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[[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. 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|'''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"
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| |  
| |  
|}
|}
[[Sensi]] [8] modes in 19edo (generator = ~9/7 = 7\19, L = 3\19, s = 2\19) The [[pergen]] is (P8, WWP5/7).
'''[[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):


{| class="wikitable"
{| class="wikitable"
|-
|-
! | scale name
! | scale name
!color name
! | Ls pattern
! | Ls pattern
! | example in C
! | example in C
Line 215: Line 216:
|-
|-
| | 1st Sensi [8]
| | 1st Sensi [8]
|1st Sepgu [8]
| | ssL ssL sL
| | ssL ssL sL
| | C Db D# E# F# G A Bb C
| | C Db D# E# F# G A Bb C
Line 220: Line 222:
|-
|-
| | 2nd Sensi [8]
| | 2nd Sensi [8]
|2nd Sepgu [8]
| | ssL sL ssL
| | ssL sL ssL
| | C Db D# E# F# G# A Bb C
| | C Db D# E# F# G# A Bb C
Line 225: Line 228:
|-
|-
| | 3rd Sensi [8]
| | 3rd Sensi [8]
|3rd Sepgu [8]
| | sL ssL ssL
| | sL ssL ssL
| | C Db Eb E# F# G# A Bb C
| | C Db Eb E# F# G# A Bb C
Line 230: Line 234:
|-
|-
| | 4th Sensi [8]
| | 4th Sensi [8]
|4th Sepgu [8]
| | sL ssL sL s
| | sL ssL sL s
| | C Db Eb E# F# G# A B C
| | C Db Eb E# F# G# A B C
Line 235: Line 240:
|-
|-
| | 5th Sensi [8]
| | 5th Sensi [8]
|5th Sepgu [8]
| | sL sL ssL s
| | sL sL ssL s
| | C Db Eb E# Gb G# A B C
| | C Db Eb E# Gb G# A B C
Line 240: Line 246:
|-
|-
| | 6th Sensi [8]
| | 6th Sensi [8]
|6th Sepgu [8]
| | Lss Lss Ls
| | Lss Lss Ls
| | C D Eb E# Gb G# A B C
| | C D Eb E# Gb G# A B C
Line 245: Line 252:
|-
|-
| | 7th Sensi [8]
| | 7th Sensi [8]
|7th Sepgu [8]
| | Lss Ls Lss
| | Lss Ls Lss
| | C D Eb E# Gb G# A# B C
| | C D Eb E# Gb G# A# B C
Line 250: Line 258:
|-
|-
| | 8th Sensi [8]
| | 8th Sensi [8]
|8th Sepgu [8]
| | Ls Lss Lss
| | Ls Lss Lss
| | C D Eb F Gb G# A# B C
| | C D Eb F Gb G# A# B C
| | F A# D Gb B Eb G# <u>'''C'''</u>
| | 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 heptatonic fifth-based 19edo notation:  
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


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.
 
The modes would follow a more regular pattern if using octotonic fourth-based notation:
 
C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G# - Hb - H - H#/Ab - A - A#/Bb - B - B# - Cb -C
 
