23edo
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[[media type="custom" key="10021317"]][[toc|flat]] ---- =<span style="background-color: #ffffff; color: #009927; font-family: 'Times New Roman',Times,serif; font-size: 113%;">23 tone equal temperament</span>= **//23-tET//**, or **//23-EDO//**, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony //[Icositrifonía]//. It has good approximations for [[5_3|5/3]], [[11_7|11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]]·[[46edo]], the larger 17-limit [[k*N subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. =<span style="font-size: 1.4em;">Intervals</span>= || <span style="color: #660000;">[[Degree]]</span> || [[Cent]]s value || Approximate Ratios* || || 0 || 0 || 1/1 || || 1 || 52.1739 || 33/32, 34/33 || || 2 || 104.3478 || 17/16, 15/16, 18/17 || || 3 || 156.5217 || 11/10, 12/11 || || 4 || 208.6957 || 9/8 || || 5 || 260.8696 || 7/6, 15/13 || || 6 || 313.0435 || 6/5 || || 7 || 365.2174 || 21/17, 26/21 || || 8 || 417.3913 || 14/11, 33/26 || || 9 || 469.5652 || 21/16, 17/13 || || 10 || 521.7391 || 65/48, 256/189 || || 11 || 573.913 || 7/5 || || 12 || 626.087 || 10/7 || || 13 || 678.2609 || 96/65, 189/128 || || 14 || 730.4348 || 32/21, 26/17 || || 15 || 782.6087 || 11/7, 52/33 || || 16 || 834.7826 || 13/8, 21/13 || || 17 || 886.9565 || 5/3 || || 18 || 939.1304 || 12/7, 26/15 || || 19 || 991.3043 || 16/9 || || 20 || 1043.4783 || 11/6, 20/11 || || 21 || 1095.6522 || 15/8, 17/9, 32/17 || || 22 || 1147.8261 || 33/17, 64/33 || || 23 || 1200 || 2/1 || *based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible. 23-EDO was proposed by ethnomusicologist [[http://en.wikipedia.org/wiki/Erich_von_Hornbostel|Erich von Hornbostel]] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe. 23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, and 7th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore unusual harmonic territory. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, and 7/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23. Like[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4_3|4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3_2|3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29|Rapoport]]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale. ==Kosmorsky's Sephiroth modes== I would argue that the most significant modes of 23 edo are those of the 2 2 2 3 2 2 3 2 2 3 scale ([[3L 7s|3L 7s fair mosh]]); This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the Harmonic21 and three add up to the Harmonic17 almost perfectly (I discovered the temperament before I even realised 23 tone fits) The chord 8:13:21:34 is a fragment of the fibonacci sequence, and 16:21:26 is harmonically symmetrical (The chord 26:29:32 is also symmetric, but only [[20edo]] and [[43edo]] approximate the Harmonic29, as well) 13:17:21 is symmetrical, and 17:21:26:32 has an interesting linear widening pattern, but more importantly, the Harmonic13 and Harmonic17 are both dual primes like 3&5 and 7&11 Thus I have named these 10 modes according to the Sephiroth as follows: 2 2 2 3 2 2 3 2 2 3 - Mode Keter 2 2 3 2 2 3 2 2 3 2 - Chesed 2 3 2 2 3 2 2 3 2 2 - Netzach 3 2 2 3 2 2 3 2 2 2 - Malkuth 2 2 3 2 2 3 2 2 2 3 - Binah 2 3 2 2 3 2 2 2 3 2 - Tiferet 3 2 2 3 2 2 2 3 2 2 - Yesod 2 2 3 2 2 2 3 2 2 3 - Chokmah 2 3 2 2 2 3 2 2 3 2 - Gevurah 3 2 2 2 3 2 2 3 2 2 - Hod -- Kosmorsky =Music= <span class="ymp-btn-page-play ymp-media-e8b030b27e37ed744bb651845d914579">[[http://home.vicnet.net.au/%7Eepoetry/family.mp3|The Family Supper]]</span> by [[Warren Burt]] <span class="ymp-btn-page-play ymp-media-ad0e9a76a742e629379c7c9667584433">[[http://www.youtube.com/watch?v=Hqst8MaRiYM|Icositriphonic Heptatonic MOS]]</span> by [[Igliashon Jones]] <span class="ymp-btn-page-play ymp-media-11d44e0c35e6cbb396811396e4304800">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20His%20Wandering%20Kinship%20with%20Ashes.mp3|His Wandering Kinship with Ashes]]</span> by Iglashion Jones <span class="ymp-btn-page-play ymp-media-09a22042e6869c062a19ee073e5d2372">[[http://www.nonoctave.com/tunes/CosmicChamber.mp3|Cosmic Chamber]]</span> by [[X. J. Scott]] <span class="ymp-btn-page-play ymp-media-1a0d476bcc56e8699d94050c4bd46398">[[http://www.nonoctave.com/tunes/Daisies.mp3|Daisies on the Beach]]</span> by X. J. Scott <span style="background-position: 100% 50%; cursor: pointer; padding-right: 10px;"><span class="ymp-btn-page-play ymp-media-c4aa76ed239a412533a11b438698b09b">[[http://www.akjmusic.com/audio/boogie_pie.mp3|Boogie Pie]]</span></span>by [[Aaron Krister Johnson]] <span class="ymp-btn-page-play ymp-media-c0b7192b09da11ecea83384b2aad9c4b">[[http://clones.soonlabel.com/public/micro/23edo/daily20110619_23edo_23_chilled.mp3|23 Chilled]]</span> by [[Chris Vaisvil]] <span class="ymp-btn-page-play ymp-media-8ca0c9a378cc362bbcbc24ba13d6974b">[[http://www.seraph.it/dep/det/DesertWinds.mp3|Desert Winds]]</span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/926007c7483e4abc5a48d582c0667947-105.html|blog entry]]) //Allegro Moderato// by Easley Blackwood = = [[image:Ciclo_Icositrifonía.png width="492" height="490" caption="Intervallic Cycle of 23 steps Equal per Octave"]] == == =Commas= 23 EDO tempers out the following commas. (Note: This assumes the val < 23 36 53 65 80 85 |.) Also note the discussion above, where there are some commas mentioned that are not in the standard comma list (e.g., 28/27). ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || ||= 135/128 ||< | -7 3 1 > ||> 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma || ||= 15625/15552 ||< | -6 -5 6 > ||> 8.11 ||= Kleisma ||= Semicomma Majeur ||= || ||= 36/35 ||< | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone ||= ||= || ||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= || ||= 4000/3969 ||< | 5 -4 3 -2 > ||> 13.47 ||= Octagar ||= ||= || ||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||= ||= || ||= 100/99 ||< | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||= ||= || ||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= || =**23 tone [[Equal Modes]]:**= || 10 10 3 || || || 9 9 5 || || || 8 8 7 || || || 7 7 7 2 || || || 7 2 7 7 || || || 6 6 6 5 || || || 6 5 6 6 || || || 5 4 5 5 4 || [[3L 2s|3L 2s (father)]] || || 5 4 5 4 5 || || || 7 1 7 7 1 || || || 7 1 7 1 7 || || || 5 5 5 5 3 || [[4L 1s|4L 1s (bug)]] || || 5 3 5 5 5 || || || 4 4 4 4 4 3 || [[5L 1s|5L 1s (Grumpy hexatonic)]] || || 4 3 4 4 4 4 || || || 5 1 5 1 5 1 5 || [[4L 3s|4L 3s (mish)]] || || 3 3 3 5 3 3 3 || [[1L 6s|1L 6s (Happy heptatonic)]] || || 4 3 3 3 3 3 4 || [[2L 5s|2L 5s (mavila, anti-diatonic)]] || || 3 4 3 3 4 3 3 || || || 3 3 4 3 3 3 4 || || || 3 3 3 4 3 3 4 || || || 3 3 3 4 3 4 3 || || || 2 5 2 5 2 5 2 || [[3L 4s|3L 4s (mosh)]] || || 4 1 4 4 1 4 4 1 || [[5L 3s|5L 3s (unfair father)]] || || 3 3 3 3 3 3 3 2 || [[7L 1s|7L 1s (Grumpy octatonic)]] || || 3 2 3 3 3 3 3 3 || || || **3 3 3 1 3 3 3 3 1** || [[7L 2s|7L 2s (mavila superdiatonic)]] || || 3 3 1 3 3 3 1 3 3 || || || 3 2 3 2 3 2 3 2 3 || [[5L 4s|5L 4s (unfair bug)]] || || 2 2 2 3 2 2 3 2 2 3 || Mode Keter || || 2 2 3 2 2 3 2 2 3 2 || Chesed || || 2 3 2 2 3 2 2 3 2 2 || Netzach || || 3 2 2 3 2 2 3 2 2 2 || Malkuth || || 2 2 3 2 2 3 2 2 2 3 || Binah || || 2 3 2 2 3 2 2 2 3 2 || Tiferet || || 3 2 2 3 2 2 2 3 2 2 || Yesod || || 2 2 3 2 2 2 3 2 2 3 || Chokmah || || 2 3 2 2 2 3 2 2 3 2 || Gevurah || || 3 2 2 2 3 2 2 3 2 2 || Hod || || **3 1 3 1 3 1 3 1 3 1 3** || || || 2 2 2 1 2 2 2 1 2 2 2 2 1 || || || 2 2 1 2 2 1 2 2 1 2 2 1 2 1 || || || **2 1 2 2 1 2 2 1 2 2 1 2 2 1** || || || 1 1 1 4 1 1 1 1 4 1 1 1 1 4 || || || 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 || || || **2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1** || || || 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 || || =Books= [[image:Libro_Icositrifónico.PNG width="242" height="294"]] =Instruments= [[image:Icositriphonic_Bass.JPG width="594" height="216"]] //An Icositriphonic Bass. 23-EDO Bass by Tútim Deft Wafil.