15edo: Difference between revisions

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**Imported revision 491201252 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-22 09:03:53 UTC</tt>.<br>
: This revision was by author [[User:clamengh|clamengh]] and made on <tt>2014-03-16 09:51:55 UTC</tt>.<br>
: The original revision id was <tt>491201252</tt>.<br>
: The original revision id was <tt>496104186</tt>.<br>
: The revision comment was: <tt></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>
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[[file:15ed2-001.svg]]
[[file:15ed2-001.svg]]


==Intervals by patent val error==
==Intervals by patent val error==  
|| Interval || Error ||
|| Interval || Error ||
|| 18/13 || -3.382 ||
|| 18/13 || -3.382 ||
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[[http://www.swordguitars.com|Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.]] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.
[[http://www.swordguitars.com|Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.]] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.


=Compositions=
=Compositions=  


**[[@http://micro.soonlabel.com/15-ET/|XA 15-ET Directory]]**
**[[@http://micro.soonlabel.com/15-ET/|XA 15-ET Directory]]**
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&lt;span class="long-title"&gt;[[@http://www.youtube.com/watch?v=kQECU5ecCd4|Portrait of insects with 15-tone equal tempered guitar music ]]&lt;/span&gt;
&lt;span class="long-title"&gt;[[@http://www.youtube.com/watch?v=kQECU5ecCd4|Portrait of insects with 15-tone equal tempered guitar music ]]&lt;/span&gt;
[[http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et|PentadecafoniCoda (15et)]] by &lt;span class="wiki_link_ext"&gt;F.F.F. Fiale&lt;/span&gt;
[[http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et|PentadecafoniCoda (15et)]] by &lt;span class="wiki_link_ext"&gt;F.F.F. Fiale&lt;/span&gt;
&lt;span class="wiki_link_ext"&gt;[[http://soonlabel.com/xenharmonic/archives/1946|Cuckoo-Rag Fugue by Claudi Meneghin]]&lt;/span&gt;
[[media type="custom" key="23473468"]]</pre></div>
[[media type="custom" key="23473468"]]</pre></div>
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&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/" rel="nofollow" target="_blank"&gt;XA 15-ET Directory&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/" rel="nofollow" target="_blank"&gt;XA 15-ET Directory&lt;/a&gt;&lt;/strong&gt;&lt;br /&gt;
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&lt;span class="long-title"&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=kQECU5ecCd4" rel="nofollow" target="_blank"&gt;Portrait of insects with 15-tone equal tempered guitar music &lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
&lt;span class="long-title"&gt;&lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=kQECU5ecCd4" rel="nofollow" target="_blank"&gt;Portrait of insects with 15-tone equal tempered guitar music &lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et" rel="nofollow"&gt;PentadecafoniCoda (15et)&lt;/a&gt; by &lt;span class="wiki_link_ext"&gt;F.F.F. Fiale&lt;/span&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et" rel="nofollow"&gt;PentadecafoniCoda (15et)&lt;/a&gt; by &lt;span class="wiki_link_ext"&gt;F.F.F. Fiale&lt;/span&gt;&lt;br /&gt;
&lt;span class="wiki_link_ext"&gt;&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1946" rel="nofollow"&gt;Cuckoo-Rag Fugue by Claudi Meneghin&lt;/a&gt;&lt;/span&gt;&lt;br /&gt;
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Revision as of 09:51, 16 March 2014

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author clamengh and made on 2014-03-16 09:51:55 UTC.
The original revision id was 496104186.
The revision comment was:

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Original Wikitext content:

[[toc]]
=15-tone equal temperament= 

From [[@http://en.wikipedia.org/wiki/15_equal_temperament|Wikipedia]]:
"In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the [[octave]] into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 [[cent]]s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of [[5edo|5 equal divisions of the octave]] (or five scales of [[3edo]])."

15-edo can be seen as a [[7-limit]] temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to [[11-limit]] intervals, so it can reasonably be described as an 11-limit temperament; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to approximate JI with 15-edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity). 15-edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, and 6/5) that has a positive [[81_80|syntonic comma]].

In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in superior harmony and xenharmony, a manageable number of tones, and the sonic fingerprint of multiples of 5-edo.


