13edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2012-12-~~06 06~~:15:42 UTC</tt>.

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2012-12-10 14:30:37 UTC</tt>.

: The original revision id was <tt>~~389720722~~</tt>.

: The original revision id was <tt>390970088</tt>.

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<h4>Original Wikitext content:</h4>

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=**Compositions**=

//[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]]// by [[http://danielthompson.blogspot.com/|Daniel Thompson]]

////[[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/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/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://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://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]]

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[[http://soundcloud.com/cityoftheasleep/farthur|Farthur]] by 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]]

//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3|Comets Over Flatland 11]]// by [[Randy Winchester]]

////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3|Comets Over Flatland 11]]//// by [[Randy Winchester]]

//[[http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3|Prelude for Synthesizer in 13 Equal]]// by [[Chris Vaisvil]]

////[[http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3|Prelude for Synthesizer in 13 Equal]]//// by [[Chris Vaisvil]]

//[[http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3|Muon Catalyzed Fusion]]// by [[Chris Vaisvil]]

////[[http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3|Muon Catalyzed Fusion]]//// by [[Chris Vaisvil]]

//[[http://www.seraph.it/dep/det/13Miles.mp3|13 Miles]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html|blog entry]])

////[[http://www.seraph.it/dep/det/13Miles.mp3|13 Miles]]//// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html|blog entry]])

//[[http://www.seraph.it/dep/det/FastAndFurious13.mp3|Fast And Furious 13]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html|blog entry]])

////[[http://www.seraph.it/dep/det/FastAndFurious13.mp3|Fast And Furious 13]]//// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html|blog entry]])

//[[http://www.seraph.it/dep/det/SazDul13.mp3|SazDul 13]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html|blog entry]])

////[[http://www.seraph.it/dep/det/SazDul13.mp3|SazDul 13]]//// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html|blog entry]])

//[[http://www.seraph.it/dep/det/Thirteenstan.mp3|Thirteenstan]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html|blog entry]])

////[[http://www.seraph.it/dep/det/Thirteenstan.mp3|Thirteenstan]]//// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html|blog entry]])

//[[http://www.seraph.it/dep/det/Concertina13.mp3|Concertina 13]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html|blog entry]])

////[[http://www.seraph.it/dep/det/Concertina13.mp3|Concertina 13]]//// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html|blog entry]])

[[@http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/|Liber Stellarum]] by [[@http://www.alvotta.net|Alfredo Votta]]

[[@http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/|Liber Stellarum]] by [[@http://www.alvotta.net|Alfredo Votta]]Mapping to Standard Keyboards

~~=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]]~~

~~=Mapping to Standard Keyboards=~~

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

~~|| 1 || 6 || 11 || 3 || 8 || (13) || 5 || 10 || 2 || 7 || 12 || 4 || 9 || 1 || Place in Chain of 730 cent intervals ||~~

~~|| X || * || || * || * || || * || || * || * || || * || || X || Marked are the octatonic scales (X=Sarnathian) ||~~

~~|| || * || || * || * || || * || || X || * || || * || * || || ||~~

~~|| || * || || X || * || || * || * || || * || || * || * || || ||~~

~~|| || * || * || || * || || * || * || || * || || X || * || || ||~~

~~|| || * || * || || * || || X || * || || * || * || || * || || ||~~

~~|| **D** || Eb || E || **F** || Gb || || **G** || Ab || **A** || Bb || B || **C** || Db || **D** || Keeps the pentatonic scale on the white keys ||~~

~~|| **A** || Bb || B || **C** || Db || || **D** || Eb || **E** || F || Gb || **G** || Ab || **A** || ||~~

~~|| **E** || F || Gb || **G** || Ab || || **A** || Bb || **B** || C || Db || **D** || Eb || **E** || ||~~

~~|| **B** || C || Db || **D** || Eb || || **E** || F || **Gb** || G || Ab || **A** || Bb || **B** || ||~~

~~|| C || Db || D || Eb || E || || F || Gb || G || Ab || A || Bb || B || C || Puts the missing key between a semitone ||~~

~~|| G || Ab || A || Bb || B || || C || Db || D || Eb || E || F || Gb || G || (if that were to be valuable in any way) ||~~

~~The archeotonic tonality is much simpler to deal with, you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C~~

~~to keep it on the white keys, with the remaining small step where it looks like it should be.~~

~~= =~~

=Commas=

13 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val < 13 21 30 36 45 48 |.)