// [[image:Icositriphonic_Guitar.PNG width="601" height="305"]] //An Icositriphonic 8-string Guitar. 23-EDO Guitar by Ron Sword.// [[image:Teclado_Icositrifónico.PNG width="591" height="326" caption="A classic visualization of an Alternative 1/3-tone Keyboard. Armodue-Hornbostel Family Musical Systems"]] Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it. [[image:playing.jpg]] Here is a still shot of the completed instrument. [[image:complette.jpg]] This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion. [[media type="youtube" key="K4iO7k152og" height="349" width="560"]]
Original HTML content:
<html><head><title>23edo</title></head><body><!-- ws:start:WikiTextMediaRule:0:<img src="http://www.wikispaces.com/site/embedthumbnail/custom/10021317?h=0&w=0" class="WikiMedia WikiMediaCustom" id="wikitext@@media@@type=&quot;custom&quot; key=&quot;10021317&quot;" title="Custom Media"/> --><script type="text/javascript" src="http://webplayer.yahooapis.com/player.js">
</script><!-- ws:end:WikiTextMediaRule:0 --><!-- ws:start:WikiTextTocRule:22:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#x23 tone equal temperament">23 tone equal temperament</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#toc4"> </a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#x23 tone Equal Modes:">23 tone Equal Modes:</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Books">Books</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Instruments">Instruments</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: -->
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<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc0"><a name="x23 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:2 --><span style="background-color: #ffffff; color: #009927; font-family: 'Times New Roman',Times,serif; font-size: 113%;">23 tone equal temperament</span></h1>
<br />
<strong><em>23-tET</em></strong>, or <strong><em>23-EDO</em></strong>, is a tempered musical system which divides the <a class="wiki_link" href="/octave">octave</a> into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony <em>[Icositrifonía]</em>. It has good approximations for <a class="wiki_link" href="/5_3">5/3</a>, <a class="wiki_link" href="/11_7">11/7</a>, 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>. If to this subgroup is added the commas of <a class="wiki_link" href="/17-limit">17-limit</a>·<a class="wiki_link" href="/46edo">46edo</a>, the larger 17-limit <a class="wiki_link" href="/k%2AN%20subgroups">2*23 subgroup</a> 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 --><span style="font-size: 1.4em;">Intervals</span></h1>
<table class="wiki_table">
<tr>
<td><span style="color: #660000;"><a class="wiki_link" href="/Degree">Degree</a></span><br />
</td>
<td><a class="wiki_link" href="/Cent">Cent</a>s value<br />
</td>
<td>Approximate <br />
Ratios*<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td>1/1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>52.1739<br />
</td>
<td>33/32, 34/33<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>104.3478<br />
</td>
<td>17/16, 15/16, 18/17<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>156.5217<br />
</td>
