Cluster MOS: Difference between revisions

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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.
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=
== Examples of cluster temperaments ===


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


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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|Mothra]]


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*[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|Rodan]]


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==Modus (of the tetracot family)==
=== Modus (of the tetracot family) ===
Main article: [[Modus|Modus]]
Main article: [[Modus|Modus]]


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* [http://sevish.com/scaleworkshop/index.htm?name=34edo%20modus&data=35.294117647058826%0A70.58823529411765%0A105.88235294117648%0A141.1764705882353%0A176.47058823529414%0A211.76470588235296%0A247.05882352941177%0A282.3529411764706%0A317.64705882352945%0A352.9411764705883%0A388.2352941176471%0A423.5294117647059%0A458.82352941176475%0A494.11764705882354%0A529.4117647058824%0A564.7058823529412%0A600.%0A635.2941176470589%0A670.5882352941177%0A705.8823529411766%0A741.1764705882354%0A776.4705882352941%0A811.764705882353%0A847.0588235294118%0A882.3529411764706%0A917.6470588235295%0A952.9411764705883%0A988.2352941176471%0A1023.529411764706%0A1058.8235294117649%0A1094.1176470588236%0A1129.4117647058824%0A1164.7058823529412%0A1200.&vert=-4&horiz=5 Play Modus in 34edo]
* [http://sevish.com/scaleworkshop/index.htm?name=34edo%20modus&data=35.294117647058826%0A70.58823529411765%0A105.88235294117648%0A141.1764705882353%0A176.47058823529414%0A211.76470588235296%0A247.05882352941177%0A282.3529411764706%0A317.64705882352945%0A352.9411764705883%0A388.2352941176471%0A423.5294117647059%0A458.82352941176475%0A494.11764705882354%0A529.4117647058824%0A564.7058823529412%0A600.%0A635.2941176470589%0A670.5882352941177%0A705.8823529411766%0A741.1764705882354%0A776.4705882352941%0A811.764705882353%0A847.0588235294118%0A882.3529411764706%0A917.6470588235295%0A952.9411764705883%0A988.2352941176471%0A1023.529411764706%0A1058.8235294117649%0A1094.1176470588236%0A1129.4117647058824%0A1164.7058823529412%0A1200.&vert=-4&horiz=5 Play Modus in 34edo]


==Miracle==
=== Miracle ===
Main article: [[Miracle|Miracle]]
Main article: [[Miracle|Miracle]]


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


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|}
|}


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


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==2.3.5.11.13 hitchcock==
=== 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.
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.


Revision as of 00:55, 1 April 2021

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 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 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. But not only that; in a cluster temperament, many different versions of each interval, differing by a chroma, these colors represent nearby JI intervals specifically (because a temmperament 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 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: 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
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