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A  '''cluster MOS''' or '''cluster scale''' is a very particular kind of [[MOS]]-based system (i.e. a system based on stacks of [[period]]s and [[generator]]s) 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 MOS''' or '''cluster scale''' is a very particular kind of [[MOS]]-based system (i.e. a system based on stacks of [[period]]s and [[generator]]s) 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|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).
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|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).
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.
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.
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== Examples of cluster MOSes ==
== Examples of cluster MOSes ==
[[4L 3s#Parasoft|Parasoft smitonic]] is a cluster MOS.
[[4L 3s #Parasoft|Parasoft smitonic]] is a cluster MOS.
== Examples of cluster temperaments ==
== Examples of cluster temperaments ==


=== Slendric ===
=== Slendric ===
Main article: [[Slendric|Slendric]]
Main article: [[Slendric]]


Chroma: 49/48~64/63
Chroma: 49/48 ~ 64/63


{| class="wikitable"
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Slendric has two quite different extensions that are both also cluster scales:
Slendric has two quite different extensions that are both also cluster scales:


====Mothra====
==== Mothra ====
Main article: [[Mothra|Mothra]]
Main article: [[Mothra]]


Chroma: 33/32~36/35~49/48~55/54~56/55~64/63
Chroma: 33/32 ~ 36/35 ~ 49/48 ~ 55/54 ~ 56/55 ~ 64/63


{| class="wikitable"
{| class="wikitable"
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| | 11/6
| | 11/6
|}
|}
*[http://sevish.com/scaleworkshop/index.htm?name=31edo%20mothra&data=38.70967741935484%0A77.41935483870968%0A116.12903225806451%0A154.83870967741936%0A193.5483870967742%0A232.25806451612902%0A270.9677419354839%0A309.6774193548387%0A348.38709677419354%0A387.0967741935484%0A425.80645161290323%0A464.51612903225805%0A503.2258064516129%0A541.9354838709678%0A580.6451612903226%0A619.3548387096774%0A658.0645161290323%0A696.7741935483871%0A735.483870967742%0A774.1935483870968%0A812.9032258064516%0A851.6129032258065%0A890.3225806451613%0A929.0322580645161%0A967.741935483871%0A1006.4516129032259%0A1045.1612903225807%0A1083.8709677419356%0A1122.5806451612902%0A1161.2903225806451%0A1200.&vert=-5&horiz=6&midi=16 Play Mothra in 31edo]
* [http://sevish.com/scaleworkshop/index.htm?name=31edo%20mothra&data=38.70967741935484%0A77.41935483870968%0A116.12903225806451%0A154.83870967741936%0A193.5483870967742%0A232.25806451612902%0A270.9677419354839%0A309.6774193548387%0A348.38709677419354%0A387.0967741935484%0A425.80645161290323%0A464.51612903225805%0A503.2258064516129%0A541.9354838709678%0A580.6451612903226%0A619.3548387096774%0A658.0645161290323%0A696.7741935483871%0A735.483870967742%0A774.1935483870968%0A812.9032258064516%0A851.6129032258065%0A890.3225806451613%0A929.0322580645161%0A967.741935483871%0A1006.4516129032259%0A1045.1612903225807%0A1083.8709677419356%0A1122.5806451612902%0A1161.2903225806451%0A1200.&vert=-5&horiz=6&midi=16 Play Mothra in 31edo]


====Rodan====
==== Rodan ====
Main article: [[Rodan|Rodan]]
Main article: [[Rodan]]


Chroma: 49/48~55/54~56/55~64/63~81/80~99/98
Chroma: 49/48 ~ 55/54 ~ 56/55 ~ 64/63 ~ 81/80 ~ 99/98


{| class="wikitable"
{| class="wikitable"
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=== Modus (of the tetracot family) ===
=== Modus (of the tetracot family) ===
Main article: [[Modus|Modus]]
Main article: [[Tetracot]] and [[Modus]]


Chroma: 40/39~45/44~55/54~66/65~81/80~121/120
Chroma: 40/39 ~ 45/44 ~ 55/54 ~ 65/64 ~ 66/65 ~ 81/80 ~ 121/120


{| class="wikitable"
{| class="wikitable"
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|-
|-
| | 2
| | 2
| | 13/11
| | 13/11~32/27
| | 6/5
| | 6/5
| | 11/9
| | 11/9~16/13
| | 5/4
| | 5/4
|-
|-
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| | 13/10
| | 13/10
| | 4/3
| | 4/3
| | 27/20
| | 27/20~15/11
| | 11/8
| | 11/8~18/13
|-
|-
| | 4
| | 4
| | 16/11
| | 13/9~16/11
| | 40/27
| | 22/15~40/27
| | 3/2
| | 3/2
| | 20/13
| | 20/13
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| | 5
| | 5
| | 8/5
| | 8/5
| | 18/11
| | 13/8~18/11
| | 5/3
| | 5/3
| | 22/13~27/16
| | 27/16~22/13
|-
|-
| | 6
| | 6
| | 16/9
| | 16/9
| | 9/5
| | 9/5~20/11
| | 11/6
| | 11/6~24/13
| | 15/8
| | 15/8
|}
|}
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=== Miracle ===
=== Miracle ===
Main article: [[Miracle|Miracle]]
Main article: [[Miracle]]


Chroma: 45/44~49/48~50/49~55/54~56/55~64/63
Chroma: 45/44 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63


{| class="wikitable"
{| class="wikitable"
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|}
|}


=== Porcupine(fish) ===
=== Porcupine ===
Main article: [[Porcupine|Porcupine]]
Main article: [[Porcupine]]


Chroma: 22/21~25/24~(26/25)~33/32~36/35~45/44~81/80
Chroma: 22/21 ~ 25/24 ~ 26/25<sup>*</sup> ~ 33/32 ~ 36/35 ~ 45/44 ~ 81/80


{| class="wikitable"
{| class="wikitable"
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| | 12/11~11/10~10/9
| | 12/11~11/10~10/9
| | 9/8~8/7
| | 9/8~8/7
| | (13/11)
| | 13/11<sup>*</sup>
|-
|-
| | 2
| | 2
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| | 6/5~11/9
| | 6/5~11/9
| | 5/4
| | 5/4
| | 9/7~(13/10)
| | 9/7~13/10<sup>*</sup>
|-
|-
| | 3
| | 3
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| | 4/3
| | 4/3
| | 11/8
| | 11/8
| | 10/7~(13/9)
| | 10/7~13/9<sup>*</sup>
|-
|-
| | 4
| | 4
| | 7/5~(18/13)
| | 7/5~18/13<sup>*</sup>
| | 16/11
| | 16/11
| | 3/2
| | 3/2
Line 337: Line 337:
|-
|-
| | 5
| | 5
| | 14/9~(20/13)
| | 14/9~20/13<sup>*</sup>
| | 8/5
| | 8/5
| | 5/3~18/11
| | 5/3~18/11
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|-
|-
| | 6
| | 6
| | (22/13)
| | 22/13<sup>*</sup>
| | 7/4~16/9
| | 7/4~16/9
| | 9/5~11/6
| | 9/5~11/6
| | 40/21~15/8
| | 40/21~15/8
|}
|}
: <sup>*</sup> 13-limit porcupinefish interpretation


=== 17-limit valentino ===
=== Valentino ===
Chroma: 49/48~55/54~56/55~64/63~65/64~85/84~119/117~128/125~143/140~153/150
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


{| class="wikitable"
{| class="wikitable"
Retrieved from "https://en.xen.wiki/w/Cluster_MOS"