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.

| | 1st Meantone[7]

| | 2nd Meantone[7]

| | 3rd Meantone[7]

| | 4th Meantone[7]

| | 5th Meantone[7]

| | 6th Meantone[7]

| | 7th Meantone[7]

 

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".

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

| | 1st Meantone[7]

| | 2nd Meantone[7]

| | 3rd Meantone[7]

| | 4th Meantone[7]

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

| | 5th Meantone[7]

| | 6th Meantone[7]

| | 7th Meantone[7]

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|§&nbsp;Rationale]]). Unlike modal UDP notation, the generator isn't always [[Chroma|chroma-positive]]. There are several disadvantages of only using chroma-positive generators. See the critique of UDP in the [[#Rationale|§&nbsp;Rationale]] section below.

| | 1st Meantone[5]

| | 2nd Meantone[5]

| | 3rd Meantone[5]

| | 4th Meantone[5]

| | 5th Meantone[5]

! | Ls pattern (assumes<br>a generator &lt; 700¢)

| | 1st Meantone[12]

| | sLsLsLL sLsLL

| | 2nd Meantone[12]

| | sLsLLsL sLsLL

| | 3rd Meantone[12]

| | sLsLLsL sLLsL

| | 4th Meantone[12]

| | sLLsLsL sLLsL

| | 5th Meantone[12]

| | sLLsLsL LsLsL

| | 6th Meantone[12]

| | LsLsLsL LsLsL

| | 7th Meantone[12]

| | LsLsLLs LsLsL

 

[[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.

| | 1st Porcupine[7]

|1st Triyo[7]

| | ssss ssL

| | C vD ^Eb F vG ^Ab Bb C

| | <u>'''C'''</u> vD ^Eb F vG ^Ab Bb

| | 2nd Porcupine[7]

|2nd Triyo[7]

| | ssss sLs

| | C vD ^Eb F vG ^Ab ^Bb C

| | ^Bb <u>'''C'''</u> vD ^Eb F vG ^Ab

| | 3rd Porcupine[7]

|3rd Triyo[7]

| | ssss Lss

| | C vD ^Eb F vG vA ^Bb C

| | vA ^Bb <u>'''C'''</u> vD ^Eb F vG

| | 4th Porcupine[7]

|4th Triyo[7]

| | sssL sss

| | C vD ^Eb F G vA ^Bb C

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

| | 5th Porcupine[7]

|5th Triyo[7]

| | ssLs sss

| | C vD ^Eb ^F G vA ^Bb C

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

| | 6th Porcupine[7]

|6th Triyo[7]

| | sLss sss

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

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

| | 7th Porcupine[7]

|7th Triyo[7]

| | Lsss sss

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

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

| | 1st Sensi[8]

| | 2nd Sensi[8]

| | ssLsL ssL

| | C ^^Db ^⁴Ebb ^³Fb vvF# v³G#  vA  ^Bb C

| | v³G# <u>'''C'''</u> ^³Fb vA ^^Db vvF# ^Bb ^⁴Ebb

| | 3rd Sensi[8]

|3rd Sepgu[8]

| | sLssL ssL

| | C ^^Db  ^Eb ^³Fb vvF# v³G#  vA  ^Bb C

| | ^Eb v³G# <u>'''C'''</u> ^³Fb vA ^^Db vvF# ^Bb

| | 4th Sensi[8]

|4th Sepgu[8]

| | sLssL sLs

| | C ^^Db  ^Eb ^³Fb vvF# v³G#  vA  vvB C

| | vvB ^Eb v³G# <u>'''C'''</u> ^³Fb vA ^^Db vvF#

| | 5th Sensi[8]

|5th Sepgu[8]

| | sLsLs sLs

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

| | ^^Gb vvB ^Eb v³G# <u>'''C'''</u> ^³Fb vA ^^Db

| | 6th Sensi[8]

|6th Sepgu[8]

| | LssLs sLs

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

| | vD ^^Gb vvB ^Eb v³G# <u>'''C'''</u> ^³Fb vA

| | 7th Sensi[8]

|7th Sepgu[8]

| | LssLs Lss

| | C   vD    ^Eb  ^³Fb ^^Gb v³G# v⁴A# vvB C

| | v⁴A# vD ^^Gb vvB ^Eb v³G# <u>'''C'''</u> ^³Fb

| | 8th Sensi[8]

|8th Sepgu[8]

| | LsLss Lss

| | C   vD    ^Eb    F   ^^Gb v³G# v⁴A# vvB C

| | F v⁴A# vD ^^Gb vvB ^Eb v³G# <u>'''C'''</u>

[[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:

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

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

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

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

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

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

* 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)

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.

