Genchain mode numbering
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 scales
MOS scales are formed from a segment of the 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.
| 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 | F C G D A E B |
| Ionian (major) | 2nd Meantone[7] | LLsL LLs | C D E F G A B C | F C G D A E B |
| Mixolydian | 3rd Meantone[7] | LLsL LsL | G A B C D E F G | F C G D A E B |
| Dorian | 4th Meantone[7] | LsLL LsL | D E F G A B C D | F C G D A E B |
| Aeolian (minor) | 5th Meantone[7] | LsLL sLL | A B C D E F G A | F C G D A E B |
| Phrygian | 6th Meantone[7] | sLLL sLL | E F G A B C D E | F C G D A E B |
| Locrian | 7th Meantone[7] | sLLs LLL | B C D E F G A B | F C G D A E B |
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 number, 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).
| 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 | C G D A E B F# |
| Ionian (major) | 2nd Meantone[7] | LLsL LLs | C D E F G A B C | F C G D A E B ---- |
| Mixolydian | 3rd Meantone[7] | LLsL LsL | C D E F G A Bb C | Bb F C G D A E ------- |
| Dorian | 4th Meantone[7] | LsLL LsL | C D Eb F G A Bb C | -------------- Eb Bb F C G D A |
| Aeolian (minor) | 5th Meantone[7] | LsLL sLL | C D Eb F G Ab Bb C | --------- Ab Eb Bb F C G D |
| Phrygian | 6th Meantone[7] | sLLL sLL | C Db Eb F G Ab Bb C | ---- Db Ab Eb Bb F C G |
| Locrian | 7th Meantone[7] | sLLs LLL | C Db Eb F Gb Ab Bb C | Gb Db Ab Eb Bb F C |
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). Unlike modal UDP notation, the generator isn't always chroma-positive. There are several disadvantages of only using chroma-positive generators. See the critique of UDP in the § Rationale section below.
Pentatonic meantone scales:
| old scale name | new scale name | Ls pattern | example in C | --------- genchain ------- |
|---|---|---|---|---|
| major pentatonic | 1st Meantone[5] | ssL sL | C D E G A C | C G D A E |
| 2nd Meantone[5] | sLs sL | C D F G A C | F C G D A -- | |
| 3rd Meantone[5] | sLs Ls | C D F G Bb C | -------- Bb F C G D | |
| minor pentatonic | 4th Meantone[5] | Lss Ls | C Eb F G Bb C | ---- Eb Bb F C G |
| 5th Meantone[5] | LsL ss | C Eb F Ab Bb C | Ab Eb Bb F C |
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.
| scale name | Ls pattern (assumes a generator < 700¢) |
example in C | genchain |
|---|---|---|---|
| 1st Meantone[12] | sLsLsLL sLsLL | C C# D D# E E# F# G G# A A# B C | C G D A E B F# C# G# D# A# E# |
| 2nd Meantone[12] | sLsLLsL sLsLL | C C# D D# E F F# G G# A A# B C | F C G D A E B F# C# G# D# A# |
| 3rd Meantone[12] | sLsLLsL sLLsL | C C# D D# E F F# G G# A Bb B C | Bb F C G D A E B F# C# G# D# |
| 4th Meantone[12] | sLLsLsL sLLsL | C C# D Eb E F F# G G# A Bb B C | Eb Bb F C G D A E B F# C# G# |
| 5th Meantone[12] | sLLsLsL LsLsL | C C# D Eb E F F# G Ab A Bb B C | Ab Eb Bb F C G D A E B F# C# |
| 6th Meantone[12] | LsLsLsL LsLsL | C Db D Eb E F F# G Ab A Bb B C | Db Ab Eb Bb F C G D A E B F# |
| 7th Meantone[12] | LsLsLLs LsLsL | C Db D Eb E F Gb G Ab A Bb B C | Gb Db Ab Eb Bb F C G D A E B |
| etc. |
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 interval is v3A1. Because the generator is a 2nd, the genchain resembles the scale.