=Intervals= 
|| Degree || Cents || Solfege
(porcupine-based) || Blackwood
"guitar notation" || Porcupine[7] || Blackwood
Decimal ||= Approximate Ratios* ||
|| 0 || 0 || do || E || G || 1 ||= 1/1 ||
|| 1 || 80 || di || E# || G# / Abb || 1# / 2b ||= 25/24, 21/20, 16/15 ||
|| 2 || 160 || ru || Gb || Gx / Ab || 2 ||= 11/10, 12/11, 10/9 ||
|| 3 || 240 || re || G || A || 3 ||= 8/7, 7/6, 9/8 ||
|| 4 || 320 || me || G# || A# / Bb || 3# / 4b ||= 6/5, 11/9 ||
|| 5 || 400 || mi || Ab || B || 4 ||= 5/4, 14/11 ||
|| 6 || 480 || fa || A || B# / Cb || 5 ||= 4/3, 9/7, 21/16 ||
|| 7 || 560 || fu || A# || C || 5# / 6b ||= 11/8, 7/5 ||
|| 8 || 640 || su || Bb || C# / Db || 6 ||= 16/11, 10/7 ||
|| 9 || 720 || sol || B || D || 7 ||= 3/2, 14/9, 32/21 ||
|| 10 || 800 || le || B# || D# / Eb || 7# / 8b ||= 8/5, 11/7 ||
|| 11 || 880 || la || Db || E || 8 ||= 5/3, 18/11 ||
|| 12 || 960 || ta || D || E# / Fb || 9 ||= 7/4, 12/7, 16/9 ||
|| 13 || 1040 || tu || D# || F || 9# / 0b ||= 20/11, 11/6, 9/5 ||
|| 14 || 1120 || ti || Eb || F# /Gb || 0 ||= 48/25, 40/21, 15/8 ||
|| 15 || 1200 || do || E || G || 1 ||= 2/1 ||
*based on treating 15-EDO as an 11-limit temperament; other approaches are possible

15-EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by [[IgliashonJones|Igliashon Jones]] in the paper "[[http://www.cityoftheasleep.com/etc/5nEDOs.pdf|Five is Not an Odd Number]]". For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see [[@Harmony in 15edo Blacksmith|Harmony in 15edo Blacksmith[10]]].

[[media type="custom" key="24802896"]]

[[file:15ed2-001.svg]]

==Intervals by patent val error== 
|| Interval || Error ||
|| 18/13 || -3.382 ||
|| 6/5 || 4.359 ||
|| 11/10 || -5.004 ||
|| 15/13 || -7.741 ||
|| 11/8 || 8.682 ||
|| 8/7 || 8.826 ||
|| 12/11 || 9.363 ||
|| 5/4 || 13.686 ||
|| 14/11 || -17.508 ||
|| 4/3 || -18.045 ||
|| 13/12 || 21.427 ||
|| 10/9 || -22.404 ||
|| 7/5 || -22.512 ||
|| 15/11 || 23.049 ||
|| 13/10 || 25.786 ||
|| 7/6 || -26.871 ||
|| 11/9 || -27.408 ||
|| 13/11 || 30.79 ||
|| 16/15 || -31.731 ||
|| 9/8 || 36.09 ||
|| 16/13 || -39.472 ||
|| 15/14 || 40.557 ||
|| 9/7 || 44.916 ||
|| 14/13 || -48.298 ||

=Notation= 

There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15-edo composition.

==Blackwood Notation== 
* **Decimal Version:** Using the nominals 1-0 (with 0 representing "10"), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).
* **Guitar Version:** On a 15-edo guitar, because the "perfect fourth" comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.

==Porcupine Notation== 
* See the main [[Porcupine Notation|porcupine notation]] page.

=Rank two temperaments= 
[[List of 15et rank two temperaments by badness]]
[[List of edo-distinct 15et rank two temperaments]]

||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\15 || [[Nautilus]]/[[valentine]] ||
|| 1 || 2\15 || [[Porcupine]]/[[opossum]] ||
|| 1 || 4\15 || [[Hanson]]/[[Keemun|keemun]]/[[orgone]] ||
|| 1 || 7\15 || [[Progress]] ||
|| 3 || 1\15 || [[Augmented]]/[[Augene|augene]] ||
|| 3 || 2\15 || [[Triforce]] ||
|| 5 || 1\15 || [[Blackwood]]/[[blacksmith]] ||

=Commas= 

15 EDO [[tempering out|tempers]] out the following [[comma]]s. (Note: This assumes the val < 15 24 35 42 52 56 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 ||< | 8 -5 > ||> 90.22 ||= Limma ||= Pythagorean Minor 2nd ||=   ||
||= 250/243 ||< | 1 -5 3 > ||> 49.17 ||= Maximal Diesis ||= Porcupine Comma ||=   ||
||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 15625/15552 ||< | -6 -5 6 > ||> 8.11 ||= Kleisma ||= Semicomma Majeur ||=   ||
||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||=   ||=   ||
||= 49/48 ||< | -4 -1 0 2 > ||> 35.70 ||= Slendro Diesis ||=   ||=   ||
||= 64/63 ||< | 6 -2 0 -1 > ||> 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||=   ||=   ||
||= 875/864 ||< | -5 -3 3 1 > ||> 21.90 ||= Keema ||=   ||=   ||
||= 126/125 ||< | 1 2 -3 1 > ||> 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 ||< | 5 -4 3 -2 > ||> 13.47 ||= Octagar ||=   ||=   ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||=   ||=   ||
||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||=   ||=   ||
||= 250047/250000 ||< | -4 6 -6 3 > ||> 0.33 ||= Landscape Comma ||=   ||=   ||
||= 100/99 ||< | 2 -2 2 0 -1 > ||> 17.40 ||= Ptolemisma ||=   ||=   ||
||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||=   ||=   ||
||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||=   ||
||= 65536/65219 ||< | 16 0 0 -2 -3 > ||> 8.39 ||= Orgonisma ||=   ||=   ||
||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||=   ||=   ||
||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||=   ||=   ||
||= 4000/3993 ||< | 5 -1 3 0 -3 > ||> 3.03 ||= Wizardharry ||=   ||=   ||
||= 3025/3024 ||< | -4 -3 2 -1 2 > ||> 0.57 ||= Lehmerisma ||=   ||=   ||
||= 91/90 ||< | -1 -2 -1 1 0 1 > ||> 19.13 ||= Superleap ||=   ||=   ||
||= 676/675 ||< | 2 -3 -2 0 0 2 > ||> 2.56 ||= Parizeksma ||=   ||=   ||


