50edo
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[[toc]] //50edo// divides the [[octave]] into 50 equal parts of precisely 24 [[cent]]s each. In the [[5-limit]], it tempers out 81/80, making it a [[meantone]] system, and in that capacity has historically has drawn some notice. In [[http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf|"Harmonics or the Philosophy of Musical Sounds"]] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the [[Target tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While [[31edo]] extends meantone with a [[7_4|7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11_8|11/8]] and [[13_8|13/8]] are nearly pure. 50 tempers out 126/125, 225/224 and 3136/3125 in the [[7-limit]], indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the [[11-limit]] and 105/104, 144/143 and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&50 temperament ([[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11|Coblack]]), and provides the optimal patent val for 11 and 13 limit [[Meantone family#Septimal%20meantone-Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], [[tel:6115295232|6115295232]]/6103515625 = |23 6 -14>, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth. =Relations= The 50edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "Golden Tone System" ([[Das Goldene Tonsystem]]) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "[[http://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y|A Theory of Evolving Tonality]]"). =Intervals= ||~ Degrees of 50edo ||~ Cents value ||~ Ratios* ||~ Generator for* || || 0 || 0 || 1/1 || || || 1 || 24 || 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168 || [[xenharmonic/Hemimean clan#Sengagen|Sengagen]] || || 2 || 48 || 33/32, 36/35, 50/49, 55/54, 64/63 || || || 3 || 72 || 21/20, 25/24, 26/25, 27/26, 28/27 || [[xenharmonic/Vishnuzmic family#Vishnu|Vishnu]] (2/oct), [[http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11|Coblack]] (5/oct) || || 4 || 96 || 22/21 || [[xenharmonic/Meantone family#Injera|Injera]] (50d val, 2/oct) || || 5 || 120 || 16/15, 15/14, 14/13 || || || 6 || 144 || 13/12, 12/11 || || || 7 || 168 || 11/10 || || || 8 || 192 || 9/8, 10/9 || || || 9 || 216 || 25/22 || [[http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&limit=2.3.5.11.13|Tremka]], [[xenharmonic/Subgroup temperaments#x2.9.7.11-Machine|Machine]] (50b val) || || 10 || 240 || 8/7, 15/13 || || || 11 || 264 || 7/6 || [[xenharmonic/Marvel temperaments#Septimin-13-limit|Septimin (13-limit)]] || || 12 || 288 || 13/11 || || || 13 || 312 || 6/5 || || || 14 || 336 || 27/22, 39/32, 40/33, 49/40 || || || 15 || 360 || 16/13, 11/9 || || || 16 || 384 || 5/4 || [[xenharmonic/Marvel temperaments#Wizard-11-limit|Wizard]] (2/oct) || || 17 || 408 || 14/11 || [[xenharmonic/Ditonmic family|Ditonic]] || || 18 || 432 || 9/7 || [[xenharmonic/Porcupine family#Hedgehog|Hedgehog]] (50cc val, 2/oct) || || 19 || 456 || 13/10 || [[xenharmonic/Starling temperaments#Bisemidim|Bisemidim]] (2/oct) || || 20 || 480 || 33/25, 55/42, 64/49 || || || 21 || 504 || 4/3 || [[xenharmonic/Meantone|Meantone]]/[[xenharmonic/Meanpop|Meanpop]] || || 22 || 528 || 15/11 || || || 23 || 552 || 11/8, 18/13 || [[xenharmonic/Chromatic pairs#Barton|Barton]], [[xenharmonic/Hemimean clan#Emka|Emka]] || || 24 || 576 || 7/5 || || || 25 || 600 || 63/44, 88/63, 78/55, 55/39 || || || 26 || 624 || 10/7 || || || 27 || 648 || 16/11, 13/9 || || || 28 || 672 || 22/15 || || || 29 || 696 || 3/2 || || || 30 || 720 || 50/33, 84/55, 49/32 || || || 31 || 744 || 20/13 || || || 32 || 768 || 14/9 || || || 33 || 792 || 11/7 || || || 34 || 816 || 8/5 || || || 35 || 840 || 13/8, 18/11 || || || 36 || 864 || 44/27, 64/39, 33/20, 80/49 || || || 37 || 888 || 5/3 || || || 38 || 912 || 22/13 || || || 39 || 936 || 12/7 || || || 40 || 960 || 7/4 || || || 41 || 984 || 44/25 || || || 42 || 1008 || 16/9, 9/5 || || || 43 || 1032 || 20/11 || || || 44 || 1056 || 24/13, 11/6 || || || 45 || 1080 || 15/8, 28/15, 13/7 || || || 46 || 1104 || 21/11 || || || 47 || 1128 || 40/21, 48/25, 25/13, 52/27, 27/14 || || || 48 || 1152 || 64/33, 35/18, 49/25, 108/55, 63/32 || || || 49 || 1176 || || || *using the 13-limit patent val except as noted ==Selected just intervals by error== The following table shows how [[Just-24|some prominent just intervals]] are represented in 50edo (ordered by absolute error). || **Interval, complement** || **Error (abs., in [[cent|cents]])** || ||= [[16_13|16/13]], [[13_8|13/8]] ||= 0.528 || ||= [[15_14|15/14]], [[28_15|28/15]] ||= 0.557 || ||= [[11_8|11/8]], [[16_11|16/11]] ||= 