| scale name | color name | Ls pattern | example in C | genchain |
|---|---|---|---|---|
| 1st Porcupine[7] | 1st Triyo[7] | ssss ssL | C vD ^Eb F vG ^Ab Bb C | C vD ^Eb F vG ^Ab Bb |
| 2nd Porcupine[7] | 2nd Triyo[7] | ssss sLs | C vD ^Eb F vG ^Ab ^Bb C | ^Bb C vD ^Eb F vG ^Ab |
| 3rd Porcupine[7] | 3rd Triyo[7] | ssss Lss | C vD ^Eb F vG vA ^Bb C | vA ^Bb C vD ^Eb F vG |
| 4th Porcupine[7] | 4th Triyo[7] | sssL sss | C vD ^Eb F G vA ^Bb C | G vA ^Bb C vD ^Eb F |
| 5th Porcupine[7] | 5th Triyo[7] | ssLs sss | C vD ^Eb ^F G vA ^Bb C | ^F G vA ^Bb C vD ^Eb |
| 6th Porcupine[7] | 6th Triyo[7] | sLss sss | C vD vE ^F G vA ^Bb C | vE ^F G vA ^Bb C vD |
| 7th Porcupine[7] | 7th Triyo[7] | Lsss sss | C D vE ^F G vA ^Bb C | D vE ^F G vA ^Bb C |
Sensi aka Sepgu has pergen (P8, ccP5/7). The ~9/7 generator is both a ^3d4 and a v4A3, and the enharmonic interval is ^7dd2.
| scale name | color name | Ls pattern | example in C | genchain |
|---|---|---|---|---|
| 1st Sensi[8] | 1st Sepgu[8] | ssLss LsL | C ^^Db ^4Ebb ^3Fb vvF# G vA ^Bb C | C ^3Fb vA ^^Db vvF# ^Bb ^4Ebb G |
| 2nd Sensi[8] | 2nd Sepgu[8] | ssLsL ssL | C ^^Db ^4Ebb ^3Fb vvF# v3G# vA ^Bb C | v3G# C ^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# C ^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# C ^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# C ^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# C ^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# C ^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# C |
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 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:
| old scale name | new scale name | Lms pattern | example in A | genchain |
|---|---|---|---|---|
| Harmonic minor | 5th Meantone[7] #7 | msmm sLs | A B C D E F G# A | F C * D A E B * * G# |
| Ascending melodic minor | 5th Meantone[7] #6 #7 | LsLL LLs | A B C D E F# G# A | C * D A E B F# * G# |
| (Major with b3) | (2nd Meantone[7] b3) | " | " | " |
| (Dorian with #7) | (4th Meantone[7] #7) | " | " | " |
| Double harmonic minor | 5th Meantone[7] #4 #7 | msLs sLs | A B C D# E F G# A | F C * * A E B * * G# D# |
| (Lydian with b3 b6) | (1st Meantone[7] b3 b6) | " | " | " |
| Double harmonic major | 2nd Meantone[7] b2 b6 | sLsm sLs | A Bb C# D E F G# A | Bb F * * D A E * * C# G# |
| (Phrygian with #3 #7) | (6th Meantone[7] #3 #7) | " | " | " |
| Hungarian gypsy minor | 5th Meantone[7] #4 | msLs smm | A B C D# E F G A | F C G * A E B * * * D# |
| Phrygian dominant | 6th Meantone[7] #3 | sLsm smm | A Bb C# D E F G A | Bb F * G D A E * * C# |
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.
The advantage of ambiguous names is that one can choose the mode number. If a piece changes from a MOS scale to a MODMOS scale, one can describe both scales with the same mode number. For example, a piece might change from D dorian to D melodic minor. In this context, melodic minor might better be described as an altered dorian scale.
