Xenharmonic Wiki

Kite's Genchain mode numbering: Difference between revisions

← Older editNewer edit →
VisualWikitext
Spt3125 (talk | contribs)
autopatrolled, Sysops
523 edits
m category maintenance, clean up formatting
TallKite (talk | contribs)
autopatrolled, Sysops
3,990 edits
major update!
Line 6: Line 6:
[[MOSScales|MOS scales]] 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.
[[MOSScales|MOS scales]] 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|Meantone]] [7], using ~3/2 as the generator:
For example, here are all the modes of [[Meantone|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.


{| class="wikitable"
{| class="wikitable"
Line 12: Line 12:
! | old scale name
! | old scale name
! | new scale name
! | new scale name
! | sL pattern
! | Ls pattern
! | example on white keys
! | example on white keys
! | genchain
! | genchain
Line 66: Line 66:
! | old scale name
! | old scale name
! | new scale name
! | new scale name
! | sL pattern
! | Ls pattern
! | example in C
! | example in C
! | ------------------- genchain ---------------
! | ------------------- genchain ---------------
Line 113: Line 113:
|}
|}


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-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-positive</u>.''' There are several disadvantages of only using chroma-positive generators. See the critique of UDP at the bottom of this page.


Pentatonic meantone scales:
Pentatonic meantone scales:
Line 121: Line 121:
! | old scale name
! | old scale name
! | new scale name
! | new scale name
! | sL pattern
! | Ls pattern
! | example in C
! | example in C
! | --------- genchain -------
! | --------- genchain -------
Line 156: Line 156:
|}
|}


Chromatic meantone scales.
12-note Meantone scales. If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | scale name
! | scale name
! | sL pattern (assumes<br>~3/2 &lt; 700¢)
! | Ls pattern (assumes<br>~3/2 &lt; 700¢)
! | example in C
! | example in C
! | genchain
! | genchain
Line 205: Line 205:
| |  
| |  
|}
|}
If the fifth were larger than 700¢, which would be the case for Superpyth[12], L and s would be interchanged.
[[Sensi]] [8] modes in 19edo (generator = ~9/7 = 7\19, L = 3\19, s = 2\19) The [[pergen]] is (P8, WWP5/7).
 
[[Sensi]] [8] modes in 19edo (generator = 3rd = ~9/7 = 7\19, L = 3\19, s = 2\19)


{| class="wikitable"
{| class="wikitable"
|-
|-
! | scale name
! | scale name
! | sL pattern
! | Ls pattern
! | example in C
! | example in C
! | genchain
! | genchain
Line 256: Line 254:
| | F A# D Gb B Eb G# <u>'''C'''</u>
| | F A# D Gb B Eb G# <u>'''C'''</u>
|}
|}
The Sensi scales are written out using the standard heptatonic fifth-based 19edo notation:
These scales might seem much more random than the meantone ones. They are written out using the standard heptatonic fifth-based 19edo notation:  


C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C
Line 266: Line 264:
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.
1st Sensi[8] would be C D E F G Hb A B C, 2nd would be C D E F G H A B C, etc.


[[Porcupine]] [7] modes in 22edo (generator = 2nd = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[ups and downs notation]]. Because the generator is a 2nd, the genchain resembles the scale.
[[Porcupine]] [7] modes in 22edo (generator = ~10/9 = 3\22, L = 4\22, s = 3\22), using [[Ups and Downs Notation|ups and downs notation]]. The pergen is (P8, P4/3). Because the generator is a 2nd, the genchain resembles the scale.


{| class="wikitable"
{| class="wikitable"
|-
|-
! | scale name
! | scale name
! | sL pattern
! | Ls pattern
! | example in C
! | example in C
! | genchain
! | genchain
Line 320: Line 318:
[[MODMOS scales]] are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The "#" symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G. A good alternative, especially for non-heptatonic and non-fifth-based scales, is to use + and - for forwards and backwards, as in 5th Meantone [7] +6 +7.
[[MODMOS scales]] are named as chromatic alterations of a MOS scale, similar to UDP notation. The ascending melodic minor scale is 5th Meantone [7] #6 #7. The "#" symbol means moved N steps forwards on the genchain, whether the generator is chroma-positive or not. This scale has the same name in 16edo, even though in 16edo, G# is actually flat of G. A good alternative, especially for non-heptatonic and non-fifth-based scales, is to use + and - for forwards and backwards, as in 5th Meantone [7] +6 +7.


