15edo: Difference between revisions

Revision as of 16:35, 20 July 2011

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author guest and made on 2011-07-20 16:35:41 UTC.

The original revision id was 242162119.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

=15 tone equal temperament= 

from //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. 

==Harmony== 

|| Degree || Cents ||= Approximate Ratios* ||
|| 0 || 0 ||= 1/1 ||
|| 1 || 80 ||= 25/24, 21/20 ||
|| 2 || 160 ||= 11/10, 12/11, 10/9 ||
|| 3 || 240 ||= 8/7, 7/6, 9/8 ||
|| 4 || 320 ||= 6/5 ||
|| 5 || 400 ||= 5/4, 14/11 ||
|| 6 || 480 ||= 4/3 ||
|| 7 || 560 ||= 11/8, 7/5 ||
|| 8 || 640 ||= 16/11, 10/7 ||
|| 9 || 720 ||= 3/2 ||
|| 10 || 800 ||= 8/5, 11/7 ||
|| 11 || 880 ||= 5/3 ||
|| 12 || 960 ||= 7/4, 12/7, 16/9 ||
|| 13 || 1040 ||= 20/11, 11/6, 9/5 ||
|| 14 || 1120 ||= 48/25, 40/21 ||
|| 15 || 1200 ||= 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]]".

==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 ||=   ||=   ||


==Additional notes== 

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 modualted 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. .


[[image:http://www.swordguitars.com/Sword_15edoclassicalsm.JPG]](//15-tone Classical Guitar by [[Ron Sword]]// / //Sword Guitars//)

=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= 
[[image:http://ronsword.com/images/Pendecaphonic_coversm.jpg width="112" height="149" link="http://www.ronsword.com"]][[http://www.ronsword.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**== 
[[https://sites.google.com/site/teamouse/home|Mizarian Porcupine Overture]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/MizarianPorcupineOverture.mp3|play]] by [[Herman Miller]] ([[http://teamouse.googlepages.com/home|Herman Miller]]) ([[Regular Temperaments#porcupine|porcupine]] chord progressions)
[[http://www.microtonalmusic.net/audio/15edostudy.mp3|Study for Bells]] 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]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/Hyperimprovisation3.3.mp3|play]] by [[Jacob Barton]] (2003)
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=145852&songID=5483130+OFOIOB|OFOIOB]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/OFOIOB.mp3|play]] by Jacob Barton
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Oldani/15%20tone%20E.T.Improvisationn.mp3|15 Tone ET Improvisationn]] by [[Norbert Oldani]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/15ET.mp3|Elegy in 15ET]] by [[Aaron Andrew Hunt]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/15ETa3fugue2.mp3|Fugue a3 in 15ET]] by Aaron Andrew Hunt
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]]

Original HTML content:

<html><head><title>15edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x15 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->15 tone equal temperament</h1>
 <br />
from <em>wikipedia</em>:<br />
&quot;&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. <br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x15 tone equal temperament-Harmony"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony</h2>
 <br />