Unlike MOS scales, adjacent MODMOS modes differ by more than one note. Harmonic minor modes:
- 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)
- 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")
- 3rd Meantone[7] #4: C D E F# G A Bb C
- 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.
| scale name | color name | Lms pattern | example in C | genchain |
|---|---|---|---|---|
| 4th Porcupine[7] #2 | 4th Triyo[7] #2 | LsmL mmm | C D ^Eb F G vA ^Bb C | D * * G vA ^Bb C * ^Eb F |
| 4th Porcupine[7] #2 b6 | 4th Triyo[7] #2 b6 | LsmL sLm | C D ^Eb F G ^Ab ^Bb C | D * * G * ^Bb C * ^Eb F* ^Ab |
| 4th Porcupine[7] b6 | 4th Triyo[7] b6 | mmmL sLm | C vD ^Eb F G ^Ab ^Bb C | G * ^Bb C vD ^Eb F * ^Ab |
| 4th Porcupine[7] b6 b7 | 4th Triyo[7] b6 b7 | mmmL smL | C vD ^Eb F G ^Ab Bb C | G * * C vD ^Eb F * ^Ab Bb |
| 5th Porcupine[7] #2 | 5th Triyo[7] #2 | LsLm mmm | C D ^Eb ^F G vA ^Bb C | D * ^F G vA ^Bb C * ^Eb |
| 6th Porcupine[7] b4 | 6th Triyo[7] b4 | mLsL mmm | C vD vE F G vA ^Bb C | vE * G vA ^Bb C vD * F |
| 7th Porcupine[7] #6 #7 | 7th Triyo[7] #6 #7 | Lmmm Lms | C D vE ^F G A vB C | A vB * D vE ^F G * * C |
| 7th Porcupine[7] #7 | 7th Triyo[7] #7 | Lmmm mLs | C D vE ^F G vA vB C | vB * D vE ^F G vA * C |
| 7th Porcupine[7] b4 #7 | 7th Triyo[7] b4 #7 | LmsL mLs | C D vE F G vA vB C | vB * D vE * G vA * C * * F |
| 7th Porcupine[7] b4 | 7th Triyo[7] b4 | LmsL mmm | C D vE F G vA ^Bb C | D vE * G vA ^Bb C * * F |
Temperaments with split octaves
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.
| scale name | color name | Ls pattern | example in C | 1st genchain | 2nd genchain |
|---|---|---|---|---|---|
| 1st Srutal[2x5] | 1st Sagugu[2x5] | ssssL-ssssL | C vC# D vD# E vF# G vG# A vA# C | C G D A E | vF# vC# vG# vD# vA# |
| 2nd Srutal[2x5] | 2nd Sagugu[2x5] | sssLs-sssLs | C vC# D vD# F vF# G vG# A vB C | F C G D A | vB vF# vC# vG# vD# |
| 3rd Srutal[2x5] | 3rd Sagugu[2x5] | ssLss-ssLss | C vC# D vE F vF# G vG# Bb vB C | Bb F C G D | vE vB vF# vC# vG# |
| 4th Srutal[2x5] | 4th Sagugu[2x5] | sLsss-sLsss | C vC# Eb vE F vF# G vA Bb vB C | Eb Bb F C G | vA vE vB vF# vC# |
| 5th Srutal[2x5] | 5th Sagugu[2x5] | Lssss-Lssss | C vD Eb vE F vF# Ab vA Bb vB C | Ab Eb Bb F 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 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).
| scale name | color name | Ls pattern | example in C | 1st chain | 2nd chain | 3rd chain |
|---|---|---|---|---|---|---|
| 1st Augmented[3x3] | 1st Trigu[3x3] | Lss-Lss-Lss | C D ^Eb vE vF# G ^Ab ^Bb vB C | C G D | vE vB vF# | ^Ab ^Eb ^Bb |
| 2nd Augmented[3x3] | 2nd Trigu[3x3] | sLs-sLs-sLs | C ^Db ^Eb vE F G ^Ab vA vB C | F C G | vA vE vB | ^Db ^Ab ^Eb |
| 3rd Augmented[3x3] | 3rd Trigu[3x3] | ssL-ssL-ssL | C ^Db vD vE F ^Gb ^Ab vA Bb C | Bb F C | vD vA vE | ^Gb ^Db ^Ab |
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.
