24edo

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Revision as of 12:43, 28 October 2010 by Wikispaces>Andrew_Heathwaite (**Imported revision 174473737 - Original comment: table with nearby JI intervals**)
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This revision was by author Andrew_Heathwaite and made on 2010-10-28 12:43:29 UTC.
The original revision id was 174473737.
The revision comment was: table with nearby JI intervals

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[[toc|flat]]
----
=Theory= 

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.

==Intervals== 

The twelve new intervals in quartertones are:

||||~ The twelve new intervals in 24edo ||||||~ some nearby JI intervals ||
||~ cents ||~ common name ||~ frequency ratio ||~ cents ||~ common name ||
||= 50 ||= subminor second, quartertone ||= [[36_35|36/35]]
[[35_34|35/34]]
[[34_33|34/33]]
[[33_32|33/32]] ||= 48.770
50.184
51.682
53.273 ||= large septimal quarter-tone (Archytas)
large 17-limit quartertone
small 17-limit quartertone
33rd harmonic ||
||= 150 ||= neutral second ||= [[12_11|12/11]] ||= 150.637 ||= large undecimal neutral second ||
||= 250 ||= supermajor second, subminor third ||= [[144|125|144/125]]
[[15_13|15/13]]
[[52_45|52/45]] ||= 244.969
247.741
250.304 ||= diminished third (6/5 x 24/25) ||
||= 350 ||= neutral third ||= [[11_9|11/9]]
[[27_22|27/22]]
[[16_13|16/13]] ||= 347.408
354.547
359.472 ||= undecimal neutral third

tridecimal neutral third ||
||= 450 ||= minor fourth, supermajor third, subfourth ||= [[22_17|22/17]]
[[35_27|35/27]]
[[13_10|13/10]] ||= 446.363
449.275
454.214 ||= 17-limit supermajor third

tridecimal subfourth ||
||= 550 ||= major fourth, superfourth ||= [[11_8|11/8]] ||= 551.318 ||= undecimal superfourth, harmonic 11th ||
||= 650 ||= minor fifth, subfifth ||= [[16_11|16/11]] ||= 648.682 ||= undecimal subfifth, 11th subharmonic ||
||= 750 ||= major fifth, subminor sixth, superfifth ||= [[20_13|20/13]]
[[54_35|54/35]]
[[17_11|17/11]] ||= 745.786
750.725
753.637 ||= tridecimal superfifth

17-limit subminor sixth ||
||= 850 ||= neutral sixth ||= [[13_8|13/8]]
[[44_27|44/27]]
[[18_11|18/11]] ||= 840.528
845.453
852.592 ||= overtone sixth, 13th harmonic

undecimal neutral sixth ||
||= 950 ||= supermajor sixth, subminor seventh ||= [[45_26|45/26]]
[[26_15|26/15]]
[[125_72|125/72]] ||= 949.696
952.259
955.031 ||=   ||
||= 1050 ||= neutral seventh ||= [[11_6|11/6]] ||= 1049.363 ||= undecimal neutral seventh ||
||= 1150 ||= minor octave, supermajor seventh ||= [[31_16|31/16]]
[[33_17|33/17]]
[[35_18|35/18]] ||= 1145.036
1148.318
1151.230 ||= 31st harmonic ||

==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
[[image:http://www.wfg.woodwind.org/n/f14_blue.gif]] = quarter-tone flat
[[image:http://www.wfg.woodwind.org/n/f34_blue.gif]] = three-quarter-tone flat

Alternately, in another notation system,
[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] can be used as a quarter-tone sharp,
††† (the horizontal line should connect all three vertical lines) can be used as a three-quartertone sharp,
[[image:http://www.dolmetsch.com/onequarterflat.gif]] = quartertone flat,
[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]] can be used as a three-quarter-tone flat.
For example, the scale 0-5-10-15-20 is written as C-D[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] (or E[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]]) F G[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] (or A[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]]) Bb.
And in persian music:
<span style="display: block; text-align: center;">[[image:http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg caption="Sori,Koron" link="24edo"]]
</span>
=Scales / Modes= 