0.682 || ||= [[13_11|13/11]], [[22_13|22/13]] ||= 1.210 || ||= [[13_10|13/10]], [[20_13|20/13]] ||= 1.786 || ||= [[5_4|5/4]], [[8_5|8/5]] ||= 2.314 || ||= [[7_6|7/6]], [[12_7|12/7]] ||= 2.871 || ||= [[11_10|11/10]], [[20_11|20/11]] ||= 2.996 || ||= [[9_7|9/7]], [[14_9|14/9]] ||= 3.084 || ||= [[6_5|6/5]], [[5_3|5/3]] ||= 3.641 || ||= [[13_12|13/12]], [[24_13|24/13]] ||= 5.427 || ||= [[4_3|4/3]], [[3_2|3/2]] ||= 5.955 || ||= [[7_5|7/5]], [[10_7|10/7]] ||= 6.512 || ||= [[12_11|12/11]], [[11_6|11/6]] ||= 6.637 || ||= [[15_13|15/13]], [[26_15|26/15]] ||= 7.741 || ||= [[16_15|16/15]], [[15_8|15/8]] ||= 8.269 || ||= [[14_13|14/13]], [[13_7|13/7]] ||= 8.298 || ||= [[8_7|8/7]], [[7_4|7/4]] ||= 8.826 || ||= [[15_11|15/11]], [[22_15|22/15]] ||= 8.951 || ||= [[14_11|14/11]], [[11_7|11/7]] ||= 9.508 || ||= [[10_9|10/9]], [[9_5|9/5]] ||= 9.596 || ||= [[18_13|18/13]], [[13_9|13/9]] ||= 11.382 || ||= [[11_9|11/9]], [[18_11|18/11]] ||= 11.408 || ||= [[9_8|9/8]], [[16_9|16/9]] ||= 11.910 || =Commas= 50 EDO tempers out the following commas. (Note: This assumes the val < 50 79 116 140 173/1 185 204 212 226/1 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2. ||~ Monzo ||~ Cents ||~ Ratio ||~ Name 1 ||~ Name 2 || || | -4 4 -1 > ||> 21.51 ||= 81/80 || Syntonic comma || Didymus comma || || | -27 -2 13 > ||> 18.17 ||= || Ditonma || || || | 23 6 -14 > ||> 3.34 ||= || Vishnu comma || || || | 1 2 -3 1 > ||> 13.79 ||= 126/125 || Starling comma || Small septimal comma || || | -5 2 2 -1 > ||> 7.71 ||= 225/224 || Septimal kleisma || Marvel comma || || | 6 0 -5 2 > ||> 6.08 ||= 3136/3125 || Hemimean || Middle second comma || || | -6 -8 2 5 > ||> 1.12 ||= || Wizma || || || |-11 2 7 -3 > ||> 1.63 ||= || Meter || || || | 11 -10 -10 10/1 > ||> 5.57 ||= || Linus || || || |-13 10 0 -1 > ||> 50.72 ||= 59049/57344 || Harrison's comma || || || | 2 3 1 -2 -1 > ||> 3.21 ||= 540/539 || Swets' comma || Swetisma || || | -3 4 -2 -2 2 > ||> 0.18 ||= 9801/9800 || Kalisma || Gauss' comma || || | 5 -1 3 0 -3 > ||> 3.03 ||= 4000/3993 || Wizardharry || Undecimal schisma || || | -7 -1 1 1 1 > ||> 4.50 ||= 385/384 || Keenanisma || Undecimal kleisma || || | -1 0 1 2 -2 > ||> 21.33 ||= 245/242 || Cassacot || || || | 2 -1 0 1 -2 1 > ||> 4.76 ||= 364/363 || Gentle comma || || || | 2 -1 -1 2 0 -1 > ||> 8.86 ||= 196/195 || Mynucuma || || || | 2 3 0 -1 1 -2 > ||> 7.30 ||= 1188/1183 || Kestrel Comma || || || | 3 0 2 0 1 -3 > ||> 2.36 ||= 2200/2197 || Petrma || Parizek comma || || | -3 1 1 1 0 -1 > ||> 16.57 ||= 105/104 || Animist comma || Small tridecimal comma || || || | 4 2 0 0 -1 -1 > ||> 12.06 ||= 144/143 || Grossma || || || | 3 -2 0 1 -1 -1 0 0 1 > ||> 1.34 ||= 1288/1287 || Triaphonisma || || =Music= [[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3|Twinkle canon – 50 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]] [[@http://soonlabel.com/xenharmonic/archives/1118|Fantasia Catalana by Claudi Meneghin]] [[http://soonlabel.com/xenharmonic/archives/1929|Fugue on the Dragnet theme by Claudi Meneghin]] =Additional reading= [[http://www.archive.org/details/harmonicsorphilo00smit|Robert Smith's book online]] [[http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html|More information about Robert Smith's temperament]] [[https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh|50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]] [[http://iamcamtaylor.wordpress.com/|iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]]
Original HTML content:
<html><head><title>50edo</title></head><body><!-- ws:start:WikiTextTocRule:12:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><div style="margin-left: 1em;"><a href="#Relations">Relations</a></div>
<!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div>
<!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><div style="margin-left: 2em;"><a href="#Intervals-Selected just intervals by error">Selected just intervals by error</a></div>
<!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div>
<!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><div style="margin-left: 1em;"><a href="#Music">Music</a></div>
<!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><div style="margin-left: 1em;"><a href="#Additional reading">Additional reading</a></div>
<!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --></div>
