Kite's Genchain mode numbering: Difference between revisions

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'''Genchain mode numbering''' provides a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[Modal UDP notation]], it starts with the convention of using ''some-temperament-name''[''some-number''] to create a generator-chain, and adds a way to number each mode uniquely.

'''Genchain mode numbering''' ('''GMN''' for short) provides a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[Modal UDP notation]], it starts with the convention of using ''some-temperament-name''[''some-number''] to create a generator-chain, and adds a way to number each mode uniquely. It also applies to abstract MOS patterns like 5L 3s.

This mode notation system was designed by [[Kite Giedraitis]].

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[[MOS scale]]s are formed from a segment of the [[periods_and_generators|generator-chain]], or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions.

For example, here are all the modes of [[Meantone]][7], using ~3/2 as the generator. ~~The~~ Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.

For example, here are all the modes of [[Meantone]][7], using ~3/2 as the generator. On this page, the Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.

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4th Meantone[7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic".

4th Meantone[7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic". The term GMN can also be read as genchain mode number, and can refer to the numbers 1st, 2nd, 3rd etc., as in "Dorian's GMN is 4".

The same seven modes, all with C as the tonic, to illustrate the difference between modes. Adjacent modes differ by only one note. The modes proceed from sharper (Lydian) to flatter (Locrian).

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[[Porcupine]] aka Triyo has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2 using [[ups and downs notation]]. The enharmonic ~~interval~~ is v3A1. Because the generator is a 2nd, the genchain resembles the scale.

[[Porcupine]] aka Triyo has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2 using [[ups and downs notation]]. The [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is v3A1. Because the generator is a 2nd, the genchain resembles the scale.

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[[Sensi]] aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^3d4 and a v4A3, and the enharmonic ~~interval~~ is ^7dd2.

[[Sensi]] aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^3d4 and a v4A3, and the [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is ^7dd2.

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|+ Sensi[8]/Sepgu[8] modes

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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.

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|vF# * vG# vD# vA# * vB#

|}

== Generalization to temperament-agnostic MOS scales ==

[[:Category:Abstract MOS patterns|Abstract MOS patterns]] like 5L 3s are not specific temperaments in which specific commas vanish. Thus there are no ratios other than the octave 2/1 (or more generally the equave 3/1 or whatever). Genchain mode numbers can be applied to these patterns. For example, 5L 3s has a generator in the 450-480¢ range. The "[8]" is redundant, so we drop it to get

* 1st 5L 3s = LLsLLsLs

* 2nd 5L 3s = LLsLsLLs

* 3rd 5L 3s = LsLLsLLs

* etc.

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

* 1st 3L 5s = ssLssLsL

* 2nd 3L 5s = ssLsLssL

* 3rd 3L 5s = sLssLssL

* etc.

For a MOS pattern with a fifth-sized generator, the fifth is still prioritized over the fourth. Otherwise the generator is the mingen.

== Rationale ==

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-'''Genchain mode numbering''' provides a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[Modal UDP notation]], it starts with the convention of using ''some-temperament-name''[''some-number''] to create a generator-chain, and adds a way to number each mode uniquely.
+'''Genchain mode numbering''' ('''GMN''' for short) provides a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[Modal UDP notation]], it starts with the convention of using ''some-temperament-name''[''some-number''] to create a generator-chain, and adds a way to number each mode uniquely. It also applies to abstract MOS patterns like 5L 3s.
 This mode notation system was designed by [[Kite Giedraitis]].
@@ Line 6: / Line 6: @@
 [[MOS scale]]s are formed from a segment of the [[periods_and_generators|generator-chain]], or genchain. The first note in the genchain is the tonic of the 1st mode, the 2nd note is the tonic of the 2nd mode, etc., somewhat analogous to harmonica positions.
-For example, here are all the modes of [[Meantone]][7], using ~3/2 as the generator. The Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.
+For example, here are all the modes of [[Meantone]][7], using ~3/2 as the generator. On this page, the Ls pattern is divided into two halves, for readability. The first half runs from the tonic to the 5th. and the second half runs from the 5th to the 8ve.
 {| class="wikitable"
@@ Line 60: / Line 60: @@
 |}
-th Meantone[7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic".
+th Meantone[7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic". The term GMN can also be read as genchain mode <u>number</u>, and can refer to the numbers 1st, 2nd, 3rd etc., as in "Dorian's GMN is 4".
 The same seven modes, all with C as the tonic, to illustrate the difference between modes. Adjacent modes differ by only one note. The modes proceed from sharper (Lydian) to flatter (Locrian).
@@ Line 211: / Line 211: @@
 |}
-[[Porcupine]] aka Triyo has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2 using [[ups and downs notation]]. The enharmonic interval is v<sup>3</sup>A1. Because the generator is a 2nd, the genchain resembles the scale.
+[[Porcupine]] aka Triyo has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2 using [[ups and downs notation]]. The [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is v<sup>3</sup>A1. Because the generator is a 2nd, the genchain resembles the scale.
 {| class="wikitable"
@@ Line 265: / Line 265: @@
 |}
-[[Sensi]] aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^<sup>3</sup>d4 and a v<sup>4</sup>A3, and the enharmonic interval is ^<sup>7</sup>dd2.
+[[Sensi]] aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^<sup>3</sup>d4 and a v<sup>4</sup>A3, and the [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is ^<sup>7</sup>dd2.
 {| class="wikitable"
 |+ Sensi[8]/Sepgu[8] modes
@@ Line 500: / Line 500: @@
 == 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 844: / Line 844: @@
 |vF# * vG# vD# vA# * vB#
 |}
+== Generalization to temperament-agnostic MOS scales ==
+[[:Category:Abstract MOS patterns|Abstract MOS patterns]] like 5L 3s are not specific temperaments in which specific commas vanish. Thus there are no ratios other than the octave 2/1 (or more generally the equave 3/1 or whatever). Genchain mode numbers can be applied to these patterns. For example, 5L 3s has a generator in the 450-480¢ range. The "[8]" is redundant, so we drop it to get
+* 1st 5L 3s = LLsLLsLs
+* 2nd 5L 3s = LLsLsLLs
+* 3rd 5L 3s = LsLLsLLs
+* etc.
+The modes of the sister MOS 3L 5s are the same, just exchange L and s:
+* 1st 3L 5s = ssLssLsL
+* 2nd 3L 5s = ssLsLssL
+* 3rd 3L 5s = sLssLssL
+* etc.
+For a MOS pattern with a fifth-sized generator, the fifth is still prioritized over the fourth. Otherwise the generator is the mingen.
 == Rationale ==