Kite's Genchain mode numbering: Difference between revisions

'''Genchain mode numbering''' ('''GMN''' for short) provides a way to name MOS, MODMOS and even non-MOS rank-2 scales and modes systematically. Like [[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. It also applies to abstract MOS patterns like 5L 3s.

This mode notation system was designed by [[Kite Giedraitis]].

[[MOS scale]]s 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]][7], using ~3/2 as the generator. On this page, 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"

! | old scale name

! | new scale name

! | Ls pattern

! | example on white keys

! | genchain

|-

| | Lydian

| | 1st Meantone[7]

| | LLLs LLs

| | F G A B C D E F

| | <u>'''F'''</u> C G D A E B

|-

| | Ionian (major)

| | 2nd Meantone[7]

| | LLsL LLs

| | C D E F G A B C

| | F <u>'''C'''</u> G D A E B

|-

| | Mixolydian

| | 3rd Meantone[7]

| | LLsL LsL

| | G A B C D E F G

| | F C <u>'''G'''</u> D A E B

|-

| | Dorian

| | 4th Meantone[7]

| | LsLL LsL

| | D E F G A B C D

| | F C G <u>'''D'''</u> A E B

|-

| | Aeolian (minor)

| | 5th Meantone[7]

| | LsLL sLL

| | A B C D E F G A

| | F C G D <u>'''A'''</u> E B

|-

| | Phrygian

| | 6th Meantone[7]

| | sLLL sLL

| | E F G A B C D E

| | F C G D A <u>'''E'''</u> B

|-

| | Locrian

| | 7th Meantone[7]

| | sLLs LLL

| | B C D E F G A B

| | F C G D A E <u>'''B'''</u>

|}

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 term GMN can also be read as genchain mode <u>number</u>, and can refer to the numbers 1st, 2nd, 3rd etc., as in "Dorian's GMN is 4".

The same seven modes, all with C as the tonic, to illustrate the difference between modes. Adjacent modes differ by only one note. The modes proceed from sharper (Lydian) to flatter (Locrian).

|+ Meantone[7] modes in C

! | old scale name

! | new scale name

! | Ls pattern

! | example in C

! | ------------------ genchain ---------------

|-

| | Lydian

| | 1st Meantone[7]

| | LLLs LLs

| | C D E F# G A B C

| style="text-align:right;" | <u>'''C'''</u> G D A E B F#

| | Ionian (major)

| | 2nd Meantone[7]

| | LLsL LLs

| | C D E F G A B C

| style="text-align:right;" | F <u>'''C'''</u> G D A E B ----

| | Mixolydian

| | 3rd Meantone[7]

| | LLsL LsL

| | C D E F G A Bb C

| style="text-align:right;" | Bb F <u>'''C'''</u> G D A E -------

| | Dorian

| | 4th Meantone[7]

| | LsLL LsL

| | C D Eb F G A Bb C

| | -------------- Eb Bb F <u>'''C'''</u> G D A

| | Aeolian (minor)

| | 5th Meantone[7]

| | LsLL sLL

| | C D Eb F G Ab Bb C

| | --------- Ab Eb Bb F <u>'''C'''</u> G D

| | Phrygian

| | 6th Meantone[7]

| | sLLL sLL

| | C Db Eb F G Ab Bb C

| | ---- Db Ab Eb Bb F <u>'''C'''</u> G

| | Locrian

| | 7th Meantone[7]

| | sLLs LLL

| | C Db Eb F Gb Ab Bb C

| | Gb Db Ab Eb Bb F <u>'''C'''</u>

|}

Pentatonic 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.

[[Porcupine]] aka Triyo has a [[pergen]] of (P8, P4/3) and a generator of ~10/9, notated as a vM2 or a ^^m2 using [[ups and downs notation]]. The [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is v³A1. Because the generator is a 2nd, the genchain resembles the scale.

![[Color notation/Temperament Names|color name]]

| | 5th Porcupine[7]

|5th Triyo[7]

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

|-

| | C D vE ^F G vA ^Bb C

| | D vE ^F G vA ^Bb <u>'''C'''</u>

[[Sensi]] aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^³d4 and a v⁴A3, and the [[Enharmonic unisons in ups and downs notation|enharmonic unison]] is ^⁷dd2.

! | scale name

|2nd Sepgu[8]

| | 3rd Sensi[8]

|3rd Sepgu[8]

|4th Sepgu[8]

|6th Sepgu[8]

| | C  vD    ^Eb  ^³Fb ^^Gb v³G#   vA  vvB C

|7th Sepgu[8]

[[MODMOS scale]]s 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 when the generator is chroma-positive, and N steps backwards when it isn't. This ensures a higher pitch. (Note that Meantone[5] is chroma-negative, more on this below.) However, an exception is made for superflat edos like 16edo when the generator is a 3/2 fifth, because in those edos, G# is actually flat of G. Another exception is when the generator is close to the "tipping point" between chroma-positive and chroma-negative. A good alternative in these and other situations, including non-heptatonic and non-fifth-generated scales, is to use + for forwards in the genchain and - for 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).  

