13edo
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[[toc|flat]] ---- =13 tone equal temperament / 13edo= 13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). || Degree || Cents ||= Approximate Ratios* || 6L1s Names || 5L3s Names || [[26edo]] names || || 0 || 0 ||= 1/1 || C || C || C || || 1 || 92.3077 ||= 22/21, 55/52, 117/110, 26/25 || C#/Db || C#/Db || Cx/Dbb || || 2 || 184.6154 ||= 10/9, 9/8, 11/10 || D || D || D || || 3 || 276.9231 ||= 13/11, 7/6 || D#/Eb || D#/Eb || Dx/Ebb || || 4 || 369.2308 ||= 5/4, 16/13, 11/9, 26/21 || E || E || E || || 5 || 461.5385 ||= 13/10, 21/16 || E#/Fb || F || Ex/Fb || || 6 || 553.84 ||= 11/8, 18/13 || F || F#/Gb || F# || || 7 || 646.15 ||= 16/11, 13/9 || F#/Gb || G || Gb || || 8 || 738.46 ||= 20/13, 32/21 || G || G#/Hb || G# || || 9 || 830.77 ||= 8/5, 13/8, 18/11, 21/13 || G#/Ab || H || Ab || || 10 || 923.08 ||= 22/13, 12/7 || A || A || A# || || 11 || 1015.38 ||= 9/5, 16/9, 20/11 || A#/Bb || A#/Bb || Bb || || 12 || 1107.69 ||= 21/11, 25/13, 104/55 || B/Cb || B || B#/Cbb || || 13 || 1200 ||= 2/1 || C/B# || C || C || *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. =Harmony in 13edo= Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad). =Scales in 13edo= Due to the prime character of the number 13, 13edo can form several xenharmonic [[MOSScales|moment of symmetry scales]]. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, & 6\13, respectively. [[image:13edo_horograms.jpg]] [[file:13edo horograms.pdf]] ~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson Another neat facet of 13-EDO is the fact that any 12-EDO scale can be "turned into" a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected. == == =Animism= The animist comma appears whenever 3*5*7=13... 26 edo approximates 3, 5, and 7 individually, however 13 edo has 21/16 (=3*7) and is also an animist temperament. In 13 edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction 0 4 5 8 9 13 pentatonic and 0 1 3 4 5 8 9 10 12 13 nonatonic =**Compositions**= [[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]] by [[http://danielthompson.blogspot.com/|Daniel Thompson]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3|Prelude in 13ET]] by [[Aaron Andrew Hunt]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3|Two-Part Invention in 13ET]] by [[Aaron Andrew Hunt]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3|Triskaidekaphobia]] by [[http://www.io.com/%7Ehmiller/music/|Herman Miller]] [[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=835265|Spikey Hair in 13tET]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3|play]] by [[Andrew Heathwaite]] [[@http://cityoftheasleep.bandcamp.com/track/broken-dream-jar|Broken Dream Jar]] by [[IgliashonJones|City of the Asleep]] [[@http://www.last.fm/music/City+of+the+Asleep/Map+of+an+Internal+Landscape/Blinding+White+Darkness|Blinding White Darkness]] by [[IgliashonJones|City of the Asleep]] [[@http://www.elvenminstrel.com/music/tuning/equal/13equal/13tet.htm|Upsidedown and Backwards: Explorations in 13-tone Equal Temperament]] by [[http://www.elvenminstrel.com/|David J. Finnamore]] =Igliashon's 13-EDO diatonic approaches= From a temperament perspective, we can probably make the "best" use of 13-EDO as a 2.5.9.11.13.21 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping <1 -1| (for 5 and 13), corresponding to the 3rd horogram above. This gives rise to "Sephiroth" modes, in which the generator is any flatly tempered 13th harmonic. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping <2 3| (for 11 and 13). This corresponds to the 2nd horogram above. This scale bears a superficial resemblance to the 9-note MOS of Orwell temperament, although its approximations to the 3rd, 5th, and 7th harmonics are much more distant than in more optimal tunings of the temperament (on the other hand, its approximations to the 11th and 13th harmonics are much better than in optimal tunings of the temperament). 