24edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~hstraub~~|~~hstraub~~]] and made on <tt>~~2010~~-11-~~10 02~~:41:40 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-30 01:38:30 UTC</tt>.<br>

: The original revision id was <tt>~~178154109~~</tt>.<br>

: The original revision id was <tt>215338052</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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----

=Theory=

The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as **quarter-tone tuning**, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and its use in theory and occasionally in practice in Arabic music.

The 24-tone equal temperament ~~is also known~~ as **quarter-tone tuning**, ~~since it evenly divides the 12~~-tone equal ~~tempered semitone~~ in ~~two. Quarter~~-~~tones~~ are the ~~most commonly~~ used ~~microtonal tuning due to its retention~~ of the ~~familiar~~ 12 ~~tones~~ and ~~since~~ it is the ~~smallest microtonal equal temperament that contains all the 12 notes~~.

==As a temperament==

The [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.

The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17.

The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):

~~==Intervals==~~

0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)

Its inversion, 0-3-6-10-14 ("minor")

0-7-14 ("neutral")

0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)

=Intervals=

The twelve new intervals in quartertones are:

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..

.. ||

~~==As a temperament==~~

The [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.

The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):

~~0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)~~

~~Its inversion, 0-3-6-10-14 ("minor")~~

~~0-7-14 ("neutral")~~

0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)

=Notation=

[[image:http://www.wfg.woodwind.org/n/s14_blue.gif]] = quarter-tone sharp

[[image:http://www.wfg.woodwind.org/n/s34_blue.gif]] = three-quarter-tone sharp

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=Instruments[[#instruments]]=

The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This [[Microtonal Keyboards#twelvenoteoctavescales|"12 note octave scales"]] approach is used in a wide part of the existing literature - see below.

[[image:http://www.swordguitars.com/Sword_quartertone_stratsm.jpg width="406" height="195"]]24-tone "1/4-tone" Guitar by Ron Sword / Sword guitars

=Compositions=

"Prométhée enchaîné" by [[http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy|Fromental Halévy]] (considered the first mainstream western orchestral composition to use quarter tones.)

"3 Hommages" by [[http://en.wikipedia.org/wiki/Georg_Friedrich_Haas|Georg Friedrich Haas]]

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[[http://www.96edo.com/24_EDO.html|About 24-EDO]]</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>24edo</title></head><body><a href="#Theory">Theory</a> | <a href="#Notation">Notation</a> | <a href="#Scales / Modes">Scales / Modes</a> | <a href="#Instruments">Instruments</a> | <a href="#Compositions">Compositions</a> | <a href="#Practical Theory / Books">Practical Theory / Books</a> | <a href="#External links">External links</a>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>24edo</title></head><body><a href="#Theory">Theory</a> | <a href="#Intervals">Intervals</a> | <a href="#Notation">Notation</a> | <a href="#Scales / Modes">Scales / Modes</a> | <a href="#Instruments">Instruments</a> | <a href="#Compositions">Compositions</a> | <a href="#Practical Theory / Books">Practical Theory / Books</a> | <a href="#External links">External links</a>

<hr />

<h1 id="toc0"><a name="Theory"></a>Theory</h1>

~~<br />~~

The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as <strong>quarter-tone tuning</strong>, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and its use in theory and occasionally in practice in Arabic music.<br />

The 24~~-tone~~ equal ~~temperament~~ is also known as <strong>quarter-tone tuning</strong>, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes.<br />

<br />

<h2 id="toc1"><a name="Theory-As a temperament"></a>As a temperament</h2>

The <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.<br />

<br />

The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 2.3.125.35.11.325.17 <a class="wiki_link" href="/just%20intonation%20subgroup">just intonation subgroup</a>, making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17.<br />

<br />

The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):<br />

<br />

0-4-8-11-14 (&quot;major&quot; chord with a 9:8 and a 11:8 above the root)<br />

Its inversion, 0-3-6-10-14 (&quot;minor&quot;)<br />

0-7-14 (&quot;neutral&quot;)<br />

0-5-10 (another kind of &quot;neutral&quot;, splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)<br />

<br />

<h2 id="~~toc1~~"><a name="~~Theory-~~Intervals"></a>Intervals</h2>

<h1 id="toc2"><a name="Intervals"></a>Intervals</h1>

~~<br />~~

The twelve new intervals in quartertones are:<br />

<br />

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</table>

~~<br />~~

<h2 id="toc2"><a name="Theory-As a temperament"></a>As a temperament</h2>

~~<br />~~

The <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.<br />

~~<br />~~

The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):<br />

~~<br />~~

~~0-4-8-11-14 (&quot;major&quot; chord with a 9:8 and a 11:8 above the root)<br />~~

~~Its inversion, 0-3-6-10-14 (&quot;minor&quot;)<br />~~

~~0-7-14 (&quot;neutral&quot;)<br />~~

0-5-10 (another kind of &quot;neutral&quot;, splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)<br />

<br />

<h1 id="toc3"><a name="Notation"></a>Notation</h1>

~~<br />~~

<img src="http://www.wfg.woodwind.org/n/s14_blue.gif" alt="external image s14_blue.gif" title="external image s14_blue.gif" /> = quarter-tone sharp<br />

