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Kite's Genchain mode numbering: Difference between revisions

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{| class="wikitable"
{| class="wikitable"
|-
|-
!
! | scale name
! | scale name
! | color name
! | color name
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! | genchain
! | genchain
|-
|-
|<nowiki>4333432 6|0 #6 #7</nowiki>
| |7th Porcupine [7] #6 #7
| |7th Porcupine [7] #6 #7
| |7th Triyo [7] #6 #7
| |7th Triyo [7] #6 #7
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| | A Bv * D Ev F^ G * * <u>'''C'''</u>
| | A Bv * D Ev F^ G * * <u>'''C'''</u>
|-
|-
|<nowiki>4333342 6|0 #7</nowiki>
|7th Porcupine [7] #7
|7th Porcupine [7] #7
|7th Triyo [7] #7
|7th Triyo [7] #7
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|Bv * D Ev F^ G Av * <u>'''C'''</u>
|Bv * D Ev F^ G Av * <u>'''C'''</u>
|-
|-
|<nowiki>4243333 4|2 #2</nowiki>
| |5th Porcupine [7] #2
| |5th Porcupine [7] #2
| |5th Triyo [7] #2
| |5th Triyo [7] #2
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| | D * F^ G Av Bb^ <u>'''C'''</u> * Eb^
| | D * F^ G Av Bb^ <u>'''C'''</u> * Eb^
|-
|-
|<nowiki>4234333 3|3 #2</nowiki>
|4th Porcupine [7] #2
|4th Porcupine [7] #2
|4th Triyo [7] #2
|4th Triyo [7] #2
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|D * * G Av Bb^ <u>'''C'''</u> * Eb^ F
|D * * G Av Bb^ <u>'''C'''</u> * Eb^ F
|-
|-
|<nowiki>4324333 6|0 b4</nowiki>
| | 7th Porcupine [7] b4
| | 7th Porcupine [7] b4
| |7th Triyo [7] b4
| |7th Triyo [7] b4
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| | D Ev * G Av Bb^ <u>'''C'''</u> * * F
| | D Ev * G Av Bb^ <u>'''C'''</u> * * F
|-
|-
|<nowiki>3424333 5|1 b4</nowiki>
| | 6th Porcupine [7] b4
| | 6th Porcupine [7] b4
| |6th Triyo [7] b4
| |6th Triyo [7] b4
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| | Ev * G Av Bb^ <u>'''C'''</u> Dv * F
| | Ev * G Av Bb^ <u>'''C'''</u> Dv * F
|-
|-
|<nowiki>3334243 3|3 b6</nowiki>
| | 4th Porcupine [7] b6
| | 4th Porcupine [7] b6
| |4th Triyo [7] b6
| |4th Triyo [7] b6
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| | G * Bb^ <u>'''C'''</u> Dv Eb^ F * Ab^
| | G * Bb^ <u>'''C'''</u> Dv Eb^ F * Ab^
|-
|-
|<nowiki>3334234 3|3 b6 b7</nowiki>
|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 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
|-
|-
|<nowiki>4324342 6|0 b4 #7</nowiki>
|7th Porcupine [7] b4 #7
|7th Porcupine [7] b4 #7
|7th Triyo [7] b4 #7
|7th Triyo [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
|-
|-
|<nowiki>4234243 3|3 #2 b6</nowiki>
|4th Porcupine [7] #2 b6
|4th Porcupine [7] #2 b6
|4th Triyo [7] #2 b6
|4th Triyo [7] #2 b6
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|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^
|-
|<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=
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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.
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.