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||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||= ||= ||</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>13edo</title></head><body><a href="#x13 tone equal temperament / 13edo">13 tone equal temperament / 13edo</a> | <a href="#Harmony in 13edo">Harmony in 13edo</a> | <a href="#Scales in 13edo">Scales in 13edo</a> | <a href="#Animism">Animism</a> | <a href="#Compositions">Compositions</a> | <a href="#Igliashon's 13-EDO diatonic approaches">Igliashon's 13-EDO diatonic approaches</a> | <a href="#Mapping to Standard Keyboards">Mapping to Standard Keyboards</a> | <a href="#toc8"> </a> | <a href="#Commas">Commas</a>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>13edo</title></head><body><a href="#x13 tone equal temperament / 13edo">13 tone equal temperament / 13edo</a> | <a href="#Harmony in 13edo">Harmony in 13edo</a> | <a href="#Scales in 13edo">Scales in 13edo</a> | <a href="#Animism">Animism</a> | <a href="#Compositions">Compositions</a> | <a href="#Commas">Commas</a>

<script type="text/javascript" src="http://webplayer.yahooapis.com/player.js">

<script type="text/javascript" src="http://webplayer.yahooapis.com/player.js">

</script>

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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 &quot;families&quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp; 6\13, respectively.

<img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" />

<img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" />

<div class="objectEmbed"><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /></a><div><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');" class="filename" title="13edo horograms.pdf">13edo horograms.pdf</a> <ul><li><a href="/file/detail/13edo%20horograms.pdf">Details</a></li><li><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf">Download</a></li><li style="color: #666">242 KB</li></ul></div></div>

<div class="objectEmbed"><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /></a><div><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');" class="filename" title="13edo horograms.pdf">13edo horograms.pdf</a> <ul><li><a href="/file/detail/13edo%20horograms.pdf">Details</a></li><li><a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf">Download</a></li><li style="color: #666">242 KB</li></ul></div></div>

~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson

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<h1 id="toc5"><a name="Compositions"></a>Compositions</h1>

<em~~><a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a~~></em> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a>

<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>

<em~~><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~~></em> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a>

<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>

<em~~><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~~></em> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a>

<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>

<em~~><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow">Triskaidekaphobia</a~~></em> by <a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/music/" rel="nofollow">Herman Miller</a>

<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>

<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=835265" rel="nofollow">Spikey Hair in 13tET</a> <em~~><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow">play</a~~></em> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a>

<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;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>

<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>

<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>

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<a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/farthur" rel="nofollow">Farthur</a> by City of the Asleep

<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>

<em~~><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3" rel="nofollow">Comets Over Flatland 11</a~~></em> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a>

<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3" rel="nofollow">Comets Over Flatland 11</a> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a>

<em~~><a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3" rel="nofollow">Prelude for Synthesizer in 13 Equal</a~~></em> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a>

<a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3" rel="nofollow">Prelude for Synthesizer in 13 Equal</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a>

<em~~><a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3" rel="nofollow">Muon Catalyzed Fusion</a~~></em> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a>

<a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3" rel="nofollow">Muon Catalyzed Fusion</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a>

<~~em><a class~~="wiki_link_ext" href="http://www.seraph.it/dep/det/13Miles.mp3" rel="nofollow">13 Miles</a~~></em~~> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html" rel="nofollow">blog entry</a>)

<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/13Miles.mp3" rel="nofollow">13 Miles</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html" rel="nofollow">blog entry</a>)

