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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:phylingual|phylingual]] and made on <tt>2012-05-12 20:18:31 UTC</tt>.<br>
: The original revision id was <tt>334262170</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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).
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.
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 14: 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.
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}}
==Slendric==  
 
== Examples of cluster MOSes ==
[[4L 3s #Parasoft|Parasoft smitonic]] is a cluster MOS.
== Examples of cluster temperaments ==
 
=== Slendric ===
Main article: [[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==
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 ~ 64/63
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 ||  ||</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Cluster temperament&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A cluster temperament (named by &lt;a class="wiki_link" href="/Keenan%20Pepper"&gt;Keenan Pepper&lt;/a&gt;) 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 (&amp;quot;diminished&amp;quot;, &amp;quot;minor&amp;quot;, &amp;quot;major&amp;quot;, &amp;quot;augmented&amp;quot;...) include many nearby JI intervals that are individually recognizable, yet conceptually grouped into the same category (or &amp;quot;cluster&amp;quot;) because they're so close.&lt;br /&gt;
&lt;br /&gt;
An example of something that is &lt;strong&gt;not&lt;/strong&gt; a cluster temperament is &lt;a class="wiki_link" href="/amity"&gt;amity&lt;/a&gt;, 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 &amp;quot;thirds&amp;quot; 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).&lt;br /&gt;
&lt;br /&gt;
Another way to describe this property is that the chroma of the near-equal MOS is a kind of &amp;quot;super-comma&amp;quot;, a set of many useful commas that are tempered to become the same, non-vanishing, interval. It should be obvious that &amp;quot;cluster temperament&amp;quot; is a vague, qualitative phrase and not mathematically well-defined.&lt;br /&gt;
&lt;br /&gt;
Rather than simply denoting one of a list of rank-2 temperaments, the phrase &amp;quot;cluster scale&amp;quot; is also associated with a compositional philosophy. It is often said that temperaments such as slendric are melodically &amp;quot;bad&amp;quot; or have &amp;quot;bad MOS structure&amp;quot;, because some MOS (in this case slendric[5]) is &amp;quot;too equal&amp;quot; and the next higher MOSes are &amp;quot;too unequal&amp;quot;. 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 &amp;quot;step&amp;quot; (a step of the quasi-equal MOS), and the &amp;quot;chroma&amp;quot;, 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.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Examples&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Examples-Slendric"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Slendric&lt;/h2&gt;
Main article: &lt;a class="wiki_link" href="/Slendric"&gt;Slendric&lt;/a&gt;&lt;br /&gt;
Chroma: 49/48~64/63&lt;br /&gt;


{| class="wikitable"
|-
! | 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
|}
* [http://sevish.com/scaleworkshop/index.htm?name=36edo%20slendric&data=33.333333333333336%0A66.66666666666667%0A100.%0A133.33333333333334%0A166.66666666666669%0A200.%0A233.33333333333334%0A266.6666666666667%0A300.%0A333.33333333333337%0A366.6666666666667%0A400.%0A433.33333333333337%0A466.6666666666667%0A500.00000000000006%0A533.3333333333334%0A566.6666666666667%0A600.%0A633.3333333333334%0A666.6666666666667%0A700.%0A733.3333333333334%0A766.6666666666667%0A800.%0A833.3333333333334%0A866.6666666666667%0A900.0000000000001%0A933.3333333333334%0A966.6666666666667%0A1000.0000000000001%0A1033.3333333333335%0A1066.6666666666667%0A1100.%0A1133.3333333333335%0A1166.6666666666667%0A1200.&vert=-6&horiz=7&midi=12 Play Slendric in 36edo]
Slendric has two quite different extensions that are both also cluster scales:


&lt;table class="wiki_table"&gt;
==== Mothra ====
    &lt;tr&gt;
Main article: [[Mothra]]
        &lt;td&gt;Steps&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Diminished&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Minor&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Major&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Augmented&amp;quot;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/27&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


Slendric has two quite different extensions that are both also cluster scales:&lt;br /&gt;
Chroma: 33/32 ~ 36/35 ~ 49/48 ~ 55/54 ~ 56/55 ~ 64/63
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="Examples-Slendric-Mothra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Mothra&lt;/h3&gt;
Main article: &lt;a class="wiki_link" href="/Mothra"&gt;Mothra&lt;/a&gt;&lt;br /&gt;
Chroma: 33/32~36/35~49/48~55/54~56/55~64/63&lt;br /&gt;


{| class="wikitable"
|-
! | 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
|}
* [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]