|||| ====24 tone modes:==== ||
|| {{**1 1 8 1 1 8 4**}} || Enharmonic Mixolydian ||
|| {{**1 8 1 1 8 4 1**}} || Enharmonic Lydian ||
|| {{**8 1 1 8 4 1 1**}} || Enharmonic Phrygian ||
|| {{**1 1 8 4 1 1 8**}} || Enharmonic Dorian ||
|| {{**1 8 4 1 1 8 1**}} || Enharmonic Hypolydian ||
|| {{**8 4 1 1 8 1 1**}} || Enharmonic Hypophrygian ||
|| {{**4 1 1 8 1 1 8**}} || Enharmonic Hypodorian ||
|| {{**2 3 5 2 3 5 4**}} || Soft Diatonic Mixolydian ||
|| {{**3 5 2 3 5 4 2**}} || Soft Diatonic Lydian ||
|| {{**5 2 3 5 4 2 3**}} || Soft Diatonic Phrygian ||
|| {{**2 3 5 4 2 3 5**}} || Soft Diatonic Dorian ||
|| {{**3 5 4 2 3 5 2**}} || Soft Diatonic Hypolydian ||
|| {{**5 4 2 3 5 2 3**}} || Soft Diatonic Hypophrygian ||
|| {{**4 2 3 5 2 3 5**}} || Soft Diatonic Hypodorian ||
|| {{**3 3 4 3 3 4 4**}} || Maqam Ouchairan-Hussaini, Bayatan, Neutral Diatonic Mixolydian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 4 3 3 4 4 3**}} || Dastgah-e Sehgah, Neutral Diatonic Lydian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 3 3 4 4 3 3**}} || Arabic Diatonic, Maqam Rast, Quasi-equal Heptatonic, Neutral Diatonic Phrygian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 3 4 4 3 3 4**}} || Maqam Hussaini, Ushaq, Neutral Diatonic Dorian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 4 4 3 3 4 3**}} || Maqam Sikah (Segah), Neutral Diatonic Hypolydian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 4 3 3 4 3 3**}} || Neutral Diatonic Hypophrygian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 3 3 4 3 3 4**}} || Miha'il Musaqa's mode: Egypt, Neutral Diatonic Hypodorian, Dastgah-e Sehgah, Maqam Nairuz - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**1 5 4 1 5 4 4**}} || Diatonic + Enharmonic Diesis Mixolydian ||
|| {{**5 4 1 5 4 4 1**}} || Diatonic + Enharmonic Diesis Lydian ||
|| {{**4 1 5 4 4 1 5**}} || Diatonic + Enharmonic Diesis Phrygian ||
|| {{**1 5 4 4 1 5 4**}} || Diatonic + Enharmonic Diesis Dorian ||
|| {{**5 4 4 1 5 4 1**}} || Diatonic + Enharmonic Diesis Hypolydian ||
|| {{**4 4 1 5 4 1 5**}} || Diatonic + Enharmonic Diesis Hypophrygian ||
|| {{**4 1 5 4 1 5 4**}} || Diatonic + Enharmonic Diesis Hypodorian ||
|| {{**1 3 6 1 3 6 4**}} || Chromatic/Enharmonic Mixolydian ||
|| {{**3 6 1 3 6 4 1**}} || Chromatic/Enharmonic Lydian ||
|| {{**6 1 3 6 4 1 3**}} || Chromatic/Enharmonic Phrygian ||
|| {{**1 3 6 4 1 3 6**}} || Chromatic/Enharmonic Dorian ||
|| {{**3 6 4 1 3 6 1**}} || Chromatic/Enharmonic Hypolydian ||
|| {{**6 4 1 3 6 1 3**}} || Chromatic/Enharmonic Hypophrygian ||
|| {{**4 1 3 6 1 3 6**}} || Chromatic/Enharmonic Hypodorian ||
|| {{**3 4 3 3 4 3 4**}} || Neutral Mixolydian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 3 3 4 3 4 3**}} || Neutral Lydian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 3 4 3 4 3 4**}} || Neutral Phrygian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 4 3 4 3 4 3**}} || Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 3 4 3 4 3 3**}} || Neutral Hypolydian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 4 3 4 3 3 4**}} || Neutral Hypophrygian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 3 4 3 3 4 3**}} || Neutral Hypodorian - [[MOSScales|MOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 5 2 4 3 5 2**}} || Athanasopoulos' Byzantine Liturgical Chromatic, Dastgah-e Chahargah ||
|| {{**4 2 4 4 3 3 4**}} || Dastgah-e Nava, Maqam Ushaq Masri ||