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

|+ Porcupine[7]/Triyo[7] MODMOS scale examples

! | example in C

|4th Porcupine[7] #2

|4th Triyo[7] #2

|C D ^Eb F G vA ^Bb C

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

|4th Porcupine[7] #2 b6

|4th Triyo[7] #2 b6

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

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

|-

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

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

|-

|4th Triyo[7] b6 b7

|C vD ^Eb F G ^Ab Bb C

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

|-

| |5th Triyo[7] #2

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

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

|-

| | C vD vE F G vA ^Bb C

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

|-

| |7th Porcupine[7] #6 #7

| |7th Triyo[7] #6 #7

| | C D vE ^F G A vB C

|-

|7th Porcupine[7] #7

|7th Triyo[7] #7

|C D vE ^F G vA vB C

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

|-

|C D vE F G vA vB C

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

|-

| | C D vE F G vA ^Bb C

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

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.

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

! | Ls pattern

| | 1st Srutal[2x5]

| | 1st Sagugu[2x5]

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

| | vF# vC# vG# vD# vA#

| | 2nd Srutal[2x5]

| | 2nd Sagugu[2x5]

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

| | vB vF# vC# vG# vD#

| | 3rd Srutal[2x5]

| | 3rd Sagugu[2x5]

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

| | vE vB vF# vC# vG#

| | 4th Srutal[2x5]

| | 4th Sagugu[2x5]

| | C vC# Eb vE F vF# G vA Bb vB C

| | vA vE vB vF# vC#

| | 5th Srutal[2x5]

| | 5th Sagugu[2x5]

| | C vD Eb vE F vF# Ab vA Bb vB C

| | vD vA vE vB vF#

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).

|-

|-

| | 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.

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

! | Ls pattern

| | 1st Diminished[4x2]

| | 1st Quadgu[4x2]

| | C ^^C# vEb ^E ^^F# G ^A vBb C

| | vEb vBb

| | ^^F# ^^C#

| | ^A ^E

| | 2nd Diminished[4x2]

|2nd Quadgu[4x2]

| | Ls-Ls-Ls-Ls

| | C ^D vEb F ^^F# vAb ^A ^^B C

| | vAb vEb

| |^^B ^^F#

| | ^D ^A

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.

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

! | Ls pattern

| | 1st Blackwood[5x2]

|1st 5edo+ya[5x2]

| | 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]

| | 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.

! | smLX pattern

| | C 2nd Meantone[8]

| | LLms mLLm

| | C 3rd Meantone[7] add #4

| | LLms mLmL

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

| | LmLs mmXm

| | A 3rd Meantone[9]

| | LLmsm Lmsm

| | A 5th Meantone[7] add #4, #7

| | LmLsm mLsm

| | F 1st Meantone[6]

| | mmL mms

| | G 2nd Meantone[7] no3

| | mLm mms

| | F 2nd Meantone[7] no4 no6

| | mmL Xs

| style="text-align:center;" | "  

| | F 1st Meantone[7] no4 no6

| | A 5th Meantone[7] no4 no7

| | msL sL

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...

== Non-heptatonic scales ==

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.

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 fourthward side is sharp (assuming a fifth &lt; 720¢). Use "+" for fifthwards and "-" for fourthwards.

* 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

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.

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

! | Ls pattern

! | 19-edo example in C

! | 19-edo genchain

| | 5th Sensi[8]

|5th Sepgu[8]

| | sLsL ssLs

| | 5th Sensi[8]

|5th Sepgu[8]

| | 5th Sensi[8] +7

|5th Sepgu[8] +7

| | sLsL sssL

| | 5th Sensi[8] +8

|5th Sepgu[8] +8

 

{| class="wikitable"

 

== Generalization to temperament-agnostic MOS scales ==

The modes of the sister MOS 3L 5s are the same, just exchange L and s:

* etc.

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

 

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

"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.

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 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.

A larger problem is that choosing the chroma-positive generator only applies to MOS and MODMOS scales, and breaks down when the length of the genchain results in a non-MOS scale. Mode Numbers notation can be applied to scales like Meantone[8], which while not a MOS, is certainly musically useful.

| | Meantone[2]

| | Meantone[3]

| | Meantone[4]

| | Meantone[5]

| | Meantone[6]

| | Meantone[7]

| | Meantone[8]

| | Meantone[9]

| | Meantone[10]

| | Meantone[11]

| | Meantone[12] if generator < 700¢