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 = ~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
!color name
! | Ls pattern
! | Ls pattern
! | example in C
! | example in C
Line 274: Line 276:
|-
|-
| | 1st Porcupine [7]
| | 1st Porcupine [7]
|1st Triyo [7]
| | ssss ssL
| | ssss ssL
| | C Dv Eb^ F Gv Ab^ Bb C
| | C Dv Eb^ F Gv Ab^ Bb C
Line 279: Line 282:
|-
|-
| | 2nd Porcupine [7]
| | 2nd Porcupine [7]
|2nd Triyo [7]
| | ssss sLs
| | ssss sLs
| | C Dv Eb^ F Gv Ab^ Bb^ C
| | C Dv Eb^ F Gv Ab^ Bb^ C
Line 284: Line 288:
|-
|-
| | 3rd Porcupine [7]
| | 3rd Porcupine [7]
|3rd Triyo [7]
| | ssss Lss
| | ssss Lss
| | C Dv Eb^ F Gv Av Bb^ C
| | C Dv Eb^ F Gv Av Bb^ C
Line 289: Line 294:
|-
|-
| | 4th Porcupine [7]
| | 4th Porcupine [7]
|4th Triyo [7]
| | sssL sss
| | sssL sss
| | C Dv Eb^ F G Av Bb^ C
| | C Dv Eb^ F G Av Bb^ C
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|-
|-
| | 5th Porcupine [7]
| | 5th Porcupine [7]
|5th Triyo [7]
| | ssLs sss
| | ssLs sss
| | C Dv Eb^ F^ G Av Bb^ C
| | C Dv Eb^ F^ G Av Bb^ C
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|-
|-
| | 6th Porcupine [7]
| | 6th Porcupine [7]
|6th Triyo [7]
| | sLss sss
| | sLss sss
| | C Dv Ev F^ G Av Bb^ C
| | C Dv Ev F^ G Av Bb^ C
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|-
|-
| | 7th Porcupine [7]
| | 7th Porcupine [7]
|7th Triyo [7]
| | Lsss sss
| | Lsss sss
| | 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>
|}
|}
Again, the modes would follow a more regular pattern if using the appropriate notation, in this case 2nd-based:
C - C# - Db - D - D# - Eb - E - E# - Fb - F - F# - Gb - G - G# - Gx/Abb - Ab - A - A# - Bb - B - B# - Cb - C
C 1st Porcupine [7] would be C D E F G Ab Bb C, 2nd would be C D E F G Ab B C, etc.
=MODMOS scales=
=MODMOS scales=


[[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.


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:
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"
|-
|-
! | old scale name
! | old scale name
! | new scale name
! | LMs pattern
! | example in A
! | example in A
! | genchain
! | genchain
! | new scale name
! | LMs pattern
|-
|-
| | Harmonic minor
| | Harmonic minor
| | 5th Meantone [7] #7
| | MsMM sLs
| | A B C D E F G# A
| | A B C D E F G# A
| | F C * D <u>'''A'''</u> E B * * G#
| | F C * D <u>'''A'''</u> E B * * G#
| | 5th Meantone [7] #7
| | MsMM sLs
|-
|-
| | Ascending melodic minor
| | Ascending melodic minor
| | 5th Meantone [7] #6 #7
| | LsLL LLs
| | A B C D E F# G# A
| | A B C D E F# G# A
| | C * D <u>'''A'''</u> E B F# * G#
| | C * D <u>'''A'''</u> E B F# * G#
| | 5th Meantone [7] #6 #7
| | LsLL LLs
|-
|-
| style="text-align:center;" | (Major with b3)
| style="text-align:center;" | (Major with b3)
| | 2nd Meantone [7] b3
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| | 2nd Meantone [7] b3
| style="text-align:center;" | "
| style="text-align:center;" | "
|-
|-
| style="text-align:center;" | (Dorian with #7)
| style="text-align:center;" | (Dorian with #7)
| | 4th Meantone [7] #7
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| | 4th Meantone [7] #7
| style="text-align:center;" | "
| style="text-align:center;" | "
|-
|-
| | Double harmonic minor
| | Double harmonic minor
| | 5th Meantone [7] #4 #7
| | MsLs sLs
| | A B C D# E F G# A
| | A B C D# E F G# A
| | F C * * <u>'''A'''</u> E B * * G# D#
| | F C * * <u>'''A'''</u> E B * * G# D#
| | 5th Meantone [7] #4 #7
| | MsLs sLs
|-
|-
| style="text-align:center;" | (Lydian with b3 b6)
| style="text-align:center;" | (Lydian with b3 b6)
| | 1st Meantone [7] b3 b6
|"
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| | 1st Meantone [7] b3 b6
|-
|-
| | Double harmonic major
| | Double harmonic major
| | 2nd Meantone [7] b2 b6
| | sLsM sLs
| | A Bb C# D E F G# A
| | A Bb C# D E F G# A
| | Bb F * * D <u>'''A'''</u> E * * C# G#
| | Bb F * * D <u>'''A'''</u> E * * C# G#
| | 2nd Meantone [7] b2 b6
| | sLsM sLs
|-
|-
| style="text-align:center;" | (Phrygian with #3 #7)
| style="text-align:center;" | (Phrygian with #3 #7)
| | 6th Meantone [7] #3 #7
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| style="text-align:center;" | "
| | 6th Meantone [7] #3 #7
| style="text-align:center;" | "
| style="text-align:center;" | "
|-
|-
| | <span style="">Hungarian gypsy </span>minor
| | <span style="">Hungarian gypsy </span>minor
| | 5th Meantone [7] #4
| | MsLs sMM
| | A B C D# E F G A
| | A B C D# E F G A
| | F C G * <u>'''A'''</u> E B * * * D#
| | F C G * <u>'''A'''</u> E B * * * D#
| | 5th Meantone [7] #4
| | MsLs sMM
|-
|-
| | Phrygian dominant
| | Phrygian dominant
| | 6th Meantone [7] #3
| | sLsM sMM
| | A Bb C# D E F G A
| | A Bb C# D E F G A
| | Bb F * G D <u>'''A'''</u> E * * C#
| | Bb F * G D <u>'''A'''</u> E * * C#
| | 6th Meantone [7] #3
| | sLsM sMM
|}
|}
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.
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.
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7th Meantone [7] #2: C D Eb F Gb Ab Bb C
7th Meantone [7] #2: C D Eb F Gb Ab Bb C