<em~~><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/FastAndFurious13.mp3" rel="nofollow">Fast And Furious 13</a~~></em> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html" rel="nofollow">blog entry</a>)

<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/FastAndFurious13.mp3" rel="nofollow">Fast And Furious 13</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html" rel="nofollow">blog entry</a>)

<em~~><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/SazDul13.mp3" rel="nofollow">SazDul 13</a~~></em> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html" rel="nofollow">blog entry</a>)

<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/SazDul13.mp3" rel="nofollow">SazDul 13</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html" rel="nofollow">blog entry</a>)

<~~em><a~~ class="wiki_link_ext" href="http://www.seraph.it/dep/det/Thirteenstan.mp3" rel="nofollow">Thirteenstan</a~~></em~~> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html" rel="nofollow">blog entry</a>)

<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Thirteenstan.mp3" rel="nofollow">Thirteenstan</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html" rel="nofollow">blog entry</a>)

<em~~><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Concertina13.mp3" rel="nofollow">Concertina 13</a~~></em> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html" rel="nofollow">blog entry</a>)

<a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Concertina13.mp3" rel="nofollow">Concertina 13</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html" rel="nofollow">blog entry</a>)

<a class="wiki_link_ext" href="http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/" rel="nofollow" target="_blank">Liber Stellarum</a> by <a class="wiki_link_ext" href="http://www.alvotta.net" rel="nofollow" target="_blank">Alfredo Votta</a>

<a class="wiki_link_ext" href="http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/" rel="nofollow" target="_blank">Liber Stellarum</a> by <a class="wiki_link_ext" href="http://www.alvotta.net" rel="nofollow" target="_blank">Alfredo Votta</a>Mapping to Standard Keyboards

~~ ~~

<h1 id="toc6"><a name="Igliashon's 13-EDO diatonic approaches"></a>Igliashon's 13-EDO diatonic approaches</h1>

~~ ~~

From a temperament perspective, we can probably make the &quot;best&quot; 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 &lt;1 -1| (for 5 and 13), corresponding to the 3rd horogram above. This gives rise to &quot;Sephiroth&quot; 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 &lt;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 &lt;2 1| (for 5 and 9) and &lt;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 &lt;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 &quot;circle of major 2nds&quot; 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 &quot;perfect 4th&quot;, giving an octave-equivalent mapping of &lt;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 &quot;circle of fifths&quot; (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 &quot;fifths&quot;. 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 &quot;Dream Cycle&quot; mythos. The 2\13-based heptatonic has been named &quot;archeotonic&quot; after the &quot;Old Ones&quot; that rule the Dreamlands, and the 5\13-based octatonic has been named &quot;oneirotonic&quot; 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 &quot;The Case for Thirteen&quot;:

<img src="/file/view/Archeotonic.png/252639498/Archeotonic.png" alt="Archeotonic.png" title="Archeotonic.png" />

<img src="/file/view/Oneirotonic.png/252639860/Oneirotonic.png" alt="Oneirotonic.png" title="Oneirotonic.png" />

<h1 id="toc7"><a name="Mapping to Standard Keyboards"></a>Mapping to Standard Keyboards</h1>

~~ ~~

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

~~ ~~

~~<table class="wiki_table">~~

~~<tr>~~

~~<td>1 ~~

~~</td>~~

~~<td>6 ~~

~~</td>~~

~~<td>11 ~~

~~</td>~~

~~<td>3 ~~

~~</td>~~

~~<td>8 ~~

~~</td>~~

~~<td>(13) ~~

~~</td>~~

~~<td>5 ~~

~~</td>~~

~~<td>10 ~~

~~</td>~~

~~<td>2 ~~

~~</td>~~

~~<td>7 ~~

~~</td>~~

~~<td>12 ~~

~~</td>~~

~~<td>4 ~~

~~</td>~~

~~<td>9 ~~

~~</td>~~

~~<td>1 ~~

~~</td>~~

~~<td>Place in Chain of 730 cent intervals ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>X ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>X ~~

