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Kite's Genchain mode numbering: Difference between revisions

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The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below in "Rationale"). '''<u>Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]</u>.''' There are several disadvantages of only using chroma-positive generators. See the critique of UDP at the bottom of this page.
The octave inverse of a generator is also a generator. To avoid ambiguity in mode numbers, the smaller of the two generators is chosen. An exception is made for 3/2, which is preferred over 4/3 for historical reasons (see below in "Rationale"). '''<u>Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]</u>.''' There are several disadvantages of only using chroma-positive generators. See the critique of UDP at the end of this article.


Pentatonic meantone scales:
Pentatonic meantone scales:
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|-
|-
! | scale name
! | scale name
! | Ls pattern (assumes<br>~3/2 &lt; 700¢)
! | Ls pattern (assumes<br>a generator &lt; 700¢)
! | example in C
! | example in C
! | genchain
! | genchain
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| |  
| |  
|}
|}
'''[[Sensi]][8] aka Sepgu''' has a ~9/7 generator. The [[pergen]] is (P8, WWP5/7). Sensi[8] modes in 19edo (gen = 7\19, L = 3\19, s = 2\19):
'''[[Porcupine]] aka Triyo''' has a [[pergen]] of (P8, P4/3) and a generator of vM2 = ~10/9. Porcupine[7] modes, using [[Ups and Downs Notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | scale name
! | scale name
!color name
![[Color notation/Temperament Names|color name]]
! | Ls pattern
! | example in C
! | genchain
|-
| | 1st Sensi[8]
|1st Sepgu[8]
| | ssL ssL sL
| | C Db D# E# F# G A Bb C
| | <u>'''C'''</u> E# A Db F# Bb D# G
|-
| | 2nd Sensi[8]
|2nd Sepgu[8]
| | ssL sL ssL
| | C Db D# E# F# G# A Bb C
| | G# <u>'''C'''</u> E# A Db F# Bb D#
|-
| | 3rd Sensi[8]
|3rd Sepgu[8]
| | sL ssL ssL
| | C Db Eb E# F# G# A Bb C
| | Eb G# <u>'''C'''</u> E# A Db F# Bb
|-
| | 4th Sensi[8]
|4th Sepgu[8]
| | sL ssL sL s
| | C Db Eb E# F# G# A B C
| | B Eb G# <u>'''C'''</u> E# A Db F#
|-
| | 5th Sensi[8]
|5th Sepgu[8]
| | sL sL ssL s
| | C Db Eb E# Gb G# A B C
| | Gb B Eb G# <u>'''C'''</u> E# A Db
|-
| | 6th Sensi[8]
|6th Sepgu[8]
| | Lss Lss Ls
| | C D Eb E# Gb G# A B C
| | D Gb B Eb G# <u>'''C'''</u> E# A
|-
| | 7th Sensi[8]
|7th Sepgu[8]
| | Lss Ls Lss
| | C D Eb E# Gb G# A# B C
| | A# D Gb B Eb G# <u>'''C'''</u> E#
|-
| | 8th Sensi[8]
|8th Sepgu[8]
| | Ls Lss Lss
| | C D Eb F Gb G# A# B C
| | F A# D Gb B Eb G# <u>'''C'''</u>
|}
These scales might seem much more random than the meantone ones. They are written out using the standard 19edo notation: C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C
 
'''[[Porcupine]] aka Triyo''' has a ~10/9 generator and a pergen of (P8, P4/3). Porcupine[7] modes in 22edo (gen = 3\22, L = 4\22, s = 3\22), using [[Ups and Downs Notation|ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.
 