'''Porcupine[7] MODMOS scales''', not including alternative names because they all modify the 3rd or the 5th.
{| class="wikitable"
|-
!
! | scale name
! | color name
! | LMs pattern
! | example in C
! | genchain
|-
|<nowiki>4333432 6|0 #6 #7</nowiki>
| |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>
|-
|<nowiki>4333342 6|0 #7</nowiki>
|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>
|-
|<nowiki>4243333 4|2 #2</nowiki>
| |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^
|-
|<nowiki>4234333 3|3 #2</nowiki>
|4th Porcupine [7] #2
|4th Triyo [7] #2
|LsmL mmm
|C D Eb^ F G Av Bb^ C
|D * * G Av Bb^ <u>'''C'''</u> * Eb^ F
|-
|<nowiki>4324333 6|0 b4</nowiki>
| | 7th Porcupine [7] b4
| |7th Triyo [7] b4
| | LmsL mmm
| | C D Ev F G Av Bb^ C
| | D Ev * G Av Bb^ <u>'''C'''</u> * * F
|-
|<nowiki>3424333 5|1 b4</nowiki>
| | 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
|-
|<nowiki>3334243 3|3 b6</nowiki>
| | 4th Porcupine [7] b6
| |4th Triyo [7] b6
| | mmmL sLm
| | C Dv Eb^ F G Ab^ Bb^ C
| | G * Bb^ <u>'''C'''</u> Dv Eb^ F * Ab^
|-
|<nowiki>3334234 3|3 b6 b7</nowiki>
|4th Porcupine [7] b6 b7
|4th Triyo [7] b6 b7
|mmmL smL
|C Dv Eb^ F G Ab^ Bb^ C
|G * * <u>'''C'''</u> Dv Eb^ F * Ab^ Bb
|-
|<nowiki>4324342 6|0 b4 #7</nowiki>
|7th Porcupine [7] b4 #7
|7th Triyo [7] b4 #7
|LmsL mLs
|C D Ev F G Av Bv C
|Bv * D Ev * G Av * <u>'''C'''</u> * * F
|-
|<nowiki>4234243 3|3 #2 b6</nowiki>
|4th Porcupine [7] #2 b6
|4th Triyo [7] #2 b6
|LsmL sLm
|C D Eb^ F G Ab^ Bb^ C
|D * * G * Bb^ <u>'''C'''</u> * Eb^ F * Ab^
|-
|<nowiki>4333324 6|0 b7</nowiki>
|7th Porcupine [7] b7
|7th Triyo [7] b7
|Lmmm msL
|C D Ev F^ G Av Bb C
|D Ev F^ G Av * <u>'''C'''</u> * * * * * Bb
|-
|<nowiki>4333234 6|0 b6 b7</nowiki>
|7th Porcupine [7] b6 b7
|7th Triyo [7] b6 b7
|Lmmm smL
|C D Ev F^ G Ab^ Bb C
|D Ev F^ G * * <u>'''C'''</u> * * * * Ab^ Bb
|}
=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. 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. 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.
 
