User:Overthink/Neutral scale theory
Neutral scales are generated by a neutral third of around 350 cents, which exists in 17edo, 24edo, and many other tunings. The generator can be taken to represent 11/9, and two generators reach 3/2, tempering out the comma 243/242. This generator leads to MOSes of size 4, 7, 10, 17, etc. Here we will mainly be considering the 7-note 3L 4s and 10-note 7L 3s MOSes.
Note: Similar theory can be used in the 2.3.13 temperament tempering out 512/507.
3L 4s (mosh)
Intervals
The 3L 4s MOS has a step pattern of LssLsLs, where L is a major second and s is a neutral second. We borrow note names from 5L 2s (diatonic) as C, D, E, F, G, A, B. We use sharp (typed as #) and flat (typed as b) to denote an increment and decrement by L-s, or a quarter tone. A table of intervals of this scale in 24edo is below. Note that other tunings can be used, but overall structures are the same.
| Note | Cents | Ratio |
|---|---|---|
| C | 0 | 1/1 |
| D | 200 | 9/8 |
| E | 350 | 11/9 |
| F | 500 | 4/3 |
| G | 700 | 3/2 |
| A | 850 | 18/11 |
| B | 1050 | 11/6 |
| C | 1200 | 2/1 |
Just like diatonic, this scale has major and minor intervals, which are found on different notes of the scale. A table of such intervals is below.
| Interval | Cents | Ratio | note on C |
|---|---|---|---|
| P1 | 0 | 1/1 | C |
| A1 | 50 | 33/32 | C# |
| m2 | 150 | 12/11 | Db |
| M2 | 200 | 9/8 | D |
| d3 | 300 | 32/27 | Eb |
| P3 | 350 | 11/9 | E |
| m4 | 500 | 4/3 | F |
| M4 | 550 | 11/8 | F# |
| m5 | 650 | 16/11 | Gb |
| M5 | 700 | 3/2 | G |
| P6 | 850 | 18/11 | A |
| A6 | 900 | 27/16 | A# |
| m7 | 1000 | 16/9 | Bb |
| M7 | 1050 | 11/6 | B |
| d8 | 1150 | 64/33 | Cb |
| P8 | 1200 | 2/1 | C |
Chords
In diatonic, chords are built by Root-3rd-5th. We can try to use the same logic here. This gives us neutral (P1-P3-M5) chords on C, E, F, G, and A, a P1-d3-m5 chord on D, and a P1-P3-m5 chord on B. This construction, however, is flawed, as none of these chords can be written with particularly simple ratios, and therefore these chords aren't very consonant.
This scale is in the 2.3.11 subgroup, with the fundamental otonal consonance being 8:9:11:12 (or one of its different voicings), which in our scale is P1-M2-M4-M5. A different construction, Root-2nd-4th-5th, is now needed. The utonal inverse of 8:9:11:12 is 1/(8:9:11:12) = 66:72:88:99, or P1-m2-m4-M5. The otonal 8:9:11:12 chord occurs on F and A, and the utonal 1/(8:9:11:12) chord occurs on E and G. Neither chord occurs on C, B, or D, with a P1-M2-m4-M5 cent chord on C, and a P1-m2-m4-m5 chord on B and D. We need names for these chords, and once again we can borrow from diatonic:
Rewrite: This is not the best construction. Chords are better constructed as a chain of three fourths (4/3 or 11/8) or fifths (3/2 or 16/11).
| Chord | Name |
|---|---|
| P1-M2-M4-M5 | Major / M |
| P1-m2-m4-M5 | Minor / m |
| P1-M2-m4-M5 | Suspended / sus |
| P1-m2-m4-m5 | Placeholder |