13edo
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[[toc|flat]] [[media type="custom" key="12670900"]] ---- =13 tone equal temperament / 13edo= 13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. It is the sixth [[prime numbers|prime]] edo, following [[11edo]] and coming before [[17edo]]. 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, 105/104, 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**= <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://www.microtonalmusic.net/audio/slowdance13edo.mp3|Slow Dance]]////</span> by [[http://danielthompson.blogspot.com/|Daniel Thompson]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2013ET.mp3|Prelude in 13ET]]////</span> by [[Aaron Andrew Hunt]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/13ET.mp3|Two-Part Invention in 13ET]]////</span> by [[Aaron Andrew Hunt]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3|Triskaidekaphobia]]////</span> 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]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3|play]]////</span> 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]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/11%20-%2011.%2013%20octave.mp3|Comets Over Flatland 11]]////</span> by [[Randy Winchester]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/13edo/20120225-midiaxe-prelude-for-synthesizer-in-13-equal.mp3|Prelude for Synthesizer in 13 Equal]]////</span> by [[Chris Vaisvil]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3|Muon Catalyzed Fusion]]////</span> by [[Chris Vaisvil]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://www.seraph.it/dep/det/13Miles.mp3|13 Miles]]////</span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html|blog entry]]) <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://www.seraph.it/dep/det/FastAndFurious13.mp3|Fast And Furious 13]]////</span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html|blog entry]]) <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://www.seraph.it/dep/det/SazDul13.mp3|SazDul 13]]////</span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html|blog entry]]) <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://www.seraph.it/dep/det/Thirteenstan.mp3|Thirteenstan]]////</span> by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html|blog entry]]) <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">////[[http://www.seraph.it/dep/det/Concertina13.mp3|Concertina 13]]////</span> 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]] =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 |.) ||~ 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 ||= ||= ||
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<html><head><title>13edo</title></head><body><!-- ws:start:WikiTextTocRule:21:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><a href="#x13 tone equal temperament / 13edo">13 tone equal temperament / 13edo</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Harmony in 13edo">Harmony in 13edo</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Scales in 13edo">Scales in 13edo</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Animism">Animism</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Igliashon's 13-EDO diatonic approaches">Igliashon's 13-EDO diatonic approaches</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Mapping to Standard Keyboards">Mapping to Standard Keyboards</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#toc8"> </a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: -->
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<!-- ws:start:WikiTextHeadingRule:1:<h1> --><h1 id="toc0"><a name="x13 tone equal temperament / 13edo"></a><!-- ws:end:WikiTextHeadingRule:1 -->13 tone equal temperament / 13edo</h1>
13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. It is the sixth <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/11edo">11edo</a> and coming before <a class="wiki_link" href="/17edo">17edo</a>. 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:3:<h1> --><h1 id="toc1"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:3 -->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:5:<h1> --><h1 id="toc2"><a name="Scales in 13edo"></a><!-- ws:end:WikiTextHeadingRule:5 -->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 />
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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:7:<h2> --><h2 id="toc3"><!-- ws:end:WikiTextHeadingRule:7 --> </h2>
<!-- ws:start:WikiTextHeadingRule:9:<h1> --><h1 id="toc4"><a name="Animism"></a><!-- ws:end:WikiTextHeadingRule:9 -->Animism</h1>
<br />
The animist comma, 105/104, 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:11:<h1> --><h1 id="toc5"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:11 --><strong>Compositions</strong></h1>