~~</td>~~

~~<td>Marked are the octatonic scales (X=Sarnathian) ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>X ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>X ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>X ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>X ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>* ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>D ~~

~~</td>~~

~~<td>Eb ~~

~~</td>~~

~~<td>E ~~

~~</td>~~

~~<td>F ~~

~~</td>~~

~~<td>Gb ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>G ~~

~~</td>~~

~~<td>Ab ~~

~~</td>~~

~~<td>A ~~

~~</td>~~

~~<td>Bb ~~

~~</td>~~

~~<td>B ~~

~~</td>~~

~~<td>C ~~

~~</td>~~

~~<td>Db ~~

~~</td>~~

~~<td>D ~~

~~</td>~~

~~<td>Keeps the pentatonic scale on the white keys ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>A ~~

~~</td>~~

~~<td>Bb ~~

~~</td>~~

~~<td>B ~~

~~</td>~~

~~<td>C ~~

~~</td>~~

~~<td>Db ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>D ~~

~~</td>~~

~~<td>Eb ~~

~~</td>~~

~~<td>E ~~

~~</td>~~

~~<td>F ~~

~~</td>~~

~~<td>Gb ~~

~~</td>~~

~~<td>G ~~

~~</td>~~

~~<td>Ab ~~

~~</td>~~

~~<td>A ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>E ~~

~~</td>~~

~~<td>F ~~

~~</td>~~

~~<td>Gb ~~

~~</td>~~

~~<td>G ~~

~~</td>~~

~~<td>Ab ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>A ~~

~~</td>~~

~~<td>Bb ~~

~~</td>~~

~~<td>B ~~

~~</td>~~

~~<td>C ~~

~~</td>~~

~~<td>Db ~~

~~</td>~~

~~<td>D ~~

~~</td>~~

~~<td>Eb ~~

~~</td>~~

~~<td>E ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>B ~~

~~</td>~~

~~<td>C ~~

~~</td>~~

~~<td>Db ~~

~~</td>~~

~~<td>D ~~

~~</td>~~

~~<td>Eb ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>E ~~

~~</td>~~

~~<td>F ~~

~~</td>~~

~~<td>Gb ~~

~~</td>~~

~~<td>G ~~

~~</td>~~

~~<td>Ab ~~

~~</td>~~

~~<td>A ~~

~~</td>~~

~~<td>Bb ~~

~~</td>~~

~~<td>B ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>C ~~

~~</td>~~

~~<td>Db ~~

~~</td>~~

~~<td>D ~~

~~</td>~~

~~<td>Eb ~~

~~</td>~~

~~<td>E ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>F ~~

~~</td>~~

~~<td>Gb ~~

~~</td>~~

~~<td>G ~~

~~</td>~~

~~<td>Ab ~~

~~</td>~~

~~<td>A ~~

~~</td>~~

~~<td>Bb ~~

~~</td>~~

~~<td>B ~~

~~</td>~~

~~<td>C ~~

~~</td>~~

~~<td>Puts the missing key between a semitone ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>G ~~

~~</td>~~

~~<td>Ab ~~

~~</td>~~

~~<td>A ~~

~~</td>~~

~~<td>Bb ~~

~~</td>~~

~~<td>B ~~

~~</td>~~

~~<td> ~~

~~</td>~~

~~<td>C ~~

~~</td>~~

~~<td>Db ~~

~~</td>~~

~~<td>D ~~

~~</td>~~

~~<td>Eb ~~

~~</td>~~

~~<td>E ~~

~~</td>~~

~~<td>F ~~

~~</td>~~

~~<td>Gb ~~

~~</td>~~

~~<td>G ~~

~~</td>~~

~~<td>(if that were to be valuable in any way) ~~

~~</td>~~

~~</tr>~~

~~</table>~~

The archeotonic tonality is much simpler to deal with, you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C

<h1 id="toc6"><a name="Commas"></a>Commas</h1>

~~to keep it on the white keys, with the remaining small step where it looks like it should be. ~~

~~<h1 id="toc8"> </h1>~~

<h1 id="~~toc9~~"><a name="Commas"></a>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 &lt; 13 21 30 36 45 48 |.)