<td>11/10, 12/11<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>208.6957<br />
</td>
<td>9/8<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>260.8696<br />
</td>
<td>7/6, 15/13<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>313.0435<br />
</td>
<td>6/5<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>365.2174<br />
</td>
<td>21/17, 26/21<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>417.3913<br />
</td>
<td>14/11, 33/26<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>469.5652<br />
</td>
<td>21/16, 17/13<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>521.7391<br />
</td>
<td>65/48, 256/189<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>573.913<br />
</td>
<td>7/5<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>626.087<br />
</td>
<td>10/7<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>678.2609<br />
</td>
<td>96/65, 189/128<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>730.4348<br />
</td>
<td>32/21, 26/17<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>782.6087<br />
</td>
<td>11/7, 52/33<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>834.7826<br />
</td>
<td>13/8, 21/13<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>886.9565<br />
</td>
<td>5/3<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>939.1304<br />
</td>
<td>12/7, 26/15<br />
</td>
</tr>
<tr>
<td>19<br />
</td>
<td>991.3043<br />
</td>
<td>16/9<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>1043.4783<br />
</td>
<td>11/6, 20/11<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>1095.6522<br />
</td>
<td>15/8, 17/9, 32/17<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>1147.8261<br />
</td>
<td>33/17, 64/33<br />
</td>
</tr>
<tr>
<td>23<br />
</td>
<td>1200<br />
</td>
<td>2/1<br />
</td>
</tr>
</table>
<br />
*based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.<br />
<br />
23-EDO was proposed by ethnomusicologist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erich_von_Hornbostel" rel="nofollow">Erich von Hornbostel</a> as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.<br />
<br />
23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, and 7th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore unusual harmonic territory. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, and 7/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23.<br />
<br />
Like<a class="wiki_link" href="/9edo"> 9-EDO</a>, <a class="wiki_link" href="/16edo">16-EDO</a>, and <a class="wiki_link" href="/25edo">25-EDO</a>, one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute <a class="wiki_link" href="/4_3">4/3</a>'s with 5/1 (related to the Armodue system). This means mapping '<a class="wiki_link" href="/3_2">3/2</a>' to 13 degrees of 23, and results in a 7 notes <a class="wiki_link" href="/2L%205s">Anti-diatonic scale</a> of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes <a class="wiki_link" href="/7L%202s">Superdiatonic scale</a> (3 3 3 1 3 3 3 3 1).<br />
<br />
However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a <a class="wiki_link" href="/7-limit">7-limit</a> temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to <a class="wiki_link" href="/13edo">13-EDO</a> and <a class="wiki_link" href="/18edo">18-EDO</a> and produces <a class="wiki_link" href="/MOSScales">MOS scales</a> of 5 and 8 notes: 5 5 4 5 4 (the <a class="wiki_link" href="/3L%202s">"anti-pentatonic"</a>) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29" rel="nofollow">Rapoport</a>/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc2"><a name="Intervals-Kosmorsky's Sephiroth modes"></a><!-- ws:end:WikiTextHeadingRule:6 -->Kosmorsky's Sephiroth modes</h2>
I would argue that the most significant modes of 23 edo are those of the 2 2 2 3 2 2 3 2 2 3 scale (<a class="wiki_link" href="/3L%207s">3L 7s fair mosh</a>);<br />