Meantone MODMOS scales, with alternative names in italics and parentheses. Alternatives that have more alterations than the original aren't listed:

| | Ascending melodic minor

| | 5th Meantone[7] #6 #7

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

| | C * D <u>'''A'''</u> E B F# * G#

| | ''(2nd Meantone[7] b3)''

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

| | ''(4th Meantone[7] #7)''

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

| | 5th Meantone[7] #4 #7

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

| | F C * * <u>'''A'''</u> E B * * G# D#

| | ''(1st Meantone[7] b3 b6)''

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

| | 2nd Meantone[7] b2 b6

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

| | Bb F * * D <u>'''A'''</u> E * * C# G#

| | ''(6th Meantone[7] #3 #7)''

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

| | A B C D# E F G A

| | F C G * <u>'''A'''</u> E B * * * D#

| | 6th Meantone[7] #3

| | A Bb C# D E F G A

| | Bb F * G D <u>'''A'''</u> E * * C#

Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:

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

* 4th Meantone[7] #4: C D Eb F# G A Bb C

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

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

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

The 3rd scale breaks the pattern to avoid an altered tonic ("3rd Meantone[7] #1"). The Bbb is "b7" not "bb7" because the 7th mode is Locrian, and Bbb is only one semitone flat of the Locrian mode's minor 7th Bb.

Ascending melodic minor modes:

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

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

* 4th Meantone[7] #7: C D Eb F G A B C

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

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

* 7th Meantone[7] #2: C D Eb F Gb Ab Bb C

Porcupine[7] aka Triyo[7] MODMOS scales, not including alternative names because they all modify the 3rd or the 5th.

! | example in C

|-

|-

|-

|-

|-

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

|-

|-

| | C D vE ^F G A vB C

| | A vB * D vE ^F G * * <u>'''C'''</u>

|-

|-

|-

If a rank-2 temperament's [[pergen]] has a split octave, the temperament has multiple genchains running in parallel. Using ups and downs notation, each genchain has its own height. There is a plain one, an up one, perhaps a down one, etc. 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 generated by the 5th and the up must be complete. The number in the brackets becomes two numbers, and the Ls pattern as written here is grouped by period, using hyphens.

[[Srutal]] aka Diaschismatic aka Sagugu has a half-8ve period of ~45/32. All five Srutal[2x5] modes. Every other scale note has a down.

|+ Srutal[2x5]/Sagugu[2x5] modes

! | [[Color notation/Temperament Names|color name]]

| | C vC# D vD# E vF# G vG# A vA# C

| | <u>'''C'''</u> G D A E

|-

| | sssLs-sssLs

| | C vC# D vD# F vF# G vG# A vB C

| | F <u>'''C'''</u> G D A

|-

| | ssLss-ssLss

| | C vC# D vE F vF# G vG# Bb vB C

| | Bb F <u>'''C'''</u> G D

|-

| | sLsss-sLsss

|-

| | Lssss-Lssss

Srutal's period is written as a vA4, but could instead be written as an ^d5. The generator is written as a P5. 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. The generator could instead be written as ~16/15 (3/2 minus a period), because that would still create the same mode numbers and thus the same scale names. The first genchain of 1st Srutal[2x5] would be C vC# D vD# E, just like the first half of the scale.

[[Augmented family|Augmented]] aka Trigu has a third-8ve period of ~5/4. The generator is ~3/2, which is equivalent to ~6/5. It could be thought of as ~16/15, but that would reverse the genchain direction and change all the mode numbers. The ~16/15 generator is not used, even though it is smaller, so that the genchain direction matches that of the pergen, which is (P8/3, P5).

{| class="wikitable"

|+ Augmented[3x3]/Trigu[3x3] modes

|-

|-

| | Lss-Lss-Lss

| style="text-align:center;" | <u>'''C'''</u> G D

|-

| | sLs-sLs-sLs

| style="text-align:center;" | F <u>'''C'''</u> G

|-

|ssL-ssL-ssL

'''[[Octatonic_scale|Diminished]] aka Quadgu''' has pergen (P8/4, P5) and a period of ~6/5. The generator is ~3/2, which is equivalent to ~5/4 or ~25/24. The generator can't be ~10/9, because that would change the mode numbers. The Diminished[4x2] scale has only two modes, because the four genchains have only two notes each. The comma is fifthward, thus the 5th is flattened, and the 32/27 minor 3rd is sharpened. Therefore the 300¢ period is narrower than a m3, and must be a vm3.

Using ~25/24 as the generator yields the same scales and mode numbers. 1st Diminished[4x2] would have genchains C – ^^C#, vEb – ^E, ^^F# – G and ^A – vBb, just like the scale.

