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36/35: Difference between revisions

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**Imported revision 244883189 - Original comment: **
Re-organize following 15/14
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| en = 36/35
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-08 14:56:59 UTC</tt>.<br>
| de = 36/35
: The original revision id was <tt>244883189</tt>.<br>
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: The revision comment was: <tt></tt><br>
{{Infobox Interval
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Name = septimal quartertone, mint comma
<h4>Original Wikitext content:</h4>
| Color name = rg1, rugu unison
<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">The //septimal quarter tone//, 36/35, is the difference between 10/9 and 8/7, 7/6 and 6/5, 5/4 and 9/7, 14/9 and 8/5, 5/3 and 12/7, and 7/4 and 9/5. It has a numerator which is both the sixth square number and the eighth triangular number, leading to it being the product of two superparticular commas both as 64/63 * 81/80 and as 66/65 * 78/77; it is also 45/44 * 176/175, 51/50 * 120/119 and 128/125 * 225/224.
| Sound = ji-36-35-csound-foscil-220hz.mp3
| Comma = yes
}}
{{Wikipedia|Septimal quarter tone}}


[[http://en.wikipedia.org/wiki/Septimal_quarter_tone]] </pre></div>
'''36/35''', the '''septimal quartertone''' is a [[7-limit]] interval with a size of about 48.8 [[cent]]s. It arises as the difference between various 7-limit [[interval quality|qualities]] of seconds, thirds, sixths, and sevenths: between supermajor and classical major, and between subminor and classical minor. These are the pairs of intervals separated by 36/35:  
<h4>Original HTML content:</h4>
* [[28/27]] and [[16/15]]
<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;36_35&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;septimal quarter tone&lt;/em&gt;, 36/35, is the difference between 10/9 and 8/7, 7/6 and 6/5, 5/4 and 9/7, 14/9 and 8/5, 5/3 and 12/7, and 7/4 and 9/5. It has a numerator which is both the sixth square number and the eighth triangular number, leading to it being the product of two superparticular commas both as 64/63 * 81/80 and as 66/65 * 78/77; it is also 45/44 * 176/175, 51/50 * 120/119 and 128/125 * 225/224. &lt;br /&gt;
* [[10/9]] and [[8/7]]
&lt;br /&gt;
* [[7/6]] and [[6/5]]
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_quarter_tone" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Septimal_quarter_tone&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
* [[5/4]] and [[9/7]]
* [[14/9]] and [[8/5]]
* [[5/3]] and [[12/7]]
* [[7/4]] and [[9/5]]
* [[15/8]] and [[27/14]]
 
It has a numerator which is both the sixth square number and the eighth [[triangular number]], leading to it being the product of two [[superparticular]] commas both as ([[64/63]])⋅([[81/80]]) and as ([[66/65]])⋅([[78/77]]); it is also ([[45/44]])⋅([[176/175]]), ([[51/50]])⋅([[120/119]]), ([[128/125]])⋅([[225/224]]), ([[50/49]])⋅([[126/125]]) and ([[56/55]])⋅([[99/98]]).
 
== Temperaments ==
When treated as a generator, it is almost a fourth of the septimal whole tone [[28/25]], differing by a [[wizma]]. This allows it to be used as a generator for a temperament of [[hemimean]] called [[sengagen]], where 5/4, 7/5 and 7/4 map to 8, 12 and 20 quartertones respectively.
 
When treated as a comma to be tempered out, it is known as the '''mint comma''', and tempering it out leads to the [[mint]] temperament. See [[Mint family]] for the family of rank-3 temperaments where it is tempered out, and [[Mint temperaments]] for the collection of rank-2 temperaments where it is tempered out.
 
== Etymology ==
The name ''mint comma'' was given by [[Mike Battaglia]] in 2012, for <u>min</u>or <u>t</u>hird because "it mixes 7/6 and 6/5 together into one minty interval"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103732.html Yahoo! Tuning Group | ''Quartonic'']</ref>. Before that, it had been known as the ''quartonic comma'', which refers to another comma today.
 
== Notation ==
=== Ben Johnston's notation ===
In [[Ben Johnston's notation]], this interval is denoted with "{{invert|7}}" (a turned "7"), and the reciprocal 35/36 with an ordinary "7". If the base note is C, then [[7/4]] is reprented by C–Bb7.  
 
=== Sagittal notation ===
In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \!) }} and is called the '''35 medium diesis''', or '''35M''' for short, because the simplest interval it notates is 35/1 = 5×7 (equiv. 35/16), as for example in C-D{{nbhsp}}{{sagittal | \!) }}. The upward version is called '''1/35M''' or '''35M up''' and is represented by {{sagittal| /|) }}.
 
== See also ==
* [[35/18]] – its [[octave complement]]
* [[35/24]] – its [[fifth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
 
== References ==
 
[[Category:Quartertone]]
[[Category:Mint]]
[[Category:Commas named for the intervals they stack]]
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