<!-- ws:end:WikiTextTocRule:19 --><em>50edo</em> divides the <a class="wiki_link" href="/octave">octave</a> into 50 equal parts of precisely 24 <a class="wiki_link" href="/cent">cent</a>s each. In the <a class="wiki_link" href="/5-limit">5-limit</a>, it tempers out 81/80, making it a <a class="wiki_link" href="/meantone">meantone</a> system, and in that capacity has historically has drawn some notice. In <a class="wiki_link_ext" href="http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf" rel="nofollow">"Harmonics or the Philosophy of Musical Sounds"</a> (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts - 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the <a class="wiki_link" href="/Target%20tunings">least squares</a> tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While <a class="wiki_link" href="/31edo">31edo</a> extends meantone with a <a class="wiki_link" href="/7_4">7/4</a> which is nearly pure, 50 has a flat 7/4 but both <a class="wiki_link" href="/11_8">11/8</a> and <a class="wiki_link" href="/13_8">13/8</a> are nearly pure.<br />
<br />
50 tempers out 126/125, 225/224 and 3136/3125 in the <a class="wiki_link" href="/7-limit">7-limit</a>, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the <a class="wiki_link" href="/11-limit">11-limit</a> and 105/104, 144/143 and 196/195 in the <a class="wiki_link" href="/13-limit">13-limit</a>, and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the 15&50 temperament (<a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11" rel="nofollow">Coblack</a>), and provides the optimal patent val for 11 and 13 limit <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone-Bimeantone">bimeantone</a>. It is also the unique equal temperament tempering out both 81/80 and the <a class="wiki_link" href="/vishnuzma">vishnuzma</a>, <a class="wiki_link" href="http://tel.wikispaces.com/6115295232">6115295232</a>/6103515625 = |23 6 -14>, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Relations"></a><!-- ws:end:WikiTextHeadingRule:0 -->Relations</h1>
The 50edo system is related to <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a> as the next approximation to the "Golden Tone System" (<a class="wiki_link" href="/Das%20Goldene%20Tonsystem">Das Goldene Tonsystem</a>) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "<a class="wiki_link_ext" href="http://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y" rel="nofollow">A Theory of Evolving Tonality</a>").<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1>
<table class="wiki_table">
<tr>
<th>Degrees of 50edo<br />
</th>
<th>Cents value<br />
</th>
<th>Ratios*<br />
</th>
<th>Generator for*<br />
</th>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>1/1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>24<br />
</td>
<td>45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hemimean%20clan#Sengagen">Sengagen</a><br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>48<br />
</td>
<td>33/32, 36/35, 50/49, 55/54, 64/63<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>72<br />
</td>
<td>21/20, 25/24, 26/25, 27/26, 28/27<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Vishnuzmic%20family#Vishnu">Vishnu</a> (2/oct), <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=15%2650&limit=11" rel="nofollow">Coblack</a> (5/oct)<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>96<br />
</td>
<td>22/21<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone%20family#Injera">Injera</a> (50d val, 2/oct)<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>120<br />
</td>
<td>16/15, 15/14, 14/13<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>144<br />
</td>
<td>13/12, 12/11<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>168<br />
</td>
<td>11/10<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>192<br />
</td>
<td>9/8, 10/9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>216<br />
</td>
<td>25/22<br />
</td>
<td><a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=50%2661p&limit=2.3.5.11.13" rel="nofollow">Tremka</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Subgroup%20temperaments#x2.9.7.11-Machine">Machine</a> (50b val)<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>240<br />
</td>
<td>8/7, 15/13<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>264<br />
</td>
<td>7/6<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20temperaments#Septimin-13-limit">Septimin (13-limit)</a><br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>288<br />
</td>
<td>13/11<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>312<br />
</td>
<td>6/5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>336<br />
</td>
<td>27/22, 39/32, 40/33, 49/40<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>360<br />
</td>