=Theory= 
[[http://sonic-arts.org/darreg/dar35.htm|The 15-Tone Scale System]] by [[Ivor Darreg]] [[http://www.webcitation.org/5xZyzKBEW|Permalink]]
[[http://www.inteas.com/Penta01.htm|The Pentadecaphonic System]]
[[http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html|15-EDO Tutorial]] by [[Brent Carson]] [[http://www.webcitation.org/5xeJYBsDg|Permalink]]

=Practical Theory / Books= 
[[http://www.swordguitars.com|Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.]] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.

=Compositions= 

**[[@http://micro.soonlabel.com/15-ET/|XA 15-ET Directory]]**

[[https://sites.google.com/site/teamouse/home|Mizarian Porcupine Overture]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/MizarianPorcupineOverture.mp3|play]]//</span></span> by [[Herman Miller]] ([[http://teamouse.googlepages.com/home|Herman Miller]]) ([[Regular Temperaments#porcupine|porcupine]] chord progressions)
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">//[[http://www.microtonalmusic.net/audio/15edostudy.mp3|Study for Bells]]//</span></span> by [[Daniel Thompson]] ([[http://danielthompson.blogspot.com/|Daniel Thompson]]) (Jan. 2007)
[[http://www.soundclick.com/bands/songInfo.cfm?bandID=145852&songID=2920478|Hyperimprovisation 3.3]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/Barton/Hyperimprovisation3.3.mp3|play]]//</span></span> by [[Jacob Barton]] (2003)
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=145852&songID=5483130+OFOIOB|OFOIOB]] <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/Barton/OFOIOB.mp3|play]]//</span></span> by Jacob Barton
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/Oldani/15%20tone%20E.T.Improvisationn.mp3|15 Tone ET Improvisationn]]//</span></span> by [[Norbert Oldani]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/15ET.mp3|Elegy in 15ET]]//</span></span> by [[Aaron Andrew Hunt]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/15ETa3fugue2.mp3|Fugue a3 in 15ET]]//</span></span> by Aaron Andrew Hunt
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/12%20-%2012.%2015%20octave.mp3|Comets Over Flatland 12]]//</span></span> by [[Randy Winchester]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/13%20-%2013.%2015%20octave.mp3|Comets Over Flatland 13]]</span></span> by [[Randy Winchester]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/16%20-%2016.%2015%20octave.mp3|Comets Over Flatland 16]]</span></span> by [[Randy Winchester]]
Study for Kyle Gann by [[http://www.akjmusic.com/works.html|Aaron K. Johnson]] (12-out-of-15)
[[http://azuma-asobi.com/Music/index.html|Rick McGowan]]: [[http://azuma-asobi.com/Music/Music-FullWorks.html|Four Ballet Scenes]]
"<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/15-ET/daily20110619_millers_porcupine_7a.mp3|Gently Playing With Miller's Porcupine]]</span></span>" by [[http://www.chrisvaisvil.com|Chris Vaisvil]] (uses Miller's Porcupine-7 mode 2 2 2 3 2 2 2)
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/15-ET/daily20110619_15edo_after_dark_in_the_pedway.mp3|After Dark on the Pedway]]</span></span> by [[http://chrisvaisvil.com/?p=969|Chris Vaisvil]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/15-ET/15_sandles.mp3|15 Sandles]]</span></span> by [[http://chrisvaisvil.com/?p=722|Chris Vaisvil]] [[http://micro.soonlabel.com/15-ET/15_sandles.mid|scordatura midi file]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.mp3|Improv on 15 EDO]]</span></span> by [[http://www.chrisvaisvil.com|Chris Vaisvil]] [[http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.mid|scordatura midi file]] and [[http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.pdf|scordatura PDF score]]
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link">[[http://micro.soonlabel.com/15-ET/20120317-synth-15edo-trolls.mp3|15 edo Trolls]]</span> by [[Chris Vaisvil]] - [[@http://chrisvaisvil.com/?p=2206|details]]
<span style="font-family: Arial,Helvetica,sans-serif;">//[[http://micro.soonlabel.com/15-ET/20120903_15edo_through_the_fire_of_the_sun.mp3|Through the Fire of the Sun (15 edo rock band)]]// by [[http://chrisvaisvil.com|Chris Vaisvil]]</span>
[[http://chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/|2-2-1-2-2-1-2-2-1 mode of 15 edo]] [[http://micro.soonlabel.com/15-ET/20130831_221of15.mp3|play]] by [[Chris Vaisvil]]
[[http://youtu.be/6TyQ9kDm2fk|Ode For Ada]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html|blog entry]])
Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)
[[file:15edo-Chords.ogg]] Some nice sounds I found in 15 EDO
[[file:happenstance15.ogg]]Sonic experiment in 15. Somewhat familiar tonality.
<span class="long-title">[[@http://www.youtube.com/watch?v=kQECU5ecCd4|Portrait of insects with 15-tone equal tempered guitar music ]]</span>
[[http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et|PentadecafoniCoda (15et)]] by <span class="wiki_link_ext">F.F.F. Fiale</span>
<span class="wiki_link_ext">[[http://soonlabel.com/xenharmonic/archives/1946|Cuckoo-Rag Fugue by Claudi Meneghin]]</span>
[[media type="custom" key="23473468"]]

