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Kite's Genchain mode numbering: Difference between revisions

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|1st Triyo[7]
|1st Triyo[7]
| | ssss ssL
| | ssss ssL
| | C Dv Eb^ F Gv Ab^ Bb C
| | C vD ^Eb F vG ^Ab Bb C
| | <u>'''C'''</u> Dv Eb^ F Gv Ab^ Bb
| | <u>'''C'''</u> vD ^Eb F vG ^Ab Bb
|-
|-
| | 2nd Porcupine[7]
| | 2nd Porcupine[7]
|2nd Triyo[7]
|2nd Triyo[7]
| | ssss sLs
| | ssss sLs
| | C Dv Eb^ F Gv Ab^ Bb^ C
| | C vD ^Eb F vG ^Ab ^Bb C
| | Bb^ <u>'''C'''</u> Dv Eb^ F Gv Ab^
| | ^Bb <u>'''C'''</u> vD ^Eb F vG ^Ab
|-
|-
| | 3rd Porcupine[7]
| | 3rd Porcupine[7]
|3rd Triyo[7]
|3rd Triyo[7]
| | ssss Lss
| | ssss Lss
| | C Dv Eb^ F Gv Av Bb^ C
| | C vD ^Eb F vG vA ^Bb C
| | Av Bb^ <u>'''C'''</u> Dv Eb^ F Gv
| | vA ^Bb <u>'''C'''</u> vD ^Eb F vG
|-
|-
| | 4th Porcupine[7]
| | 4th Porcupine[7]
|4th Triyo[7]
|4th Triyo[7]
| | sssL sss
| | sssL sss
| | C Dv Eb^ F G Av Bb^ C
| | C vD ^Eb F G vA ^Bb C
| | G Av Bb^ <u>'''C'''</u> Dv Eb^ F
| | G vA ^Bb <u>'''C'''</u> vD ^Eb F
|-
|-
| | 5th Porcupine[7]
| | 5th Porcupine[7]
|5th Triyo[7]
|5th Triyo[7]
| | ssLs sss
| | ssLs sss
| | C Dv Eb^ F^ G Av Bb^ C
| | C vD ^Eb ^F G vA ^Bb C
| style="text-align:center;" | F^ G Av Bb^ <u>'''C'''</u> Dv Eb^
| style="text-align:center;" | ^F G vA ^Bb <u>'''C'''</u> vD ^Eb
|-
|-
| | 6th Porcupine[7]
| | 6th Porcupine[7]
|6th Triyo[7]
|6th Triyo[7]
| | sLss sss
| | sLss sss
| | C Dv Ev F^ G Av Bb^ C
| | C vD vE ^F G vA ^Bb C
| | Ev F^ G Av Bb^ <u>'''C'''</u> Dv
| | vE ^F G vA ^Bb <u>'''C'''</u> vD
|-
|-
| | 7th Porcupine[7]
| | 7th Porcupine[7]
|7th Triyo[7]
|7th Triyo[7]
| | Lsss sss
| | Lsss sss
| | C D Ev F^ G Av Bb^ C
| | C D vE ^F G vA ^Bb C
| | D Ev F^ G Av Bb^ <u>'''C'''</u>
| | D vE ^F G vA ^Bb <u>'''C'''</u>
|}
|}
'''[[Sensi]] aka Sepgu''' has pergen (P8, WWP5/7). 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[8] modes:
'''[[Sensi]] aka Sepgu''' has pergen (P8, WWP5/7). 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[8] modes:
Line 429: Line 429:
|4th Triyo[7] #2
|4th Triyo[7] #2
|LsmL mmm
|LsmL mmm
|C D Eb^ F G Av Bb^ C
|C D ^Eb F G vA ^Bb C
|D * * G Av Bb^ <u>'''C'''</u> * Eb^ F
|D * * G vA ^Bb <u>'''C'''</u> * ^Eb F
|-
|-
|4th Porcupine[7] #2 b6
|4th Porcupine[7] #2 b6
|4th Triyo[7] #2 b6
|4th Triyo[7] #2 b6
|LsmL sLm
|LsmL sLm
|C D Eb^ F G Ab^ Bb^ C
|C D ^Eb F G ^Ab ^Bb C
|D * * G * Bb^ <u>'''C'''</u> * Eb^ F * Ab^
|D * * G * ^Bb <u>'''C'''</u> * ^Eb F* ^Ab
|-
|-
| | 4th Porcupine[7] b6
| | 4th Porcupine[7] b6
| |4th Triyo[7] b6
| |4th Triyo[7] b6
| | mmmL sLm
| | mmmL sLm
| | C Dv Eb^ F G Ab^ Bb^ C
| | C vD ^Eb F G ^Ab ^Bb C
| | G * Bb^ <u>'''C'''</u> Dv Eb^ F * Ab^
| | G * ^Bb <u>'''C'''</u> vD ^Eb F * ^Ab
|-
|-
|4th Porcupine[7] b6 b7
|4th Porcupine[7] b6 b7
|4th Triyo[7] b6 b7
|4th Triyo[7] b6 b7
|mmmL smL
|mmmL smL
|C Dv Eb^ F G Ab^ Bb C
|C vD ^Eb F G ^Ab Bb C
|G * * <u>'''C'''</u> Dv Eb^ F * Ab^ Bb
|G * * <u>'''C'''</u> vD ^Eb F * ^Ab Bb
|-
|-
