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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
'''81/80'''
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on <tt>2016-08-24 02:10:36 UTC</tt>.<br>
: The original revision id was <tt>590042524</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">**81/80**
|-4 4 -1&gt;
|-4 4 -1&gt;
21.506290 cents
21.506290 cents


The **syntonic** or **Didymus comma** (frequency ratio **81/80**) is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit]]. Like [[16_15|16/15]], [[625_624|625/624]], [[2401_2400|2401/2400]] and [[4096_4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10_9|10/9]] and [[9_8|9/8]], the product of which is the just major third, [[5_4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. [[105edo]] tempers it out, while [[15edo|3edo]] does not.
The '''syntonic''' or '''Didymus comma''' (frequency ratio '''81/80''') is the smallest [[superparticular|superparticular interval]] which belongs to the [[5-limit|5-limit]]. Like [[16/15|16/15]], [[625/624|625/624]], [[2401/2400|2401/2400]] and [[4096/4095|4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between [[10/9|10/9]] and [[9/8|9/8]], the product of which is the just major third, [[5/4|5/4]]. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. [[105edo|105edo]] tempers it out, while [[15edo|3edo]] does not.


Tempering out 81/80 gives a tuning for the [[tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone family|meantone temperament]].
Tempering out 81/80 gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone_family|meantone temperament]].


Youtube video of "[[http://www.youtube.com/watch?v=IpWiEWFRGAY|Five senses of 81/80]]", demonstratory video by Jacob Barton.
Youtube video of "[http://www.youtube.com/watch?v=IpWiEWFRGAY Five senses of 81/80]", demonstratory video by Jacob Barton.


According to [[http://untwelve.org/interviews/golden.html|this interview]], Monroe Golden's //Incongruity// uses just-intonation chord progressions that exploit this comma.
According to [http://untwelve.org/interviews/golden.html this interview], Monroe Golden's ''Incongruity'' uses just-intonation chord progressions that exploit this comma.