2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of <2 1| (for 5 and 9) and <2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating a full 2.5.9.11.13 pentad, which it achieves in the space of 5 generators via the octave-equivalent mapping <2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it as a tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for 13-EDO notation, leading to a notation very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart (in the 6L1s column). It can be thought of as a "circle of major 2nds" rather than a circle of 5ths. For the 2.9.21 subgroup, we can use the 5\13 generator, the closest thing 13-EDO has to a "perfect 4th", giving an octave-equivalent mapping of <3 1| and MOS scales corresponding to the 4th horogram above. The 8-note MOS scale of 5L3s, independently discovered by Easley Blackwood Jr, Paul Rapoport, and Erv Wilson (among others), is excellent for melody, being somewhat similar to the 12-TET diatonic scale but with an extra semitone added. It is also a conceivable basis for 13-EDO notation, using a modified "circle of fifths" (8\13, the octave inversion of 5\13) including an H: B#-G#-D#-A#-F#-C#-H-E-B-G-D-A-F-C-Hb-Eb-Bb-Gb-Db-Ab, which when arranged in order of ascending pitch within the octave gives the 5L3s names in the above interval chart. This notation has the advantage of preserving some familiar features: diatonic semitones still occur between B and C and E and F, and the dyads E-B, G-D, D-A, and F-C (and associated accidentals) sound approximately like "fifths". Also, the 5L3s scale on C somewhat approximates a 12-TET C major scale (if H is omitted). To facilitate discussion of these scales, Igliashon has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos. The 2\13-based heptatonic has been named "archeotonic" after the "Old Ones" that rule the Dreamlands, and the 5\13-based octatonic has been named "oneirotonic" after the Dreamlands themselves. Modes of the archeotonic are named after the individual Old Ones themselves; modes of the oneirotonic are named after cities in the Dreamlands. See the charts of modes of the two scales below, excerpted from Igliashon's forthcoming paper "The Case for Thirteen": [[image:Archeotonic.png]] [[image:Oneirotonic.png]] =Commas= 13 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val < 13 21 30 36 45 48 |.) ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || ||= 2109375/2097152 ||< | -21 3 7 > ||> 10.06 ||= Semicomma ||= Fokker Comma ||= || ||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||= ||= || ||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= || ||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma || ||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||= ||= || ||= 3125/3087 ||< | 0 -2 5 -3 > ||> 21.18 ||= Gariboh ||= ||= || ||= 3136/3125 ||< | 6 0 -5 2 > ||> 6.08 ||= Hemimean ||= ||= || ||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||= ||= || ||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= ||
Original HTML content:
<html><head><title>13edo</title></head><body><!-- ws:start:WikiTextTocRule:16:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#x13 tone equal temperament / 13edo">13 tone equal temperament / 13edo</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Harmony in 13edo">Harmony in 13edo</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Scales in 13edo">Scales in 13edo</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Animism">Animism</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Igliashon's 13-EDO diatonic approaches">Igliashon's 13-EDO diatonic approaches</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: -->
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<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x13 tone equal temperament / 13edo"></a><!-- ws:end:WikiTextHeadingRule:0 -->13 tone equal temperament / 13edo</h1>
13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo).<br />
<table class="wiki_table">
<tr>
<td>Degree<br />
</td>
<td>Cents<br />
</td>
<td style="text-align: center;">Approximate Ratios*<br />
</td>
<td>6L1s Names<br />
</td>
<td>5L3s Names<br />
</td>
<td><a class="wiki_link" href="/26edo">26edo</a> names<br />
</td>
</tr>
<tr>
<td>0<br />
</td>
<td>0<br />
</td>
<td style="text-align: center;">1/1<br />
</td>
<td>C<br />
</td>
<td>C<br />
</td>
<td>C<br />
</td>
</tr>
<tr>
<td>1<br />
</td>