<td>16/13, 11/9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>384<br />
</td>
<td>5/4<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20temperaments#Wizard-11-limit">Wizard</a> (2/oct)<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>408<br />
</td>
<td>14/11<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ditonmic%20family">Ditonic</a><br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>432<br />
</td>
<td>9/7<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Porcupine%20family#Hedgehog">Hedgehog</a> (50cc val, 2/oct)<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>456<br />
</td>
<td>13/10<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Starling%20temperaments#Bisemidim">Bisemidim</a> (2/oct)<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>480<br />
</td>
<td>33/25, 55/42, 64/49<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>504<br />
</td>
<td>4/3<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone">Meantone</a>/<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meanpop">Meanpop</a><br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>528<br />
</td>
<td>15/11<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>552<br />
</td>
<td>11/8, 18/13<br />
</td>
<td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chromatic%20pairs#Barton">Barton</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Hemimean%20clan#Emka">Emka</a><br />
</td>
</tr>
<tr>
<td>24<br />
</td>
<td>576<br />
</td>
<td>7/5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>25<br />
</td>
<td>600<br />
</td>
<td>63/44, 88/63, 78/55, 55/39<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>26<br />
</td>
<td>624<br />
</td>
<td>10/7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>27<br />
</td>
<td>648<br />
</td>
<td>16/11, 13/9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>28<br />
</td>
<td>672<br />
</td>
<td>22/15<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>29<br />
</td>
<td>696<br />
</td>
<td>3/2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>30<br />
</td>
<td>720<br />
</td>
<td>50/33, 84/55, 49/32<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>31<br />
</td>
<td>744<br />
</td>
<td>20/13<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>32<br />
</td>
<td>768<br />
</td>
<td>14/9<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>33<br />
</td>
<td>792<br />
</td>
<td>11/7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>34<br />
</td>
<td>816<br />
</td>
<td>8/5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>35<br />
</td>
<td>840<br />
</td>
<td>13/8, 18/11<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>36<br />
</td>
<td>864<br />
</td>
<td>44/27, 64/39, 33/20, 80/49<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>37<br />
</td>
<td>888<br />
</td>
<td>5/3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>38<br />
</td>
<td>912<br />
</td>
<td>22/13<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>39<br />
</td>
<td>936<br />
</td>
<td>12/7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>40<br />
</td>
<td>960<br />
</td>
<td>7/4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>41<br />
</td>
<td>984<br />
</td>
<td>44/25<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>42<br />
</td>
<td>1008<br />
</td>
<td>16/9, 9/5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>43<br />
</td>
<td>1032<br />
</td>
<td>20/11<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>44<br />
</td>
<td>1056<br />
</td>
<td>24/13, 11/6<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>45<br />
</td>
<td>1080<br />
</td>
<td>15/8, 28/15, 13/7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>46<br />
</td>
<td>1104<br />
</td>
<td>21/11<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>47<br />
</td>
<td>1128<br />
</td>
<td>40/21, 48/25, 25/13, 52/27, 27/14<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>48<br />
</td>
<td>1152<br />
</td>
<td>64/33, 35/18, 49/25, 108/55, 63/32<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>49<br />
</td>
<td>1176<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
</table>
*using the 13-limit patent val except as noted<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Intervals-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:4 -->Selected just intervals by error</h2>
The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 50edo (ordered by absolute error).<br />
<table class="wiki_table">
<tr>
<td><strong>Interval, complement</strong><br />
</td>