| scale name | color name | Ls pattern | example in C | 1st chain | 2nd chain | 3rd chain | 4th chain |
|---|---|---|---|---|---|---|---|
| 1st Diminished[4x2] | 1st Quadgu[4x2] | sL-sL-sL-sL | C ^^C# vEb ^E ^^F# G ^A vBb C | C G | 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 | F 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 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.
| scale name | 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 | C-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 | ^Ab-C, ^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.
| scale | genchain | name | smLX pattern |
|---|---|---|---|
| octotonic: | (assumes 3/2 < 700¢) | ||
| C D E F F# G A B C | F C G D A E B F# | C 2nd Meantone[8] | LLms mLLm |
| C D E F F# G A Bb C | Bb F C G D A E * F# | C 3rd Meantone[7] add #4 | LLms mLmL |
| A B C D D# E F G# A | F C * D A E B * * G# D# | A 5th Meantone[7] #7 add #4 | LmLs mmXm |
| nonatonic: | (X = extra large) | ||
| A B C# D D# E F# G G# A | G D A E B F# C# G# D# | A 3rd Meantone[9] | LLmsm Lmsm |
| A B C D D# E F G G# A | F C G D A E B * * G# D# | A 5th Meantone[7] add #4, #7 | LmLsm mLsm |
| hexatonic: | |||
| F G A C D E F | F C G D A E | F 1st Meantone[6] | mmL mms |
| G A C D E F# G | C G D A E * F# | G 2nd Meantone[7] no3 | mLm mms |
| pentatonic: | |||
| F G A C E F | F C G * A E | F 2nd Meantone[7] no4 no6 | mmL Xs |
| " | " | F 1st Meantone[7] no4 no6 | " |
| A B C E F A | F C * * A E B | A 5th Meantone[7] no4 no7 | msL sL |
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...
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.
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 < 720¢). Use "+" for fifthwards and "-" for fourthwards.
Using J K L M N for note names, and arbitrarily centering the genchain on L, we get this genchain:
...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:
- L 3rd Meantone[5] add -2, +5
- L 2nd 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. 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.
| notation | scale name | color name | Ls pattern | 19-edo example in C | 19-edo genchain |
|---|---|---|---|---|---|
| heptatonic | 5th Sensi[8] | 5th Sepgu[8] | sLsL ssLs | C Db Eb E# Gb G# A B C | Gb B Eb G# C E# A Db |
| octotonic | 5th Sensi[8] | 5th Sepgu[8] | " | C D E# F G# H A B# C | G# B# E# H C F A D |
| heptatonic | 5th Sensi[8] +7 | 5th Sepgu[8] +7 | sLsL sssL | C Db Eb E# Gb G# A Bb C | Gb * Eb G# C E# A Db * Bb |
| octotonic | 5th Sensi[8] +8 | 5th Sepgu[8] +8 | " | C D E# F G# H A B C | G# * E# H C F A D * B |
Heptatonic fifth-based notation:
C - C# - Db - D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C
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
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.
