Kite's Genchain mode numbering: Difference between revisions

=MOS Scales=

'''Mode Numbers''' provide a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[Modal_UDP_Notation|Modal UDP notation]], it starts with the convention of using ''some-temperament-name''[''some-number''] to create a generator-chain, and adds a way to number each mode uniquely.

[[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.

4th Meantone[7] is spoken as "fourth meantone heptatonic", or possibly "fourth meantone seven". If in D, as above, it would be "D fourth meantone heptatonic".

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.

! | Ls pattern (assumes<br>~3/2 &lt; 700¢)

| | sLsL sLL sLsLL

| | sLsL LsL sLsLL

| | sLsL LsL sLLsL

| | sLLs LsL sLLsL

| | sLLs LsL LsLsL

| | LsLs LsL LsLsL

| | LsLs LLs LsLsL

 

'''[[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.

!color name

| | C Dv Eb^ F Gv Ab^ Bb C

| | <u>'''C'''</u> Dv Eb^ F Gv Ab^ Bb

| | C Dv Eb^ F Gv Ab^ Bb^ C

| | Bb^ <u>'''C'''</u> Dv Eb^ F Gv Ab^

| | C Dv Eb^ F Gv Av Bb^ C

| | Av Bb^ <u>'''C'''</u> Dv Eb^ F Gv

| | C Dv Eb^ F G Av Bb^ C

| | G Av Bb^ <u>'''C'''</u> Dv Eb^ F

| | C Dv Eb^ F^ G Av Bb^ C

| style="text-align:center;" | F^ G Av Bb^ <u>'''C'''</u> Dv Eb^

| | C Dv Ev F^ G Av Bb^ C

| | Ev F^ G Av Bb^ <u>'''C'''</u> Dv

| | C D Ev F^ G Av Bb^ C

| | D Ev F^ G Av Bb^ <u>'''C'''</u>

[[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:

! | LMs pattern

| | MsMM sLs

| style="text-align:center;" | (Major with b3)

| | 2nd Meantone[7] b3

| style="text-align:center;" | (Dorian with #7)

| | 4th Meantone[7] #7

| | MsLs sLs

| style="text-align:center;" | (Lydian with b3 b6)

| | 1st Meantone[7] b3 b6

| | sLsM sLs

| style="text-align:center;" | (Phrygian with #3 #7)

| | 6th Meantone[7] #3 #7

| | MsLs sMM

| | sLsM sMM

1st Meantone[7] #2: C D# E F# G A B C <br>

2nd Meantone[7] #:5 C D E F G# A B C<br>

7th Meantone[7] b4 b7: C Db Eb Fb Gb Ab Bbb C (breaks the pattern, 7th mode not 3rd mode)<br>

4th Meantone[7] #4: C D Eb F# G A Bb C<br>

5th Meantone[7] #7: C D Eb F G Ab B C (harmonic minor)<br>

6th Meantone[7] #3: C Db E F G Ab Bb C (phrygian dominant)<br>

7th Meantone[7] #6: C Db Eb F Gb A Bb C

1st Meantone[7] #5: C D E F# G# A B C<br>

7th Meantone[7] b4: C Db Eb Fb Gb Ab Bb C (avoid "2nd Meantone[7] #1")<br>

3rd Meantone[7] #4: C D E F# G A Bb C<br>

4th Meantone[7] #7: C D Eb F G A B C<br>

5th Meantone[7] #3: C D E F G Ab Bb C<br>

6th Meantone[7] #6: C Db Eb F G A Bb C<br>

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.

! | color name

! | LMs pattern

|C D Eb^ F G Av Bb^ C

|D * * G Av Bb^ <u>'''C'''</u> * Eb^ F

| | 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

| | C Dv Ev F G Av Bb^ C

| | Ev * G Av Bb^ <u>'''C'''</u> Dv * F

| | C Dv Eb^ F G Ab^ Bb^ C

| | G * Bb^ <u>'''C'''</u> Dv Eb^ F * Ab^

|C Dv Eb^ F G Ab^ Bb C

|G * * <u>'''C'''</u> Dv Eb^ F * Ab^ Bb

|C D Ev F G Av Bv C

|Bv * D Ev * G Av * <u>'''C'''</u> * * F

|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^

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.