F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5
 
F1 --- C2 --- G2 --- D3 --- A3 --- E4 --- B4
 
F0 --- C1 --- G1 --- D2 --- A2 --- E3 --- B3
 
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 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#v4 -- G#v4 -- D#v5 -- A#v5
 
C3 ----- G3 ----- D4 ----- A4 ----- E5
 
F#v2 -- C#v3 -- G#v3 -- D#v4 -- A#v4
 
C2 ----- G2 ------ D3 ----- A3 ----- E3
 
F#v1 -- C#v2 -- G#v2 -- D#v3 -- A#v3
 
C1 ----- G1 ------ D2 ----- A2 ----- E2
 
which octave-reduces to two genchains:
 
F#v -- C#v -- G#v -- D#v -- A#v
 
C ----- G ------ D ----- A ----- E
 
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
 
C ------ G ------ D ----- A ----- E
 
It would also be valid to exchange the two rows:
 
C ------ G ------ D ----- A ----- E
 
Gbv -- Dbv -- Abv -- Ebv -- Bbv
 
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. 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#v -- G --- G#v -- A --- A#v
 
C --- C#v -- D --- D#v -- E


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


{| class="wikitable"
{| class="wikitable"
|-
|-
! | scale name
! | scale name
!color name
! | Ls pattern
! | Ls pattern
! | example in C
! | example in C
Line 478: Line 528:
|-
|-
| | 1st Srutal [10]
| | 1st Srutal [10]
|1st Sagugu [10]
| | ssssL-ssssL
| | ssssL-ssssL
| | C C#v D D#v E F#v G G#v A A#v C
| | C C#v D D#v E F#v G G#v A A#v C
Line 484: Line 535:
|-
|-
| | 2nd Srutal [10]
| | 2nd Srutal [10]
|2nd Sagugu [10]
| | sssLs-sssLs
| | sssLs-sssLs
| | C C#v D D#v F F#v G G#v A Bv C
| | C C#v D D#v F F#v G G#v A Bv C
Line 490: Line 542:
|-
|-
| | 3rd Srutal [10]
| | 3rd Srutal [10]
|3rd Sagugu [10]
| | ssLss-ssLss
| | ssLss-ssLss
| | C C#v D Ev F F#v G G#v Bb Bv C
| | C C#v D Ev F F#v G G#v Bb Bv C
Line 496: Line 549:
|-
|-
| | 4th Srutal [10]
| | 4th Srutal [10]
|4th Sagugu [10]
| | sLsss-sLsss
| | sLsss-sLsss
| | C C#v Eb Ev F F#v G Av Bb Bv C
| | C C#v Eb Ev F F#v G Av Bb Bv C
Line 502: Line 556:
|-
|-
| | 5th Srutal [10]
| | 5th Srutal [10]
|5th Sagugu [10]
| | Lssss-Lssss
| | Lssss-Lssss
| | C Dv Eb Ev F F#v Ab Av Bb Bv C
| | C Dv Eb Ev F F#v Ab Av Bb Bv C
Line 508: Line 563:
|}
|}


The [[Octatonic_scale|Diminished]] [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.
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.


Gb^^ ----- Db^^
'''[[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:


Eb^ ------- Bb^
Ebv ------- Bbv


C ---------- G
C ---------- G


Av --------- Ev
A^ --------- E^


The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. "Up" means "a quarter-octave minus a ~32/27".
F#^^ ----- C#^^


Using ~25/24 as the generator yields the same scales and mode numbers:
Using ~25/24 as the generator yields the same scales and mode numbers:


Gb^^ ----- G
Ebv ------- E^
 
Eb^ ------- Ev
 
C ---------- Db^^
 
Av --------- Bb^
 
In color notation, the diminished comma 648/625 is g<span style="vertical-align: super;">4</span>2. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8].
 