MODMOS scales of split-octave temperaments are named as usual:
| scale name | color name | Ls pattern | example in C | 1st genchain | 2nd genchain |
|---|---|---|---|---|---|
| 1st Srutal[2x5] | 1st Sagugu[2x5] | ssssL-ssssL | C vC# D vD# E vF# G vG# A vA# C | C G D A E | vF# vC# vG# vD# vA# |
| 1st Srutal[2x5] b2 b5 | 1st Sagugu[2x5] b2 b5 | sLmmL-sLmmL | C vB# D vD# E vF# F# vG# A vA# C | C * D A E * F# | vF# * vG# vD# vA# * vB# |
| 1st Srutal[2x5] b2 | 1st Sagugu[2x5] b2 | sLmmL-mmmmL | C vB# D vD# E vF# G vG# A vA# C | C G D A E | vF# * vG# vD# vA# * vB# |
Generalization to temperament-agnostic MOS scales
Abstract MOS patterns like 5L 3s are not specific temperaments in which specific commas vanish. Thus there are no ratios other than the octave 2/1 (or more generally the equave 3/1 or whatever). Genchain mode numbers can be applied to these patterns. For example, 5L 3s has a generator in the 450-480¢ range. The "[8]" is redundant, so we drop it to get
- 1st 5L 3s = LLsLLsLs
- 2nd 5L 3s = LLsLsLLs
- 3rd 5L 3s = LsLLsLLs
- etc.
The modes of the sister MOS 3L 5s are the same, just exchange L and s:
- 1st 3L 5s = ssLssLsL
- 2nd 3L 5s = ssLsLssL
- 3rd 3L 5s = sLssLssL
- etc.
In this context, when no specific temperament is named, the choice of generator follows slightly different rules. Prioritizing 3/2 over 4/3 doesn't make any sense, because the generator doesn't have an actual ratio. So the generator is always chosen to be as small as possible, i.e. less than half of a period. Thus while Meantone[7]'s generator is 3/2, i.e. a 5th, 5L 2s's generator is in the 480-514¢ range, i.e. a 4th.
Rationale
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.
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 disadvantage of genchain-based numbering is that the mode numbers are harder to relate to the scale. However this is arguably an advantage, because in the course of learning to relate the mode numbers, one internalizes the genchain.
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)." — [1]
"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." — [2]
"Meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths." — [3]
"In this system the perfect fifth is flattened by one quarter of a syntonic comma." — [4]
"The term "well temperament" or "good temperament" usually means some sort of irregular temperament in which the tempered fifths are of different sizes." — [5]
"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?
See below.
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.
| scale | UDP generator | UDP genchain | Mode Numbers generator | Mode Numbers genchain |
|---|---|---|---|---|
| Meantone[5] in 31edo | 4/3 | E A D G C | 3/2 | C G D A E |
| Meantone[7] in 31edo | 3/2 | C G D A E B F# | 3/2 | C G D A E B F# |
| Meantone[12] in 31edo | 4/3 | E# A# D# G# C# F#
B E A D G C |
3/2 | C G D A E B F# C# G#
D# A# E# |
| Meantone[19] in 31edo | 3/2 | C G D A E B F# C#
G# D# A# E# B# FxCx Gx Dx Ax Ex |
3/2 | C G D A E B F# C# G#
D# A# E# B# Fx Cx Gx Dx Ax Ex |
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.
| scale | UDP genchain | Mode Numbers genchain |
|---|---|---|
| Meantone[2] | C G | C G |
| Meantone[3] | D G C | C G D |
| Meantone[4] | ??? | C G D A |
| Meantone[5] | E A D G C | C G D A E |
| Meantone[6] | ??? | G C D A E B |
| Meantone[7] | C G D A E B F# | C G D A E B F# |
| Meantone[8] | ??? | C G D A E B F# C# |
| Meantone[9] | ??? | C G D A E B F# C# G# |
| Meantone[10] | ??? | C G D A E B F# C# G# D# |
| Meantone[11] | ??? | C G D A E B F# C# G# D# A# |
| Meantone[12] if generator < 700¢ | E# A# D# G# C# F# B E A D G C | C G D A E B F# C# G# D# A# E# |
| Meantone[12] if generator > 700¢ | C G D A E B F# C# G# D# A# E# | C G D A E B F# C# G# D# A# E# |
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 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.
Furthermore, UDP uses the more mathematical zero-based numbering and Mode Numbers notation uses the more intuitive one-based counting. UDP is mathematician-oriented whereas Mode Numbers notation is musician-oriented.