This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the Harmonic21<br />
and three add up to the Harmonic17 almost perfectly (I discovered the temperament before I even realised 23 tone fits)<br />
The chord 8:13:21:34 is a fragment of the fibonacci sequence, and 16:21:26 is harmonically symmetrical<br />
(The chord 26:29:32 is also symmetric, but only <a class="wiki_link" href="/20edo">20edo</a> and <a class="wiki_link" href="/43edo">43edo</a> approximate the Harmonic29, as well)<br />
13:17:21 is symmetrical, and 17:21:26:32 has an interesting linear widening pattern, but more importantly,<br />
the Harmonic13 and Harmonic17 are both dual primes like 3&5 and 7&11<br />
<br />
Thus I have named these 10 modes according to the Sephiroth as follows:<br />
2 2 2 3 2 2 3 2 2 3 - Mode Keter<br />
2 2 3 2 2 3 2 2 3 2 - Chesed<br />
2 3 2 2 3 2 2 3 2 2 - Netzach<br />
3 2 2 3 2 2 3 2 2 2 - Malkuth<br />
2 2 3 2 2 3 2 2 2 3 - Binah<br />
2 3 2 2 3 2 2 2 3 2 - Tiferet<br />
3 2 2 3 2 2 2 3 2 2 - Yesod<br />
2 2 3 2 2 2 3 2 2 3 - Chokmah<br />
2 3 2 2 2 3 2 2 3 2 - Gevurah<br />
3 2 2 2 3 2 2 3 2 2 - Hod<br />
<br />
-- Kosmorsky<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc3"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:8 -->Music</h1>
<span class="ymp-btn-page-play ymp-media-e8b030b27e37ed744bb651845d914579"><a class="wiki_link_ext" href="http://home.vicnet.net.au/%7Eepoetry/family.mp3" rel="nofollow">The Family Supper</a></span> by <a class="wiki_link" href="/Warren%20Burt">Warren Burt</a><br />
<span class="ymp-btn-page-play ymp-media-ad0e9a76a742e629379c7c9667584433"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Hqst8MaRiYM" rel="nofollow">Icositriphonic Heptatonic MOS</a></span> by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Jones</a><br />
<span class="ymp-btn-page-play ymp-media-11d44e0c35e6cbb396811396e4304800"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20His%20Wandering%20Kinship%20with%20Ashes.mp3" rel="nofollow">His Wandering Kinship with Ashes</a></span> by Iglashion Jones<br />
<span class="ymp-btn-page-play ymp-media-09a22042e6869c062a19ee073e5d2372"><a class="wiki_link_ext" href="http://www.nonoctave.com/tunes/CosmicChamber.mp3" rel="nofollow">Cosmic Chamber</a></span> by <a class="wiki_link" href="/X.%20J.%20Scott">X. J. Scott</a><br />
<span class="ymp-btn-page-play ymp-media-1a0d476bcc56e8699d94050c4bd46398"><a class="wiki_link_ext" href="http://www.nonoctave.com/tunes/Daisies.mp3" rel="nofollow">Daisies on the Beach</a></span> by X. J. Scott<br />
<span style="background-position: 100% 50%; cursor: pointer; padding-right: 10px;"><span class="ymp-btn-page-play ymp-media-c4aa76ed239a412533a11b438698b09b"><a class="wiki_link_ext" href="http://www.akjmusic.com/audio/boogie_pie.mp3" rel="nofollow">Boogie Pie</a></span></span>by <a class="wiki_link" href="/Aaron%20Krister%20Johnson">Aaron Krister Johnson</a><br />
<span class="ymp-btn-page-play ymp-media-c0b7192b09da11ecea83384b2aad9c4b"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/23edo/daily20110619_23edo_23_chilled.mp3" rel="nofollow">23 Chilled</a></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<span class="ymp-btn-page-play ymp-media-8ca0c9a378cc362bbcbc24ba13d6974b"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/DesertWinds.mp3" rel="nofollow">Desert Winds</a></span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/926007c7483e4abc5a48d582c0667947-105.html" rel="nofollow">blog entry</a>)<br />
<em>Allegro Moderato</em> by Easley Blackwood<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc4"><!-- ws:end:WikiTextHeadingRule:10 --> </h1>