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<html><head><title>15edo</title></head><body><!-- ws:start:WikiTextTocRule:24:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --><div style="margin-left: 1em;"><a href="#x15-tone equal temperament">15-tone equal temperament</a></div>
<!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div>
<!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><div style="margin-left: 2em;"><a href="#Intervals-Intervals by patent val error">Intervals by patent val error</a></div>
<!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 1em;"><a href="#Notation">Notation</a></div>
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 2em;"><a href="#Notation-Blackwood Notation">Blackwood Notation</a></div>
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#Notation-Porcupine Notation">Porcupine Notation</a></div>
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div>
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div>
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Theory">Theory</a></div>
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 1em;"><a href="#Practical Theory / Books">Practical Theory / Books</a></div>
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div>
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --></div>
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="x15-tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:2 -->15-tone equal temperament</h1>
 <br />
From <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/15_equal_temperament" rel="nofollow" target="_blank">Wikipedia</a>:<br />
&quot;In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the <a class="wiki_link" href="/octave">octave</a> into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 <a class="wiki_link" href="/cent">cent</a>s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of <a class="wiki_link" href="/5edo">5 equal divisions of the octave</a> (or five scales of <a class="wiki_link" href="/3edo">3edo</a>).&quot;<br />
<br />
15-edo can be seen as a <a class="wiki_link" href="/7-limit">7-limit</a> temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to <a class="wiki_link" href="/11-limit">11-limit</a> intervals, so it can reasonably be described as an 11-limit temperament; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to approximate JI with 15-edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity). 15-edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, and 6/5) that has a positive <a class="wiki_link" href="/81_80">syntonic comma</a>.<br />
<br />
In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in superior harmony and xenharmony, a manageable number of tones, and the sonic fingerprint of multiples of 5-edo.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents<br />
</td>
        <td>Solfege<br />
(porcupine-based)<br />
</td>
        <td>Blackwood<br />
&quot;guitar notation&quot;<br />
</td>
        <td>Porcupine[7]<br />
</td>
        <td>Blackwood<br />
Decimal<br />
</td>
        <td style="text-align: center;">Approximate Ratios*<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>do<br />
</td>
        <td>E<br />
</td>
        <td>G<br />
</td>
        <td>1<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>80<br />
</td>
        <td>di<br />
</td>
        <td>E#<br />
</td>
        <td>G# / Abb<br />
</td>
        <td>1# / 2b<br />
</td>
        <td style="text-align: center;">25/24, 21/20, 16/15<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>160<br />
</td>
        <td>ru<br />
</td>
        <td>Gb<br />
</td>
        <td>Gx / Ab<br />
</td>
        <td>2<br />
</td>
        <td style="text-align: center;">11/10, 12/11, 10/9<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>240<br />
</td>
        <td>re<br />
</td>
        <td>G<br />
</td>
        <td>A<br />
</td>
        <td>3<br />
</td>
        <td style="text-align: center;">8/7, 7/6, 9/8<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>320<br />
</td>
        <td>me<br />
</td>
        <td>G#<br />
</td>
        <td>A# / Bb<br />
</td>
        <td>3# / 4b<br />
</td>
        <td style="text-align: center;">6/5, 11/9<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>400<br />
</td>
        <td>mi<br />
</td>
        <td>Ab<br />
</td>
        <td>B<br />
</td>
        <td>4<br />
</td>
        <td style="text-align: center;">5/4, 14/11<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>480<br />
</td>
        <td>fa<br />
</td>
        <td>A<br />
</td>
        <td>B# / Cb<br />
</td>
        <td>5<br />
</td>
        <td style="text-align: center;">4/3, 9/7, 21/16<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>560<br />
</td>
        <td>fu<br />
</td>
        <td>A#<br />
</td>
        <td>C<br />
</td>
        <td>5# / 6b<br />
</td>
        <td style="text-align: center;">11/8, 7/5<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>640<br />
</td>
        <td>su<br />
</td>
        <td>Bb<br />
</td>
        <td>C# / Db<br />
</td>
        <td>6<br />
</td>
        <td style="text-align: center;">16/11, 10/7<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>720<br />
</td>
        <td>sol<br />
</td>
        <td>B<br />
</td>
        <td>D<br />
</td>
        <td>7<br />
</td>
        <td style="text-align: center;">3/2, 14/9, 32/21<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>800<br />
</td>
        <td>le<br />
</td>
        <td>B#<br />
</td>
        <td>D# / Eb<br />
</td>
        <td>7# / 8b<br />
</td>
        <td style="text-align: center;">8/5, 11/7<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>880<br />
</td>
        <td>la<br />
</td>
        <td>Db<br />
</td>
        <td>E<br />
</td>
        <td>8<br />
</td>
        <td style="text-align: center;">5/3, 18/11<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>960<br />
</td>
        <td>ta<br />
</td>
        <td>D<br />
</td>
        <td>E# / Fb<br />
</td>
        <td>9<br />
</td>
        <td style="text-align: center;">7/4, 12/7, 16/9<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>1040<br />
</td>
        <td>tu<br />
</td>
        <td>D#<br />
</td>
        <td>F<br />
</td>
        <td>9# / 0b<br />
</td>
        <td style="text-align: center;">20/11, 11/6, 9/5<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>1120<br />
</td>
        <td>ti<br />
</td>
        <td>Eb<br />
</td>
        <td>F# /Gb<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;">48/25, 40/21, 15/8<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>1200<br />
</td>
        <td>do<br />
</td>
        <td>E<br />
</td>
        <td>G<br />
</td>
        <td>1<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
    </tr>
</table>