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2012-12-06 06:15:42 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2012-12-10 14:30:37 UTC</tt>.<br>
-: The original revision id was <tt>389720722</tt>.<br>
+: The original revision id was <tt>390970088</tt>.<br>
-: The revision comment was: <tt>changed link to cenobyte's latest upload in this wiki</tt><br>
+: The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 55: / Line 55: @@
 =**Compositions**=
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]]//&lt;/span&gt; by [[http://danielthompson.blogspot.com/|Daniel Thompson]]
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]]////&lt;/span&gt; by [[http://danielthompson.blogspot.com/|Daniel Thompson]]
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3|Prelude in 13ET]]//&lt;/span&gt; by [[Aaron Andrew Hunt]]
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3|Prelude in 13ET]]////&lt;/span&gt; by [[Aaron Andrew Hunt]]
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3|Two-Part Invention in 13ET]]//&lt;/span&gt; by [[Aaron Andrew Hunt]]
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3|Two-Part Invention in 13ET]]////&lt;/span&gt; by [[Aaron Andrew Hunt]]
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3|Triskaidekaphobia]]//&lt;/span&gt; by [[http://www.io.com/%7Ehmiller/music/|Herman Miller]]
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3|Triskaidekaphobia]]////&lt;/span&gt; by [[http://www.io.com/%7Ehmiller/music/|Herman Miller]]
-[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=835265|Spikey Hair in 13tET]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3|play]]//&lt;/span&gt; by [[Andrew Heathwaite]]
+[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=835265|Spikey Hair in 13tET]] &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3|play]]////&lt;/span&gt; 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]]
@@ Line 66: / Line 66: @@
 [[http://soundcloud.com/cityoftheasleep/farthur|Farthur]] by 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]]
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3|Comets Over Flatland 11]]//&lt;/span&gt; by [[Randy Winchester]]
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3|Comets Over Flatland 11]]////&lt;/span&gt; by [[Randy Winchester]]
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3|Prelude for Synthesizer in 13 Equal]]//&lt;/span&gt; by [[Chris Vaisvil]]
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3|Prelude for Synthesizer in 13 Equal]]////&lt;/span&gt; by [[Chris Vaisvil]]
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3|Muon Catalyzed Fusion]]//&lt;/span&gt; by [[Chris Vaisvil]]
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3|Muon Catalyzed Fusion]]////&lt;/span&gt; by [[Chris Vaisvil]]
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://www.seraph.it/dep/det/13Miles.mp3|13 Miles]]//&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html|blog entry]])
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://www.seraph.it/dep/det/13Miles.mp3|13 Miles]]////&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html|blog entry]])
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://www.seraph.it/dep/det/FastAndFurious13.mp3|Fast And Furious 13]]//&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html|blog entry]])
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://www.seraph.it/dep/det/FastAndFurious13.mp3|Fast And Furious 13]]////&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html|blog entry]])
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://www.seraph.it/dep/det/SazDul13.mp3|SazDul 13]]//&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html|blog entry]])
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://www.seraph.it/dep/det/SazDul13.mp3|SazDul 13]]////&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html|blog entry]])
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://www.seraph.it/dep/det/Thirteenstan.mp3|Thirteenstan]]//&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html|blog entry]])
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://www.seraph.it/dep/det/Thirteenstan.mp3|Thirteenstan]]////&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html|blog entry]])
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;//[[http://www.seraph.it/dep/det/Concertina13.mp3|Concertina 13]]//&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html|blog entry]])
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;////[[http://www.seraph.it/dep/det/Concertina13.mp3|Concertina 13]]////&lt;/span&gt; by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html|blog entry]])
-[[@http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/|Liber Stellarum]] by [[@http://www.alvotta.net|Alfredo Votta]]
+[[@http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/|Liber Stellarum]] by [[@http://www.alvotta.net|Alfredo Votta]]Mapping to Standard Keyboards
-=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 &lt;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 &lt;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).
-.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 &lt;2 1| (for 5 and 9) and &lt;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 &lt;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 &lt;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]]