<td>92.3077<br />
</td>
<td style="text-align: center;">22/21, 55/52, 117/110, 26/25<br />
</td>
<td>C#/Db<br />
</td>
<td>C#/Db<br />
</td>
<td>Cx/Dbb<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>184.6154<br />
</td>
<td style="text-align: center;">10/9, 9/8, 11/10<br />
</td>
<td>D<br />
</td>
<td>D<br />
</td>
<td>D<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>276.9231<br />
</td>
<td style="text-align: center;">13/11, 7/6<br />
</td>
<td>D#/Eb<br />
</td>
<td>D#/Eb<br />
</td>
<td>Dx/Ebb<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>369.2308<br />
</td>
<td style="text-align: center;">5/4, 16/13, 11/9, 26/21<br />
</td>
<td>E<br />
</td>
<td>E<br />
</td>
<td>E<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>461.5385<br />
</td>
<td style="text-align: center;">13/10, 21/16<br />
</td>
<td>E#/Fb<br />
</td>
<td>F<br />
</td>
<td>Ex/Fb<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>553.84<br />
</td>
<td style="text-align: center;">11/8, 18/13<br />
</td>
<td>F<br />
</td>
<td>F#/Gb<br />
</td>
<td>F#<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>646.15<br />
</td>
<td style="text-align: center;">16/11, 13/9<br />
</td>
<td>F#/Gb<br />
</td>
<td>G<br />
</td>
<td>Gb<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>738.46<br />
</td>
<td style="text-align: center;">20/13, 32/21<br />
</td>
<td>G<br />
</td>
<td>G#/Hb<br />
</td>
<td>G#<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>830.77<br />
</td>
<td style="text-align: center;">8/5, 13/8, 18/11, 21/13<br />
</td>
<td>G#/Ab<br />
</td>
<td>H<br />
</td>
<td>Ab<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>923.08<br />
</td>
<td style="text-align: center;">22/13, 12/7<br />
</td>
<td>A<br />
</td>
<td>A<br />
</td>
<td>A#<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>1015.38<br />
</td>
<td style="text-align: center;">9/5, 16/9, 20/11<br />
</td>
<td>A#/Bb<br />
</td>
<td>A#/Bb<br />
</td>
<td>Bb<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>1107.69<br />
</td>
<td style="text-align: center;">21/11, 25/13, 104/55<br />
</td>
<td>B/Cb<br />
</td>
<td>B<br />
</td>
<td>B#/Cbb<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>1200<br />
</td>
<td style="text-align: center;">2/1<br />
</td>
<td>C/B#<br />
</td>
<td>C<br />
</td>
<td>C<br />
</td>
</tr>
</table>
*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h1>
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.<br />
<br />
The 2.9.5.11.13 subgroup has commas 45/44, 65/64 and 81/80, leading to a linear temperament with POTE generator 185.728 cents, quite close to 2\13. Use this as a generator, and at 7 notes (6L1s) two full pentads are available (as well as two more 4:5:9:11 tetrad, and one 4:5:9:13 tetrad).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Scales in 13edo"></a><!-- ws:end:WikiTextHeadingRule:4 -->Scales in 13edo</h1>
Due to the prime character of the number 13, 13edo can form several xenharmonic <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a>. The diagram below shows five "families" of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, & 6\13, respectively.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:381:<img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="" title="" /> --><img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /><!-- ws:end:WikiTextLocalImageRule:381 --><br />
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~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson<br />
<br />
Another neat facet of 13-EDO is the fact that any 12-EDO scale can be "turned into" a 13-EDO scale by either adding an extra semitone, or turning an existent semitone into a whole-tone. Because of this, melody in 13-EDO can be quite mind-bending and uncanny, and phrases that begin in a familiar way quickly lead to something totally unexpected.<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><!-- ws:end:WikiTextHeadingRule:6 --> </h2>
<!-- ws:start:WikiTextHeadingRule:8:<h1> --><h1 id="toc4"><a name="Animism"></a><!-- ws:end:WikiTextHeadingRule:8 -->Animism</h1>
<br />