*based on treating 15-EDO as an 11-limit temperament; other approaches are possible<br />
<br />
15-EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a> in the paper &quot;<a class="wiki_link_ext" href="http://www.cityoftheasleep.com/etc/5nEDOs.pdf" rel="nofollow">Five is Not an Odd Number</a>&quot;. For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see <a class="wiki_link" href="/Harmony%20in%2015edo%20Blacksmith" target="_blank">Harmony in 15edo Blacksmith[10</a>].<br />
<br />
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc2"><a name="Intervals-Intervals by patent val error"></a><!-- ws:end:WikiTextHeadingRule:6 -->Intervals by patent val error</h2>
 

<table class="wiki_table">
    <tr>
        <td>Interval<br />
</td>
        <td>Error<br />
</td>
    </tr>
    <tr>
        <td>18/13<br />
</td>
        <td>-3.382<br />
</td>
    </tr>
    <tr>
        <td>6/5<br />
</td>
        <td>4.359<br />
</td>
    </tr>
    <tr>
        <td>11/10<br />
</td>
        <td>-5.004<br />
</td>
    </tr>
    <tr>
        <td>15/13<br />
</td>
        <td>-7.741<br />
</td>
    </tr>
    <tr>
        <td>11/8<br />
</td>
        <td>8.682<br />
</td>
    </tr>
    <tr>
        <td>8/7<br />
</td>
        <td>8.826<br />
</td>
    </tr>
    <tr>
        <td>12/11<br />
</td>
        <td>9.363<br />
</td>
    </tr>
    <tr>
        <td>5/4<br />
</td>
        <td>13.686<br />
</td>
    </tr>
    <tr>
        <td>14/11<br />
</td>
        <td>-17.508<br />
</td>
    </tr>
    <tr>
        <td>4/3<br />
</td>
        <td>-18.045<br />
</td>
    </tr>
    <tr>
        <td>13/12<br />
</td>
        <td>21.427<br />
</td>
    </tr>
    <tr>
        <td>10/9<br />
</td>
        <td>-22.404<br />
</td>
    </tr>
    <tr>
        <td>7/5<br />
</td>
        <td>-22.512<br />
</td>
    </tr>
    <tr>
        <td>15/11<br />
</td>
        <td>23.049<br />
</td>
    </tr>
    <tr>
        <td>13/10<br />
</td>
        <td>25.786<br />
</td>
    </tr>
    <tr>
        <td>7/6<br />
</td>
        <td>-26.871<br />
</td>
    </tr>
    <tr>
        <td>11/9<br />
</td>
        <td>-27.408<br />
</td>
    </tr>
    <tr>
        <td>13/11<br />
</td>
        <td>30.79<br />
</td>
    </tr>
    <tr>
        <td>16/15<br />
</td>
        <td>-31.731<br />
</td>
    </tr>
    <tr>
        <td>9/8<br />
</td>
        <td>36.09<br />
</td>
    </tr>
    <tr>
        <td>16/13<br />
</td>
        <td>-39.472<br />
</td>
    </tr>
    <tr>
        <td>15/14<br />
</td>
        <td>40.557<br />
</td>
    </tr>
    <tr>
        <td>9/7<br />
</td>
        <td>44.916<br />
</td>
    </tr>
    <tr>
        <td>14/13<br />
</td>
        <td>-48.298<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:8 -->Notation</h1>
 <br />
There are a variety of ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the &quot;main focus&quot; of 15-edo composition.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc4"><a name="Notation-Blackwood Notation"></a><!-- ws:end:WikiTextHeadingRule:10 -->Blackwood Notation</h2>
 <ul><li><strong>Decimal Version:</strong> Using the nominals 1-0 (with 0 representing &quot;10&quot;), one of the three chains of 5-edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).</li><li><strong>Guitar Version:</strong> On a 15-edo guitar, because the &quot;perfect fourth&quot; comes from 5-edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the circle of fourths on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15-edo on the guitar, since 5-edo provides a useful set of 3-limit landmarks (or &quot;perfect fourths&quot; and &quot;perfect fifths&quot;) that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc5"><a name="Notation-Porcupine Notation"></a><!-- ws:end:WikiTextHeadingRule:12 -->Porcupine Notation</h2>
 <ul><li>See the main <a class="wiki_link" href="/Porcupine%20Notation">porcupine notation</a> page.</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc6"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:14 -->Rank two temperaments</h1>
 <a class="wiki_link" href="/List%20of%2015et%20rank%20two%20temperaments%20by%20badness">List of 15et rank two temperaments by badness</a><br />
<a class="wiki_link" href="/List%20of%20edo-distinct%2015et%20rank%20two%20temperaments">List of edo-distinct 15et rank two temperaments</a><br />
<br />


