Cluster MOS
A cluster MOS or cluster scale is a very particular kind of MOS-based system (i.e. a system based on stacks of periods and generators) whose generator is quite near a rational fraction of an octave. Therefore some MOS generated by the generator is quasi-equal (which should be reasonably sized for it to be a good cluster MOS, usually between 5 and 10 notes per octave). But not only that; in a cluster temperament, the different versions of each interval, differing by a chroma ("diminished", "minor", "major", "augmented"...) include many nearby interval colors that are individually recognizable, yet conceptually grouped into the same category (or "cluster") because they're so close.
A cluster temperament (named by Keenan Pepper) is a rank-2 regular temperament interpretation of a cluster MOS. This means that in a cluster temperament, many different versions of each interval, differing by a chroma, these colors represent nearby JI intervals specifically (because a temperament is a JI interpretation of MOS generator chains separated by the period).
An example of something that is not a cluster temperament is amity, because although the amity generator is within 4 cents of 2\7, making amity[7] near equal, amity is too complex of a temperament and most of the intervals differing by a chroma do not represent simple JI intervals at all. For example, the list of amity "thirds" includes ...6/5 (339.5) (363.2) 5/4... where the intervals given in cents are not representable as simple JI intervals (243/200 and 100/81 are about as simple as you can get).
Another way to describe this property is that the chroma of the near-equal MOS is a kind of "super-comma", a set of many useful commas that are tempered to become the same, non-vanishing, interval. It should be obvious that "cluster temperament" is a vague, qualitative phrase and not mathematically well-defined.
Rather than simply denoting one of a list of rank-2 temperaments, the phrase "cluster scale" is also associated with a compositional philosophy. It is often said that temperaments such as slendric are melodically "bad" or have "bad MOS structure", because some MOS (in this case slendric[5]) is "too equal" and the next higher MOSes are "too unequal". But this can be thought of as a feature rather than a bug. Rather than forcing them into the MOS framework, one can think of cluster scales as having two hierarchical levels of melodic structure: the "step" (a step of the quasi-equal MOS), and the "chroma", and a chroma is so much smaller than a step that the steps seldom go out of order no matter how many chromas are involved.
Examples of cluster MOSes
Parasoft smitonic is a cluster MOS.
Examples of cluster temperaments
Slendric
Main article: Slendric
Chroma: 49/48 ~ 64/63
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" |
|---|---|---|---|---|
| 1 | 9/8 | 8/7 | 7/6 | 32/27 |
| 2 | 9/7 | 21/16 | 4/3 | |
| 3 | 3/2 | 32/21 | 14/9 | |
| 4 | 27/16 | 12/7 | 7/4 | 16/9 |
Slendric has two quite different extensions that are both also cluster scales:
Mothra
Main article: Mothra
Chroma: 33/32 ~ 36/35 ~ 49/48 ~ 55/54 ~ 56/55 ~ 64/63
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" | ||
|---|---|---|---|---|---|---|
| 1 | 12/11 | 10/9~9/8 | 8/7 | 7/6 | 6/5 | 11/9 |
| 2 | 5/4 | 14/11~9/7 | 21/16 | 4/3 | 11/8 | 7/5 |