'''[[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:


Ebv ------- Bbv
Ebv ------- Bbv
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A^ --------- E^
A^ --------- E^


F#^^ ----- C#^^
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:
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A^ --------- Bbv
A^ --------- Bbv


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


Both Diminished [8] modes, using ups and downs:
Both Diminished [8] modes, using ups and downs:
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|2nd 5-edo+ya[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^ F F^ G Ab^ A Bb^ C
| style="text-align:center;" | Ab^-<u>'''C'''</u>, Bb^-D, C^-E, Eb^-G, F^-A
| style="text-align:center;" | Ab^-<u>'''C'''</u>, Bb^-D, C^-F, Eb^-G, F^-A
|}
|}


=Other rank-2 scales=
=Other rank-2 scales=


These are scales that are neither MOS nor MODMOS. Some scales have too many or too few notes. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc.
These are scales that are neither MOS nor MODMOS. Some scales have too many or too few notes. If they have an unbroken genchain, they can be named Meantone [6], Meantone [8], etc. But if there are chromatic alterations, and the genchain has gaps, there's no clear way to number the notes, and no clear way to name the scale. Such a scale must be named as a MOS scale with notes added or removed, using "add" and "no", analogous to chord names. As with MODMOS scales, there is often more than one name for a scale. Meantone examples:
 
However if there are chromatic alterations, and the genchain has gaps, there's no clear way to number the notes, and no clear way to name the scale. Such a scale must be named as a MOS scale with notes added or removed, using "add" and "no", analogous to chord names. As with MODMOS scales, there is often more than one name for a scale.


{| class="wikitable"
{| class="wikitable"
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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 being used.


The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Because the pentatonic fifth is chroma-negative, the fifthward side of the genchain is flat and the fourthwards side is sharp (assuming a fifth &lt; 720¢). Use "+" for fifthwards and "-" for fourthwards.
The scale of 8\13 fifths A C B D F E G A mentioned above can't be notated with fifth-based heptatonic and requires pentatonic notation. Because the pentatonic fifth is chroma-negative, the fifthward side of the genchain is flat and the fourthwards side is sharp (assuming a fifth &lt; 720¢). Use "+" for fifthwards and "-" for fourthwards.
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L 4th Meantone [5] add +2, +5
L 4th Meantone [5] add +2, +5


Sensi is a good example because it's nether heptatonic nor fifth-generated. Here's a Sensi [8] MOS and MODMOS in both heptatonic and octotonic notation. The generator, a heptatonic 3rd or octotonic 4th, is chroma-negative. In 19edo, generator = 7\19, L = 3\19, and s = 2\19.
Sensi is a good example because it's nether heptatonic nor fifth-generated. Below is a Sensi [8] MOS and a Sensi [8] MODMOS, each in both heptatonic and octotonic notation. The generator, a heptatonic 3rd or octotonic 4th, is chroma-negative. In 19edo, generator = 7\19, L = 3\19, and s = 2\19.


{| class="wikitable"
{| class="wikitable"
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! | notation
! | notation
! | scale name
! | scale name
!color name
! | sL pattern
! | sL pattern
! | example in C
! | example in C
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| | heptatonic
| | heptatonic
| | 5th Sensi [8]
| | 5th Sensi [8]
|5th Sepgu [8]
| | sL sL ssLs
| | sL sL ssLs
| | C Db Eb E# Gb G# A B C
| | C Db Eb E# Gb G# A B C
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| | octotonic
| | octotonic
| | 5th Sensi [8]
| | 5th Sensi [8]
|5th Sepgu [8]
| style="text-align:center;" | "
| style="text-align:center;" | "
| | C D E# F G# H A B# C
| | C D E# F G# H A B# C
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| | heptatonic
| | heptatonic
| | 5th Sensi [8] +7
| | 5th Sensi [8] +7
|5th Sepgu [8] +7
| | sL sL sssL
| | sL sL sssL
| | C Db Eb E# Gb G# A Bb C
| | C Db Eb E# Gb G# A Bb C
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| | octotonic
| | octotonic
| | 5th Sensi [8] +8
| | 5th Sensi [8] +8
|5th Sepgu [8] +8
| style="text-align:center;" | "
| style="text-align:center;" | "
| | C D E# F G# H A B C
| | C D E# F G# H A B C
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'''Why not number the modes in the order they occur in the scale?'''
'''Why not number the modes in the order they occur in the scale?'''