<table class="wiki_table">
    <tr>
        <th>Periods<br />
per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Temperaments<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\15<br />
</td>
        <td><a class="wiki_link" href="/Nautilus">Nautilus</a>/<a class="wiki_link" href="/valentine">valentine</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>2\15<br />
</td>
        <td><a class="wiki_link" href="/Porcupine">Porcupine</a>/<a class="wiki_link" href="/opossum">opossum</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>4\15<br />
</td>
        <td><a class="wiki_link" href="/Hanson">Hanson</a>/<a class="wiki_link" href="/Keemun">keemun</a>/<a class="wiki_link" href="/orgone">orgone</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>7\15<br />
</td>
        <td><a class="wiki_link" href="/Progress">Progress</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>1\15<br />
</td>
        <td><a class="wiki_link" href="/Augmented">Augmented</a>/<a class="wiki_link" href="/Augene">augene</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>2\15<br />
</td>
        <td><a class="wiki_link" href="/Triforce">Triforce</a><br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>1\15<br />
</td>
        <td><a class="wiki_link" href="/Blackwood">Blackwood</a>/<a class="wiki_link" href="/blacksmith">blacksmith</a><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc7"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:16 -->Commas</h1>
 <br />
15 EDO <a class="wiki_link" href="/tempering%20out">tempers</a> out the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; 15 24 35 42 52 56 |.)<br />
<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;">256/243<br />
</td>
        <td style="text-align: left;">| 8 -5 &gt;<br />
</td>
        <td style="text-align: right;">90.22<br />
</td>
        <td style="text-align: center;">Limma<br />
</td>
        <td style="text-align: center;">Pythagorean Minor 2nd<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">250/243<br />
</td>
        <td style="text-align: left;">| 1 -5 3 &gt;<br />
</td>
        <td style="text-align: right;">49.17<br />
</td>
        <td style="text-align: center;">Maximal Diesis<br />
</td>
        <td style="text-align: center;">Porcupine Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">128/125<br />
</td>
        <td style="text-align: left;">| 7 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">41.06<br />
</td>
        <td style="text-align: center;">Diesis<br />
</td>
        <td style="text-align: center;">Augmented Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15625/15552<br />
</td>
        <td style="text-align: left;">| -6 -5 6 &gt;<br />
</td>
        <td style="text-align: right;">8.11<br />
</td>
        <td style="text-align: center;">Kleisma<br />
</td>
        <td style="text-align: center;">Semicomma Majeur<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1000<br />
</td>
        <td style="text-align: left;">| -3 1 -3 3 &gt;<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;">49/48<br />
</td>
        <td style="text-align: left;">| -4 -1 0 2 &gt;<br />
</td>
        <td style="text-align: right;">35.70<br />
</td>
        <td style="text-align: center;">Slendro Diesis<br />
</td>
        <td style="text-align: center;"><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 &gt;<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 &gt;<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;">875/864<br />
</td>
        <td style="text-align: left;">| -5 -3 3 1 &gt;<br />
</td>
        <td style="text-align: right;">21.90<br />
</td>
        <td style="text-align: center;">Keema<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">126/125<br />
</td>
        <td style="text-align: left;">| 1 2 -3 1 &gt;<br />
</td>
        <td style="text-align: right;">13.79<br />
</td>
        <td style="text-align: center;">Septimal Semicomma<br />
</td>
        <td style="text-align: center;">Starling Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4000/3969<br />
</td>
        <td style="text-align: left;">| 5 -4 3 -2 &gt;<br />
</td>
        <td style="text-align: right;">13.47<br />
</td>
        <td style="text-align: center;">Octagar<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: left;">| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: right;">8.43<br />
</td>
        <td style="text-align: center;">Gamelisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6144/6125<br />
</td>
        <td style="text-align: left;">| 11 1 -3 -2 &gt;<br />
</td>
        <td style="text-align: right;">5.36<br />
</td>
        <td style="text-align: center;">Porwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">250047/250000<br />
</td>
        <td style="text-align: left;">| -4 6 -6 3 &gt;<br />
</td>
        <td style="text-align: right;">0.33<br />
</td>
        <td style="text-align: center;">Landscape Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">100/99<br />
</td>
        <td style="text-align: left;">| 2 -2 2 0 -1 &gt;<br />
</td>
        <td style="text-align: right;">17.40<br />
</td>
        <td style="text-align: center;">Ptolemisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">121/120<br />
</td>
        <td style="text-align: left;">| -3 -1 -1 0 2 &gt;<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;">176/175<br />
</td>
