Rastmic chords
Rastmic chords are essentially tempered chords tempered by the rastma, 243/242, in the 2.3.11 subgroup in the 11-odd-limit.
The count of chords is 3 triads, 7 tetrads, 5 pentads, and 1 hexad, for a total of 16.
There are three rastmic triads: the classic neutral triad,
- 1–11/9–3/2 with steps of 11/9, 11/9, 4/3;
and an inversely related pair of triads,
- 1–9/8–11/6 with steps of 9/8, 18/11, 12/11, and its inverse
- 1–9/8–11/9 with steps of 9/8, 12/11, 18/11.
Rastmic tetrads are seven in number: three palindromic tetrads,
- 1–11/9–3/2–11/6 with steps of 11/9, 11/9, 11/9, 12/11 (→ neutral tetrad);
- 1–3/2–18/11–11/6 with steps of 3/2, 12/11, 9/8, 12/11;
- 1–9/8–11/9–11/8 with steps of 9/8, 12/11, 9/8, 16/11;
and two inversely related pairs of tetrads,
- 1–11/9–11/8–3/2 with steps of 11/9, 9/8, 12/11, 4/3, and its inverse
- 1–12/11–11/9–3/2 with steps of 12/11, 9/8, 11/9, 4/3;
- 1–9/8–11/9–3/2 with steps of 9/8, 12/11, 11/9, 4/3, and its inverse
- 1–11/9–4/3–3/2 with steps of 11/9, 12/11, 9/8, 4/3.
There are five rastmic pentads: the palindromic pentad,
- 1–9/8–11/9–3/2–11/6 with steps of 9/8, 12/11, 11/9, 11/9, 12/11;
and two inversely related pairs of pentads,
- 1–9/8–11/8–3/2–11/6 with steps of 9/8, 11/9, 12/11, 11/9, 12/11, and its inverse
- 1–12/11–4/3–3/2–18/11 with steps of 12/11, 11/9, 9/8, 12/11, 11/9;
- 1–9/8–11/9–11/8–3/2 with steps of 9/8, 12/11, 9/8, 12/11, 4/3, and its inverse
- 1–12/11–11/9–4/3–3/2 with steps of 12/11, 9/8, 12/11, 9/8, 4/3.
There is also a unique rastmic hexad:
- 1–9/8–11/9–11/8–3/2–11/6 with steps of 9/8, 12/11, 9/8, 12/11, 11/9, 12/11.
Equal temperaments with rastmic chords include 10, 17, 24, 31, 41, 58, 72, 130, 202, 736be, 938be, 1075be, 1116be, 1277be and 1318be.
| Chord by Pitches | Chord by Intervals |
|---|---|
| 1–11/9–3/2 | 11/9, 11/9, 4/3 |
| 1–9/8–11/6 | 9/8, 18/11, 2/11 |
| 1–9/8–11/9 | 9/8, 12/11, 18/11 |
| 1–11/9–3/2–11/6 | 11/9, 11/9, 11/9, 12/11 |
| 1–3/2–18/11–11/6 | 3/2, 12/11, 9/8, 12/11 |
| 1–9/8–11/9–11/8 | 9/8, 12/11, 9/8, 16/11 |
| 1–11/9–11/8–3/2 | 11/9, 9/8, 12/11, 4/3 |
| 1–12/11–11/9–3/2 | 12/11, 9/8, 11/9, 4/3 |
| 1–9/8–11/9–3/2 | 9/8, 12/11, 11/9, 4/3 |
| 1–11/9–4/3–3/2 | 11/9, 12/11, 9/8, 4/3 |
| 1–9/8–11/9–3/2–11/6 | 9/8, 12/11, 11/9, 11/9, 12/11 |
| 1–9/8–11/8–3/2–11/6 | 9/8, 11/9, 12/11, 11/9, 12/11 |
| 1–12/11–4/3–3/2–18/11 | 12/11, 11/9, 9/8, 12/11, 11/9 |
| 1–9/8–11/9–11/8–3/2 | 9/8, 12/11, 9/8, 12/11, 4/3 |
| 1–12/11–11/9–4/3–3/2 | 12/11, 9/8, 12/11, 9/8, 4/3 |
| 1–9/8–11/9–11/8–3/2–11/6 | 9/8, 12/11, 9/8, 12/11, 11/9, 12/11 |
Rastgross heptad
Rastmic chords can be extended to the neutralized diatonic scale, LsLsLss, which is Neutral[7]. In neutral, with the neutral third (~11/9) as generator, the rastmic hexad is a chain of five neutral thirds rather than the six which give Neutral[7], which therefore has two rastmic hexads and of course many more smaller rastmic chords.
In the 2.3.11.13 subgroup, this scale is interpreted as an essentially tempered heptad, the rastgross heptad, tempered by 144/143 (grossma) and 243/242. This heptad is 1–9/8–11/9–11/8–3/2–22/13–11/6 with steps of 9/8, 12/11, 9/8, 12/11, 9/8, 12/11, 12/11 (→ rastgross1).