<!-- ws:start:WikiTextLocalImageRule:649:<img src="/file/view/Ciclo_Icositrifon%C3%ADa.png/255957914/492x490/Ciclo_Icositrifon%C3%ADa.png" alt="Intervallic Cycle of 23 steps Equal per Octave" title="Intervallic Cycle of 23 steps Equal per Octave" style="height: 490px; width: 492px;" /> --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Ciclo_Icositrifon%C3%ADa.png/255957914/492x490/Ciclo_Icositrifon%C3%ADa.png" alt="Ciclo_Icositrifonía.png" title="Ciclo_Icositrifonía.png" style="height: 490px; width: 492px;" /></td></tr><tr><td class="imageCaption">Intervallic Cycle of 23 steps Equal per Octave</td></tr></table><!-- ws:end:WikiTextLocalImageRule:649 --><br />
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc5"><!-- ws:end:WikiTextHeadingRule:12 --> </h2>
<!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc6"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:14 -->Commas</h1>
23 EDO tempers out the following commas. (Note: This assumes the val < 23 36 53 65 80 85 |.) Also note the discussion above, where there are some commas mentioned that are not in the standard comma list (e.g., 28/27).<br />
<table class="wiki_table">
<tr>
<th>Comma<br />
</th>
<th>Monzo<br />
</th>
<th>Value (Cents)<br />
</th>
<th>Name 1<br />
</th>
<th>Name 2<br />
</th>
<th>Name 3<br />
</th>
</tr>
<tr>
<td style="text-align: center;">135/128<br />
</td>
<td style="text-align: left;">| -7 3 1 ><br />
</td>
<td style="text-align: right;">92.18<br />
</td>
<td style="text-align: center;">Major Chroma<br />
</td>
<td style="text-align: center;">Major Limma<br />
</td>
<td style="text-align: center;">Pelogic Comma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">15625/15552<br />
</td>
<td style="text-align: left;">| -6 -5 6 ><br />
</td>
<td style="text-align: right;">8.11<br />
</td>
<td style="text-align: center;">Kleisma<br />
</td>
<td style="text-align: center;">Semicomma Majeur<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">36/35<br />
</td>
<td style="text-align: left;">| 2 2 -1 -1 ><br />
</td>
<td style="text-align: right;">48.77<br />
</td>
<td style="text-align: center;">Septimal Quarter Tone<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">525/512<br />
</td>
<td style="text-align: left;">| -9 1 2 1 ><br />
</td>
<td style="text-align: right;">43.41<br />
</td>
<td style="text-align: center;">Avicennma<br />
</td>
<td style="text-align: center;">Avicenna's Enharmonic Diesis<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4000/3969<br />
</td>
<td style="text-align: left;">| 5 -4 3 -2 ><br />
</td>
<td style="text-align: right;">13.47<br />
</td>
<td style="text-align: center;">Octagar<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">6144/6125<br />
</td>
<td style="text-align: left;">| 11 1 -3 -2 ><br />
</td>
<td style="text-align: right;">5.36<br />
</td>
<td style="text-align: center;">Porwell<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">100/99<br />
</td>
<td style="text-align: left;">| 2 -2 2 0 -1 ><br />
</td>
<td style="text-align: right;">17.40<br />
</td>
<td style="text-align: center;">Ptolemisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">441/440<br />
</td>
<td style="text-align: left;">| -3 2 -1 2 -1 ><br />
</td>
<td style="text-align: right;">3.93<br />
</td>
<td style="text-align: center;">Werckisma<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc7"><a name="x23 tone Equal Modes:"></a><!-- ws:end:WikiTextHeadingRule:16 --><strong>23 tone <a class="wiki_link" href="/Equal%20Modes">Equal Modes</a>:</strong></h1>
<br />
<table class="wiki_table">
<tr>