-=Mapping to Standard Keyboards=
-The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.
-|| 1 || 6 || 11 || 3 || 8 || (13) || 5 || 10 || 2 || 7 || 12 || 4 || 9 || 1 || Place in Chain of 730 cent intervals ||
-|| X || * ||   || * || * ||   || * ||   || * || * ||   || * ||   || X || Marked are the octatonic scales (X=Sarnathian) ||
-||   || * ||   || * || * ||   || * ||   || X || * ||   || * || * ||   ||   ||
-||   || * ||   || X || * ||   || * || * ||   || * ||   || * || * ||   ||   ||
-||   || * || * ||   || * ||   || * || * ||   || * ||   || X || * ||   ||   ||
-||   || * || * ||   || * ||   || X || * ||   || * || * ||   || * ||   ||   ||
-|| **D** || Eb || E || **F** || Gb ||   || **G** || Ab || **A** || Bb || B || **C** || Db || **D** || Keeps the pentatonic scale on the white keys ||
-|| **A** || Bb || B || **C** || Db ||   || **D** || Eb || **E** || F || Gb || **G** || Ab || **A** ||   ||
-|| **E** || F || Gb || **G** || Ab ||   || **A** || Bb || **B** || C || Db || **D** || Eb || **E** ||   ||
-|| **B** || C || Db || **D** || Eb ||   || **E** || F || **Gb** || G || Ab || **A** || Bb || **B** ||   ||
-|| C || Db || D || Eb || E ||   || F || Gb || G || Ab || A || Bb || B || C || Puts the missing key between a semitone ||
-|| G || Ab || A || Bb || B ||   || C || Db || D || Eb || E || F || Gb || G || (if that were to be valuable in any way) ||
-The archeotonic tonality is much simpler to deal with, you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C
-to keep it on the white keys, with the remaining small step where it looks like it should be.
-= =
 =Commas=
 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val &lt; 13 21 30 36 45 48 |.)
@@ Line 120: / Line 89: @@
 ||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=   ||=   ||</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;13edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:21:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;a href="#x13 tone equal temperament / 13edo"&gt;13 tone equal temperament / 13edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Harmony in 13edo"&gt;Harmony in 13edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Scales in 13edo"&gt;Scales in 13edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Animism"&gt;Animism&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Igliashon's 13-EDO diatonic approaches"&gt;Igliashon's 13-EDO diatonic approaches&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Mapping to Standard Keyboards"&gt;Mapping to Standard Keyboards&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#toc8"&gt; &lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;13edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:15:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;a href="#x13 tone equal temperament / 13edo"&gt;13 tone equal temperament / 13edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Harmony in 13edo"&gt;Harmony in 13edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Scales in 13edo"&gt;Scales in 13edo&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Animism"&gt;Animism&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/12670900?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;12670900&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&gt;
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@@ Line 353: / Line 322: @@
   Due to the prime character of the number 13, 13edo can form several xenharmonic &lt;a class="wiki_link" href="/MOSScales"&gt;moment of symmetry scales&lt;/a&gt;. The diagram below shows five &amp;quot;families&amp;quot; of MOS scales: those generated by making a chain of 2\13 (two degrees of 13edo), 3\13, 4\13, 5\13, &amp;amp; 6\13, respectively.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:774:&amp;lt;img src=&amp;quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:774 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:379:&amp;lt;img src=&amp;quot;/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/13edo_horograms.jpg/104015789/13edo_horograms.jpg" alt="13edo_horograms.jpg" title="13edo_horograms.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:379 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextFileRule:778:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/13edo%20horograms.pdf?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@13edo horograms.pdf&amp;quot; title=&amp;quot;File: 13edo horograms.pdf&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="13edo horograms.pdf" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf" onclick="ws.common.trackFileLink('/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf');" class="filename" title="13edo horograms.pdf"&gt;13edo horograms.pdf&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/13edo%20horograms.pdf"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/13edo%20horograms.pdf/104047129/13edo%20horograms.pdf"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;242 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:778 --&gt;&lt;br /&gt;
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 ~diagram by Andrew Heathwaite, based on horograms pioneered by Erv Wilson&lt;br /&gt;
 &lt;br /&gt;
@@ Line 370: / Line 339: @@
 &lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;&lt;strong&gt;Compositions&lt;/strong&gt;&lt;/h1&gt;
