Cluster temperament

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A cluster temperament (named by Keenan Pepper) is a very particular kind of rank-2 temperament whose generator is quite near a rational fraction of an octave. Therefore some MOS of the temperament is quasi-equal (which should be reasonably sized for it to be a good cluster temperament, 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 JI intervals that are individually recognizable, yet conceptually grouped into the same category (or "cluster") because they're so close.

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

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: Modus

Chroma: 40/39~45/44~55/54~66/65~81/80~121/120

Steps "Diminished" "Minor" "Major" "Augmented"
1 16/15 13/12~12/11 11/10~10/9 9/8
2 13/11 6/5 11/9 5/4
3 13/10 4/3 27/20 11/8
4 16/11 40/27 3/2 20/13
5 8/5 18/11 5/3 22/13~27/16
6 16/9 9/5 11/6 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(fish)

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

17-limit valentino

Chroma: 49/48~55/54~56/55~64/63~65/64~85/84~119/117~128/125~143/140~153/150

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