        <td style="text-align: left;">| 4 0 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.86<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">65536/65219<br />
</td>
        <td style="text-align: left;">| 16 0 0 -2 -3 &gt;<br />
</td>
        <td style="text-align: right;">8.39<br />
</td>
        <td style="text-align: center;">Orgonisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td style="text-align: left;">| -7 -1 1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">4.50<br />
</td>
        <td style="text-align: center;">Keenanisma<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 &gt;<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>
    <tr>
        <td style="text-align: center;">4000/3993<br />
</td>
        <td style="text-align: left;">| 5 -1 3 0 -3 &gt;<br />
</td>
        <td style="text-align: right;">3.03<br />
</td>
        <td style="text-align: center;">Wizardharry<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3025/3024<br />
</td>
        <td style="text-align: left;">| -4 -3 2 -1 2 &gt;<br />
</td>
        <td style="text-align: right;">0.57<br />
</td>
        <td style="text-align: center;">Lehmerisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td style="text-align: left;">| -1 -2 -1 1 0 1 &gt;<br />
</td>
        <td style="text-align: right;">19.13<br />
</td>
        <td style="text-align: center;">Superleap<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">676/675<br />
</td>
        <td style="text-align: left;">| 2 -3 -2 0 0 2 &gt;<br />
</td>
        <td style="text-align: right;">2.56<br />
</td>
        <td style="text-align: center;">Parizeksma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc8"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:18 -->Theory</h1>
 <a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar35.htm" rel="nofollow">The 15-Tone Scale System</a> by <a class="wiki_link" href="/Ivor%20Darreg">Ivor Darreg</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xZyzKBEW" rel="nofollow">Permalink</a><br />
<a class="wiki_link_ext" href="http://www.inteas.com/Penta01.htm" rel="nofollow">The Pentadecaphonic System</a><br />
<a class="wiki_link_ext" href="http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html" rel="nofollow">15-EDO Tutorial</a> by <a class="wiki_link" href="/Brent%20Carson">Brent Carson</a> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeJYBsDg" rel="nofollow">Permalink</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc9"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:20 -->Practical Theory / Books</h1>
 <a class="wiki_link_ext" href="http://www.swordguitars.com" rel="nofollow">Sword, Ronald. &quot;Pendecaphonic Scales for Guitar&quot; IAAA Press, UK-USA. First Ed: June 2009.</a> - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc10"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:22 -->Compositions</h1>
 <br />
<strong><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/" rel="nofollow" target="_blank">XA 15-ET Directory</a></strong><br />
<br />
<a class="wiki_link_ext" href="https://sites.google.com/site/teamouse/home" rel="nofollow">Mizarian Porcupine Overture</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/MizarianPorcupineOverture.mp3" rel="nofollow">play</a></em></span></span> by <a class="wiki_link" href="/Herman%20Miller">Herman Miller</a> (<a class="wiki_link_ext" href="http://teamouse.googlepages.com/home" rel="nofollow">Herman Miller</a>) (<a class="wiki_link" href="/Regular%20Temperaments#porcupine">porcupine</a> chord progressions)<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/15edostudy.mp3" rel="nofollow">Study for Bells</a></em></span></span> by <a class="wiki_link" href="/Daniel%20Thompson">Daniel Thompson</a> (<a class="wiki_link_ext" href="http://danielthompson.blogspot.com/" rel="nofollow">Daniel Thompson</a>) (Jan. 2007)<br />
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/songInfo.cfm?bandID=145852&amp;songID=2920478" rel="nofollow">Hyperimprovisation 3.3</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/Hyperimprovisation3.3.mp3" rel="nofollow">play</a></em></span></span> by <a class="wiki_link" href="/Jacob%20Barton">Jacob Barton</a> (2003)<br />
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=145852&amp;songID=5483130+OFOIOB" rel="nofollow">OFOIOB</a> <span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/OFOIOB.mp3" rel="nofollow">play</a></em></span></span> by Jacob Barton<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/15%20tone%20E.T.Improvisationn.mp3" rel="nofollow">15 Tone ET Improvisationn</a></em></span></span> by <a class="wiki_link" href="/Norbert%20Oldani">Norbert Oldani</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/15ET.mp3" rel="nofollow">Elegy in 15ET</a></em></span></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"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/15ETa3fugue2.mp3" rel="nofollow">Fugue a3 in 15ET</a></em></span></span> by Aaron Andrew Hunt<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"><em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/12%20-%2012.%2015%20octave.mp3" rel="nofollow">Comets Over Flatland 12</a></em></span></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"><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/13%20-%2013.%2015%20octave.mp3" rel="nofollow">Comets Over Flatland 13</a></span></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"><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/16%20-%2016.%2015%20octave.mp3" rel="nofollow">Comets Over Flatland 16</a></span></span> by <a class="wiki_link" href="/Randy%20Winchester">Randy Winchester</a><br />