=Relations to other Superparticular Ratios=  
=Relations to other Superparticular Ratios=
Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations r1 * r2 or r2 / r1 of 81/80, where r1 and r2 are other superparticular ratios.
Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations r1 * r2 or r2 / r1 of 81/80, where r1 and r2 are other superparticular ratios.
Names in brackets refer to 7-limit [[Meantone family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus rank three family|Didymus family]] that temper out the respective ratios as commas.
||~ Limit ||~ r1 * r2 ||~ r2 / r1 ||
|| 5 || - || 9/8 * 9/10 ||
|| 7 || 126/125 * 225/224 (septimal meantone) || 21/20 * 27/28 (sharptone), 36/35 * 63/64 (dominant) ||
|| 11 || 99/98 * 441/440 (euterpe), 121/120 * 243/242 (urania) || 33/32 * 54/55 (thalia), 45/44 * 99/100 (calliope) ||
|| 13 || 91/90 * 729/728, 105/104 * 351/350 || 27/26 * 39/40, 65/64 * 324/325, 66/65 * 351/352, 78/77 * 2079/2080 ||
|| 17 || 85/84 * 1701/1700 || 51/50 * 135/136 ||
|| 19 || 96/95 * 513/512, 153/152 * 171/170 || 57/56 * 189/190, 76/75 * 1215/1216, 77/76 * 1539/1540 ||
|| 23 || 161/160 * 162/161 || 69/68 * 459/460 ||
|| 29 || 117/116 * 261/260 || - ||
|| 31 || 93/92 * 621/620 || 63/62 * 279/280 ||
|| 37 || 111/110 * 297/296 || 75/74 * 999/1000 ||
|| 41 || 82/81 * 6561/6560 || 41/40 * 81/82 ||
|| 43 || 86/85 * 1377/1376, 87/86 * 1161/1160, 129/128 * 216/215 || - ||
|| 47 || 141/140 * 189/188 || - ||
|| 53 || - || 54/53 * 159/160 ||
|| 59 || - || - ||
|| 61 || - || 61/60 * 243/244 ||
|| 67 || 135/134 * 201/200 || - ||
|| 71 || - || 71/70 * 567/568, 72/71 * 639/640 ||
|| 73 || - || 73/72 * 729/730 ||
|| 79 || - || 79/78 * 3159/3160, 80/79 * 6399/6400 ||
|| 83 || 83/82 * 3321/3320, 84/83 * 2241/2240 || - ||
|| 89 || 89/88 * 891/890, 90/89 * 801/800 || - ||
|| 97 || 97/96 * 486/485 || - ||
|| 101 || 101/100 * 405/404 || - ||
|| 103 || - || - ||
|| 107 || 108/107 * 321/320 || - ||
==External Links==
[[http://en.wikipedia.org/wiki/Syntonic_comma]]</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;81_80&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;81/80&lt;/strong&gt;&lt;br /&gt;
|-4 4 -1&amp;gt;&lt;br /&gt;
21.506290 cents&lt;br /&gt;
&lt;br /&gt;
The &lt;strong&gt;syntonic&lt;/strong&gt; or &lt;strong&gt;Didymus comma&lt;/strong&gt; (frequency ratio &lt;strong&gt;81/80&lt;/strong&gt;) is the smallest &lt;a class="wiki_link" href="/superparticular"&gt;superparticular interval&lt;/a&gt; which belongs to the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt;. Like &lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/625_624"&gt;625/624&lt;/a&gt;, &lt;a class="wiki_link" href="/2401_2400"&gt;2401/2400&lt;/a&gt; and &lt;a class="wiki_link" href="/4096_4095"&gt;4096/4095&lt;/a&gt; it has a fourth power as a numerator. Fourth powers are squares, and any comma with a square numerator is the ratio between two larger successive superparticular intervals; it is in fact the difference between &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; and &lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, the product of which is the just major third, &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;. That the numerator is a fourth power entails that the larger of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2. &lt;a class="wiki_link" href="/105edo"&gt;105edo&lt;/a&gt; tempers it out, while &lt;a class="wiki_link" href="/15edo"&gt;3edo&lt;/a&gt; does not.&lt;br /&gt;
&lt;br /&gt;
Tempering out 81/80 gives a tuning for the &lt;a class="wiki_link" href="/tone"&gt;whole tone&lt;/a&gt; which is intermediate between 10/9 and 9/8, and leads to &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone temperament&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
Youtube video of &amp;quot;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=IpWiEWFRGAY" rel="nofollow"&gt;Five senses of 81/80&lt;/a&gt;&amp;quot;, demonstratory video by Jacob Barton.&lt;br /&gt;
&lt;br /&gt;
According to &lt;a class="wiki_link_ext" href="http://untwelve.org/interviews/golden.html" rel="nofollow"&gt;this interview&lt;/a&gt;, Monroe Golden's &lt;em&gt;Incongruity&lt;/em&gt; uses just-intonation chord progressions that exploit this comma.&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="Relations to other Superparticular Ratios"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Relations to other Superparticular Ratios&lt;/h1&gt;
Superparticular ratios, like 81/80, can be expressed as products or quotients of other superparticular ratios. Following is a list of such representations r1 * r2 or r2 / r1 of 81/80, where r1 and r2 are other superparticular ratios.&lt;br /&gt;
Names in brackets refer to 7-limit &lt;a class="wiki_link" href="/Meantone%20family"&gt;meantone&lt;/a&gt; extensions, or 11-limit rank three temperaments from the &lt;a class="wiki_link" href="/Didymus%20rank%20three%20family"&gt;Didymus family&lt;/a&gt; that temper out the respective ratios as commas.&lt;br /&gt;


Names in brackets refer to 7-limit [[Meantone_family|meantone]] extensions, or 11-limit rank three temperaments from the [[Didymus_rank_three_family|Didymus family]] that temper out the respective ratios as commas.