<td>10 10 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>9 9 5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>8 8 7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7 7 7 2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7 2 7 7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6 6 6 5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>6 5 6 6<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>5 4 5 5 4<br />
</td>
<td><a class="wiki_link" href="/3L%202s">3L 2s (father)</a><br />
</td>
</tr>
<tr>
<td>5 4 5 4 5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7 1 7 7 1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>7 1 7 1 7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>5 5 5 5 3<br />
</td>
<td><a class="wiki_link" href="/4L%201s">4L 1s (bug)</a><br />
</td>
</tr>
<tr>
<td>5 3 5 5 5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>4 4 4 4 4 3<br />
</td>
<td><a class="wiki_link" href="/5L%201s">5L 1s (Grumpy hexatonic)</a><br />
</td>
</tr>
<tr>
<td>4 3 4 4 4 4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>5 1 5 1 5 1 5<br />
</td>
<td><a class="wiki_link" href="/4L%203s">4L 3s (mish)</a><br />
</td>
</tr>
<tr>
<td>3 3 3 5 3 3 3<br />
</td>
<td><a class="wiki_link" href="/1L%206s">1L 6s (Happy heptatonic)</a><br />
</td>
</tr>
<tr>
<td>4 3 3 3 3 3 4<br />
</td>
<td><a class="wiki_link" href="/2L%205s">2L 5s (mavila, anti-diatonic)</a><br />
</td>
</tr>
<tr>
<td>3 4 3 3 4 3 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3 3 4 3 3 3 4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3 3 3 4 3 3 4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3 3 3 4 3 4 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2 5 2 5 2 5 2<br />
</td>
<td><a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
</tr>
<tr>
<td>4 1 4 4 1 4 4 1<br />
</td>
<td><a class="wiki_link" href="/5L%203s">5L 3s (unfair father)</a><br />
</td>
</tr>
<tr>
<td>3 3 3 3 3 3 3 2<br />
</td>
<td><a class="wiki_link" href="/7L%201s">7L 1s (Grumpy octatonic)</a><br />
</td>
</tr>
<tr>
<td>3 2 3 3 3 3 3 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><strong>3 3 3 1 3 3 3 3 1</strong><br />
</td>
<td><a class="wiki_link" href="/7L%202s">7L 2s (mavila superdiatonic)</a><br />
</td>
</tr>
<tr>
<td>3 3 1 3 3 3 1 3 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>3 2 3 2 3 2 3 2 3<br />
</td>
<td><a class="wiki_link" href="/5L%204s">5L 4s (unfair bug)</a><br />
</td>
</tr>
<tr>
<td>2 2 2 3 2 2 3 2 2 3<br />
</td>
<td>Mode Keter<br />
</td>
</tr>
<tr>
<td>2 2 3 2 2 3 2 2 3 2<br />
</td>
<td>Chesed<br />
</td>
</tr>
<tr>
<td>2 3 2 2 3 2 2 3 2 2<br />
</td>
<td>Netzach<br />
</td>
</tr>
<tr>
<td>3 2 2 3 2 2 3 2 2 2<br />
</td>
<td>Malkuth<br />
</td>
</tr>
<tr>
<td>2 2 3 2 2 3 2 2 2 3<br />
</td>
<td>Binah<br />
</td>
</tr>
<tr>
<td>2 3 2 2 3 2 2 2 3 2<br />
</td>
<td>Tiferet<br />
</td>
</tr>
<tr>
<td>3 2 2 3 2 2 2 3 2 2<br />
</td>
<td>Yesod<br />
</td>
</tr>
<tr>
<td>2 2 3 2 2 2 3 2 2 3<br />
</td>
<td>Chokmah<br />
</td>
</tr>
<tr>
<td>2 3 2 2 2 3 2 2 3 2<br />
</td>
<td>Gevurah<br />
</td>
</tr>
<tr>
<td>3 2 2 2 3 2 2 3 2 2<br />
</td>
<td>Hod<br />
</td>
</tr>
<tr>
<td><strong>3 1 3 1 3 1 3 1 3 1 3</strong><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2 2 2 1 2 2 2 1 2 2 2 2 1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2 2 1 2 2 1 2 2 1 2 2 1 2 1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><strong>2 1 2 2 1 2 2 1 2 2 1 2 2 1</strong><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1 1 1 4 1 1 1 1 4 1 1 1 1 4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2 1 2 1 2 1 2 1 2 1 2 1 2 1 2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><strong>2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1</strong><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1<br />
</td>
<td><br />
</td>
</tr>
</table>
<br />
<!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc8"><a name="Books"></a><!-- ws:end:WikiTextHeadingRule:18 -->Books</h1>