Study for Kyle Gann by <a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow">Aaron K. Johnson</a> (12-out-of-15)<br />
<a class="wiki_link_ext" href="http://azuma-asobi.com/Music/index.html" rel="nofollow">Rick McGowan</a>: <a class="wiki_link_ext" href="http://azuma-asobi.com/Music/Music-FullWorks.html" rel="nofollow">Four Ballet Scenes</a><br />
&quot;<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/15-ET/daily20110619_millers_porcupine_7a.mp3" rel="nofollow">Gently Playing With Miller's Porcupine</a></span></span>&quot; by <a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow">Chris Vaisvil</a> (uses Miller's Porcupine-7 mode 2 2 2 3 2 2 2)<br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/15-ET/daily20110619_15edo_after_dark_in_the_pedway.mp3" rel="nofollow">After Dark on the Pedway</a></span></span> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=969" rel="nofollow">Chris Vaisvil</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/15-ET/15_sandles.mp3" rel="nofollow">15 Sandles</a></span></span> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=722" rel="nofollow">Chris Vaisvil</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/15_sandles.mid" rel="nofollow">scordatura midi file</a><br />
<span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"><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/15-ET/Improv_on_15edo/Improv_on_15.mp3" rel="nofollow">Improv on 15 EDO</a></span></span> by <a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow">Chris Vaisvil</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.mid" rel="nofollow">scordatura midi file</a> and <a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/Improv_on_15edo/Improv_on_15.pdf" rel="nofollow">scordatura PDF score</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/15-ET/20120317-synth-15edo-trolls.mp3" rel="nofollow">15 edo Trolls</a></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a> - <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2206" rel="nofollow" target="_blank">details</a><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><em><a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/20120903_15edo_through_the_fire_of_the_sun.mp3" rel="nofollow">Through the Fire of the Sun (15 edo rock band)</a></em> by <a class="wiki_link_ext" href="http://chrisvaisvil.com" rel="nofollow">Chris Vaisvil</a></span><br />
<a class="wiki_link_ext" href="http://chrisvaisvil.com/2-2-1-2-2-1-2-2-1-mode-of-15-edo/" rel="nofollow">2-2-1-2-2-1-2-2-1 mode of 15 edo</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/15-ET/20130831_221of15.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://youtu.be/6TyQ9kDm2fk" rel="nofollow">Ode For Ada</a> by <a class="wiki_link" href="/Carlo%20Serafini">Carlo Serafini</a> (<a class="wiki_link_ext" href="http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html" rel="nofollow">blog entry</a>)<br />
Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)<br />
<!-- ws:start:WikiTextFileRule:892:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/15edo-Chords.ogg?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@15edo-Chords.ogg&quot; title=&quot;File: 15edo-Chords.ogg&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="15edo-Chords.ogg" /></a><div><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg" onclick="ws.common.trackFileLink('/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg');" class="filename" title="15edo-Chords.ogg">15edo-Chords.ogg</a><br /><ul><li><a href="/file/detail/15edo-Chords.ogg">Details</a></li><li><a href="/file/view/15edo-Chords.ogg/252472844/15edo-Chords.ogg">Download</a></li><li style="color: #666">2 MB</li></ul></div></div><!-- ws:end:WikiTextFileRule:892 --> Some nice sounds I found in 15 EDO<br />
<!-- ws:start:WikiTextFileRule:893:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/happenstance15.ogg?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@happenstance15.ogg&quot; title=&quot;File: happenstance15.ogg&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="happenstance15.ogg" /></a><div><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg" onclick="ws.common.trackFileLink('/file/view/happenstance15.ogg/257380320/happenstance15.ogg');" class="filename" title="happenstance15.ogg">happenstance15.ogg</a><br /><ul><li><a href="/file/detail/happenstance15.ogg">Details</a></li><li><a href="/file/view/happenstance15.ogg/257380320/happenstance15.ogg">Download</a></li><li style="color: #666">6 MB</li></ul></div></div><!-- ws:end:WikiTextFileRule:893 -->Sonic experiment in 15. Somewhat familiar tonality.<br />
<span class="long-title"><a class="wiki_link_ext" href="http://www.youtube.com/watch?v=kQECU5ecCd4" rel="nofollow" target="_blank">Portrait of insects with 15-tone equal tempered guitar music </a></span><br />
<a class="wiki_link_ext" href="http://www.reverbnation.com/ffffiale/song/3073160-pentadecafonicoda-15et" rel="nofollow">PentadecafoniCoda (15et)</a> by <span class="wiki_link_ext">F.F.F. Fiale</span><br />
<span class="wiki_link_ext"><a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/1946" rel="nofollow">Cuckoo-Rag Fugue by Claudi Meneghin</a></span><br />
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