&lt;table class="wiki_table"&gt;
{| class="wikitable"
    &lt;tr&gt;
|-
        &lt;th&gt;Limit&lt;br /&gt;
! | Limit
&lt;/th&gt;
! | r1 * r2
        &lt;th&gt;r1 * r2&lt;br /&gt;
! | r2 / r1
&lt;/th&gt;
|-
        &lt;th&gt;r2 / r1&lt;br /&gt;
| | 5
&lt;/th&gt;
| | -
    &lt;/tr&gt;
| | 9/8 * 9/10
    &lt;tr&gt;
|-
        &lt;td&gt;5&lt;br /&gt;
| | 7
&lt;/td&gt;
| | 126/125 * 225/224 (septimal meantone)
        &lt;td&gt;-&lt;br /&gt;
| | 21/20 * 27/28 (sharptone), 36/35 * 63/64 (dominant)
&lt;/td&gt;
|-
        &lt;td&gt;9/8 * 9/10&lt;br /&gt;
| | 11
&lt;/td&gt;
| | 99/98 * 441/440 (euterpe), 121/120 * 243/242 (urania)
    &lt;/tr&gt;
| | 33/32 * 54/55 (thalia), 45/44 * 99/100 (calliope)
    &lt;tr&gt;
|-
        &lt;td&gt;7&lt;br /&gt;
| | 13
&lt;/td&gt;
| | 91/90 * 729/728, 105/104 * 351/350
        &lt;td&gt;126/125 * 225/224 (septimal meantone)&lt;br /&gt;
| | 27/26 * 39/40, 65/64 * 324/325, 66/65 * 351/352, 78/77 * 2079/2080
&lt;/td&gt;
|-
        &lt;td&gt;21/20 * 27/28 (sharptone), 36/35 * 63/64 (dominant)&lt;br /&gt;
| | 17
&lt;/td&gt;
| | 85/84 * 1701/1700
    &lt;/tr&gt;
| | 51/50 * 135/136
    &lt;tr&gt;
|-
        &lt;td&gt;11&lt;br /&gt;
| | 19
&lt;/td&gt;
| | 96/95 * 513/512, 153/152 * 171/170
        &lt;td&gt;99/98 * 441/440 (euterpe), 121/120 * 243/242 (urania)&lt;br /&gt;
| | 57/56 * 189/190, 76/75 * 1215/1216, 77/76 * 1539/1540
&lt;/td&gt;
|-
        &lt;td&gt;33/32 * 54/55 (thalia), 45/44 * 99/100 (calliope)&lt;br /&gt;
| | 23
&lt;/td&gt;
| | 161/160 * 162/161
    &lt;/tr&gt;
| | 69/68 * 459/460
    &lt;tr&gt;
|-
        &lt;td&gt;13&lt;br /&gt;
| | 29
&lt;/td&gt;
| | 117/116 * 261/260
        &lt;td&gt;91/90 * 729/728, 105/104 * 351/350&lt;br /&gt;
| | -
&lt;/td&gt;
|-
        &lt;td&gt;27/26 * 39/40, 65/64 * 324/325, 66/65 * 351/352, 78/77 * 2079/2080&lt;br /&gt;
| | 31
&lt;/td&gt;
| | 93/92 * 621/620
    &lt;/tr&gt;
| | 63/62 * 279/280
    &lt;tr&gt;
|-
        &lt;td&gt;17&lt;br /&gt;
| | 37
&lt;/td&gt;
| | 111/110 * 297/296
        &lt;td&gt;85/84 * 1701/1700&lt;br /&gt;
| | 75/74 * 999/1000
&lt;/td&gt;
|-
        &lt;td&gt;51/50 * 135/136&lt;br /&gt;
| | 41
&lt;/td&gt;
| | 82/81 * 6561/6560
    &lt;/tr&gt;
| | 41/40 * 81/82
    &lt;tr&gt;
|-
        &lt;td&gt;19&lt;br /&gt;
| | 43
&lt;/td&gt;
| | 86/85 * 1377/1376, 87/86 * 1161/1160, 129/128 * 216/215
        &lt;td&gt;96/95 * 513/512, 153/152 * 171/170&lt;br /&gt;
| | -
&lt;/td&gt;
|-
        &lt;td&gt;57/56 * 189/190, 76/75 * 1215/1216, 77/76 * 1539/1540&lt;br /&gt;
| | 47
&lt;/td&gt;
| | 141/140 * 189/188
    &lt;/tr&gt;
| | -
    &lt;tr&gt;
|-
        &lt;td&gt;23&lt;br /&gt;
| | 53
&lt;/td&gt;
| | -
        &lt;td&gt;161/160 * 162/161&lt;br /&gt;
| | 54/53 * 159/160
&lt;/td&gt;
|-
        &lt;td&gt;69/68 * 459/460&lt;br /&gt;
| | 59
&lt;/td&gt;
| | -
    &lt;/tr&gt;
| | -
    &lt;tr&gt;
|-
        &lt;td&gt;29&lt;br /&gt;
| | 61
&lt;/td&gt;
| | -
        &lt;td&gt;117/116 * 261/260&lt;br /&gt;
| | 61/60 * 243/244
&lt;/td&gt;
|-
        &lt;td&gt;-&lt;br /&gt;
| | 67
&lt;/td&gt;
| | 135/134 * 201/200
    &lt;/tr&gt;
| | -
    &lt;tr&gt;
|-
        &lt;td&gt;31&lt;br /&gt;
| | 71
&lt;/td&gt;
| | -
        &lt;td&gt;93/92 * 621/620&lt;br /&gt;
| | 71/70 * 567/568, 72/71 * 639/640
&lt;/td&gt;
|-
        &lt;td&gt;63/62 * 279/280&lt;br /&gt;
| | 73
&lt;/td&gt;
| | -