The animist comma appears whenever 3*5*7=13... 26 edo approximates 3, 5, and 7 individually, however 13 edo has 21/16 (=3*7) and is also an animist temperament. In 13 edo, the 5th harmonic is tuned so flatly that 5/4 = 16/13, leading to some interesting identities. So two scales stand out through this construction<br />
<br />
0 4 5 8 9 13 pentatonic<br />
and<br />
0 1 3 4 5 8 9 10 12 13 nonatonic<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h1> --><h1 id="toc5"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:10 --><strong>Compositions</strong></h1>
<br />
<a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow">Prelude in 13ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3" rel="nofollow">Two-Part Invention in 13ET</a> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow">Triskaidekaphobia</a> by <a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/music/" rel="nofollow">Herman Miller</a><br />
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&songID=835265" rel="nofollow">Spikey Hair in 13tET</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a><br />
<a class="wiki_link_ext" href="http://cityoftheasleep.bandcamp.com/track/broken-dream-jar" rel="nofollow" target="_blank">Broken Dream Jar</a> by <a class="wiki_link" href="/IgliashonJones">City of the Asleep</a><br />
<a class="wiki_link_ext" href="http://www.last.fm/music/City+of+the+Asleep/Map+of+an+Internal+Landscape/Blinding+White+Darkness" rel="nofollow" target="_blank">Blinding White Darkness</a> by <a class="wiki_link" href="/IgliashonJones">City of the Asleep</a><br />
<a class="wiki_link_ext" href="http://www.elvenminstrel.com/music/tuning/equal/13equal/13tet.htm" rel="nofollow" target="_blank">Upsidedown and Backwards: Explorations in 13-tone Equal Temperament</a> by <a class="wiki_link_ext" href="http://www.elvenminstrel.com/" rel="nofollow">David J. Finnamore</a><br />
<!-- ws:start:WikiTextHeadingRule:12:<h1> --><h1 id="toc6"><a name="Igliashon's 13-EDO diatonic approaches"></a><!-- ws:end:WikiTextHeadingRule:12 -->Igliashon's 13-EDO diatonic approaches</h1>
<br />
From a temperament perspective, we can probably make the "best" use of 13-EDO as a 2.5.9.11.13.21 subgroup, but assuming our goal is to make reasonably-tonal, triad-based music, we might prefer to think in terms of subsets of this subgroup. The simplest and most accurately-tuned subsets are 2.5.9, 2.5.11, 2.5.13, 2.11.13, and 2.9.21, and for each of these, there is a corresponding MOS generator that is maximally-efficient at producing the desired triad. For 2.5.13, the simplest generator is 4\13, with an octave-equivalent mapping <1 -1| (for 5 and 13), corresponding to the 3rd horogram above. This gives rise to "Sephiroth" modes, in which the generator is any flatly tempered 13th harmonic. For 2.11.13, the simplest generator is 3\13, with an octave-equivalent mapping <2 3| (for 11 and 13). This corresponds to the 2nd horogram above. This scale bears a superficial resemblance to the 9-note MOS of Orwell temperament, although its approximations to the 3rd, 5th, and 7th harmonics are much more distant than in more optimal tunings of the temperament (on the other hand, its approximations to the 11th and 13th harmonics are much better than in optimal tunings of the temperament).<br />
<br />
2.5.9 and 2.5.11 are both best-served by the 2\13 generator, corresponding to the 1st horogram above, having the (octave-equivalent) mappings of <2 1| (for 5 and 9) and <2 3| (for 5 and 11). This generator incidentally is also the most efficient at generating a full 2.5.9.11.13 pentad, which it achieves in the space of 5 generators via the octave-equivalent mapping <2 1 3 -2|. Being that this scale is the most well-supplied with the greatest number of target triads, we might want to consider it as a tonal basis for 13-EDO, analogous to the diatonic scale in 12-TET. We could, conveniently enough, use the 7-note MOS scale as a basis for 13-EDO notation, leading to a notation very much like 12-TET except for the insertion of an additional accidental between E and F, as in the above interval chart (in the 6L1s column). It can be thought of as a "circle of major 2nds" rather than a circle of 5ths.<br />
<br />