<img src="http://www.wfg.woodwind.org/n/s34_blue.gif" alt="external image s34_blue.gif" title="external image s34_blue.gif" /> = three-quarter-tone sharp<br />

<img src="http://www.wfg.woodwind.org/n/f14_blue.gif" alt="external image f14_blue.gif" title="external image f14_blue.gif" /> = quarter-tone flat<br />

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<br />

<h1 id="toc6"><a name="Instruments"></a>Instruments<a name="instruments"></a></h1>

~~<br />~~

The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This <a class="wiki_link" href="/Microtonal%20Keyboards#twelvenoteoctavescales">&quot;12 note octave scales&quot;</a> approach is used in a wide part of the existing literature - see below.<br />

<img src="http://www.swordguitars.com/Sword_quartertone_stratsm.jpg" alt="external image Sword_quartertone_stratsm.jpg" title="external image Sword_quartertone_stratsm.jpg" style="height: 195px; width: 406px;" />24-tone &quot;1/4-tone&quot; Guitar by Ron Sword / Sword guitars<br />

<br />

<h1 id="toc7"><a name="Compositions"></a>Compositions</h1>

~~<br />~~

&quot;Prométhée enchaîné&quot; by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy" rel="nofollow">Fromental Halévy</a> (considered the first mainstream western orchestral composition to use quarter tones.)<br />

&quot;3 Hommages&quot; by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Georg_Friedrich_Haas" rel="nofollow">Georg Friedrich Haas</a><br />

<a class="wiki_link_ext" href="http://www.96edo.com/My_Music.html" rel="nofollow">Some microtonal music in 24-EDO</a> by <a class="wiki_link_ext" href="http://www.96edo.com/About_me.html" rel="nofollow">Shaahin Mohajeri</a><br />