| |5th Porcupine[7] #2
| |5th Porcupine[7] #2
| |5th Triyo[7] #2
| |5th Triyo[7] #2
| | LsLm mmm
| | LsLm mmm
| | C D Eb^ F^ G Av Bb^ C
| | C D ^Eb ^F G vA ^Bb C
| | D * F^ G Av Bb^ <u>'''C'''</u> * Eb^
| | D * ^F G vA ^Bb <u>'''C'''</u> * ^Eb
|-
|-
| | 6th Porcupine[7] b4
| | 6th Porcupine[7] b4
| |6th Triyo[7] b4
| |6th Triyo[7] b4
| | mLsL mmm
| | mLsL mmm
| | C Dv Ev F G Av Bb^ C
| | C vD vE F G vA ^Bb C
| | Ev * G Av Bb^ <u>'''C'''</u> Dv * F
| | vE * G vA ^Bb <u>'''C'''</u> vD * F
|-
|-
| |7th Porcupine[7] #6 #7
| |7th Porcupine[7] #6 #7
| |7th Triyo[7] #6 #7
| |7th Triyo[7] #6 #7
| |Lmmm Lms
| |Lmmm Lms
| | C D Ev F^ G A Bv C
| | C D vE ^F G A vB C
| | A Bv * D Ev F^ G * * <u>'''C'''</u>
| | A vB * D vE ^F G * * <u>'''C'''</u>
|-
|-
|7th Porcupine[7] #7
|7th Porcupine[7] #7
|7th Triyo[7] #7
|7th Triyo[7] #7
|Lmmm mLs
|Lmmm mLs
|C D Ev F^ G Av Bv C
|C D vE ^F G vA vB C
|Bv * D Ev F^ G Av * <u>'''C'''</u>
|vB * D vE ^F G vA * <u>'''C'''</u>
|-
|-
|7th Porcupine[7] b4 #7
|7th Porcupine[7] b4 #7
|7th Triyo[7] b4 #7
|7th Triyo[7] b4 #7
|LmsL mLs
|LmsL mLs
|C D Ev F G Av Bv C
|C D vE F G vA vB C
|Bv * D Ev * G Av * <u>'''C'''</u> * * F
|vB * D vE * G vA * <u>'''C'''</u> * * F
|-
|-
| | 7th Porcupine[7] b4
| | 7th Porcupine[7] b4
| |7th Triyo[7] b4
| |7th Triyo[7] b4
| | LmsL mmm
| | LmsL mmm
| | C D Ev F G Av Bb^ C
| | C D vE F G vA ^Bb C
| | D Ev * G Av Bb^ <u>'''C'''</u> * * F
| | D vE * G vA ^Bb <u>'''C'''</u> * * F
|}
|}
=Temperaments with split octaves=
=Temperaments with split octaves=
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| | 1st Sagugu[2x5]
| | 1st Sagugu[2x5]
| | ssssL-ssssL
| | ssssL-ssssL
| | C C#v D D#v E F#v G G#v A A#v C
| | C vC# D vD# E vF# G vG# A vA# C
| | <u>'''C'''</u> G D A E
| | <u>'''C'''</u> G D A E
| | F#v C#v G#v D#v A#v
| | vF# vC# vG# vD# vA#
|-
|-
| | 2nd Srutal[2x5]
| | 2nd Srutal[2x5]
| | 2nd Sagugu[2x5]
| | 2nd Sagugu[2x5]
| | sssLs-sssLs
| | sssLs-sssLs
| | C C#v D D#v F F#v G G#v A Bv C
| | C vC# D vD# F vF# G vG# A vB C
| | F <u>'''C'''</u> G D A
| | F <u>'''C'''</u> G D A
| | Bv F#v C#v G#v D#v
| | vB vF# vC# vG# vD#
|-
|-
| | 3rd Srutal[2x5]
| | 3rd Srutal[2x5]
| | 3rd Sagugu[2x5]
| | 3rd Sagugu[2x5]
| | ssLss-ssLss
| | ssLss-ssLss
| | C C#v D Ev F F#v G G#v Bb Bv C
| | C vC# D vE F vF# G vG# Bb vB C
| | Bb F <u>'''C'''</u> G D
| | Bb F <u>'''C'''</u> G D
| | Ev Bv F#v C#v G#v
| | vE vB vF# vC# vG#
|-
|-
| | 4th Srutal[2x5]
| | 4th Srutal[2x5]
| | 4th Sagugu[2x5]
| | 4th Sagugu[2x5]
| | sLsss-sLsss
| | sLsss-sLsss
| | C C#v Eb Ev F F#v G Av Bb Bv C
| | C vC# Eb vE F vF# G vA Bb vB C
| | Eb Bb F <u>'''C'''</u> G
| | Eb Bb F <u>'''C'''</u> G
| | Av Ev Bv F#v C#v
| | vA vE vB vF# vC#
|-
|-
| | 5th Srutal[2x5]
| | 5th Srutal[2x5]
| | 5th Sagugu[2x5]
| | 5th Sagugu[2x5]
| | Lssss-Lssss
| | Lssss-Lssss
| | C Dv Eb Ev F F#v Ab Av Bb Bv C
| | C vD Eb vE F vF# Ab vA Bb vB C
| | Ab Eb Bb F <u>'''C'''</u>
| | Ab Eb Bb F <u>'''C'''</u>
| | Dv Av Ev Bv F#v
| | vD vA vE vB vF#
|}
|}