For the 2.9.21 subgroup, we can use the 5\13 generator, the closest thing 13-EDO has to a "perfect 4th", giving an octave-equivalent mapping of <3 1| and MOS scales corresponding to the 4th horogram above. The 8-note MOS scale of 5L3s, independently discovered by Easley Blackwood Jr, Paul Rapoport, and Erv Wilson (among others), is excellent for melody, being somewhat similar to the 12-TET diatonic scale but with an extra semitone added. It is also a conceivable basis for 13-EDO notation, using a modified "circle of fifths" (8\13, the octave inversion of 5\13) including an H: B#-G#-D#-A#-F#-C#-H-E-B-G-D-A-F-C-Hb-Eb-Bb-Gb-Db-Ab, which when arranged in order of ascending pitch within the octave gives the 5L3s names in the above interval chart. This notation has the advantage of preserving some familiar features: diatonic semitones still occur between B and C and E and F, and the dyads E-B, G-D, D-A, and F-C (and associated accidentals) sound approximately like "fifths". Also, the 5L3s scale on C somewhat approximates a 12-TET C major scale (if H is omitted).<br />
<br />
To facilitate discussion of these scales, Igliashon has ascribed them names based on H.P. Lovecraft's "Dream Cycle" mythos. The 2\13-based heptatonic has been named "archeotonic" after the "Old Ones" that rule the Dreamlands, and the 5\13-based octatonic has been named "oneirotonic" after the Dreamlands themselves. Modes of the archeotonic are named after the individual Old Ones themselves; modes of the oneirotonic are named after cities in the Dreamlands. See the charts of modes of the two scales below, excerpted from Igliashon's forthcoming paper "The Case for Thirteen":<br />
<!-- ws:start:WikiTextLocalImageRule:382:<img src="/file/view/Archeotonic.png/252639498/Archeotonic.png" alt="" title="" /> --><img src="/file/view/Archeotonic.png/252639498/Archeotonic.png" alt="Archeotonic.png" title="Archeotonic.png" /><!-- ws:end:WikiTextLocalImageRule:382 --><br />
<!-- ws:start:WikiTextLocalImageRule:383:<img src="/file/view/Oneirotonic.png/252639860/Oneirotonic.png" alt="" title="" /> --><img src="/file/view/Oneirotonic.png/252639860/Oneirotonic.png" alt="Oneirotonic.png" title="Oneirotonic.png" /><!-- ws:end:WikiTextLocalImageRule:383 --><br />
<!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:14 -->Commas</h1>
13 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val < 13 21 30 36 45 48 |.)<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;">2109375/2097152<br />
</td>
<td style="text-align: left;">| -21 3 7 ><br />
</td>
<td style="text-align: right;">10.06<br />
</td>
<td style="text-align: center;">Semicomma<br />
</td>
<td style="text-align: center;">Fokker Comma<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">1029/1000<br />
</td>
<td style="text-align: left;">| -3 1 -3 3 ><br />
</td>
<td style="text-align: right;">49.49<br />
</td>
<td style="text-align: center;">Keega<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;">64/63<br />
</td>
<td style="text-align: left;">| 6 -2 0 -1 ><br />
</td>
<td style="text-align: right;">27.26<br />
</td>
<td style="text-align: center;">Septimal Comma<br />
</td>
<td style="text-align: center;">Archytas' Comma<br />
</td>
<td style="text-align: center;">Leipziger Komma<br />
</td>
</tr>
<tr>
<td style="text-align: center;">64827/64000<br />
</td>
<td style="text-align: left;">| -9 3 -3 4 ><br />
</td>
<td style="text-align: right;">22.23<br />
</td>
<td style="text-align: center;">Squalentine<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3125/3087<br />
</td>
<td style="text-align: left;">| 0 -2 5 -3 ><br />
</td>
<td style="text-align: right;">21.18<br />
</td>
<td style="text-align: center;">Gariboh<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3136/3125<br />
</td>
<td style="text-align: left;">| 6 0 -5 2 ><br />
</td>
<td style="text-align: right;">6.08<br />
</td>
<td style="text-align: center;">Hemimean<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">121/120<br />
</td>
<td style="text-align: left;">| -3 -1 -1 0 2 ><br />
</td>
<td style="text-align: right;">14.37<br />
</td>
<td style="text-align: center;">Biyatisma<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>
</body></html>