|| {{**2 7 1 4 2 7 1**}} || Second plagal Byzantine Liturgical mode ||
|| {{**5 5 4 5 5**}} || Quasi-equal Pentatonic - [[MOSScales|MOS]] of type [[4L 1s|4L 1s (bug)]] ||
|| {{**3 3 4 4 2 4 4**}} || Maqam 'Ushshaq Turki, Urfa, Isfahan, Dastgah-e Shur ||
|| {{**3 3 4 4 2 1 3 4**}} || Maqam Bayati ||
|| {{**3 3 4 4 4 2 4**}} || Maqam Nahfat ||
|| {{**3 3 2 6 2 4 4**}} || Maqam Saba ||
|| {{**3 3 2 6 2 4 2 2**}} || Maqam Saba ||
|| {{**3 3 2 6 2 6 2**}} || Maqam Sabr Jadid ||
|| {{**3 3 2 2 4 3 3 4**}} || Maqam Mansuri ||
|| {{**4 3 3 4 2 6 2**}} || Maqam Suznak (Soznak) ||
|| {{**4 3 3 4 4 2 1 3**}} || Maqam Rast, Dilkashidah, Dilnishin ||
|| {{**4 3 3 4 4 4 2**}} || Maqam Mahur ||
|| {{**4 3 1 2 1 3 4 2 1 3**}} || Maqam Yakah ||
|| {{**4 3 1 2 4 4 2 4**}} || Maqam Suzidil 'ara ||
|| {{**3 3 4 2 6 2 4**}} || Maqam Qarjighar, Bayati Shuri ||
|| {{**3 4 2 2 2 2 2 4 3**}} || Maqam Huzam ||
|| {{**3 4 2 6 2 4 3**}} || Maqam Hizam (Huzam, El Houzam), Rahat al Arouah ||
|| {{**3 4 2 6 2 4 2 1**}} || Maqam Rahat al-Arwah ||
|| {{**3 4 3 3 4 4 2 1**}} || Maqam Iraq ||
|| {{**2 6 2 4 2 1 3 4**}} || Maqam Hijaz ||
|| {{**3 4 4 2 1 3 4 3**}} || Maqam Musta'ar ||
|| {{**3 4 4 4 2 4 2 1**}} || Maqam Farahnak ||
|| {{**3 4 3 3 2 6 2 1**}} || Maqam Bastanikar, Tarz Nuin ||
|| {{**4 2 6 2 4 2 1 3**}} || Maqam Farah Faza, Maqam Nakriz ||
|| {{**4 2 1 3 2 2 2 2 3 1 2**}} || Maqam Hayyan ||
|| {{**4 2 1 3 2 2 4 2 1 3**}} || Maqam Basandida ||
|| {{**4 4 2 1 3 2 2 4 2**}} || Maqam Shawq Afza ||
|| {{**2 1 3 2 2 4 2 4 2 2**}} || Maqam Shawq Tarab ||
|| {{**3 1 2 4 4 2 4 4**}} || Maqam Jabburi ||
|| {{**2 4 4 4 3 3 4**}} || Maqam Nawa ||
|| {{**2 5 3 4 2 5 3**}} || Maqam Higaz-kar ||
|| {{**3 4 4 2 4 4 3**}} || Maqam Su'ar, Naghmeh Abuata, Naghmeh Afshari ||
|| {{**4 4 2 4 4 3 3**}} || Maqam Jahargah (Jiharkah), Naghmeh Bayat-e Tork, Naghmeh Dashti ||
|| {{**3 5 2 4 2 4 4**}} || Dastgah-e Homayun ||
|| {{**4 2 4 4 3 5 2**}} || Naghmeh Esfahan ||
|| {{**3 6 1 5 2 6 1**}} || Maqam 'Awg 'ara (Aug-ara) ||
|| {{**4 1 5 4 2 6 2**}} || Maqam Buselik ||
|| {{**4 2 6 2 2 5 3**}} || Maqam Neuter ||
|| {{**2 8 3 6 5**}} || Anchihoye: Ethiopia ||
|| {{**4 3 3 4 4 2 4**}} || Dance scale of Yi people: China ||
|| {{**4 4 2 3 1 4 6**}} || Daniel-mode of Spanish-Arab Jews ||
|| {{**1 2 2 2 2 2 2 1 2 2 2 2 2**}} || de Vries 13-tone - [[MOSScales|MOS]] of type [[11L 2s]] ||
|| {{**4 1 4 1 4 1 4 1 4**}} || de Vries 9-tone - [[MOSScales|MOS]] of type [[5L 4s|5L 4s (unfair bug)]] ||
|| {{**9 2 2 9 2**}} || de Vries 5-tone - [[MOSScales|MOS]] of type [[2L 3s|2L 3s (classic pentatonic)]] ||
|| {{**3 7 4 3 7**}} || Spondeion ||
|| {{**1 1 8 4 2 8**}} || Spondeiakos ||
|| {{**1 2 3 4 4 1 2 3 4**}} || Progressive Enneatonic ||
|| {{**2 3 2 3 2 2 3 2 3 2**}} || Young Decatonic - [[MOSScales|MOS]] of type [[4L 5s|4L 5s (fair bug)]] ||
|| {{**2 2 1 2 2 2 1 2 2 1 2 2 2 1**}} || Young Half-Octave Diatonic - [[MOSScales|MOS]] of type [[10L 4s]] ||
|| {{**3 1 3 3 3 1 3 3 1 3**}} || Breed Decatonic - [[MOSScales|MOS]] of type [[7L 3s|7L 3s (unfair mosh)]] ||
|| {{**2 3 2 2 3 2 3 2 2 3**}} || Oljare Decatonic - [[MOSScales|MOS]] of type [[4L 6s|4L 6s (fair bicycle)]] ||
|| {{**2 2 2 2 2 1 2 2 2 2 2 2 1**}} || Agmon Diatonic DS5 - [[MOSScales|MOS]] of type [[11L 2s]] ||
|| **<span style="font-family: 'Courier New',Courier,monospace;">4 4 2 4 1 3 4 2</span>** || Giancarlo Dalmonte's octatonic scale (see [[http://www.ottavanota.info/|http://www.ottavanota.info]]) ||