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[2x5] 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[2x5] would be C vC# D vD# E, just like the first half of the scale.


'''[[Augmented family|Augmented]] aka Trigu''' has a third-8ve period of ~5/4. The generator is ~3/2, which is equivalent to ~6/5. It could be thought of as ~16/15, but that would reverse the genchain direction and change all the mode numbers. The ~16/15 generator is not used, even though it is smaller, so that the genchain direction matches that of the pergen, which is (P8/3, P5).
'''[[Augmented family|Augmented]] aka Trigu''' has a third-8ve period of ~5/4. The generator is ~3/2, which is equivalent to ~6/5. It could be thought of as ~16/15, but that would reverse the genchain direction and change all the mode numbers. The ~16/15 generator is not used, even though it is smaller, so that the genchain direction matches that of the pergen, which is (P8/3, P5).
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| | 1st Trigu[3x3]
| | 1st Trigu[3x3]
| | Lss-Lss-Lss
| | Lss-Lss-Lss
| | C D Eb^ Ev F#v G Ab^ Bb^ Bv C
| | C D ^Eb vE vF# G ^Ab ^Bb vB C
| style="text-align:center;" | <u>'''C'''</u> G D
| style="text-align:center;" | <u>'''C'''</u> G D
| | Ev Bv F#v
| | vE vB vF#
| | Ab^ Eb^ Bb^
| | ^Ab ^Eb ^Bb
|-
|-
| | 2nd Augmented[3x3]
| | 2nd Augmented[3x3]
|2nd Trigu[3x3]
|2nd Trigu[3x3]
| | sLs-sLs-sLs
| | sLs-sLs-sLs
| | C Db^ Eb^ Ev F G Ab^ Av Bv C
| | C ^Db ^Eb vE F G ^Ab vA vB C
| style="text-align:center;" | F <u>'''C'''</u> G
| style="text-align:center;" | F <u>'''C'''</u> G
| | Av Ev Bv
| | vA vE vB
| | Db^ Ab^ Eb^
| | ^Db ^Ab ^Eb
|-
|-
|3rd Augmented[3x3]
|3rd Augmented[3x3]
|3rd Trigu[3x3]
|3rd Trigu[3x3]
|ssL-ssL-ssL
|ssL-ssL-ssL
|C Db^ Eb^ Ev F G Ab^ Av Bv C
|C ^Db vD vE F ^Gb ^Ab vA Bb C
|Bb F <u>'''C'''</u>  
|Bb F <u>'''C'''</u>  
|Dv Av Ev
|vD vA vE
|Gb^ Db^ Ab^
|^Gb ^Db ^Ab
|}
|}