| 3 | 10/7 | 16/11 | 3/2 | 32/21 | 14/9~11/7 | 8/5 |
| 4 | 18/11 | 5/3 | 12/7 | 7/4 | 16/9~9/5 | 11/6 |
Rodan
Main article: Rodan
Chroma: 49/48 ~ 55/54 ~ 56/55 ~ 64/63 ~ 81/80 ~ 99/98
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" | ||||
|---|---|---|---|---|---|---|---|---|
| 1 | 12/11 | 10/9 | 9/8 | 8/7 | 7/6 | 32/27 | 6/5 | 11/9 |
| 2 | 5/4 | 14/11 | 9/7 | 21/16 | 4/3 | 27/20 | 11/8 | 7/5 |
| 3 | 10/7 | 16/11 | 40/27 | 3/2 | 32/21 | 14/9 | 11/7 | 8/5 |
| 4 | 18/11 | 5/3 | 27/16 | 12/7 | 7/4 | 16/9 | 9/5 | 11/6 |
Modus (of the tetracot family)
Main article: Tetracot and Modus
Chroma: 40/39 ~ 45/44 ~ 55/54 ~ 65/64 ~ 66/65 ~ 81/80 ~ 121/120
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" |
|---|---|---|---|---|
| 1 | 16/15 | 13/12~12/11 | 11/10~10/9 | 9/8 |
| 2 | 13/11~32/27 | 6/5 | 11/9~16/13 | 5/4 |
| 3 | 13/10 | 4/3 | 27/20~15/11 | 11/8~18/13 |
| 4 | 13/9~16/11 | 22/15~40/27 | 3/2 | 20/13 |
| 5 | 8/5 | 13/8~18/11 | 5/3 | 27/16~22/13 |
| 6 | 16/9 | 9/5~20/11 | 11/6~24/13 | 15/8 |
Miracle
Main article: Miracle
Chroma: 45/44 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" | ||
|---|---|---|---|---|---|---|
| 1 | 22/21~21/20 | 16/15~15/14 | 12/11 | 10/9 | ||
| 2 | 11/10 | 9/8 | 8/7 | 7/6 | 32/27 | |
| 3 | 6/5 | 11/9 | 5/4 | 14/11 | ||
| 4 | 9/7 | 21/16 | 4/3 | |||
| 5 | 11/8 | 7/5 | 10/7 | 16/11 | ||
| 6 | 3/2 | 32/21 | 14/9 | |||
| 7 | 11/7 | 8/5 | 18/11 | 5/3 | ||
| 8 | 27/16 | 12/7 | 7/4 | 16/9 | 20/11 | |
| 9 | 9/5 | 11/6 | 15/8 | 21/11 |
Porcupine
Main article: Porcupine
Chroma: 22/21 ~ 25/24 ~ 26/25* ~ 33/32 ~ 36/35 ~ 45/44 ~ 81/80
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" |
|---|---|---|---|---|
| 1 | 21/20~16/15 | 12/11~11/10~10/9 | 9/8~8/7 | 13/11* |
| 2 | 7/6 | 6/5~11/9 | 5/4 | 9/7~13/10* |
| 3 | 14/11 | 4/3 | 11/8 | 10/7~13/9* |
| 4 | 7/5~18/13* | 16/11 | 3/2 | 11/7 |
| 5 | 14/9~20/13* | 8/5 | 5/3~18/11 | 12/7 |
| 6 | 22/13* | 7/4~16/9 | 9/5~11/6 | 40/21~15/8 |
- * 13-limit porcupinefish interpretation
Valentino
Chroma: 49/48 ~ 51/50 ~ 52/51 ~ 55/54 ~ 56/55 ~ 64/63 ~ 65/64 ~ 77/75 ~ 85/84 ~ 119/117 ~ 128/125 ~ 143/140
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" | ||
|---|---|---|---|---|---|---|
| 1 | 36/35 | 21/20~25/24 | 17/16~16/15 | 13/12 | ||
| 2 | 14/13 | 12/11~11/10 | 10/9 | 17/15 | 20/17 | |
| 3 | 9/8 | 8/7 | 7/6 | 32/27 | ||
| 4 | 13/11 | 6/5 | 11/9~17/14 | |||
| 5 | 16/13~21/17 | 5/4 | 14/11 | 13/10 | 27/20 | |
| 6 | 9/7 | 21/16~17/13 | 4/3 | 34/25 | ||
| 7 | 27/20 | 11/8~15/11 | 7/5 | 17/12 | ||
| 8 | 24/17 | 10/7 | 16/11~22/15 | 40/27 | ||
| 9 | 25/17 | 3/2 | 26/17~32/21 | 14/9 | ||
| 10 | 40/27 | 20/13 | 11/7 | 8/5 | 13/8~34/21 | |
| 11 | 18/11~28/17 | 5/3 | 22/13 | |||
| 12 | 27/16 | 12/7 | 7/4 | 16/9 | ||
| 13 | 17/10 | 30/14 | 9/5 | 11/6~20/11 | 13/7 | |
| 14 | 24/13 | 15/8 | 40/21~48/25 | 35/18 |
2.3.5.11.13 hitchcock
Unlike amity itself, this 2.3.5.11.13 amity extension is a cluster temperament because the intervals between 6/5 and 5/4 are mapped to 11/9 and 16/13.
| Steps | "Diminished" | "Minor" | "Major" | "Augmented" | ||
|---|---|---|---|---|---|---|
| 1 | 13/12 | 12/11~11/10 | 10/9 | 9/8 | ||
| 2 | 13/11 | 6/5 | 11/9 | 16/13 | 5/4 | |
| 3 | 13/10 | 4/3 | 27/20 | 11/8 | 18/13 | |
| 4 | 13/9 | 16/11 | 40/27 | 3/2 | 20/13 | |
| 5 | 8/5 | 13/8 | 18/11 | 5/3 | 22/13 | |
| 6 | 16/9 | 9/5 | 11/6 | 24/13 |