<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents<br />
</td>
        <td style="text-align: center;">Approximate Ratios*<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>80<br />
</td>
        <td style="text-align: center;">25/24, 21/20<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>160<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 style="text-align: center;">8/7, 7/6, 9/8<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>320<br />
</td>
        <td style="text-align: center;">6/5<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>400<br />
</td>
        <td style="text-align: center;">5/4, 14/11<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>480<br />
</td>
        <td style="text-align: center;">4/3<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>560<br />
</td>
        <td style="text-align: center;">11/8, 7/5<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>640<br />
</td>
        <td style="text-align: center;">16/11, 10/7<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>720<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>800<br />
</td>
        <td style="text-align: center;">8/5, 11/7<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>880<br />
</td>
        <td style="text-align: center;">5/3<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>960<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 style="text-align: center;">20/11, 11/6, 9/5<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>1120<br />
</td>
        <td style="text-align: center;">48/25, 40/21<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>1200<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;.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x15 tone equal temperament-Commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->Commas</h2>
 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:6:&lt;h2&gt; --><h2 id="toc3"><a name="x15 tone equal temperament-Additional notes"></a><!-- ws:end:WikiTextHeadingRule:6 -->Additional notes</h2>
 <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 modualted 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:WikiTextRemoteImageRule:504:&lt;img src=&quot;http://www.swordguitars.com/Sword_15edoclassicalsm.JPG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://www.swordguitars.com/Sword_15edoclassicalsm.JPG" alt="external image Sword_15edoclassicalsm.JPG" title="external image Sword_15edoclassicalsm.JPG" /><!-- ws:end:WikiTextRemoteImageRule:504 -->(<em>15-tone Classical Guitar by <a class="wiki_link" href="/Ron%20Sword">Ron Sword</a></em> / <em>Sword Guitars</em>)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:8 -->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:10:&lt;h1&gt; --><h1 id="toc5"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:10 -->Practical Theory / Books</h1>
 <!-- ws:start:WikiTextRemoteImageRule:506:&lt;a href=&quot;http://www.ronsword.com&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://ronsword.com/images/Pendecaphonic_coversm.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 149px; width: 112px;&quot; /&gt;&lt;/a&gt; --><a href="http://www.ronsword.com" rel="nofollow"><img src="http://ronsword.com/images/Pendecaphonic_coversm.jpg" alt="external image Pendecaphonic_coversm.jpg" title="external image Pendecaphonic_coversm.jpg" style="height: 149px; width: 112px;" /></a><!-- ws:end:WikiTextRemoteImageRule:506 --><a class="wiki_link_ext" href="http://www.ronsword.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 />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Practical Theory / Books-Compositions"></a><!-- ws:end:WikiTextHeadingRule:12 --><strong>Compositions</strong></h2>
 <a class="wiki_link_ext" href="https://sites.google.com/site/teamouse/home" rel="nofollow">Mizarian Porcupine Overture</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/MizarianPorcupineOverture.mp3" rel="nofollow">play</a> 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 />
<a class="wiki_link_ext" href="http://www.microtonalmusic.net/audio/15edostudy.mp3" rel="nofollow">Study for Bells</a> 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> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/Hyperimprovisation3.3.mp3" rel="nofollow">play</a> 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> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/OFOIOB.mp3" rel="nofollow">play</a> by Jacob Barton<br />
<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> by <a class="wiki_link" href="/Norbert%20Oldani">Norbert Oldani</a><br />
<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> by <a class="wiki_link" href="/Aaron%20Andrew%20Hunt">Aaron Andrew Hunt</a><br />
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/15ETa3fugue2.mp3" rel="nofollow">Fugue a3 in 15ET</a> by Aaron Andrew Hunt<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></body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2011-07-18 18:04:24 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2011-07-20 16:35:41 UTC</tt>.<br>
-: The original revision id was <tt>241839323</tt>.<br>
+: The original revision id was <tt>242162119</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=15 tone equal temperament=
--edo can be seen as a [[7-limit]] tuning because of its ability to approximate some septimal intervals, but it also contains some approximations to [[11-limit]] intervals.
 from //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]])."
+-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.
 ==Harmony==
-|| Degree || Cents || Approximate Ratios* ||
+|| Degree || Cents ||= Approximate Ratios* ||
-|| 0 || 0 || 1/1 ||
+|| 0 || 0 ||= 1/1 ||
-|| 1 || 80 ||   ||
+|| 1 || 80 ||= 25/24, 21/20 ||
