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>== [[image:Ciclo_Icositrifonía.png width="486" height="486" caption="23-ED2 Cycle chart" link="Harmony of 23edo"]] || <span style="color: #660000;">[[Degree]]</span> || [[Cent]]s value ||= Approximate Ratios* ||= Armodue Notation || || 0 || 0 ||= 1/1 ||= 1 || || 1 || 52.1739 ||= 33/32, 34/33 ||= 2b || || 2 || 104.3478 ||= 17/16, 16/15, 18/17 ||= 1# || || 3 || 156.5217 ||= 11/10, 12/11, 35/32 ||= 2 || || 4· || 208.6957 ||= 9/8, 44/39 ||= 3b || || 5 || 260.8696 ||= 7/6, 15/13, 29/25 ||= 2# || || 6 || 313.0435 ||= 6/5 ||= 3 || || 7· || 365.2174 ||= 16/13, 21/17, 26/21 ||= 4b || || 8 || 417.3913 ||= 14/11, 33/26 ||= 3# || || 9 || 469.5652 ||= 21/16, 17/13 ||= 4 || || 10· || 521.7391 ||= 23/17, 88/65, 256/189 ||= 5 || || 11 || 573.913 ||= 7/5, 32/23, 46/33 ||= 6b || || 12 || 626.087 ||= 10/7, 23/16, 33/23 ||= 5# || || 13· || 678.2609 ||= 34/23, 65/44, 189/128 ||= 6 || || 14 || 730.4348 ||= 32/21, 26/17 ||= 7b || || 15 || 782.6087 ||= 11/7, 52/33 ||= 6# || || 16· || 834.7826 ||= 13/8, 34/21, 21/13 ||= 7 || || 17 || 886.9565 ||= 5/3 ||= 8b || || 18 || 939.1304 ||= 12/7, 26/15, 50/29 ||= 7# || || 19· || 991.3043 ||= 16/9, 39/22 ||= 8 || || 20 || 1043.4783 ||= 11/6, 20/11, 64/35 ||= 9b || || 21 || 1095.6522 ||= 15/8, 17/9, 32/17 ||= 8# || || 22 || 1147.8261 ||= 33/17, 64/33 ||= 9 || || 23·· || 1200 ||= 2/1 ||= 1 || *based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible. The chart below shows some of the [[MOSScales|Moment of Symmetry (MOS)]] modes of [[Mavila]] available in 23edo, mainly Pentatonic(5-note), anti-diatonic(7-note), 9- and 16-note MOSs: [[image:23edoMavilaMOS.jpg]] 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, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23. See [[Harmony of 23edo|here]] for more details. As with[[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). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#. 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 21st harmonic and three add up to the 17th harmonic almost perfectly. Interestingly, the chord 8:13:21:34 is a fragment of the fibonacci sequence. Notated in ascending (standard) form. 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 =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]]) <span class="ymp-btn-page-play ymp-media-2b9e61a2a9e271622ebe4a1311677936">[[http://www.seraph.it/dep/det/23Laments.mp3|23 Laments]]</span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/b2bf6f252efd467ee36ecc332a4872ac-106.html|blog entry]]) //Allegro Moderato// by Easley Blackwood =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="608" height="353" caption="Armodue-Hornbostel 1/3-tone keyboard prototype"]] 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"]]
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<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">
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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>
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<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 />
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<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc1"><a name="x23 tone equal temperament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 --><span style="font-size: 1.4em;">Intervals</span></h2>
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<!-- ws:start:WikiTextLocalImageRule:694:<a href="/Harmony%20of%2023edo"><img src="/file/view/Ciclo_Icositrifon%C3%ADa.png/255957914/486x486/Ciclo_Icositrifon%C3%ADa.png" alt="23-ED2 Cycle chart" title="23-ED2 Cycle chart" style="height: 486px; width: 486px;" /></a> --><table class="captionBox"><tr><td class="captionedImage"><a href="/Harmony%20of%2023edo"><img src="/file/view/Ciclo_Icositrifon%C3%ADa.png/255957914/486x486/Ciclo_Icositrifon%C3%ADa.png" alt="Ciclo_Icositrifonía.png" title="Ciclo_Icositrifonía.png" style="height: 486px; width: 486px;" /></a></td></tr><tr><td class="imageCaption">23-ED2 Cycle chart</td></tr></table><!-- ws:end:WikiTextLocalImageRule:694 --><br />
<br />
<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 style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
<td style="text-align: center;">Armodue<br />
Notation<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>52.1739<br />
</td>
<td style="text-align: center;">33/32, 34/33<br />
</td>
<td style="text-align: center;">2b<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>104.3478<br />
</td>
<td style="text-align: center;">17/16, 16/15, 18/17<br />
</td>
<td style="text-align: center;">1#<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>156.5217<br />
</td>
<td style="text-align: center;">11/10, 12/11, 35/32<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
<tr>
<td>4·<br />
</td>
<td>208.6957<br />
</td>
<td style="text-align: center;">9/8, 44/39<br />
</td>
<td style="text-align: center;">3b<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>260.8696<br />
</td>
<td style="text-align: center;">7/6, 15/13, 29/25<br />
</td>
<td style="text-align: center;">2#<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>313.0435<br />
</td>
<td style="text-align: center;">6/5<br />
</td>
<td style="text-align: center;">3<br />
</td>
</tr>
<tr>
<td>7·<br />
</td>
<td>365.2174<br />
</td>
<td style="text-align: center;">16/13, 21/17, 26/21<br />
</td>
<td style="text-align: center;">4b<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>417.3913<br />
</td>
<td style="text-align: center;">14/11, 33/26<br />
</td>
<td style="text-align: center;">3#<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>469.5652<br />