&lt;table class="wiki_table"&gt;
==== Rodan ====
    &lt;tr&gt;
Main article: [[Rodan]]
        &lt;td&gt;Steps&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Diminished&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Minor&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Major&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Augmented&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9~9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11~9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9~11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9~9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Examples-Slendric-Rodan"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Rodan&lt;/h3&gt;
Chroma: 49/48 ~ 55/54 ~ 56/55 ~ 64/63 ~ 81/80 ~ 99/98
Main article: &lt;a class="wiki_link" href="/Rodan"&gt;Rodan&lt;/a&gt;&lt;br /&gt;
Chroma: 49/48~55/54~56/55~64/63~81/80~99/98&lt;br /&gt;


{| class="wikitable"
|-
! | 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
|}


&lt;table class="wiki_table"&gt;
=== Modus (of the tetracot family) ===
    &lt;tr&gt;
Main article: [[Tetracot]] and [[Modus]]
        &lt;td&gt;Steps&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Diminished&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Minor&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Major&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Augmented&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Examples-Modus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Modus&lt;/h2&gt;
Chroma: 40/39 ~ 45/44 ~ 55/54 ~ 65/64 ~ 66/65 ~ 81/80 ~ 121/120
Main article: &lt;a class="wiki_link" href="/Modus"&gt;Modus&lt;/a&gt;&lt;br /&gt;
Chroma: 40/39~45/44~55/54~66/65~81/80~121/120&lt;br /&gt;


{| class="wikitable"
|-
! | Steps
! | "Diminished"
! | "Minor"
! | "Major"
! | "Augmented"
|-
| | 1
| | 16/15
| | 13/12~12/11
| | 11/10~10/9
| | 9/8
|-
| | 2
| | 13/11~32/27
| | 6/5
| | 11/9~16/13
| | 5/4
|-
| | 3
| | 13/10
| | 4/3
| | 27/20~15/11
| | 11/8~18/13
|-
| | 4
| | 13/9~16/11
| | 22/15~40/27
| | 3/2
| | 20/13
|-
| | 5
| | 8/5
| | 13/8~18/11
| | 5/3
| | 27/16~22/13
|-
| | 6
| | 16/9
| | 9/5~20/11
| | 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]


&lt;table class="wiki_table"&gt;
=== Miracle ===
    &lt;tr&gt;
Main article: [[Miracle]]
        &lt;td&gt;Steps&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Diminished&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Minor&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Major&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Augmented&amp;quot;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/12~12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/10~10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/13&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/13~27/16&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Examples-Miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Miracle&lt;/h2&gt;
Chroma: 45/44 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63
Main article: &lt;a class="wiki_link" href="/Miracle"&gt;Miracle&lt;/a&gt;&lt;br /&gt;
Chroma: 45/44~49/48~50/49~55/54~56/55~64/63&lt;br /&gt;


{| class="wikitable"
|-
! | 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
| |
|}


&lt;table class="wiki_table"&gt;
=== Porcupine ===
    &lt;tr&gt;
Main article: [[Porcupine]]
        &lt;td&gt;Steps&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Diminished&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Minor&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Major&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Augmented&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/21~21/20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/15~15/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/11&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Examples-Porcupine(fish)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Porcupine(fish)&lt;/h2&gt;
Chroma: 22/21 ~ 25/24 ~ 26/25<sup>*</sup> ~ 33/32 ~ 36/35 ~ 45/44 ~ 81/80
Main article: &lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt;&lt;br /&gt;
Chroma: 22/21~25/24~(26/25)~33/32~36/35~45/44~81/80&lt;br /&gt;


{| class="wikitable"
|-
! | Steps
! | "Diminished"
! | "Minor"
! | "Major"
! | "Augmented"
|-
| | 1
| | 21/20~16/15
| | 12/11~11/10~10/9
| | 9/8~8/7
| | 13/11<sup>*</sup>
|-
| | 2
| | 7/6
| | 6/5~11/9
| | 5/4
| | 9/7~13/10<sup>*</sup>
|-
| | 3
| | 14/11
| | 4/3
| | 11/8
| | 10/7~13/9<sup>*</sup>
|-
| | 4
| | 7/5~18/13<sup>*</sup>
| | 16/11
| | 3/2
| | 11/7
|-
| | 5
| | 14/9~20/13<sup>*</sup>
| | 8/5
| | 5/3~18/11
| | 12/7
|-
| | 6
| | 22/13<sup>*</sup>
| | 7/4~16/9
| | 9/5~11/6
| | 40/21~15/8
|}
: <sup>*</sup> 13-limit porcupinefish interpretation


&lt;table class="wiki_table"&gt;
=== Valentino ===
    &lt;tr&gt;
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
        &lt;td&gt;Steps&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Diminished&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Minor&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Major&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;quot;Augmented&amp;quot;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/20~16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11~11/10~10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8~8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(13/11)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5~11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7~(13/10)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7~(13/9)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5~(18/13)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9~(20/13)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3~18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(22/13)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4~16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5~11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/21~15/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
{| class="wikitable"
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Examples-17-limit valentino"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;17-limit valentino&lt;/h2&gt;
|-
Chroma: 49/48~55/54~56/55~64/63~65/64~85/84~119/117~128/125~143/140~153/150&lt;br /&gt;
! | 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.