@@ 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:hstraub|hstraub]] and made on <tt>2010-11-10 02:41:40 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-30 01:38:30 UTC</tt>.<br>
-: The original revision id was <tt>178154109</tt>.<br>
+: The original revision id was <tt>215338052</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>
@@ Line 9: / Line 9: @@
 ----
 =Theory=
+The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as **quarter-tone tuning**, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and its use in theory and occasionally in practice in Arabic music.
-The 24-tone equal temperament is also known as **quarter-tone tuning**, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes.
+==As a temperament==
+The [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.
+The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17.
+The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):
-==Intervals==
+-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
+Its inversion, 0-3-6-10-14 ("minor")
+-7-14 ("neutral")
+-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)
+=Intervals=
 The twelve new intervals in quartertones are:
@@ Line 81: / Line 91: @@
 ..
 .. ||
-==As a temperament==
-The [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.
-The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):
--4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
-Its inversion, 0-3-6-10-14 ("minor")
--7-14 ("neutral")
--5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)
 =Notation=
 [[image:http://www.wfg.woodwind.org/n/s14_blue.gif]] = quarter-tone sharp
 [[image:http://www.wfg.woodwind.org/n/s34_blue.gif]] = three-quarter-tone sharp
@@ Line 211: / Line 209: @@
 =Instruments[[#instruments]]=
 The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This [[Microtonal Keyboards#twelvenoteoctavescales|"12 note octave scales"]] approach is used in a wide part of the existing literature - see below.
 [[image:http://www.swordguitars.com/Sword_quartertone_stratsm.jpg width="406" height="195"]]24-tone "1/4-tone" Guitar by Ron Sword / Sword guitars
 =Compositions=
 "Prométhée enchaîné" by [[http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy|Fromental Halévy]] (considered the first mainstream western orchestral composition to use quarter tones.)
 "3 Hommages" by [[http://en.wikipedia.org/wiki/Georg_Friedrich_Haas|Georg Friedrich Haas]]
@@ Line 230: / Line 226: @@
 [[http://www.96edo.com/24_EDO.html|About 24-EDO]]</pre></div>
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-<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;24edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Theory"&gt;Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Notation"&gt;Notation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Scales / Modes"&gt;Scales / Modes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Instruments"&gt;Instruments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Practical Theory / Books"&gt;Practical Theory / Books&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#External links"&gt;External links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;24edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Theory"&gt;Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Notation"&gt;Notation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Scales / Modes"&gt;Scales / Modes&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Instruments"&gt;Instruments&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt; | &lt;a href="#Practical Theory / Books"&gt;Practical Theory / Books&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt; | &lt;a href="#External links"&gt;External links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;hr /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Theory&lt;/h1&gt;
-  &lt;br /&gt;
+  The 24edo system divides the octave into 24 equal parts of exactly 50 cents each. It is also known as &lt;strong&gt;quarter-tone tuning&lt;/strong&gt;, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes, and its use in theory and occasionally in practice in Arabic music.&lt;br /&gt;
-The 24-tone equal temperament is also known as &lt;strong&gt;quarter-tone tuning&lt;/strong&gt;, since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones and since it is the smallest microtonal equal temperament that contains all the 12 notes.&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="Theory-As a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;As a temperament&lt;/h2&gt;
+ The &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit&lt;/a&gt; approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.&lt;br /&gt;
+&lt;br /&gt;
+The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 2.3.125.35.11.325.17 &lt;a class="wiki_link" href="/just%20intonation%20subgroup"&gt;just intonation subgroup&lt;/a&gt;, making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17.&lt;br /&gt;
+&lt;br /&gt;
+The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):&lt;br /&gt;
+&lt;br /&gt;
+-4-8-11-14 (&amp;quot;major&amp;quot; chord with a 9:8 and a 11:8 above the root)&lt;br /&gt;
+Its inversion, 0-3-6-10-14 (&amp;quot;minor&amp;quot;)&lt;br /&gt;
+-7-14 (&amp;quot;neutral&amp;quot;)&lt;br /&gt;
+-5-10 (another kind of &amp;quot;neutral&amp;quot;, splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)&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="Theory-Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Intervals&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals&lt;/h1&gt;
-  &lt;br /&gt;
+  The twelve new intervals in quartertones are:&lt;br /&gt;
-The twelve new intervals in quartertones are:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 458: / Line 464: @@
 &lt;/table&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Theory-As a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;As a temperament&lt;/h2&gt;
- &lt;br /&gt;
-The &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit&lt;/a&gt; approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.&lt;br /&gt;
-&lt;br /&gt;
-The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):&lt;br /&gt;
-&lt;br /&gt;
--4-8-11-14 (&amp;quot;major&amp;quot; chord with a 9:8 and a 11:8 above the root)&lt;br /&gt;
-Its inversion, 0-3-6-10-14 (&amp;quot;minor&amp;quot;)&lt;br /&gt;
--7-14 (&amp;quot;neutral&amp;quot;)&lt;br /&gt;
--5-10 (another kind of &amp;quot;neutral&amp;quot;, splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24, that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24, a heptatonic scale close to several Arabic scales.)&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Notation&lt;/h1&gt;
-  &lt;br /&gt;
+  &lt;!-- ws:start:WikiTextRemoteImageRule:786:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/s14_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/s14_blue.gif" alt="external image s14_blue.gif" title="external image s14_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:786 --&gt; = quarter-tone sharp&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:786:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/s14_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/s14_blue.gif" alt="external image s14_blue.gif" title="external image s14_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:786 --&gt; = quarter-tone sharp&lt;br /&gt;
 &lt;!-- ws:start:WikiTextRemoteImageRule:787:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/s34_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/s34_blue.gif" alt="external image s34_blue.gif" title="external image s34_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:787 --&gt; = three-quarter-tone sharp&lt;br /&gt;
 &lt;!-- ws:start:WikiTextRemoteImageRule:788:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/f14_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/f14_blue.gif" alt="external image f14_blue.gif" title="external image f14_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:788 --&gt; = quarter-tone flat&lt;br /&gt;
@@ Line 1,081: / Line 1,075: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Instruments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Instruments&lt;!-- ws:start:WikiTextAnchorRule:32:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@instruments&amp;quot; title=&amp;quot;Anchor: instruments&amp;quot;/&amp;gt; --&gt;&lt;a name="instruments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:32 --&gt;&lt;/h1&gt;
-  &lt;br /&gt;
+  The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This &lt;a class="wiki_link" href="/Microtonal%20Keyboards#twelvenoteoctavescales"&gt;&amp;quot;12 note octave scales&amp;quot;&lt;/a&gt; approach is used in a wide part of the existing literature - see below.&lt;br /&gt;
-The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This &lt;a class="wiki_link" href="/Microtonal%20Keyboards#twelvenoteoctavescales"&gt;&amp;quot;12 note octave scales&amp;quot;&lt;/a&gt; approach is used in a wide part of the existing literature - see below.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextRemoteImageRule:799:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/Sword_quartertone_stratsm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 195px; width: 406px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/Sword_quartertone_stratsm.jpg" alt="external image Sword_quartertone_stratsm.jpg" title="external image Sword_quartertone_stratsm.jpg" style="height: 195px; width: 406px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:799 --&gt;24-tone &amp;quot;1/4-tone&amp;quot; Guitar by Ron Sword / Sword guitars&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Compositions&lt;/h1&gt;
-  &lt;br /&gt;
+  &amp;quot;Prométhée enchaîné&amp;quot; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy" rel="nofollow"&gt;Fromental Halévy&lt;/a&gt; (considered the first mainstream western orchestral composition to use quarter tones.)&lt;br /&gt;
-&amp;quot;Prométhée enchaîné&amp;quot; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy" rel="nofollow"&gt;Fromental Halévy&lt;/a&gt; (considered the first mainstream western orchestral composition to use quarter tones.)&lt;br /&gt;
 &amp;quot;3 Hommages&amp;quot; by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Georg_Friedrich_Haas" rel="nofollow"&gt;Georg Friedrich Haas&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.96edo.com/My_Music.html" rel="nofollow"&gt;Some microtonal music in 24-EDO&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.96edo.com/About_me.html" rel="nofollow"&gt;Shaahin Mohajeri&lt;/a&gt;&lt;br /&gt;