[[Blackwood|'''Blackwood''']] '''aka Sawa+ya''' has a fifth-octave period of 240¢. The generator is a just 5/4 = 386¢. There are only two [[Blackwood]][5x2] modes. Ups and downs indicate the generator, not the period.

|+ Blackwood[5x2]/5edo+ya[5x2]

|-

! | scale name

![[Color notation/Temperament Names|color name]]

! | Ls pattern

! | example in C

! | genchains

|-

| | 1st Blackwood[5x2]

|1st 5edo+ya[5x2]

| | Ls-Ls-Ls-Ls-Ls

| | C vC# D vE F vF# G vA A vB C

| style="text-align:center;" | <u>'''C'''</u>-vE, D-vF#, F-vA, G-vB, A-vC#

|-

| | 2nd Blackwood[5x2]

|2nd 5edo+ya[5x2]

| | sL-sL-sL-sL-sL

| | 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

|}

{| class="wikitable"

|+ Non-MOS/MODMOS Meantone examples

|-

|-

|-

|-

|-

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

| | F C * D <u>'''A'''</u> E B * * G# D#

|-

|-

|-

|-

|-

|-

| | G A C D E F# G

| | C <u>'''G'''</u> D A E * F#

|-

|-

|-

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

| style="text-align:center;" | "   

|-

| | F C * * <u>'''A'''</u> E B

Even 7-note scales can be non-MOS and non-MODMOS. For example, A C D D# E F G# A. The genchain is F C * D A E * * * G# D#. The name requires alterations, adds and drops: A 5th Meantone[7] #7 no2 add #4.

Another possibility is a scale that would be MOS, but the generator is too sharp or flat. For example, a genchain F C G D A E B of 8\13 fifths makes an out-of-order scale A C B D F E G A. This scale is best named as Meantone[5] with added notes: Which brings us to...

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 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 needs to specify which type of numbering is being used.

...5# 3# 1# 4# 2# 5 3 1 4 2 5b 3b 1b 4b 2b bb5...

...-K -N -L -J -M K N L J M +K +N +L +J +M ++K...

and these standard modes:

* L 1st Meantone[5] = L M +N J +K L

* L 2nd Meantone[5] = L M N J +K L

* L 3rd Meantone[5] = L M N J K L

* L 4th Meantone[5] = L -M N J K L

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

The A C B D F E G A scale becomes L M -M N J +K K L, which has 3 possible names:

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.

|+ Sensi[8]/Sepgu[8] MOS and MODMOS examples

! | 19-edo example in C

| | Gb B Eb G# <u>'''C'''</u> E# A Db

| | C D E# F G# H A B# C

| | G# B# E# H <u>'''C'''</u> F A D

| | C Db Eb E# Gb G# A Bb C

| | Gb * Eb G# <u>'''C'''</u> E# A Db * Bb

|5th Sepgu[8] +8

| | C D E# F G# H A B C

| | G# * E# H <u>'''C'''</u> F A D * B

Heptatonic fifth-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

The heptatonic-notated MODMOS has "+7" because B is the 7th letter from C. Likewise octotonic has "+8" because with H, B is the 8th letter.

* 1st 5L 3s = LLsLLsLs

* 2nd 5L 3s = LLsLsLLs

* 3rd 5L 3s = LsLLsLLs

* 1st 3L 5s = ssLssLsL

* 2nd 3L 5s = ssLsLssL

* 3rd 3L 5s = sLssLssL

* etc.

For a MOS pattern with a fifth-sized generator, the fifth is still prioritized over the fourth. Otherwise the generator is the mingen.

'''Why not number the modes in the order they occur in the scale?'''

Scale-based numbering would order the modes 1st = Ionian, 2nd = Dorian, 3rd = Phrygian, etc.

'''Why make an exception for 3/2 vs 4/3 as the generator?'''

There are centuries of established thought that the fifth, not the fourth, generates the Pythagorean, meantone and well tempered scales, as these quotes show (emphasis added):

"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 ''wise'' consistency, it wouldn't reduce memorization, because it's well known that the generator is historically 3/2.

'''Then why not always choose the larger of the two generators?'''

Interval arithmetic is easier with smaller intervals. It's easier to add up stacked 2nds than stacked 7ths. Also, when the generator is a 2nd, the genchain is often identical to the scale, simplifying mode numbering. (See Porcupine[7] above.)

'''Why not always choose the chroma-positive generator?'''

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

One problem with [[modal UDP notation]] 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.

|+ Comparison of meantone MOS scales in UDP and Mode Numbers

|-

! | scale

! | UDP generator

! | UDP genchain

! | Mode Numbers generator

! | Mode Numbers genchain

|-

| | Meantone[5] in 31edo

| style="text-align:center;" | 4/3

| | E A D G C

| style="text-align:center;" | 3/2

| | C G D A E

|-

| | Meantone[7] in 31edo

| style="text-align:center;" | 3/2

| | C G D A E B F#

| style="text-align:center;" | 3/2

| | C G D A E B F#

|-

| | Meantone[12] in 31edo

| style="text-align:center;" | 4/3

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

| style="text-align:center;" | 3/2

| | Meantone[19] in 31edo

| style="text-align:center;" | 3/2

G# D# A# E# B#

| style="text-align:center;" | 3/2

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