</td>
<td style="text-align: center;">21/16, 17/13<br />
</td>
<td style="text-align: center;">4<br />
</td>
</tr>
<tr>
<td>10·<br />
</td>
<td>521.7391<br />
</td>
<td style="text-align: center;">23/17, 88/65, 256/189<br />
</td>
<td style="text-align: center;">5<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>573.913<br />
</td>
<td style="text-align: center;">7/5, 32/23, 46/33<br />
</td>
<td style="text-align: center;">6b<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>626.087<br />
</td>
<td style="text-align: center;">10/7, 23/16, 33/23<br />
</td>
<td style="text-align: center;">5#<br />
</td>
</tr>
<tr>
<td>13·<br />
</td>
<td>678.2609<br />
</td>
<td style="text-align: center;">34/23, 65/44, 189/128<br />
</td>
<td style="text-align: center;">6<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>730.4348<br />
</td>
<td style="text-align: center;">32/21, 26/17<br />
</td>
<td style="text-align: center;">7b<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>782.6087<br />
</td>
<td style="text-align: center;">11/7, 52/33<br />
</td>
<td style="text-align: center;">6#<br />
</td>
</tr>
<tr>
<td>16·<br />
</td>
<td>834.7826<br />
</td>
<td style="text-align: center;">13/8, 34/21, 21/13<br />
</td>
<td style="text-align: center;">7<br />
</td>
</tr>
<tr>
<td>17<br />
</td>
<td>886.9565<br />
</td>
<td style="text-align: center;">5/3<br />
</td>
<td style="text-align: center;">8b<br />
</td>
</tr>
<tr>
<td>18<br />
</td>
<td>939.1304<br />
</td>
<td style="text-align: center;">12/7, 26/15, 50/29<br />
</td>
<td style="text-align: center;">7#<br />
</td>
</tr>
<tr>
<td>19·<br />
</td>
<td>991.3043<br />
</td>
<td style="text-align: center;">16/9, 39/22<br />
</td>
<td style="text-align: center;">8<br />
</td>
</tr>
<tr>
<td>20<br />
</td>
<td>1043.4783<br />
</td>
<td style="text-align: center;">11/6, 20/11, 64/35<br />
</td>
<td style="text-align: center;">9b<br />
</td>
</tr>
<tr>
<td>21<br />
</td>
<td>1095.6522<br />
</td>
<td style="text-align: center;">15/8, 17/9, 32/17<br />
</td>
<td style="text-align: center;">8#<br />
</td>
</tr>
<tr>
<td>22<br />
</td>
<td>1147.8261<br />
</td>
<td style="text-align: center;">33/17, 64/33<br />
</td>
<td style="text-align: center;">9<br />
</td>
</tr>
<tr>
<td>23··<br />
</td>
<td>1200<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
<td style="text-align: center;">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 />
The chart below shows some of the <a class="wiki_link" href="/MOSScales">Moment of Symmetry (MOS)</a> modes of <a class="wiki_link" href="/Mavila">Mavila</a> available in 23edo, mainly Pentatonic(5-note), anti-diatonic(7-note), 9- and 16-note MOSs:<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:695:<img src="/file/view/23edoMavilaMOS.jpg/288679838/23edoMavilaMOS.jpg" alt="" title="" /> --><img src="/file/view/23edoMavilaMOS.jpg/288679838/23edoMavilaMOS.jpg" alt="23edoMavilaMOS.jpg" title="23edoMavilaMOS.jpg" /><!-- ws:end:WikiTextLocalImageRule:695 --><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, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 13, 17, 21, and 23. See <a class="wiki_link" href="/Harmony%20of%2023edo">here</a> for more details.<br />
<br />
As with<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). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.<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="x23 tone equal temperament-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>); This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the 21st harmonic and three add up to the 17th harmonic almost perfectly. Interestingly, the chord 8:13:21:34 is a fragment of the fibonacci sequence.<br />
<br />
Notated in ascending (standard) form. 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 />
<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 />
<span class="ymp-btn-page-play ymp-media-2b9e61a2a9e271622ebe4a1311677936"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/23Laments.mp3" rel="nofollow">23 Laments</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/b2bf6f252efd467ee36ecc332a4872ac-106.html" rel="nofollow">blog entry</a>)<br />
<em>Allegro Moderato</em> by Easley Blackwood<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc4"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:10 -->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:12:<h1> --><h1 id="toc5"><a name="x23 tone Equal Modes:"></a><!-- ws:end:WikiTextHeadingRule:12 --><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:14:<h1> --><h1 id="toc6"><a name="Books"></a><!-- ws:end:WikiTextHeadingRule:14 -->Books</h1>
<!-- ws:start:WikiTextLocalImageRule:696:<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:696 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc7"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:16 -->Instruments</h1>
<!-- ws:start:WikiTextLocalImageRule:697:<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:697 --><br />
<em>An Icositriphonic Bass. 23-EDO Bass by Tútim Deft Wafil.</em><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:698:<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:698 --><br />
<em>An Icositriphonic 8-string Guitar. 23-EDO Guitar by Ron Sword.</em><br />
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Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it.<br />
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Here is a still shot of the completed instrument.<br />
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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 />
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