=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]]
[[http://www.96edo.com/My_Music.html|Some microtonal music in 24-EDO]] by [[http://www.96edo.com/About_me.html|Shaahin Mohajeri]]
[[http://en.wikipedia.org/wiki/List_of_quarter_tone_pieces|List of quartertone pieces on Wikipedia]]

=Practical Theory / Books= 

[[image:http://ronsword.com/images/24_tet_Coversm.jpg width="179" height="236"]]**"Icosakaiteraphonic Scales for Guitar"** - A Book for Twenty-Four Equal Divisions of the Octave on guitar, or 'Quarter-tones'. Features a practical approach to understanding the tuning, and over 550 Scale Examples on Nine-String finger board charts, which allows for both symmetrical tuning visualization and standard guitar tuning- helpful for bassists and large range guitarists as well. Includes MOS, DE, and *all* the Scales / Modes from the list above.

=External links= 
[[http://tonalsoft.com/enc/q/quarter-tone.aspx|quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory]]
[[http://www.96edo.com/24_EDO.html|About 24-EDO]]

Original HTML content:

<html><head><title>24edo</title></head><body><!-- ws:start:WikiTextTocRule:20:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --><a href="#Theory">Theory</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Notation">Notation</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Scales / Modes">Scales / Modes</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Instruments">Instruments</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: -->
<!-- ws:end:WikiTextTocRule:31 --><hr />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Theory"></a><!-- ws:end:WikiTextHeadingRule:0 -->Theory</h1>
 <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 />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Theory-Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h2>
 <br />
The twelve new intervals in quartertones are:<br />
<br />