-|| 2 || 160 || 11/10 ||
+|| 2 || 160 ||= 11/10, 12/11, 10/9 ||
-|| 3 || 240 || 8/7, 7/6 ||
+|| 3 || 240 ||= 8/7, 7/6, 9/8 ||
-|| 4 || 320 || 6/5 ||
+|| 4 || 320 ||= 6/5 ||
-|| 5 || 400 || 5/4, 14/11 ||
+|| 5 || 400 ||= 5/4, 14/11 ||
-|| 6 || 480 || 4/3 ||
+|| 6 || 480 ||= 4/3 ||
-|| 7 || 560 || 11/8, 7/5 ||
+|| 7 || 560 ||= 11/8, 7/5 ||
-|| 8 || 640 || 16/11, 107 ||
+|| 8 || 640 ||= 16/11, 10/7 ||
-|| 9 || 720 || 3/2 ||
+|| 9 || 720 ||= 3/2 ||
-|| 10 || 800 || 8/5, 11/7 ||
+|| 10 || 800 ||= 8/5, 11/7 ||
-|| 11 || 880 || 5/3 ||
+|| 11 || 880 ||= 5/3 ||
-|| 12 || 960 || 7/4, 12/7 ||
+|| 12 || 960 ||= 7/4, 12/7, 16/9 ||
-|| 13 || 1040 || 20/11 ||
+|| 13 || 1040 ||= 20/11, 11/6, 9/5 ||
-|| 14 || 1120 ||   ||
+|| 14 || 1120 ||= 48/25, 40/21 ||
-|| 15 || 1200 || 2/1 ||
+|| 15 || 1200 ||= 2/1 ||
 *based on treating 15-EDO as an 11-limit temperament; other approaches are possible
@@ Line 93: / Line 93: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;15edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x15 tone equal temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;15 tone equal temperament&lt;/h1&gt;
   &lt;br /&gt;
--edo can be seen as a &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; tuning because of its ability to approximate some septimal intervals, but it also contains some approximations to &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; intervals.&lt;br /&gt;
-&lt;br /&gt;
 from &lt;em&gt;wikipedia&lt;/em&gt;:&lt;br /&gt;
 &amp;quot;&amp;quot;In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of &lt;a class="wiki_link" href="/5edo"&gt;5 equal divisions of the octave&lt;/a&gt; (or five scales of &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt;).&amp;quot;&lt;br /&gt;
+&lt;br /&gt;
+-edo can be seen as a &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; 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. &lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x15 tone equal temperament-Harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harmony&lt;/h2&gt;
@@ Line 108: / Line 108: @@
          &lt;td&gt;Cents&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Approximate Ratios*&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Approximate Ratios*&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 116: / Line 116: @@
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1/1&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 124: / Line 124: @@
          &lt;td&gt;80&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;25/24, 21/20&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 132: / Line 132: @@
          &lt;td&gt;160&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;11/10&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;11/10, 12/11, 10/9&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 140: / Line 140: @@
          &lt;td&gt;240&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8/7, 7/6&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;8/7, 7/6, 9/8&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 148: / Line 148: @@
          &lt;td&gt;320&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;6/5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;6/5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 156: / Line 156: @@
          &lt;td&gt;400&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5/4, 14/11&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;5/4, 14/11&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 164: / Line 164: @@
          &lt;td&gt;480&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;4/3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;4/3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 172: / Line 172: @@
          &lt;td&gt;560&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;11/8, 7/5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;11/8, 7/5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 180: / Line 180: @@
          &lt;td&gt;640&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;16/11, 107&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;16/11, 10/7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 188: / Line 188: @@
          &lt;td&gt;720&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3/2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;3/2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 196: / Line 196: @@
          &lt;td&gt;800&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8/5, 11/7&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;8/5, 11/7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 204: / Line 204: @@
          &lt;td&gt;880&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5/3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;5/3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 212: / Line 212: @@
          &lt;td&gt;960&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7/4, 12/7&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;7/4, 12/7, 16/9&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 220: / Line 220: @@
          &lt;td&gt;1040&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;20/11&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;20/11, 11/6, 9/5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 228: / Line 228: @@
          &lt;td&gt;1120&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;48/25, 40/21&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 236: / Line 236: @@
          &lt;td&gt;1200&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2/1&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;

15edo: Difference between revisions

Revision as of 16:35, 20 July 2011

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

Navigation menu

Search