<td><strong>Error (abs., in <a class="wiki_link" href="/cent">cents</a>)</strong><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_13">16/13</a>, <a class="wiki_link" href="/13_8">13/8</a><br />
</td>
<td style="text-align: center;">0.528<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_14">15/14</a>, <a class="wiki_link" href="/28_15">28/15</a><br />
</td>
<td style="text-align: center;">0.557<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br />
</td>
<td style="text-align: center;">0.682<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_11">13/11</a>, <a class="wiki_link" href="/22_13">22/13</a><br />
</td>
<td style="text-align: center;">1.210<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_10">13/10</a>, <a class="wiki_link" href="/20_13">20/13</a><br />
</td>
<td style="text-align: center;">1.786<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br />
</td>
<td style="text-align: center;">2.314<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br />
</td>
<td style="text-align: center;">2.871<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_10">11/10</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
<td style="text-align: center;">2.996<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br />
</td>
<td style="text-align: center;">3.084<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br />
</td>
<td style="text-align: center;">3.641<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/13_12">13/12</a>, <a class="wiki_link" href="/24_13">24/13</a><br />
</td>
<td style="text-align: center;">5.427<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br />
</td>
<td style="text-align: center;">5.955<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br />
</td>
<td style="text-align: center;">6.512<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/12_11">12/11</a>, <a class="wiki_link" href="/11_6">11/6</a><br />
</td>
<td style="text-align: center;">6.637<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_13">15/13</a>, <a class="wiki_link" href="/26_15">26/15</a><br />
</td>
<td style="text-align: center;">7.741<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_8">15/8</a><br />
</td>
<td style="text-align: center;">8.269<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_13">14/13</a>, <a class="wiki_link" href="/13_7">13/7</a><br />
</td>
<td style="text-align: center;">8.298<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br />
</td>
<td style="text-align: center;">8.826<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/22_15">22/15</a><br />
</td>
<td style="text-align: center;">8.951<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/11_7">11/7</a><br />
</td>
<td style="text-align: center;">9.508<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br />
</td>
<td style="text-align: center;">9.596<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/18_13">18/13</a>, <a class="wiki_link" href="/13_9">13/9</a><br />
</td>
<td style="text-align: center;">11.382<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br />
</td>
<td style="text-align: center;">11.408<br />
</td>
</tr>
<tr>
<td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br />
</td>
<td style="text-align: center;">11.910<br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h1>
50 EDO tempers out the following commas. (Note: This assumes the val < 50 79 116 140 173/1 185 204 212 226/1 |, comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.<br />
<table class="wiki_table">
<tr>
<th>Monzo<br />
</th>
<th>Cents<br />
</th>
<th>Ratio<br />
</th>
<th>Name 1<br />
</th>
<th>Name 2<br />
</th>
</tr>
<tr>
<td>| -4 4 -1 ><br />
</td>
<td style="text-align: right;">21.51<br />
</td>
<td style="text-align: center;">81/80<br />
</td>
<td>Syntonic comma<br />
</td>
<td>Didymus comma<br />
</td>
</tr>
<tr>
<td>| -27 -2 13 ><br />
</td>
<td style="text-align: right;">18.17<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>Ditonma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 23 6 -14 ><br />
</td>
<td style="text-align: right;">3.34<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>Vishnu comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 1 2 -3 1 ><br />
</td>
<td style="text-align: right;">13.79<br />
</td>
<td style="text-align: center;">126/125<br />
</td>
<td>Starling comma<br />
</td>
<td>Small septimal comma<br />
</td>
</tr>
<tr>
<td>| -5 2 2 -1 ><br />
</td>
<td style="text-align: right;">7.71<br />
</td>
<td style="text-align: center;">225/224<br />
</td>
<td>Septimal kleisma<br />
</td>
<td>Marvel comma<br />
</td>
</tr>
<tr>
<td>| 6 0 -5 2 ><br />