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;td&gt;Steps&lt;br /&gt;
! | Steps
&lt;/td&gt;
! |
        &lt;td&gt;&lt;br /&gt;
! | "Diminished"
&lt;/td&gt;
! | "Minor"
        &lt;td&gt;&amp;quot;Diminished&amp;quot;&lt;br /&gt;
! | "Major"
&lt;/td&gt;
! | "Augmented"
        &lt;td&gt;&amp;quot;Minor&amp;quot;&lt;br /&gt;
! |
&lt;/td&gt;
|-
        &lt;td&gt;&amp;quot;Major&amp;quot;&lt;br /&gt;
| | 1
&lt;/td&gt;
| |
        &lt;td&gt;&amp;quot;Augmented&amp;quot;&lt;br /&gt;
| | 13/12
&lt;/td&gt;
| | 12/11~11/10
        &lt;td&gt;&lt;br /&gt;
| | 10/9
&lt;/td&gt;
| | 9/8
    &lt;/tr&gt;
| |
    &lt;tr&gt;
|-
        &lt;td&gt;1&lt;br /&gt;
| | 2
&lt;/td&gt;
| | 13/11
        &lt;td&gt;&lt;br /&gt;
| | 6/5
&lt;/td&gt;
| | 11/9
        &lt;td&gt;36/35&lt;br /&gt;
| | 16/13
&lt;/td&gt;
| | 5/4
        &lt;td&gt;21/20~25/24&lt;br /&gt;
| |
&lt;/td&gt;
|-
        &lt;td&gt;17/16~16/15&lt;br /&gt;
| | 3
&lt;/td&gt;
| | 13/10
        &lt;td&gt;13/12&lt;br /&gt;
| |
&lt;/td&gt;
| | 4/3
        &lt;td&gt;&lt;br /&gt;
| | 27/20
&lt;/td&gt;
| | 11/8
    &lt;/tr&gt;
| | 18/13
    &lt;tr&gt;
|-
        &lt;td&gt;2&lt;br /&gt;
| | 4
&lt;/td&gt;
| | 13/9
        &lt;td&gt;&lt;br /&gt;
| | 16/11
&lt;/td&gt;
| | 40/27
        &lt;td&gt;14/13&lt;br /&gt;
| | 3/2
&lt;/td&gt;
| |
        &lt;td&gt;12/11~11/10&lt;br /&gt;
| | 20/13
&lt;/td&gt;
|-
        &lt;td&gt;10/9&lt;br /&gt;
| | 5
&lt;/td&gt;
| |
        &lt;td&gt;17/15&lt;br /&gt;
| | 8/5
&lt;/td&gt;
| | 13/8
        &lt;td&gt;20/17&lt;br /&gt;
| | 18/11
&lt;/td&gt;
| | 5/3
    &lt;/tr&gt;
| | 22/13
    &lt;tr&gt;
|-
        &lt;td&gt;3&lt;br /&gt;
| | 6
&lt;/td&gt;
| |
        &lt;td&gt;&lt;br /&gt;
| | 16/9
&lt;/td&gt;
| | 9/5
        &lt;td&gt;9/8&lt;br /&gt;
| | 11/6
&lt;/td&gt;
| | 24/13
        &lt;td&gt;8/7&lt;br /&gt;
| |
&lt;/td&gt;
|}
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;32/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9~17/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/13~21/17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/20&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21/16~17/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;34/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8~15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;17/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;24/17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/11~22/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;25/17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;26/17~32/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/8~34/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11~28/17&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;17/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;30/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6~20/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;24/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40/21~48/25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;35/18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Category:Rank 2]]
[[Category:MOS scales]]

Latest revision as of 14:55, 13 June 2025

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.

Todo: add more detail

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

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: Tetracot and Modus

Chroma: 40/39 ~ 45/44 ~ 55/54 ~ 65/64 ~ 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 13/11~32/27 6/5 11/9~16/13 5/4
3 13/10 4/3 27/20~15/11 11/8~18/13
4 13/9~16/11 22/15~40/27 3/2 20/13
5 8/5 13/8~18/11 5/3 27/16~22/13
6 16/9 9/5~20/11 11/6~24/13 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

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
* 13-limit porcupinefish interpretation

Valentino

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

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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