    &lt;/tr&gt;
| | 73/72 * 729/730
    &lt;tr&gt;
|-
        &lt;td&gt;37&lt;br /&gt;
| | 79
&lt;/td&gt;
| | -
        &lt;td&gt;111/110 * 297/296&lt;br /&gt;
| | 79/78 * 3159/3160, 80/79 * 6399/6400
&lt;/td&gt;
|-
        &lt;td&gt;75/74 * 999/1000&lt;br /&gt;
| | 83
&lt;/td&gt;
| | 83/82 * 3321/3320, 84/83 * 2241/2240
    &lt;/tr&gt;
| | -
    &lt;tr&gt;
|-
        &lt;td&gt;41&lt;br /&gt;
| | 89
&lt;/td&gt;
| | 89/88 * 891/890, 90/89 * 801/800
        &lt;td&gt;82/81 * 6561/6560&lt;br /&gt;
| | -
&lt;/td&gt;
|-
        &lt;td&gt;41/40 * 81/82&lt;br /&gt;
| | 97
&lt;/td&gt;
| | 97/96 * 486/485
    &lt;/tr&gt;
| | -
    &lt;tr&gt;
|-
        &lt;td&gt;43&lt;br /&gt;
| | 101
&lt;/td&gt;
| | 101/100 * 405/404
        &lt;td&gt;86/85 * 1377/1376, 87/86 * 1161/1160, 129/128 * 216/215&lt;br /&gt;
| | -
&lt;/td&gt;
|-
        &lt;td&gt;-&lt;br /&gt;
| | 103
&lt;/td&gt;
| | -
    &lt;/tr&gt;
| | -
    &lt;tr&gt;
|-
        &lt;td&gt;47&lt;br /&gt;
| | 107
&lt;/td&gt;
| | 108/107 * 321/320
        &lt;td&gt;141/140 * 189/188&lt;br /&gt;
| | -
&lt;/td&gt;
|}
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;54/53 * 159/160&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;59&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;61&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;61/60 * 243/244&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;67&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;135/134 * 201/200&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;71&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;71/70 * 567/568, 72/71 * 639/640&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;73&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;73/72 * 729/730&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;79&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;79/78 * 3159/3160, 80/79 * 6399/6400&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;83&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;83/82 * 3321/3320, 84/83 * 2241/2240&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;89&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;89/88 * 891/890, 90/89 * 801/800&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;97&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;97/96 * 486/485&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;101&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;101/100 * 405/404&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;103&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;107&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;108/107 * 321/320&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;-&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
==External Links==
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Relations to other Superparticular Ratios-External Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;External Links&lt;/h2&gt;
[http://en.wikipedia.org/wiki/Syntonic_comma http://en.wikipedia.org/wiki/Syntonic_comma]      [[Category:5-limit]]
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Syntonic_comma&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Category:comma]]
[[Category:definition]]
[[Category:interval]]
[[Category:superparticular]]
[[Category:syntonic]]
Retrieved from "https://en.xen.wiki/w/81/80"