MODMOS names are ambiguous. The ascending melodic minor scale could also be written as 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). Here are some Meantone MODMOS scales, with alternate names included only if they don't have more alterations than the original:


{| class="wikitable"
{| class="wikitable"
Line 328: Line 326:
! | genchain
! | genchain
! | new scale name
! | new scale name
! | sML pattern
! | LMs pattern
|-
|-
| | Harmonic minor
| | Harmonic minor
Line 342: Line 340:
| | LsLL LLs
| | LsLL LLs
|-
|-
| style="text-align:center;" | "
| style="text-align:center;" | (Major with b3)
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
Line 348: Line 346:
| style="text-align:center;" | "
| style="text-align:center;" | "
|-
|-
| style="text-align:center;" | "
| style="text-align:center;" | (Dorian with #7)
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
Line 360: Line 358:
| | MsLs sLs
| | MsLs sLs
|-
|-
| style="text-align:center;" | (Lydian with b3 b6)
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| | 1st Meantone [7] b3 b6
| | 1st Meantone [7] b3 b6 |=| "
|-
|-
| | Double harmonic major
| | Double harmonic major
Line 371: Line 369:
| | sLsM sLs
| | sLsM sLs
|-
|-
| style="text-align:center;" | "
| style="text-align:center;" | (Phrygian with #3 #7)
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
Line 389: Line 387:
| | sLsM sMM
| | sLsM sMM
|}
|}
As can be seen from the genchains, or from the sML patterns, the harmonic minor and the phrygian dominant are modes of each other, as are the double harmonic minor and the double harmonic major. Unfortunately the scale names do not indicate this.
As can be seen from the genchains, or from the LMs patterns, the harmonic minor and the phrygian dominant are modes of each other, as are the double harmonic minor and the double harmonic major. Unfortunately the scale names do not indicate this.


The advantage of ambiguous names is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from D dorian to D melodic minor. In this context, melodic minor might better be described as an altered dorian scale.
The advantage of ambiguous names is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from D dorian to D melodic minor. In this context, melodic minor might better be described as an altered dorian scale.
Line 427: Line 425:
7th Meantone [7] #2: C D Eb F Gb Ab Bb C
7th Meantone [7] #2: C D Eb F Gb Ab Bb C


=Fractional-octave periods=
=Temperaments with split octaves=


Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
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
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5
Line 437: Line 435:
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3


When the period is an octave, the genweb octave-reduces to a single horizontal genchain:
Because the period is an octave, the genweb octave-reduces to a single horizontal genchain:


F --- C --- G --- D --- A --- E --- B
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, [[Diaschismic_family|Srutal]] [10] might look like this:
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:


F^3 --- C^4 --- G^4 --- D^5 --- A^5
F#^3 -- C#^4 -- G#^4 -- D#^5 -- A#^5


C3 ---- G3 ----- D4 ---- A4 ---- E5
C3 ----- G3 ----- D4 ----- A4 ----- E5


F^2 --- C^3 --- G^3 --- D^4 --- A^4
F#^2 -- C#^3 -- G#^3 -- D#^4 -- A#^4


C2 ---- G2 ----- D3 ---- A3 ---- E3
C2 ----- G2 ------ D3 ----- A3 ----- E3


F^1 --- C^2 --- G^2 --- D^3 --- A^3
F#^1 -- C#^2 -- G#^2 -- D#^3 -- A#^3


C1 ---- G1 ----- D2 ---- A2 ---- E2
C1 ----- G1 ------ D2 ----- A2 ----- E2


which octave-reduces to two genchains:
which octave-reduces to two genchains:


F^ --- C^ --- G^ --- D^ --- A^
F#^ -- C#^ -- G#^ -- D#^ -- A#^
 
C ---- G ----- D ---- A ---- E
 
Moving up from C to F^ moves up a half-octave. [[Ups_and_Downs_Notation|Ups and downs]] are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of "up" here is "a half-octave minus a fourth", with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave.


Srutal is only compatible with even-numbered frameworks. In order to preserve the primary meaning of ups and downs, which is up or down one key or fret, this notation is limited to frameworks in which the 4th is one key less than half an octave. These are 10, 12, 14, 16 and 18b, but not 18, 20, 22, 24, etc. For those frameworks, use double-ups.
C ----- G ------ D ----- A ----- E


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#^ 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.


Gv --- Dv --- Av --- Ev --- Bv
Gbv -- Dbv -- Abv -- Ebv -- Bbv


C ----- G ----- D ---- A ---- E
C ------ G ------ D ----- A ----- E


It would also be valid to exchange the two rows:
It would also be valid to exchange the two rows:


C ----- G ----- D ---- A ---- E
C ------ G ------ D ----- A ----- E
 
Gv --- Dv --- Av --- Ev --- Bv


Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.
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 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.


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:
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:


F^ -- G --- G^ -- A --- A^
F#^ -- G --- G#^ -- A --- A#^


C --- C^ -- D --- D^ -- E
C --- C#^ -- D --- D#^ -- E


Another alternative is to use [[Kite's_color_notation|color notation]]. The srutal 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]:
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#
yF# --- yC# --- yG# --- yD# --- yA#
Retrieved from "https://en.xen.wiki/w/Kite%27s_Genchain_mode_numbering"