   &lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow"&gt;Slow Dance&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow"&gt;Daniel Thompson&lt;/a&gt;&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow"&gt;Slow Dance&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow"&gt;Daniel Thompson&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow"&gt;Prelude in 13ET&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3" rel="nofollow"&gt;Prelude in 13ET&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3" rel="nofollow"&gt;Two-Part Invention in 13ET&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3" rel="nofollow"&gt;Two-Part Invention in 13ET&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&gt;Aaron Andrew Hunt&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow"&gt;Triskaidekaphobia&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/music/" rel="nofollow"&gt;Herman Miller&lt;/a&gt;&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow"&gt;Triskaidekaphobia&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/music/" rel="nofollow"&gt;Herman Miller&lt;/a&gt;&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=835265" rel="nofollow"&gt;Spikey Hair in 13tET&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=835265" rel="nofollow"&gt;Spikey Hair in 13tET&lt;/a&gt; &lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow"&gt;play&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://cityoftheasleep.bandcamp.com/track/broken-dream-jar" rel="nofollow" target="_blank"&gt;Broken Dream Jar&lt;/a&gt; by &lt;a class="wiki_link" href="/IgliashonJones"&gt;City of the Asleep&lt;/a&gt;&lt;br /&gt;
 &lt;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"&gt;Blinding White Darkness&lt;/a&gt; by &lt;a class="wiki_link" href="/IgliashonJones"&gt;City of the Asleep&lt;/a&gt;&lt;br /&gt;
@@ Line 381: / Line 350: @@
 &lt;a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/farthur" rel="nofollow"&gt;Farthur&lt;/a&gt; by City of the Asleep&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.elvenminstrel.com/music/tuning/equal/13equal/13tet.htm" rel="nofollow" target="_blank"&gt;Upsidedown and Backwards: Explorations in 13-tone Equal Temperament&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.elvenminstrel.com/" rel="nofollow"&gt;David J. Finnamore&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3" rel="nofollow"&gt;Comets Over Flatland 11&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3" rel="nofollow"&gt;Comets Over Flatland 11&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3" rel="nofollow"&gt;Prelude for Synthesizer in 13 Equal&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3" rel="nofollow"&gt;Prelude for Synthesizer in 13 Equal&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3" rel="nofollow"&gt;Muon Catalyzed Fusion&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3" rel="nofollow"&gt;Muon Catalyzed Fusion&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/13Miles.mp3" rel="nofollow"&gt;13 Miles&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/13Miles.mp3" rel="nofollow"&gt;13 Miles&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/FastAndFurious13.mp3" rel="nofollow"&gt;Fast And Furious 13&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/FastAndFurious13.mp3" rel="nofollow"&gt;Fast And Furious 13&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/SazDul13.mp3" rel="nofollow"&gt;SazDul 13&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/SazDul13.mp3" rel="nofollow"&gt;SazDul 13&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Thirteenstan.mp3" rel="nofollow"&gt;Thirteenstan&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Thirteenstan.mp3" rel="nofollow"&gt;Thirteenstan&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
-&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Concertina13.mp3" rel="nofollow"&gt;Concertina 13&lt;/a&gt;&lt;/em&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
+&lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&gt;&lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Concertina13.mp3" rel="nofollow"&gt;Concertina 13&lt;/a&gt;&lt;/span&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/" rel="nofollow" target="_blank"&gt;Liber Stellarum&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.alvotta.net" rel="nofollow" target="_blank"&gt;Alfredo Votta&lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://soundcloud.com/xenvotta/sets/votta-liber-stellarum-for/" rel="nofollow" target="_blank"&gt;Liber Stellarum&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.alvotta.net" rel="nofollow" target="_blank"&gt;Alfredo Votta&lt;/a&gt;Mapping to Standard Keyboards&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Igliashon's 13-EDO diatonic approaches"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Igliashon's 13-EDO diatonic approaches&lt;/h1&gt;
- &lt;br /&gt;
-From a temperament perspective, we can probably make the &amp;quot;best&amp;quot; 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 &amp;lt;1 -1| (for 5 and 13), corresponding to the 3rd horogram above. This gives rise to &amp;quot;Sephiroth&amp;quot; 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 &amp;lt;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).&lt;br /&gt;
-&lt;br /&gt;
-.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 &amp;lt;2 1| (for 5 and 9) and &amp;lt;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 &amp;lt;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 &amp;quot;circle of major 2nds&amp;quot; rather than a circle of 5ths.&lt;br /&gt;
