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. Multiple genchains occur because a rank-2 genchain is actually a 2-dimensional lattice with vertical periods and horizontal generators that's been octave-reduced,. For example, here's Meantone's non-octave-reduced lattice, with vertical octaves and horizontal fifths:
+If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is actually a 2-dimensional lattice with vertical periods and horizontal generators that's been octave-reduced. For example, here's Meantone's non-octave-reduced lattice, with vertical octaves and horizontal fifths:
 F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5
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 F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3
-Because the period is an octave, the genweb octave-reduces to a single horizontal genchain:
+Because the period is an octave, the lattice octave-reduces to a single horizontal genchain:
 F --- C --- G --- D --- A --- E --- B
-But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, the unreduced lattice of [[Diaschismic_family|Diaschismatic]] [10] might look like this:
+But if the period is a half-octave, the lattice has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, the unreduced lattice of [[Diaschismic_family|Shrutal]] [10] might look like this:
-F#^3 -- C#^4 -- G#^4 -- D#^5 -- A#^5
+F#^3 -- C#v4 -- G#v4 -- D#v5 -- A#v5
 C3 ----- G3 ----- D4 ----- A4 ----- E5
-F#^2 -- C#^3 -- G#^3 -- D#^4 -- A#^4
+F#v2 -- C#v3 -- G#v3 -- D#v4 -- A#v4
 C2 ----- G2 ------ D3 ----- A3 ----- E3
-F#^1 -- C#^2 -- G#^2 -- D#^3 -- A#^3
+F#v1 -- C#v2 -- G#v2 -- D#v3 -- A#v3
 C1 ----- G1 ------ D2 ----- A2 ----- E2
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 which octave-reduces to two genchains:
-F#^ -- C#^ -- G#^ -- D#^ -- A#^
+F#v -- C#v -- G#v -- D#v -- A#v
 C ----- G ------ D ----- A ----- E
-Moving from C to F#^ moves up or down a half-octave. See the [[pergen]] page for an explanation of the notation. It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.
+Moving from C to F#v moves up or down a half-octave. See the [[pergen]] page for an explanation of the notation. It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.
 Gbv -- Dbv -- Abv -- Ebv -- Bbv
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 Gbv -- Dbv -- Abv -- Ebv -- Bbv
-In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.
+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 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. A generator plus or minus a period is still a generator. Diaschismatic's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:
+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. A generator plus or minus a period is still a generator. Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:
-F#^ -- G --- G#^ -- A --- A#^
+F#v -- G --- G#v -- A --- A#v
-C --- C#^ -- D --- D#^ -- E
+C --- C#v -- D --- D#v -- E
-Another alternative is to use [[Kite's_color_notation|color notation]]. The diaschismatic comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:
-yF# --- yC# --- yG# --- yD# --- yA#
-wC ---- wG ---- wD ---- wA ---- wE
-As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period:
-wC ---- wG ---- wD ----- wA ---- wE
-gGb --- gDb --- gAb --- gEb --- gBb
 All five Srutal [10] modes, using ups and downs. Every other scale note has an up.
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 |-
 ! | scale name
-! | sL pattern
+! | Ls pattern
 ! | example in C
 ! | 1st genchain
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 | | 1st Srutal [10]
 | | ssssL-ssssL
-| | C C^ D D^ E F^ G G^ A A^ C
+| | C C#v D D#v E F#v G G#v A A#v C
 | | <u>'''C'''</u> G D A E
-| | F^ C^ G^ D^ A^
+| | F#v C#v G#v D#v A#v
 |-
 | | 2nd Srutal [10]
 | | sssLs-sssLs
-| | C C^ D D^ F F^ G G^ A Bb^ C
+| | C C#v D D#v F F#v G G#v A Bv C
 | | F <u>'''C'''</u> G D A
-| | Bb^ F^ C^ G^ D^
+| | Bv F#v C#v G#v D#v
 |-
 | | 3rd Srutal [10]
 | | ssLss-ssLss
-| | C C^ D Eb^ F F^ G G^ Bb Bb^ C
+| | C C#v D Ev F F#v G G#v Bb Bv C
 | | Bb F <u>'''C'''</u> G D
-| | Eb^ Bb^ F^ C^ G^
+| | Ev Bv F#v C#v G#v
 |-
 | | 4th Srutal [10]
 | | sLsss-sLsss
-| | C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C
+| | C C#v Eb Ev F F#v G Av Bb Bv C
 | | Eb Bb F <u>'''C'''</u> G
-| | Ab^ Eb^ Bb^ F^ C^
+| | Av Ev Bv F#v C#v
 |-
 | | 5th Srutal [10]
 | | Lssss-Lssss
-| | C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C
+| | C Dv Eb Ev F F#v Ab Av Bb Bv C
 | | Ab Eb Bb F <u>'''C'''</u>
-| | Db^ Ab^ Eb^ Bb^ F^
+| | Dv Av Ev Bv F#v
 |}