<table class="wiki_table">
    <tr>
        <th colspan="2">The twelve new intervals in 24edo<br />
</th>
        <th colspan="3">some nearby JI intervals<br />
</th>
    </tr>
    <tr>
        <th>cents<br />
</th>
        <th>common name<br />
</th>
        <th>frequency ratio<br />
</th>
        <th>cents<br />
</th>
        <th>common name<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">50<br />
</td>
        <td style="text-align: center;">subminor second, quartertone<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/36_35">36/35</a><br />
<a class="wiki_link" href="/35_34">35/34</a><br />
<a class="wiki_link" href="/34_33">34/33</a><br />
<a class="wiki_link" href="/33_32">33/32</a><br />
</td>
        <td style="text-align: center;">48.770<br />
50.184<br />
51.682<br />
53.273<br />
</td>
        <td style="text-align: center;">large septimal quarter-tone (Archytas)<br />
large 17-limit quartertone<br />
small 17-limit quartertone<br />
33rd harmonic<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">150<br />
</td>
        <td style="text-align: center;">neutral second<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/12_11">12/11</a><br />
</td>
        <td style="text-align: center;">150.637<br />
</td>
        <td style="text-align: center;">large undecimal neutral second<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">250<br />
</td>
        <td style="text-align: center;">supermajor second, subminor third<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/144">125|144/125</a><br />
<a class="wiki_link" href="/15_13">15/13</a><br />
<a class="wiki_link" href="/52_45">52/45</a><br />
</td>
        <td style="text-align: center;">244.969<br />
247.741<br />
250.304<br />
</td>
        <td style="text-align: center;">diminished third (6/5 x 24/25)<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">350<br />
</td>
        <td style="text-align: center;">neutral third<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a><br />
<a class="wiki_link" href="/27_22">27/22</a><br />
<a class="wiki_link" href="/16_13">16/13</a><br />
</td>
        <td style="text-align: center;">347.408<br />
354.547<br />
359.472<br />
</td>
        <td style="text-align: center;">undecimal neutral third<br />
<br />
tridecimal neutral third<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">450<br />
</td>
        <td style="text-align: center;">minor fourth, supermajor third, subfourth<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/22_17">22/17</a><br />
<a class="wiki_link" href="/35_27">35/27</a><br />
<a class="wiki_link" href="/13_10">13/10</a><br />
</td>
        <td style="text-align: center;">446.363<br />
449.275<br />
454.214<br />
</td>
        <td style="text-align: center;">17-limit supermajor third<br />
<br />
tridecimal subfourth<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">550<br />
</td>
        <td style="text-align: center;">major fourth, superfourth<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a><br />
</td>
        <td style="text-align: center;">551.318<br />
</td>
        <td style="text-align: center;">undecimal superfourth, harmonic 11th<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">650<br />
</td>
        <td style="text-align: center;">minor fifth, subfifth<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/16_11">16/11</a><br />
</td>
        <td style="text-align: center;">648.682<br />
</td>
        <td style="text-align: center;">undecimal subfifth, 11th subharmonic<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">750<br />
</td>
        <td style="text-align: center;">major fifth, subminor sixth, superfifth<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/20_13">20/13</a><br />
<a class="wiki_link" href="/54_35">54/35</a><br />
<a class="wiki_link" href="/17_11">17/11</a><br />
</td>
        <td style="text-align: center;">745.786<br />
750.725<br />
753.637<br />
</td>
        <td style="text-align: center;">tridecimal superfifth<br />
<br />
17-limit subminor sixth<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">850<br />
</td>
        <td style="text-align: center;">neutral sixth<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/13_8">13/8</a><br />
<a class="wiki_link" href="/44_27">44/27</a><br />
<a class="wiki_link" href="/18_11">18/11</a><br />
</td>
        <td style="text-align: center;">840.528<br />
845.453<br />
852.592<br />
</td>
        <td style="text-align: center;">overtone sixth, 13th harmonic<br />