<!-- ws:start:WikiTextLocalImageRule:650:<img src="/file/view/Libro_Icositrif%C3%B3nico.PNG/163031733/242x294/Libro_Icositrif%C3%B3nico.PNG" alt="" title="" style="height: 294px; width: 242px;" /> --><img src="/file/view/Libro_Icositrif%C3%B3nico.PNG/163031733/242x294/Libro_Icositrif%C3%B3nico.PNG" alt="Libro_Icositrifónico.PNG" title="Libro_Icositrifónico.PNG" style="height: 294px; width: 242px;" /><!-- ws:end:WikiTextLocalImageRule:650 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc9"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:20 -->Instruments</h1>
<!-- ws:start:WikiTextLocalImageRule:651:<img src="/file/view/Icositriphonic_Bass.JPG/206711470/594x216/Icositriphonic_Bass.JPG" alt="" title="" style="height: 216px; width: 594px;" /> --><img src="/file/view/Icositriphonic_Bass.JPG/206711470/594x216/Icositriphonic_Bass.JPG" alt="Icositriphonic_Bass.JPG" title="Icositriphonic_Bass.JPG" style="height: 216px; width: 594px;" /><!-- ws:end:WikiTextLocalImageRule:651 --><br />
<em>An Icositriphonic Bass. 23-EDO Bass by Tútim Deft Wafil.</em><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:652:<img src="/file/view/Icositriphonic_Guitar.PNG/206712964/601x305/Icositriphonic_Guitar.PNG" alt="" title="" style="height: 305px; width: 601px;" /> --><img src="/file/view/Icositriphonic_Guitar.PNG/206712964/601x305/Icositriphonic_Guitar.PNG" alt="Icositriphonic_Guitar.PNG" title="Icositriphonic_Guitar.PNG" style="height: 305px; width: 601px;" /><!-- ws:end:WikiTextLocalImageRule:652 --><br />
<em>An Icositriphonic 8-string Guitar. 23-EDO Guitar by Ron Sword.</em><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:653:<img src="/file/view/Teclado_Icositrif%C3%B3nico.PNG/258408436/591x326/Teclado_Icositrif%C3%B3nico.PNG" alt="A classic visualization of an Alternative 1/3-tone Keyboard. Armodue-Hornbostel Family Musical Systems" title="A classic visualization of an Alternative 1/3-tone Keyboard. Armodue-Hornbostel Family Musical Systems" style="height: 326px; width: 591px;" /> --><table class="captionBox"><tr><td class="captionedImage"><img src="/file/view/Teclado_Icositrif%C3%B3nico.PNG/258408436/591x326/Teclado_Icositrif%C3%B3nico.PNG" alt="Teclado_Icositrifónico.PNG" title="Teclado_Icositrifónico.PNG" style="height: 326px; width: 591px;" /></td></tr><tr><td class="imageCaption">A classic visualization of an Alternative 1/3-tone Keyboard. Armodue-Hornbostel Family Musical Systems</td></tr></table><!-- ws:end:WikiTextLocalImageRule:653 --><br />
<br />
<br />
Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it.<br />
<!-- ws:start:WikiTextLocalImageRule:654:<img src="/file/view/playing.jpg/241873781/playing.jpg" alt="" title="" /> --><img src="/file/view/playing.jpg/241873781/playing.jpg" alt="playing.jpg" title="playing.jpg" /><!-- ws:end:WikiTextLocalImageRule:654 --><br />
Here is a still shot of the completed instrument.<br />
<!-- ws:start:WikiTextLocalImageRule:655:<img src="/file/view/complette.jpg/241873831/complette.jpg" alt="" title="" /> --><img src="/file/view/complette.jpg/241873831/complette.jpg" alt="complette.jpg" title="complette.jpg" /><!-- ws:end:WikiTextLocalImageRule:655 --><br />
This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion.<br />
<!-- ws:start:WikiTextMediaRule:1:<img src="http://www.wikispaces.com/site/embedthumbnail/youtube/K4iO7k152og?h=349&w=560" class="WikiMedia WikiMediaYoutube" id="wikitext@@media@@type=&quot;youtube&quot; key=&quot;K4iO7k152og&quot; height=&quot;349&quot; width=&quot;560&quot;" title="YouTube Video"height="349" width="560"/> --><iframe width="560" height="349" src="//www.youtube.com/embed/K4iO7k152og" frameborder="0" allowfullscreen></iframe><!-- ws:end:WikiTextMediaRule:1 --></body></html>