</td>
<td style="text-align: right;">6.08<br />
</td>
<td style="text-align: center;">3136/3125<br />
</td>
<td>Hemimean<br />
</td>
<td>Middle second comma<br />
</td>
</tr>
<tr>
<td>| -6 -8 2 5 ><br />
</td>
<td style="text-align: right;">1.12<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>Wizma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>|-11 2 7 -3 ><br />
</td>
<td style="text-align: right;">1.63<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>Meter<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 11 -10 -10 10/1 ><br />
</td>
<td style="text-align: right;">5.57<br />
</td>
<td style="text-align: center;"><br />
</td>
<td>Linus<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>|-13 10 0 -1 ><br />
</td>
<td style="text-align: right;">50.72<br />
</td>
<td style="text-align: center;">59049/57344<br />
</td>
<td>Harrison's comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 2 3 1 -2 -1 ><br />
</td>
<td style="text-align: right;">3.21<br />
</td>
<td style="text-align: center;">540/539<br />
</td>
<td>Swets' comma<br />
</td>
<td>Swetisma<br />
</td>
</tr>
<tr>
<td>| -3 4 -2 -2 2 ><br />
</td>
<td style="text-align: right;">0.18<br />
</td>
<td style="text-align: center;">9801/9800<br />
</td>
<td>Kalisma<br />
</td>
<td>Gauss' comma<br />
</td>
</tr>
<tr>
<td>| 5 -1 3 0 -3 ><br />
</td>
<td style="text-align: right;">3.03<br />
</td>
<td style="text-align: center;">4000/3993<br />
</td>
<td>Wizardharry<br />
</td>
<td>Undecimal schisma<br />
</td>
</tr>
<tr>
<td>| -7 -1 1 1 1 ><br />
</td>
<td style="text-align: right;">4.50<br />
</td>
<td style="text-align: center;">385/384<br />
</td>
<td>Keenanisma<br />
</td>
<td>Undecimal kleisma<br />
</td>
</tr>
<tr>
<td>| -1 0 1 2 -2 ><br />
</td>
<td style="text-align: right;">21.33<br />
</td>
<td style="text-align: center;">245/242<br />
</td>
<td>Cassacot<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 2 -1 0 1 -2 1 ><br />
</td>
<td style="text-align: right;">4.76<br />
</td>
<td style="text-align: center;">364/363<br />
</td>
<td>Gentle comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 2 -1 -1 2 0 -1 ><br />
</td>
<td style="text-align: right;">8.86<br />
</td>
<td style="text-align: center;">196/195<br />
</td>
<td>Mynucuma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 2 3 0 -1 1 -2 ><br />
</td>
<td style="text-align: right;">7.30<br />
</td>
<td style="text-align: center;">1188/1183<br />
</td>
<td>Kestrel Comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 3 0 2 0 1 -3 ><br />
</td>
<td style="text-align: right;">2.36<br />
</td>
<td style="text-align: center;">2200/2197<br />
</td>
<td>Petrma<br />
</td>
<td>Parizek comma<br />
</td>
</tr>
<tr>
<td>| -3 1 1 1 0 -1 ><br />
</td>
<td style="text-align: right;">16.57<br />
</td>
<td style="text-align: center;">105/104<br />
</td>
<td>Animist comma<br />
</td>
<td>Small tridecimal comma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 4 2 0 0 -1 -1 ><br />
</td>
<td style="text-align: right;">12.06<br />
</td>
<td style="text-align: center;">144/143<br />
</td>
<td>Grossma<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>| 3 -2 0 1 -1 -1 0 0 1 ><br />
</td>
<td style="text-align: right;">1.34<br />
</td>
<td style="text-align: center;">1288/1287<br />
</td>
<td>Triaphonisma<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:8 -->Music</h1>
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow">Twinkle canon – 50 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br />
<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1118" rel="nofollow" target="_blank">Fantasia Catalana by Claudi Meneghin</a><br />
<a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1929" rel="nofollow">Fugue on the Dragnet theme by Claudi Meneghin</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Additional reading"></a><!-- ws:end:WikiTextHeadingRule:10 -->Additional reading</h1>
<a class="wiki_link_ext" href="http://www.archive.org/details/harmonicsorphilo00smit" rel="nofollow">Robert Smith's book online</a><br />
<a class="wiki_link_ext" href="http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html" rel="nofollow">More information about Robert Smith's temperament</a><br />
<br />
<a class="wiki_link_ext" href="https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh" rel="nofollow">50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor</a><br />
<a class="wiki_link_ext" href="http://iamcamtaylor.wordpress.com/" rel="nofollow">iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor</a></body></html>