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| | 1st Quadgu[4x2]
| | 1st Quadgu[4x2]
| | sL-sL-sL-sL
| | sL-sL-sL-sL
| | C C#^^ Ebv E^ F#^^ G A^ Bbv C
| | C ^^C# vEb ^E ^^F# G ^A vBb C
| style="text-align:center;" | <u>'''C'''</u> G
| style="text-align:center;" | <u>'''C'''</u> G
| | Ebv Bbv
| | vEb vBb
| | F#^^ C#^^
| | ^^F# ^^C#
| | A^ E^
| | ^A ^E
|-
|-
| | 2nd Diminished[4x2]
| | 2nd Diminished[4x2]
|2nd Quadgu[4x2]
|2nd Quadgu[4x2]
| | Ls-Ls-Ls-Ls
| | Ls-Ls-Ls-Ls
| | C D^ Ebv F F#^^ Abv A^ B^^ C
| | C ^D vEb F ^^F# vAb ^A ^^B C
| style="text-align:center;" | F <u>'''C'''</u>
| style="text-align:center;" | F <u>'''C'''</u>
| | Abv Ebv
| | vAb vEb
| |B^^ F#^^
| |^^B ^^F#
| | D^ A^
| | ^D ^A
|}
|}


Using ~25/24 as the generator yields the same scales and mode numbers. 1st Diminished[4x2] would have genchains  C – C#^^, Ebv – E^, F#^^ – G and A^ – Bbv, just like the scale.
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 5-edo+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. Both Blackwood modes, using ups and downs:
[[Blackwood|'''Blackwood''']] '''aka 5-edo+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. Both Blackwood modes, using ups and downs:
Line 625: Line 625:
|1st 5edo+ya[5x2]
|1st 5edo+ya[5x2]
| | Ls-Ls-Ls-Ls-Ls
| | Ls-Ls-Ls-Ls-Ls
| | C C#v D Ev F F#v G Av A Bv C
| | C vC# D vE F vF# G vA A vB C
| style="text-align:center;" | <u>'''C'''</u>-Ev, D-F#v, F-Av, G-Bv, A-C#v
| style="text-align:center;" | <u>'''C'''</u>-vE, D-vF#, F-vA, G-vB, A-vC#
|-
|-
| | 2nd Blackwood[5x2]
| | 2nd Blackwood[5x2]
|2nd 5-edo+ya[5x2]
|2nd 5-edo+ya[5x2]
| | sL-sL-sL-sL-sL
| | sL-sL-sL-sL-sL
| | C C^ D Eb^ F F^ G Ab^ A Bb^ C
| | C ^C D ^Eb F ^F G ^Ab A ^Bb C
| style="text-align:center;" | Ab^-<u>'''C'''</u>, Bb^-D, C^-F, Eb^-G, F^-A
| style="text-align:center;" | ^Ab-<u>'''C'''</u>, ^Bb-D, ^C-F, ^Eb-G, ^F-A
|}
|}


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| | 1st Sagugu[2x5]
| | 1st Sagugu[2x5]
| | ssssL-ssssL
| | ssssL-ssssL
| | C C#v D D#v E F#v G G#v A A#v C
| | C vC# D vD# E vF# G vG# A vA# C
| | <u>'''C'''</u> G D A E
| | <u>'''C'''</u> G D A E
| | F#v C#v G#v D#v A#v
| | vF# vC# vG# vD# vA#
|-
|-
|1st Srutal[2x5] b2 b5
|1st Srutal[2x5] b2 b5
|1st Sagugu[2x5] b2 b5
|1st Sagugu[2x5] b2 b5
|sLmmL-sLmmL
|sLmmL-sLmmL
|C B#v D D#v E F#v F# G#v A A#v C
|C vB# D vD# E vF# F# vG# A vA# C
|<u>'''C'''</u> * D A E * F#
|<u>'''C'''</u> * D A E * F#
|F#v * G#v D#v A#v * B#v
|vF# * vG# vD# vA# * vB#
|-
|-
|1st Srutal[2x5] b2
|1st Srutal[2x5] b2
|1st Sagugu[2x5] b2
|1st Sagugu[2x5] b2
|sLmmL-mmmmL
|sLmmL-mmmmL
|C B#v D D#v E F#v G G#v A A#v C
|C vB# D vD# E vF# G vG# A vA# C
|<u>'''C'''</u> G D A E
|<u>'''C'''</u> G D A E
|F#v * G#v D#v A#v * B#v
|vF# * vG# vD# vA# * vB#
|-
| | 1st Srutal[2x5]
| | 1st Sagugu[2x5]
| | ssssL-ssssL
| | C C#v D Ev E F#v G G#v Bb A#v C
| | Bb * <u>'''C'''</u> G D * E
| | Ev * F#v C#v G#v *A#v
|}
|}


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