81/80
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**81/80** |-4 4 -1> 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. [[55edo]] tempers it out, while [[15edo]] 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]]. 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. =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. 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]]
Original HTML content:
<html><head><title>81_80</title></head><body><strong>81/80</strong><br />
|-4 4 -1><br />
21.506290 cents<br />
<br />
The <strong>syntonic</strong> or <strong>Didymus comma</strong> (frequency ratio <strong>81/80</strong>) is the smallest <a class="wiki_link" href="/superparticular">superparticular interval</a> which belongs to the <a class="wiki_link" href="/5-limit">5-limit</a>. Like <a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/625_624">625/624</a>, <a class="wiki_link" href="/2401_2400">2401/2400</a> and <a class="wiki_link" href="/4096_4095">4096/4095</a> 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 <a class="wiki_link" href="/10_9">10/9</a> and <a class="wiki_link" href="/9_8">9/8</a>, the product of which is the just major third, <a class="wiki_link" href="/5_4">5/4</a>. 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. <a class="wiki_link" href="/55edo">55edo</a> tempers it out, while <a class="wiki_link" href="/15edo">15edo</a> does not.<br />
<br />
Tempering out 81/80 gives a tuning for the <a class="wiki_link" href="/tone">whole tone</a> which is intermediate between 10/9 and 9/8, and leads to <a class="wiki_link" href="/Meantone%20family">meantone temperament</a>.<br />
<br />
Youtube video of "<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=IpWiEWFRGAY" rel="nofollow">Five senses of 81/80</a>", demonstratory video by Jacob Barton.<br />
<br />
According to <a class="wiki_link_ext" href="http://untwelve.org/interviews/golden.html" rel="nofollow">this interview</a>, Monroe Golden's <em>Incongruity</em> uses just-intonation chord progressions that exploit this comma.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Relations to other Superparticular Ratios"></a><!-- ws:end:WikiTextHeadingRule:0 -->Relations to other Superparticular Ratios</h1>
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.<br />
Names in brackets refer to 7-limit <a class="wiki_link" href="/Meantone%20family">meantone</a> extensions, or 11-limit rank three temperaments from the <a class="wiki_link" href="/Didymus%20rank%20three%20family">Didymus family</a> that temper out the respective ratios as commas.<br />
<table class="wiki_table">
<tr>
<th>Limit<br />
</th>
<th>r1 * r2<br />
</th>
<th>r2 / r1<br />
</th>
</tr>
<tr>
<td>5<br />
</td>
<td>-<br />
</td>
<td>9/8 * 9/10<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>126/125 * 225/224 (septimal meantone)<br />
</td>
<td>21/20 * 27/28 (sharptone), 36/35 * 63/64 (dominant)<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>99/98 * 441/440 (euterpe), 121/120 * 243/242 (urania)<br />
</td>
<td>33/32 * 54/55 (thalia), 45/44 * 99/100 (calliope)<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>91/90 * 729/728, 105/104 * 351/350<br />
</td>
<td>27/26 * 39/40, 65/64 * 324/325, 66/65 * 351/352, 78/77 * 2079/2080<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>85/84 * 1701/1700<br />
</td>
<td>51/50 * 135/136<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>96/95 * 513/512, 153/152 * 171/170<br />
</td>
<td>57/56 * 189/190, 76/75 * 1215/1216, 77/76 * 1539/1540<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>161/160 * 162/161<br />
</td>
<td>69/68 * 459/460<br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>117/116 * 261/260<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>93/92 * 621/620<br />
</td>
<td>63/62 * 279/280<br />
</td>
</tr>
<tr>
<td>37<br />
</td>
<td>111/110 * 297/296<br />
</td>
<td>75/74 * 999/1000<br />
</td>
</tr>
<tr>
<td>41<br />
</td>
<td>82/81 * 6561/6560<br />
</td>
<td>41/40 * 81/82<br />
</td>
</tr>
<tr>
<td>43<br />
</td>
<td>86/85 * 1377/1376, 87/86 * 1161/1160, 129/128 * 216/215<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>47<br />
</td>
<td>141/140 * 189/188<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>53<br />
</td>
<td>-<br />
</td>
<td>54/53 * 159/160<br />
</td>
</tr>
<tr>
<td>59<br />
</td>
<td>-<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>61<br />
</td>
<td>-<br />
</td>
<td>61/60 * 243/244<br />
</td>
</tr>
<tr>
<td>67<br />
</td>
<td>135/134 * 201/200<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>71<br />
</td>
<td>-<br />
</td>
<td>71/70 * 567/568, 72/71 * 639/640<br />
</td>
</tr>
<tr>
<td>73<br />
</td>
<td>-<br />
</td>
<td>73/72 * 729/730<br />
</td>
</tr>
<tr>
<td>79<br />
</td>
<td>-<br />
</td>
<td>79/78 * 3159/3160, 80/79 * 6399/6400<br />
</td>
</tr>
<tr>
<td>83<br />
</td>
<td>83/82 * 3321/3320, 84/83 * 2241/2240<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>89<br />
</td>
<td>89/88 * 891/890, 90/89 * 801/800<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>97<br />
</td>
<td>97/96 * 486/485<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>101<br />
</td>
<td>101/100 * 405/404<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>103<br />
</td>
<td>-<br />
</td>
<td>-<br />
</td>
</tr>
<tr>
<td>107<br />
</td>
<td>108/107 * 321/320<br />
</td>
<td>-<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Relations to other Superparticular Ratios-External Links"></a><!-- ws:end:WikiTextHeadingRule:2 -->External Links</h2>
<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">http://en.wikipedia.org/wiki/Syntonic_comma</a></body></html>