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

• Play Slendric in 36edo

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

• Play Mothra in 31edo

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

• Play Modus in 34edo

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