<br />
undecimal neutral sixth<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">950<br />
</td>
        <td style="text-align: center;">supermajor sixth, subminor seventh<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/45_26">45/26</a><br />
<a class="wiki_link" href="/26_15">26/15</a><br />
<a class="wiki_link" href="/125_72">125/72</a><br />
</td>
        <td style="text-align: center;">949.696<br />
952.259<br />
955.031<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1050<br />
</td>
        <td style="text-align: center;">neutral seventh<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/11_6">11/6</a><br />
</td>
        <td style="text-align: center;">1049.363<br />
</td>
        <td style="text-align: center;">undecimal neutral seventh<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1150<br />
</td>
        <td style="text-align: center;">minor octave, supermajor seventh<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/31_16">31/16</a><br />
<a class="wiki_link" href="/33_17">33/17</a><br />
<a class="wiki_link" href="/35_18">35/18</a><br />
</td>
        <td style="text-align: center;">1145.036<br />
1148.318<br />
1151.230<br />
</td>
        <td style="text-align: center;">31st harmonic<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Theory-As a temperament"></a><!-- ws:end:WikiTextHeadingRule:4 -->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 />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:6 -->Notation</h1>
 <br />
<!-- ws:start:WikiTextRemoteImageRule:786:&lt;img src=&quot;http://www.wfg.woodwind.org/n/s14_blue.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://www.wfg.woodwind.org/n/s14_blue.gif" alt="external image s14_blue.gif" title="external image s14_blue.gif" /><!-- ws:end:WikiTextRemoteImageRule:786 --> = quarter-tone sharp<br />
<!-- ws:start:WikiTextRemoteImageRule:787:&lt;img src=&quot;http://www.wfg.woodwind.org/n/s34_blue.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://www.wfg.woodwind.org/n/s34_blue.gif" alt="external image s34_blue.gif" title="external image s34_blue.gif" /><!-- ws:end:WikiTextRemoteImageRule:787 --> = three-quarter-tone sharp<br />
<!-- ws:start:WikiTextRemoteImageRule:788:&lt;img src=&quot;http://www.wfg.woodwind.org/n/f14_blue.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://www.wfg.woodwind.org/n/f14_blue.gif" alt="external image f14_blue.gif" title="external image f14_blue.gif" /><!-- ws:end:WikiTextRemoteImageRule:788 --> = quarter-tone flat<br />
<!-- ws:start:WikiTextRemoteImageRule:789:&lt;img src=&quot;http://www.wfg.woodwind.org/n/f34_blue.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://www.wfg.woodwind.org/n/f34_blue.gif" alt="external image f34_blue.gif" title="external image f34_blue.gif" /><!-- ws:end:WikiTextRemoteImageRule:789 --> = three-quarter-tone flat<br />
<br />
Alternately, in another notation system,<br />
<!-- ws:start:WikiTextRemoteImageRule:790:&lt;img src=&quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /><!-- ws:end:WikiTextRemoteImageRule:790 --> can be used as a quarter-tone sharp,<br />
††† (the horizontal line should connect all three vertical lines) can be used as a three-quartertone sharp,<br />
<!-- ws:start:WikiTextRemoteImageRule:791:&lt;img src=&quot;http://www.dolmetsch.com/onequarterflat.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://www.dolmetsch.com/onequarterflat.gif" alt="external image onequarterflat.gif" title="external image onequarterflat.gif" /><!-- ws:end:WikiTextRemoteImageRule:791 --> = quartertone flat,<br />
<!-- ws:start:WikiTextRemoteImageRule:792:&lt;img src=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png" alt="external image 9px-Arabic_music_notation_half_flat.svg.png" title="external image 9px-Arabic_music_notation_half_flat.svg.png" /><!-- ws:end:WikiTextRemoteImageRule:792 --> can be used as a three-quarter-tone flat.<br />
For example, the scale 0-5-10-15-20 is written as C-D<!-- ws:start:WikiTextRemoteImageRule:793:&lt;img src=&quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /><!-- ws:end:WikiTextRemoteImageRule:793 --> (or E<!-- ws:start:WikiTextRemoteImageRule:794:&lt;img src=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png" alt="external image 9px-Arabic_music_notation_half_flat.svg.png" title="external image 9px-Arabic_music_notation_half_flat.svg.png" /><!-- ws:end:WikiTextRemoteImageRule:794 -->) F G<!-- ws:start:WikiTextRemoteImageRule:795:&lt;img src=&quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /><!-- ws:end:WikiTextRemoteImageRule:795 --> (or A<!-- ws:start:WikiTextRemoteImageRule:796:&lt;img src=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png" alt="external image 9px-Arabic_music_notation_half_flat.svg.png" title="external image 9px-Arabic_music_notation_half_flat.svg.png" /><!-- ws:end:WikiTextRemoteImageRule:796 -->) Bb.<br />