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

! | color name

| | 1st Srutal[10]

| | 1st Sagugu[10]

| | C C#v D D#v E F#v G G#v A A#v C

| | F#v C#v G#v D#v A#v

| | 2nd Srutal[10]

| | 2nd Sagugu[10]

| | C C#v D D#v F F#v G G#v A Bv C

| | Bv F#v C#v G#v D#v

| | 3rd Srutal[10]

| | 3rd Sagugu[10]

| | C C#v D Ev F F#v G G#v Bb Bv C

| | Ev Bv F#v C#v G#v

| | 4th Srutal[10]

| | 4th Sagugu[10]

| | C C#v Eb Ev F F#v G Av Bb Bv C

| | Av Ev Bv F#v C#v

| | 5th Srutal[10]

| | 5th Sagugu[10]

| | C Dv Eb Ev F F#v Ab Av Bb Bv C

| | Dv Av Ev Bv F#v

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:

Ebv ------- Bbv

 

 

 

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

 

 

 

 

 

Both Diminished[8] modes, using ups and downs:

! | color name

! | sL pattern

| | 1st Diminished[8]

| | 1st Quadgu[8]

| | sLsL sLsL

| | C C#^^ Ebv E^ F#^^ G A^ Bbv C

| | Ebv Bbv

| | F#^^ C#^^

| | A^ E^

| | 2nd Diminished[8]

|2nd Quadgu[8]

| | LsLs LsLs

| | C D^ Ebv F F#^^ Abv A^ B^^ C

| | Abv Ebv

| |B^^ F#^^

| | D^ A^

[[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:

F ------ Av

 

 

 

 

 

Both Blackwood modes, using ups and downs:

!color name

! | sL pattern

| | 1st Blackwood[10]

|1st 5edo+ya[10]

| | C C#v D Ev F F#v G Av A Bv C

| style="text-align:center;" | <u>'''C'''</u>-Ev, D-F#v, F-Av, G-Bv, A-C#v

| | 2nd Blackwood[10]

|2nd 5-edo+ya[10]

| | 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

=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. 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:

! | sMLX pattern

| | LLMs MLLM

| | LLMs MLML

| | LMLs MMXM

| | LLMsM LMsM

| | LMLsM MLsM

| | MML MMs

| | MLM MMs

| | MML Xs

| style="text-align:center;" | "

| | MsL sL

=Non-heptatonic Scales=

 

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.

 

L 1st Meantone[5] = L M +N J +K L<br>

L 2nd Meantone[5] = L M N J +K L<br>

L 3rd Meantone[5] = L M N J K L<br>

L 4th Meantone[5] = L -M N J K L<br>

L 5th Meantone[5] = L -M N -J K L

 

L 3rd Meantone[5] add -2, +5<br>

L 2nd Meantone[5] add -2, -5<br>

L 4th Meantone[5] add +2, +5

!color name

! | sL pattern

! | example in C

! | genchain

| | sL sL ssLs

| | sL sL sssL

"Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio '''3:2''' (i.e., the untempered perfect '''fifth''')." -- [https://en.wikipedia.org/wiki/Pythagorean_tuning]

"The syntonic temperament is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect '''fifth'''." -- [https://en.wikipedia.org/wiki/Syntonic_temperament]

"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect '''fifths'''." -- [https://en.wikipedia.org/wiki/Meantone_temperament]

"In this system the perfect '''fifth''' is flattened by one quarter of a syntonic comma." -- [https://en.wikipedia.org/wiki/Quarter-comma_meantone]

"The term "well temperament" or "good temperament" usually means some sort of irregular temperament in which the tempered '''fifths''' are of different sizes." -- [https://en.wikipedia.org/wiki/Well_temperament]

"A foolish consistency is the hobgoblin of little minds". To choose 4/3 over 3/2 merely for the sake of consistency would be pointless. Unlike a <u>wise</u> consistency, it wouldn't reduce memorization, because it's well known that the generator is historically 3/2.

'''Why not just use UDP notation?'''

One problem with [[Modal_UDP_Notation|UDP]] is that avoiding chroma-negative generators causes the genchain to reverse direction frequently as you lengthen or shorten it, which affects the mode names. If exploring the various MOS's of a temperament, one has to constantly check the genchain direction. In Mode Numbers notation, the direction is unchanging.

| | Meantone[12] if generator &lt; 700¢

| | Meantone[12] if generator &gt; 700¢

Furthermore, UDP uses the more mathematical [https://en.wikipedia.org/wiki/Zero-based_numbering zero-based counting] and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.

 

 

[[Naming_Rank-2_Scales#Jake Freivald method|http://xenharmonic.wikispaces.com/Naming+Rank-2+Scales#Jake%20Freivald%20method]]