ggGb ----- ggDb


gEb ------- gBb
C ---------- C#^^


wC -------- wG
A^ --------- Bbv


yA --------- yE
F#^^ ----- G


Both Diminished [8] modes, using ups and downs:
Both Diminished [8] modes, using ups and downs:
Line 545: Line 590:
|-
|-
! | scale name
! | scale name
!color name
! | sL pattern
! | sL pattern
! | example in C
! | example in C
Line 552: Line 598:
! | 4th chain
! | 4th chain
|-
|-
| | 1st Diminished[ 8]
| | 1st Diminished[8]
|1st Quadgu[8]
| | sLsL sLsL
| | sLsL sLsL
| | C Db^^ Eb^ Ev Gb^^ G Av Bb^ C
| | C C#^^ Ebv E^ F#^^ G A^ Bbv C
| style="text-align:center;" | <u>'''C'''</u> G
| style="text-align:center;" | <u>'''C'''</u> G
| | Eb^ Bb^
| | Ebv Bbv
| | Gb^^ Db^^
| | F#^^ C#^^
| | Av Ev
| | A^ E^
|-
|-
| | 2nd Diminished [8]
| | 2nd Diminished [8]
|2nd Quadgu[8]
| | LsLs LsLs
| | LsLs LsLs
| | C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C
| | C D^ Ebv F F#^^ Abv A^ B^^ C
| style="text-align:center;" | F <u>'''C'''</u>
| style="text-align:center;" | F <u>'''C'''</u>
| | Ab^ Eb^
| | Abv Ebv
| | Cb^^ Gb^^
| |B^^ F#^^
| | Dv Av
| | D^ A^
|}
|}


There are only two [[Blackwood]] [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height":
[[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]] [10] modes. The lattice can be expressed using a 3\5 period. Ups and downs indicate the generator, not the period:
 
E^^ ------- G#^^
 
D^ -------- F#^
 
C ---------- E
 
Bbv ------- Fv
 
Gvv ------- Dvv
 
Ups and downs could indicate the generator instead of the period:


F ------ Av
F ------ Av
Line 592: Line 628:


G ------ Bv
G ------ Bv
Assuming octave equivalence, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2.
F ------ Av
C ------ Ev
G ------ Bv
D ------ F#v
A ------ C#v
In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y.
wF ------ yA
wC ------ yE
wG ------ yB
wD ------ yF#
wA ------ yC#


Both Blackwood modes, using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4":
Both Blackwood modes, using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4":
Line 622: Line 634:
|-
|-
! | scale name
! | scale name
!color name
! | sL pattern
! | sL pattern
! | example in C
! | example in C
Line 627: Line 640:
|-
|-
| | 1st Blackwood [10]
| | 1st Blackwood [10]
|1st 5edo+ya[10]
| | Ls-Ls-Ls-Ls-Ls
| | Ls-Ls-Ls-Ls-Ls
| | C C#v D Ev F F#v G Av A Bv C
| | C C#v D Ev F F#v G Av A Bv C
Line 632: Line 646:
|-
|-
| | 2nd Blackwood [10]
| | 2nd Blackwood [10]
|2nd 5-edo+ya[10]
| | sL-sL-sL-sL-sL
| | sL-sL-sL-sL-sL
| | C C^ D Eb^ E F^ G Ab^ A Bb^ C
| | C C^ D Eb^ E F^ G Ab^ A Bb^ C
Line 732: Line 747:
=Non-heptatonic Scales=
=Non-heptatonic Scales=


As long as we stick to MOS scales, terms like Meantone [5] or Meantone {6} are fine. But when we alter, add or drop notes, we need to define what something like "#5" means in a pentatonic or hexatonic context.
As long as we stick to MOS scales, terms like Meantone [5] or Meantone [6] are fine. But when we alter, add or drop notes, we need to define what something like "#5" means in a pentatonic or hexatonic context.


If the scale is written using heptatonically using 7 note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone [5] #5. If discussing scales in the abstract without reference to any note names, one need to specify which type of numbering is bering used.
If the scale is written using heptatonically using 7 note names, the degree numbers are heptatonic. C D E G A# is written 1st Meantone [5] #6. If the scale were written pentatonically using 5 note names, perhaps J K L M #N, it would be 1st Meantone [5] #5. If discussing scales in the abstract without reference to any note names, one need to specify which type of numbering is bering used.
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