{| class="wikitable"
|-
! | scale name
!color name
! | Ls pattern
! | Ls pattern
! | example in C
! | example in C
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| | C D Ev F^ G Av Bb^ C
| | C D Ev F^ G Av Bb^ C
| | D Ev F^ G Av Bb^ <u>'''C'''</u>
| | D Ev F^ G Av Bb^ <u>'''C'''</u>
|}
'''[[Sensi]] aka Sepgu''' has pergen (P8, WWP5/7). The ~9/7 generator is a ^<sup>3</sup>d4 = v<sup>4</sup>A3, and the enharmonic is a ^<sup>7</sup>dd2. Sensi[8] modes:
{| class="wikitable"
|-
! | scale name
![[Color notation/Temperament Names|color name]]
! | Ls pattern
! | example in C
! | genchain
|-
| | 1st Sensi[8]
|1st Sepgu[8]
| | ssL ssL sL
| | C ^^Db ^<sup>4</sup>Ebb ^<sup>3</sup>Fb vvF#    G    vA ^Bb C
| | <u>'''C'''</u> ^<sup>3</sup>Fb vA ^^Db vvF# ^Bb ^<sup>4</sup>Ebb G
|-
| | 2nd Sensi[8]
|2nd Sepgu[8]
| | ssL sL ssL
| | C ^^Db ^<sup>4</sup>Ebb ^<sup>3</sup>Fb vvF# v<sup>3</sup>G#  vA  ^Bb C
| | v<sup>3</sup>G# <u>'''C'''</u> ^<sup>3</sup>Fb vA ^^Db vvF# ^Bb ^<sup>4</sup>Ebb
|-
| | 3rd Sensi[8]
|3rd Sepgu[8]
| | sL ssL ssL
| | C ^^Db  ^Eb  ^<sup>3</sup>Fb vvF# v<sup>3</sup>G#  vA  ^Bb C
| | ^Eb v<sup>3</sup>G# <u>'''C'''</u> ^<sup>3</sup>Fb vA ^^Db vvF# ^Bb
|-
| | 4th Sensi[8]
|4th Sepgu[8]
| | sL ssL sL s
| | C ^^Db  ^Eb  ^<sup>3</sup>Fb vvF# v<sup>3</sup>G#  vA  vvB C
| | vvB ^Eb v<sup>3</sup>G# <u>'''C'''</u> ^<sup>3</sup>Fb vA ^^Db vvF#
|-
| | 5th Sensi[8]
|5th Sepgu[8]
| | sL sL ssL s
| | C ^^Db  ^Eb  ^<sup>3</sup>Fb ^^Gb v<sup>3</sup>G#  vA  vvB C
| | ^^Gb vvB ^Eb v<sup>3</sup>G# <u>'''C'''</u> ^<sup>3</sup>Fb vA ^^Db
|-
| | 6th Sensi[8]
|6th Sepgu[8]
| | Lss Lss Ls
| | C  vD    ^Eb  ^<sup>3</sup>Fb ^^Gb v<sup>3</sup>G#  vA  vvB C
| | vD ^^Gb vvB ^Eb v<sup>3</sup>G# <u>'''C'''</u> ^<sup>3</sup>Fb vA
|-
| | 7th Sensi[8]
|7th Sepgu[8]
| | Lss Ls Lss
| | C  vD    ^Eb  ^<sup>3</sup>Fb ^^Gb v<sup>3</sup>G# v<sup>4</sup>A# vvB C
| | v<sup>4</sup>A# vD ^^Gb vvB ^Eb v<sup>3</sup>G# <u>'''C'''</u> ^<sup>3</sup>Fb
|-
| | 8th Sensi[8]
|8th Sepgu[8]
| | Ls Lss Lss
| | C  vD    ^Eb    F  ^^Gb v<sup>3</sup>G# v<sup>4</sup>A# vvB C
| | F v<sup>4</sup>A# vD ^^Gb vvB ^Eb v<sup>3</sup>G# <u>'''C'''</u>
|}
|}
=MODMOS scales=
=MODMOS scales=
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A MODMOS scale can have alternate names. The ascending melodic minor scale could also be called 2nd Meantone[7] b3 (major scale with a minor 3rd), or as 4th Meantone[7] #7 (dorian with a major 7th).  
A MODMOS scale can have alternate names. The ascending melodic minor scale could also be called 2nd Meantone[7] b3 (major scale with a minor 3rd), or as 4th Meantone[7] #7 (dorian with a major 7th).  


'''Meantone''' MODMOS scales, with alternate names included only if they don't have more alterations than the original:
'''Meantone''' MODMOS scales, with alternative names included only if they don't have more alterations than the original:


{| class="wikitable"
{| class="wikitable"
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|-
|-
! | scale name
! | scale name
! | color name
! | [[Color notation/Temperament Names|color name]]
! | LMs pattern
! | LMs pattern
! | example in C
! | example in C
! | genchain
! | genchain
|-
| |7th Porcupine[7] #6 #7
| |7th Triyo[7] #6 #7
| |Lmmm Lms
| | C D Ev F^ G A Bv C
| | A Bv * D Ev F^ G * * <u>'''C'''</u>
|-
|7th Porcupine[7] #7
|7th Triyo[7] #7
|Lmmm mLs
|C D Ev F^ G Av Bv C
|Bv * D Ev F^ G Av * <u>'''C'''</u>
|-
| |5th Porcupine[7] #2
| |5th Triyo[7] #2
| | LsLm mmm
| | C D Eb^ F^ G Av Bb^ C
| | D * F^ G Av Bb^ <u>'''C'''</u> * Eb^
|-
|-
|4th Porcupine[7] #2
|4th Porcupine[7] #2
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|D * * G Av Bb^ <u>'''C'''</u> * Eb^ F
|D * * G Av Bb^ <u>'''C'''</u> * Eb^ F
|-
|-
| | 7th Porcupine[7] b4
|4th Porcupine[7] #2 b6
| |7th Triyo[7] b4
|4th Triyo[7] #2 b6
| | LmsL mmm
|LsmL sLm
| | C D Ev F G Av Bb^ C
|C D Eb^ F G Ab^ Bb^ C
| | D Ev * G Av Bb^ <u>'''C'''</u> * * F
|D * * G * Bb^ <u>'''C'''</u> * Eb^ F * Ab^
|-
| | 6th Porcupine[7] b4
| |6th Triyo[7] b4
| | mLsL mmm
| | C Dv Ev F G Av Bb^ C
| | Ev * G Av Bb^ <u>'''C'''</u> Dv * F
|-
|-
| | 4th Porcupine[7] b6
| | 4th Porcupine[7] b6
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|C Dv Eb^ F G Ab^ Bb C
|C Dv Eb^ F G Ab^ Bb C
|G * * <u>'''C'''</u> Dv Eb^ F * Ab^ Bb
|G * * <u>'''C'''</u> Dv Eb^ F * Ab^ Bb
|-
| |5th Porcupine[7] #2
| |5th Triyo[7] #2
| | LsLm mmm
| | C D Eb^ F^ G Av Bb^ C
| | D * F^ G Av Bb^ <u>'''C'''</u> * Eb^
|-
| | 6th Porcupine[7] b4
| |6th Triyo[7] b4
| | mLsL mmm
| | C Dv Ev F G Av Bb^ C
| | Ev * G Av Bb^ <u>'''C'''</u> Dv * F
|-
| |7th Porcupine[7] #6 #7
| |7th Triyo[7] #6 #7
| |Lmmm Lms
| | C D Ev F^ G A Bv C
| | A Bv * D Ev F^ G * * <u>'''C'''</u>
|-
|7th Porcupine[7] #7
|7th Triyo[7] #7
|Lmmm mLs
|C D Ev F^ G Av Bv C
|Bv * D Ev F^ G Av * <u>'''C'''</u>
|-
|-
|7th Porcupine[7] b4 #7
|7th Porcupine[7] b4 #7
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|Bv * D Ev * G Av * <u>'''C'''</u> * * F
|Bv * D Ev * G Av * <u>'''C'''</u> * * F
|-
|-
|4th Porcupine[7] #2 b6
| | 7th Porcupine[7] b4
|4th Triyo[7] #2 b6
| |7th Triyo[7] b4
|LsmL sLm
| | LmsL mmm
|C D Eb^ F G Ab^ Bb^ C
| | C D Ev F G Av Bb^ C
|D * * G * Bb^ <u>'''C'''</u> * Eb^ F * Ab^
| | D Ev * G Av Bb^ <u>'''C'''</u> * * F
|}
|}
=Temperaments with split octaves=
=Temperaments with split octaves=


If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel. 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.
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.


'''[[Srutal]] aka Sagugu''' has a half-8ve period. All five Srutal[10] modes, using ups and downs. Every other scale note has a down.
'''[[Srutal]] aka Diaschismatic aka Sagugu''' has a half-8ve period. All five Srutal[10] modes. Every other scale note has a down.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | scale name
! | scale name
! | color name
! | [[Color notation/Temperament Names|color name]]
! | Ls pattern
! | Ls pattern
! | example in C
! | example in C
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|}
|}


The generator is written as a 5th. If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. Srutal's generator could also be thought of as ~16/15, because that would still create the same mode numbers and thus the same scale names. The first genchain of 1st Srutal[10] would be C C#v D D#v E, just like the first half of the scale.
Srutal's period is written as a vA4, but could instead be written as an ^d5. The generator is written as a P5. If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. The generator could instead be written as ~16/15 (3/2 minus a period), because that would still create the same mode numbers and thus the same scale names. The first genchain of 1st Srutal[10] would be C C#v D D#v E, just like the first half of the scale.


'''[[Octatonic_scale|Diminished]] aka Quadgu''' has a quarter-8ve period. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24 or even ~9/5. The Diminished[8] scale has only two modes, because there are four very short genchains of only two notes. The comma is fifthward, so the 5th is flattened, and the 32/27 minor 3rd is > 300¢. Therefore the 300¢ period is narrower than a m3, and must be a vm3. The four genchains:
'''[[Octatonic_scale|Diminished]] aka Quadgu''' has a quarter-8ve period. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24 or even ~9/5. The Diminished[8] scale has only two modes, because there are four very short genchains of only two notes. The comma is fifthward, so the 5th is flattened, and the 32/27 minor 3rd is > 300¢. Therefore the 300¢ period is narrower than a m3, and must be a vm3. The four genchains:
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|-
|-
! | scale name
! | scale name
! | color name
! | [[Color notation/Temperament Names|color name]]
! | sL pattern
! | sL pattern
! | example in C
! | example in C
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|-
|-
! | scale name
! | scale name
!color name
![[Color notation/Temperament Names|color name]]
! | sL pattern
! | sL pattern
! | example in C
! | example in C
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