And in persian music:<br />
<span style="display: block; text-align: center;"><!-- ws:start:WikiTextRemoteImageRule:798:&lt;a href=&quot;/24edo&quot;&gt;&lt;img src=&quot;http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg&quot; alt=&quot;Sori,Koron&quot; title=&quot;Sori,Koron&quot; /&gt;&lt;/a&gt; --><table class="captionBox"><tr><td class="captionedImage"><a href="/24edo"><img src="http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg" alt="Sori,Koron" title="Sori,Koron" /></a></td></tr><tr><td class="imageCaption">Sori,Koron</td></tr></table><!-- ws:end:WikiTextRemoteImageRule:798 --><br />
</span><br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Scales / Modes"></a><!-- ws:end:WikiTextHeadingRule:8 -->Scales / Modes</h1>
 <br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:10:&lt;h4&gt; --><h4 id="toc5"><a name="Scales / Modes---24 tone modes:"></a><!-- ws:end:WikiTextHeadingRule:10 -->24 tone modes:</h4>
</td>
    </tr>
    <tr>
        <td><tt><strong>1 1 8 1 1 8 4</strong></tt><br />
</td>
        <td>Enharmonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 8 1 1 8 4 1</strong></tt><br />
</td>
        <td>Enharmonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 1 1 8 4 1 1</strong></tt><br />
</td>
        <td>Enharmonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 1 8 4 1 1 8</strong></tt><br />
</td>
        <td>Enharmonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 8 4 1 1 8 1</strong></tt><br />
</td>
        <td>Enharmonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 4 1 1 8 1 1</strong></tt><br />
</td>
        <td>Enharmonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 1 8 1 1 8</strong></tt><br />
</td>
        <td>Enharmonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 5 2 3 5 4</strong></tt><br />
</td>
        <td>Soft Diatonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 2 3 5 4 2</strong></tt><br />
</td>
        <td>Soft Diatonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 3 5 4 2 3</strong></tt><br />
</td>
        <td>Soft Diatonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 5 4 2 3 5</strong></tt><br />
</td>
        <td>Soft Diatonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 4 2 3 5 2</strong></tt><br />
</td>
        <td>Soft Diatonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 2 3 5 2 3</strong></tt><br />
</td>
        <td>Soft Diatonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 3 5 2 3 5</strong></tt><br />
</td>
        <td>Soft Diatonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 3 3 4 4</strong></tt><br />
</td>
        <td>Maqam Ouchairan-Hussaini, Bayatan, Neutral Diatonic Mixolydian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 4 4 3</strong></tt><br />
</td>
        <td>Dastgah-e Sehgah, Neutral Diatonic Lydian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 3 4 4 3 3</strong></tt><br />
</td>
        <td>Arabic Diatonic, Maqam Rast, Quasi-equal Heptatonic, Neutral Diatonic Phrygian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 4 3 3 4</strong></tt><br />
</td>
        <td>Maqam Hussaini, Ushaq, Neutral Diatonic Dorian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 4 3 3 4 3</strong></tt><br />
</td>
        <td>Maqam Sikah (Segah), Neutral Diatonic Hypolydian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 3 3 4 3 3</strong></tt><br />
</td>
        <td>Neutral Diatonic Hypophrygian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 3 4 3 3 4</strong></tt><br />
</td>
        <td>Miha'il Musaqa's mode: Egypt, Neutral Diatonic Hypodorian, Dastgah-e Sehgah, Maqam Nairuz - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 5 4 1 5 4 4</strong></tt><br />
</td>
        <td>Diatonic + Enharmonic Diesis Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 1 5 4 4 1</strong></tt><br />
</td>
        <td>Diatonic + Enharmonic Diesis Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 5 4 4 1 5</strong></tt><br />
</td>
        <td>Diatonic + Enharmonic Diesis Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 5 4 4 1 5 4</strong></tt><br />
