15edo: Difference between revisions

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[[image:15 tone keyboard.png width="800" height="333"]]

==Summary==  

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  15 Equal or 15 EDO is a tuning which divides the octave into 15 equally spaced pitches. It can be thought of as three sets of 5-EDO which do not connect by fifths. The fifth at 720 cents is quite wide yet still useable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. The perfect fifth of 15 EDO returns to the octave if stacked five times which is radically different than a meantone system.&lt;br /&gt;

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A recommended method to the notation of 15-edo by some is a system based on porcupine[8] in which eight nominals form the base diatonic scale. In this sense, the &amp;quot;quill&amp;quot; is the name given to the two step interval (160c) of 15-edo while the &amp;quot;small quill&amp;quot; (80c) is the chroma of 15-edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15-edo into a seven nominal based framework.&lt;br /&gt;

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  The following table shows how &lt;a class="wiki_link" href="/Just-24"&gt;some prominent just intervals&lt;/a&gt; are represented in 15edo (ordered by absolute error).&lt;br /&gt;

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There are a variety of other 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 &amp;quot;main focus&amp;quot; of 15-edo composition.&lt;br /&gt;

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  &lt;ul&gt;&lt;li&gt;&lt;strong&gt;Decimal Version:&lt;/strong&gt; Using the nominals 1-0 (with 0 representing &amp;quot;10&amp;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).&lt;/li&gt;&lt;li&gt;&lt;strong&gt;Guitar Version:&lt;/strong&gt; On a 15-edo guitar, because the &amp;quot;perfect fourth&amp;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 &amp;quot;perfect fourths&amp;quot; and &amp;quot;perfect fifths&amp;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.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

  &lt;ul&gt;&lt;li&gt;See the main &lt;a class="wiki_link" href="/Porcupine%20Notation"&gt;porcupine notation&lt;/a&gt; page.&lt;/li&gt;&lt;/ul&gt;Porcupine notation bases porcupine[8] LLLLLLLs scale using eight nominals α β χ δ ε φ γ η. Others have proposed ABCDEFGHA but conflicts with european notation have caused many to reject this approach. Thus greek letters can be used in place with a close resemblance to the spelling of ABCDEFGHA.&lt;br /&gt;

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  &lt;a class="wiki_link" href="/List%20of%2015et%20rank%20two%20temperaments%20by%20badness"&gt;List of 15et rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;

&lt;a class="wiki_link" href="/List%20of%20edo-distinct%2015et%20rank%20two%20temperaments"&gt;List of edo-distinct 15et rank two temperaments&lt;/a&gt;&lt;br /&gt;

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15 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers&lt;/a&gt; out the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 15 24 35 42 52 56 |.)&lt;br /&gt;

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  &lt;a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar35.htm" rel="nofollow"&gt;The 15-Tone Scale System&lt;/a&gt; by &lt;a class="wiki_link" href="/Ivor%20Darreg"&gt;Ivor Darreg&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xZyzKBEW" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;br /&gt;

&lt;a class="wiki_link_ext" href="http://www.inteas.com/Penta01.htm" rel="nofollow"&gt;The Pentadecaphonic System&lt;/a&gt;&lt;br /&gt;

&lt;a class="wiki_link_ext" href="http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html" rel="nofollow"&gt;15-EDO Tutorial&lt;/a&gt; by &lt;a class="wiki_link" href="/Brent%20Carson"&gt;Brent Carson&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xeJYBsDg" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;br /&gt;

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  &lt;a class="wiki_link_ext" href="http://www.swordguitars.com" rel="nofollow"&gt;Sword, Ronald. &amp;quot;Pendecaphonic Scales for Guitar&amp;quot; IAAA Press, UK-USA. First Ed: June 2009.&lt;/a&gt; - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.&lt;br /&gt;

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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;a class="wiki_link_ext" href="http://youtu.be/6TyQ9kDm2fk" rel="nofollow"&gt;Ode For Ada&lt;/a&gt; by &lt;a class="wiki_link" href="/Carlo%20Serafini"&gt;Carlo Serafini&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html" rel="nofollow"&gt;blog entry&lt;/a&gt;)&lt;br /&gt;

Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)&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;