<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/slowdance13edo.mp3" rel="nofollow">Slow Dance</a></span> by <a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><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></span> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><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></span> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/triskaidekaphobia.mp3" rel="nofollow">Triskaidekaphobia</a></span> 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> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin13tet.mp3" rel="nofollow">play</a></span> 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 />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><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></span> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><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></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://micro.soonlabel.com/13edo/muon_catalyzed_fusion_13_edo.mp3" rel="nofollow">Muon Catalyzed Fusion</a></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/13Miles.mp3" rel="nofollow">13 Miles</a></span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/ca5de720a48c401bcab8fc82c3b81ddc-152.html" rel="nofollow">blog entry</a>)<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/FastAndFurious13.mp3" rel="nofollow">Fast And Furious 13</a></span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/d2849db485fe3de42e769dad4db3a6ac-153.html" rel="nofollow">blog entry</a>)<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/SazDul13.mp3" rel="nofollow">SazDul 13</a></span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/bb5f0dcff57ba4fc56fcf50f54f10e4b-154.html" rel="nofollow">blog entry</a>)<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Thirteenstan.mp3" rel="nofollow">Thirteenstan</a></span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/2cd2657e7fc17cdbe44dcd5fc255a951-155.html" rel="nofollow">blog entry</a>)<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Concertina13.mp3" rel="nofollow">Concertina 13</a></span> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/3e06afae9b0565b3bc0f39581683cfa7-156.html" rel="nofollow">blog entry</a>)<br />
<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><br />
<br />
<!-- ws:start:WikiTextHeadingRule:13:<h1> --><h1 id="toc6"><a name="Igliashon's 13-EDO diatonic approaches"></a><!-- ws:end:WikiTextHeadingRule:13 -->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:775:<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:775 --><br />
<!-- ws:start:WikiTextLocalImageRule:776:<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:776 --><br />
<!-- ws:start:WikiTextHeadingRule:15:<h1> --><h1 id="toc7"><a name="Mapping to Standard Keyboards"></a><!-- ws:end:WikiTextHeadingRule:15 -->Mapping to Standard Keyboards</h1>
<br />
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.<br />
<br />
<table class="wiki_table">
<tr>
<td>1<br />
</td>
<td>6<br />
</td>
<td>11<br />
</td>
<td>3<br />
</td>
<td>8<br />
</td>
<td>(13)<br />
</td>
<td>5<br />
</td>
<td>10<br />
</td>
<td>2<br />
</td>
<td>7<br />
</td>
<td>12<br />
</td>
<td>4<br />
</td>
<td>9<br />
</td>
<td>1<br />
</td>
<td>Place in Chain of 730 cent intervals<br />
</td>
</tr>
<tr>
<td>X<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>X<br />
</td>
<td>Marked are the octatonic scales (X=Sarnathian)<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>X<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>X<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>X<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>X<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td>*<br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><strong>D</strong><br />
</td>
<td>Eb<br />
</td>
<td>E<br />
</td>
<td><strong>F</strong><br />
</td>
<td>Gb<br />
</td>
<td><br />
</td>
<td><strong>G</strong><br />
</td>
<td>Ab<br />
</td>
<td><strong>A</strong><br />
</td>
<td>Bb<br />
</td>
<td>B<br />
</td>
<td><strong>C</strong><br />
</td>
<td>Db<br />
</td>
<td><strong>D</strong><br />
</td>
<td>Keeps the pentatonic scale on the white keys<br />
</td>
</tr>
<tr>
<td><strong>A</strong><br />
</td>
<td>Bb<br />
</td>
<td>B<br />
</td>
<td><strong>C</strong><br />
</td>
<td>Db<br />
</td>
<td><br />
</td>
<td><strong>D</strong><br />
</td>
<td>Eb<br />
</td>
<td><strong>E</strong><br />
</td>
<td>F<br />
</td>
<td>Gb<br />
</td>
<td><strong>G</strong><br />
</td>
<td>Ab<br />
</td>
<td><strong>A</strong><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><strong>E</strong><br />
</td>
<td>F<br />
</td>
<td>Gb<br />
</td>
<td><strong>G</strong><br />
</td>
<td>Ab<br />
</td>
<td><br />
</td>
<td><strong>A</strong><br />
</td>
<td>Bb<br />
</td>
<td><strong>B</strong><br />
</td>
<td>C<br />
</td>
<td>Db<br />
</td>
<td><strong>D</strong><br />
</td>
<td>Eb<br />
</td>
<td><strong>E</strong><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><strong>B</strong><br />
</td>
<td>C<br />
</td>
<td>Db<br />
</td>
<td><strong>D</strong><br />
</td>
<td>Eb<br />
</td>
<td><br />
</td>
<td><strong>E</strong><br />
</td>
<td>F<br />
</td>
<td><strong>Gb</strong><br />
</td>
<td>G<br />
</td>
<td>Ab<br />
</td>
<td><strong>A</strong><br />
</td>
<td>Bb<br />
</td>
<td><strong>B</strong><br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>C<br />
</td>
<td>Db<br />
</td>
<td>D<br />
</td>
<td>Eb<br />
</td>
<td>E<br />
</td>
<td><br />
</td>
<td>F<br />
</td>
<td>Gb<br />
</td>
<td>G<br />
</td>
<td>Ab<br />
</td>
<td>A<br />
</td>
<td>Bb<br />
</td>
<td>B<br />
</td>
<td>C<br />
</td>
<td>Puts the missing key between a semitone<br />
</td>
</tr>
<tr>
<td>G<br />
</td>
<td>Ab<br />
</td>
<td>A<br />
</td>
<td>Bb<br />
</td>
<td>B<br />
</td>
<td><br />
</td>
<td>C<br />
</td>
<td>Db<br />
</td>
<td>D<br />
</td>
<td>Eb<br />
</td>
<td>E<br />
</td>
<td>F<br />
</td>
<td>Gb<br />
</td>
<td>G<br />
</td>
<td>(if that were to be valuable in any way)<br />
</td>
</tr>
</table>
<br />
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<br />
to keep it on the white keys, with the remaining small step where it looks like it should be.<br />
<!-- ws:start:WikiTextHeadingRule:17:<h1> --><h1 id="toc8"><!-- ws:end:WikiTextHeadingRule:17 --> </h1>
<!-- ws:start:WikiTextHeadingRule:19:<h1> --><h1 id="toc9"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:19 -->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>