-&lt;br /&gt;
-For the 2.9.21 subgroup, we can use the 5\13 generator, the closest thing 13-EDO has to a &amp;quot;perfect 4th&amp;quot;, giving an octave-equivalent mapping of &amp;lt;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 &amp;quot;circle of fifths&amp;quot; (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 &amp;quot;fifths&amp;quot;. Also, the 5L3s scale on C somewhat approximates a 12-TET C major scale (if H is omitted).&lt;br /&gt;
-&lt;br /&gt;
-To facilitate discussion of these scales, Igliashon has ascribed them names based on H.P. Lovecraft's &amp;quot;Dream Cycle&amp;quot; mythos. The 2\13-based heptatonic has been named &amp;quot;archeotonic&amp;quot; after the &amp;quot;Old Ones&amp;quot; that rule the Dreamlands, and the 5\13-based octatonic has been named &amp;quot;oneirotonic&amp;quot; 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 &amp;quot;The Case for Thirteen&amp;quot;:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:775:&amp;lt;img src=&amp;quot;/file/view/Archeotonic.png/252639498/Archeotonic.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Archeotonic.png/252639498/Archeotonic.png" alt="Archeotonic.png" title="Archeotonic.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:775 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:776:&amp;lt;img src=&amp;quot;/file/view/Oneirotonic.png/252639860/Oneirotonic.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Oneirotonic.png/252639860/Oneirotonic.png" alt="Oneirotonic.png" title="Oneirotonic.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:776 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Mapping to Standard Keyboards"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Mapping to Standard Keyboards&lt;/h1&gt;
- &lt;br /&gt;
-The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.&lt;br /&gt;
-&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;(13)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;12&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Place in Chain of 730 cent intervals&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;X&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;X&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Marked are the octatonic scales (X=Sarnathian)&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;X&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;X&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;X&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;X&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;*&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;E&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Gb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;B&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Db&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Keeps the pentatonic scale on the white keys&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;B&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Db&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;E&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Gb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;strong&gt;E&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Gb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;B&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Db&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;E&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;strong&gt;B&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Db&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;E&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;Gb&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;B&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Db&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;D&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;E&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Gb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;A&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;B&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Puts the missing key between a semitone&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;A&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;B&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Db&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;D&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;E&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;F&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Gb&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;(if that were to be valuable in any way)&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
 &lt;br /&gt;
-The archeotonic tonality is much simpler to deal with, you just leave out a tone and remember which one. Although, for diatonic use it may be more convenient to put the missing tone between E/F or B/C&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;Commas&lt;/h1&gt;
-to keep it on the white keys, with the remaining small step where it looks like it should be.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt; &lt;/h1&gt;
- &lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;Commas&lt;/h1&gt;
 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 13 21 30 36 45 48 |.)&lt;br /&gt;