Kite's Genchain mode numbering: Difference between revisions

'''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.

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.

| | 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".

| | 1st Meantone [7]

| | 2nd Meantone [7]

| | 3rd Meantone [7]

| | 4th Meantone [7]

| | 5th Meantone [7]

| | 6th Meantone [7]

| | 7th Meantone [7]

| | 1st Meantone [5]

| | 2nd Meantone [5]

| | 3rd Meantone [5]

| | 4th Meantone [5]

| | 5th Meantone [5]

| | 1st Meantone [12]

| | 2nd Meantone [12]

| | 3rd Meantone [12]

| | 4th Meantone [12]

| | 5th Meantone [12]

| | 6th Meantone [12]

| | 7th Meantone [12]

'''[[Sensi]] [8] aka Sepgu''' has a ~9/7 generator. The [[pergen]] is (P8, WWP5/7). Sensi[8] modes in 19edo (gen = 7\19, L = 3\19, s = 2\19):  

| | 1st Sensi [8]

|1st Sepgu [8]

| | 2nd Sensi [8]

|2nd Sepgu [8]

| | 3rd Sensi [8]

|3rd Sepgu [8]

| | 4th Sensi [8]

|4th Sepgu [8]

| | 5th Sensi [8]

|5th Sepgu [8]

| | 6th Sensi [8]

|6th Sepgu [8]

| | 7th Sensi [8]

|7th Sepgu [8]

| | 8th Sensi [8]

|8th Sepgu [8]

| | 1st Porcupine [7]

|1st Triyo [7]

| | 2nd Porcupine [7]

|2nd Triyo [7]

| | 3rd Porcupine [7]

|3rd Triyo [7]

| | 4th Porcupine [7]

|4th Triyo [7]

| | 5th Porcupine [7]

|5th Triyo [7]

| | 6th Porcupine [7]

|6th Triyo [7]

| | 7th Porcupine [7]

|7th Triyo [7]

[[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:

| | 5th Meantone [7] #7

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

| | 2nd Meantone [7] b3

| | 4th Meantone [7] #7

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

| | 1st Meantone [7] b3 b6

| | 2nd Meantone [7] b2 b6

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

| | 5th Meantone [7] #4

| | 6th Meantone [7] #3

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

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<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

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

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

|7th Porcupine [7] #7

|7th Triyo [7] #7

| |5th Porcupine [7] #2

| |5th Triyo [7] #2

|4th Porcupine [7] #2

|4th Triyo [7] #2

| | 7th Porcupine [7] b4

| |7th Triyo [7] b4

| | 6th Porcupine [7] b4

| |6th Triyo [7] b4

| | 4th Porcupine [7] b6

| |4th Triyo [7] b6

|4th Porcupine [7] b6 b7

|4th Triyo [7] b6 b7

|7th Porcupine [7] b4 #7

|7th Triyo [7] b4 #7

|4th Porcupine [7] #2 b6

|4th Triyo [7] #2 b6

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

| | 1st Srutal [10]

| | 1st Sagugu [10]

| | 2nd Srutal [10]

| | 2nd Sagugu [10]

| | 3rd Srutal [10]

| | 3rd Sagugu [10]

| | 4th Srutal [10]

| | 4th Sagugu [10]

| | 5th Srutal [10]

| | 5th Sagugu [10]

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

| | 2nd Diminished [8]

[[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:

| | 1st Blackwood [10]

| | 2nd Blackwood [10]

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:

| | C 2nd Meantone [8]

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

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

| | A 5th Meantone [7] #7 add #4 no6

| | A 3rd Meantone [9]

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

| | F 1st Meantone [6]

| | G 2nd Meantone [7] no3

| | F 2nd Meantone [7] no4 no6

| | F 1st Meantone [7] no4 no6

| | A 5th Meantone [7] no4 no7

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 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.

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

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.

| | 5th Sensi [8]

|5th Sepgu [8]

| | 5th Sensi [8]

|5th Sepgu [8]

| | 5th Sensi [8] +7

|5th Sepgu [8] +7

| | 5th Sensi [8] +8

|5th Sepgu [8] +8

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.)

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 &lt; 700¢

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

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.