Scale-based numbering would order the modes Ionian, Dorian, Phrygian, etc.
Scale-based numbering would order the modes 1st = Ionian, 2nd = Dorian, 3rd = Phrygian, etc.
 
<u>Genchain-based</u>: if the Meantone[7] genchain were notated 1 2 3 4 5 6 7, the Lydian scale would be 1 3 5 7 2 4 6 1, and the major scale would be 2 4 6 1 3 5 7 2.
 
<u>Scale-based</u>: if the Meantone[7] major scale were notated 1 2 3 4 5 6 7 1, the genchain would be 4 1 5 2 6 3 7.


The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The modes are ordered in a spectrum, and the 1st and last modes are always the two most extreme. For MOS scales, adjacent modes differ by only one note.
The advantage of genchain-based numbering is that similar modes are grouped together, and the structure of the temperament is better shown. The modes are ordered in a spectrum, and the 1st and last modes are always the two most extreme. For MOS scales, adjacent modes differ by only one note.
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An even larger problem is that the notation is overly tuning-dependent. Meantone [12] generated by 701¢ has a different genchain than Meantone [12] generated by 699¢, so slight differences in tempering result in different mode names. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example "Dominant 8|3" could mean either "4th Dominant [12]" or "9th Dominant [12]". Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] or Mohajira [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.
An even larger problem is that the notation is overly tuning-dependent. Meantone [12] generated by 701¢ has a different genchain than Meantone [12] generated by 699¢, so slight differences in tempering result in different mode names. One might address this problem by reasonably constraining meantone's fifth to be less than 700¢. Likewise one could constrain Superpyth [12]'s fifth to be more than 700¢. But this approach fails with Dominant meantone, which tempers out both 81/80 and 64/63, and in which the fifth can reasonably be either more or less than 700¢. This makes every single UDP mode of Dominant [12] ambiguous. For example "Dominant 8|3" could mean either "4th Dominant [12]" or "9th Dominant [12]". Something similar happens with Meantone [19]. If the fifth is greater than 694¢ = 11\19, the generator is 3/2, but if less than 694¢, it's 4/3. This makes every UDP mode of Meantone [19] ambiguous. Another example is Dicot [7] or Mohajira [7] when the neutral 3rd generator is greater or less than 2\7 = 343¢. Another example is Semaphore [5]'s generator of ~8/7 or ~7/6 if near 1\5 = 240¢. In general, this ambiguity arises whenever the generator of an N-note MOS ranges from slightly flat of any N-edo interval to slightly sharp of it.


Three other problems with UDP are more issues of taste. The most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And fractional-period temperaments, e.g. Srutal [10] 6|2(2), use four numbers where only two are needed.
Three other problems with UDP are more issues of taste. The most important piece of information, the number of notes in the scale, is hidden by UDP notation. It must be calculated by adding together the up, down, and period numbers (and the period number is often omitted). For example, to determine that Meantone 5|1 is heptatonic, one must add the 5, the 1 and the omitted 1. If the number of notes is indicated with brackets, e.g. Meantone [7] 5|1, then three numbers are used where only two are needed. And split-octave temperaments, e.g. Srutal [10] 6|2(2), use four numbers where only two are needed.


Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C.
Also, when comparing different MOS's of a temperament, with Mode Numbers notation but not with UDP, the Nth mode of the smaller MOS is always a subset of the Nth mode of the larger MOS. For example, Meantone [5] is generated by 3/2, not 4/3 as with UDP. Because Meantone [5] and Meantone [7] have the same generator, C 2nd Meantone [5] = C D F G A C is a subset of C 2nd Meantone [7] = C D E F G A B C. But using UDP, C Meantone 3|1 = C Eb F G Bb C isn't a subset of C Meantone 5|1 = C D E F G A B C.
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