+* 2nd 3L 5s = ssLsLssL
-* 3rd 3s 5L = sLssLssL
+* 3rd 3L 5s = sLssLssL
 * etc.
-In this context, when no specific temperament is named, the choice of generator follows slightly different rules. Prioritizing 3/2 over 4/3 doesn't make any sense, because the generator doesn't have an actual ratio. Thus while Meantone[7]'s generator is 3/2, i.e. a 5th, 5L 2s's generator is in the 480-514¢ range, i.e. a 4th.
+In this context, when no specific temperament is named, the choice of generator follows slightly different rules. Prioritizing 3/2 over 4/3 doesn't make any sense, because the generator doesn't have an actual ratio. So the generator is always chosen to be as small as possible, i.e. less than half of a period. Thus while Meantone[7]'s generator is 3/2, i.e. a 5th, 5L 2s's generator is in the 480-514¢ range, i.e. a 4th.
 == Rationale ==