Cluster MOS: Difference between revisions

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A cluster ~~temperament (named by [[Keenan_Pepper|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.

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.

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).

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.

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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.

=Examples=

== Examples of cluster MOSes ==

[[4L 3s #Parasoft|Parasoft smitonic]] is a cluster MOS.

== 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"

|-

| | Steps

! | Steps

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

|-

| | 1

Line 49:

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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"

|-

| | Steps

! | Steps

| |

! |

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

| |

! |

|-

| | 1

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| | 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"

|-

| | Steps

! | Steps

| |

! |

| |

! |

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

| |

! |

| |

! |

|-

| | 1

Line 154:

Line 160:

| | 9/5

| | 11/6

|}

==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"

|-

| | Steps

! | Steps

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

|-

| | 1

Line 176:

Line 182:

|-

| | 2

| | 13/11

| | 13/11~32/27

| | 6/5

| | 11/9

| | 11/9~16/13

| | 5/4

|-

Line 184:

Line 190:

| | 13/10

| | 4/3

| | 27/20

| | 27/20~15/11

| | 11/8

| | 11/8~18/13

|-

| | 4

| | 16/11

| | 13/9~16/11

| | 40/27

| | 22/15~40/27

| | 3/2

| | 20/13

Line 195:

Line 201:

| | 5

| | 8/5

| | 18/11

| | 13/8~18/11

| | 5/3

| | 22/13~~~27/16~~

| | 27/16~22/13

|-

| | 6

| | 16/9

| | 9/5

| | 9/5~20/11

| | 11/6

| | 11/6~24/13

| | 15/8

|}

* [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]]

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"

|-

| | Steps

! | Steps

| |

! |

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

| |

! |

|-

| | 1

Line 295:

Line 301:

|}

==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* ~ 33/32 ~ 36/35 ~ 45/44 ~ 81/80

{| class="wikitable"

|-

| | Steps

! | Steps

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

|-

| | 1

Line 312:

Line 318:

| | 12/11~11/10~10/9

| | 9/8~8/7

| | (13/11)

| | 13/11*

|-

| | 2

Line 318:

Line 324:

| | 6/5~11/9

| | 5/4

| | 9/7~(13/10)

| | 9/7~13/10*

|-

| | 3

Line 324:

Line 330:

| | 4/3

| | 11/8

| | 10/7~(13/9)

| | 10/7~13/9*

|-

| | 4

| | 7/5~(18/13)

| | 7/5~18/13*

| | 16/11

| | 3/2

Line 333:

Line 339:

|-

| | 5

| | 14/9~(20/13)

| | 14/9~20/13*

| | 8/5

| | 5/3~18/11

Line 339:

Line 345:

|-

| | 6

| | (22/13)

| | 22/13*

| | 7/4~16/9

| | 9/5~11/6

| | 40/21~15/8

|}

: * 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"

|-

| | Steps

! | Steps

| |

! |

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

| |

! |

|-

| | 1

Line 471:

Line 478:

|}

==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.

{| class="wikitable"

|-

| | Steps

! | Steps

| |

! |

| | "Diminished"

! | "Diminished"

| | "Minor"

! | "Minor"

| | "Major"

! | "Major"

| | "Augmented"

! | "Augmented"

| |

! |

|-

| | 1

Line 532:

Line 539:

| |

|}

[[Category:Rank 2]]

[[Category:MOS scales]]

@@ Line 1: / Line 1: @@
-A cluster temperament (named by [[Keenan_Pepper|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.
+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.
-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).
+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.
@@ Line 7: / Line 9: @@
 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=
+{{todo|add more detail|inline=1}}
+== Examples of cluster MOSes ==
+[[4L 3s #Parasoft|Parasoft smitonic]] is a cluster MOS.
+== 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"
 |-
-| | Steps
+! | Steps
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
 |-
 | | 1
@@ Line 49: / Line 55: @@
 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"
 |-
-| | Steps
+! | Steps
-| |
+! |
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
-| |
+! |
 |-
 | | 1
@@ Line 96: / Line 102: @@
 | | 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"
 |-
-| | Steps
+! | Steps
-| |
+! |
-| |
+! |
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
-| |
+! |
-| |
+! |
 |-
 | | 1
@@ Line 154: / Line 160: @@
 | | 9/5
 | | 11/6
 |}
-==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"
 |-
-| | Steps
+! | Steps
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
 |-
 | | 1
@@ Line 176: / Line 182: @@
 |-
 | | 2
-| | 13/11
+| | 13/11~32/27
 | | 6/5
-| | 11/9
+| | 11/9~16/13
 | | 5/4
 |-
@@ Line 184: / Line 190: @@
 | | 13/10
 | | 4/3
-| | 27/20
+| | 27/20~15/11
-| | 11/8
+| | 11/8~18/13
 |-
 | | 4
-| | 16/11
+| | 13/9~16/11
-| | 40/27
+| | 22/15~40/27
 | | 3/2
 | | 20/13
@@ Line 195: / Line 201: @@
 | | 5
 | | 8/5
-| | 18/11
+| | 13/8~18/11
 | | 5/3
-| | 22/13~27/16
+| | 27/16~22/13
 |-
 | | 6
 | | 16/9
-| | 9/5
+| | 9/5~20/11
-| | 11/6
+| | 11/6~24/13
 | | 15/8
 |}
 * [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]]
-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"
 |-
-| | Steps
+! | Steps
-| |
+! |
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
-| |
+! |
 |-
 | | 1
@@ Line 295: / Line 301: @@
 |}
-==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"
 |-
-| | Steps
+! | Steps
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
 |-
 | | 1
@@ Line 312: / Line 318: @@
 | | 12/11~11/10~10/9
 | | 9/8~8/7
-| | (13/11)
+| | 13/11<sup>*</sup>
 |-
 | | 2
@@ Line 318: / Line 324: @@
 | | 6/5~11/9
 | | 5/4
-| | 9/7~(13/10)
+| | 9/7~13/10<sup>*</sup>
 |-
 | | 3
@@ Line 324: / Line 330: @@
 | | 4/3
 | | 11/8
-| | 10/7~(13/9)
+| | 10/7~13/9<sup>*</sup>
 |-
 | | 4
-| | 7/5~(18/13)
+| | 7/5~18/13<sup>*</sup>
 | | 16/11
 | | 3/2
@@ Line 333: / Line 339: @@
 |-
 | | 5
-| | 14/9~(20/13)
+| | 14/9~20/13<sup>*</sup>
 | | 8/5
 | | 5/3~18/11
@@ Line 339: / Line 345: @@
 |-
 | | 6
-| | (22/13)
+| | 22/13<sup>*</sup>
 | | 7/4~16/9
 | | 9/5~11/6
 | | 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"
 |-
-| | Steps
+! | Steps
-| |
+! |
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
-| |
+! |
 |-
 | | 1
@@ Line 471: / Line 478: @@
 |}
-==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.
 {| class="wikitable"
 |-
-| | Steps
+! | Steps
-| |
+! |
-| | "Diminished"
+! | "Diminished"
-| | "Minor"
+! | "Minor"
-| | "Major"
+! | "Major"
-| | "Augmented"
+! | "Augmented"
-| |
+! |
 |-
 | | 1
@@ Line 532: / Line 539: @@
 | |
 |}
+[[Category:Rank 2]]
+[[Category:MOS scales]]