</td>
        <td>Diatonic + Enharmonic Diesis Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 4 1 5 4 1</strong></tt><br />
</td>
        <td>Diatonic + Enharmonic Diesis Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 1 5 4 1 5</strong></tt><br />
</td>
        <td>Diatonic + Enharmonic Diesis Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 5 4 1 5 4</strong></tt><br />
</td>
        <td>Diatonic + Enharmonic Diesis Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 3 6 1 3 6 4</strong></tt><br />
</td>
        <td>Chromatic/Enharmonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 6 1 3 6 4 1</strong></tt><br />
</td>
        <td>Chromatic/Enharmonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 1 3 6 4 1 3</strong></tt><br />
</td>
        <td>Chromatic/Enharmonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 3 6 4 1 3 6</strong></tt><br />
</td>
        <td>Chromatic/Enharmonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 6 4 1 3 6 1</strong></tt><br />
</td>
        <td>Chromatic/Enharmonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 4 1 3 6 1 3</strong></tt><br />
</td>
        <td>Chromatic/Enharmonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 3 6 1 3 6</strong></tt><br />
</td>
        <td>Chromatic/Enharmonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 4 3 4</strong></tt><br />
</td>
        <td>Neutral Mixolydian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 3 4 3 4 3</strong></tt><br />
</td>
        <td>Neutral Lydian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 3 4 3 4</strong></tt><br />
</td>
        <td>Neutral Phrygian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 4 3 4 3</strong></tt><br />
</td>
        <td>Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 4 3 4 3 3</strong></tt><br />
</td>
        <td>Neutral Hypolydian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 4 3 3 4</strong></tt><br />
</td>
        <td>Neutral Hypophrygian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 4 3 3 4 3</strong></tt><br />
</td>
        <td>Neutral Hypodorian - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/3L%204s">3L 4s (mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 2 4 3 5 2</strong></tt><br />
</td>
        <td>Athanasopoulos' Byzantine Liturgical Chromatic, Dastgah-e Chahargah<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 4 4 3 3 4</strong></tt><br />
</td>
        <td>Dastgah-e Nava, Maqam Ushaq Masri<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 7 1 4 2 7 1</strong></tt><br />
</td>
        <td>Second plagal Byzantine Liturgical mode<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 4 5 5</strong></tt><br />
</td>
        <td>Quasi-equal Pentatonic - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%201s">4L 1s (bug)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 4 2 4 4</strong></tt><br />
</td>
        <td>Maqam 'Ushshaq Turki, Urfa, Isfahan, Dastgah-e Shur<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 4 2 1 3 4</strong></tt><br />
</td>
        <td>Maqam Bayati<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 4 4 2 4</strong></tt><br />
</td>
        <td>Maqam Nahfat<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 2 6 2 4 4</strong></tt><br />
</td>
        <td>Maqam Saba<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 2 6 2 4 2 2</strong></tt><br />
</td>
        <td>Maqam Saba<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 2 6 2 6 2</strong></tt><br />
</td>
        <td>Maqam Sabr Jadid<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 2 2 4 3 3 4</strong></tt><br />
</td>
        <td>Maqam Mansuri<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 3 4 2 6 2</strong></tt><br />
</td>
        <td>Maqam Suznak (Soznak)<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 3 4 4 2 1 3</strong></tt><br />
</td>
        <td>Maqam Rast, Dilkashidah, Dilnishin<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 3 4 4 4 2</strong></tt><br />
</td>
        <td>Maqam Mahur<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 1 2 1 3 4 2 1 3</strong></tt><br />
</td>
        <td>Maqam Yakah<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 1 2 4 4 2 4</strong></tt><br />
</td>
        <td>Maqam Suzidil 'ara<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 2 6 2 4</strong></tt><br />
</td>
        <td>Maqam Qarjighar, Bayati Shuri<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 2 2 2 2 2 4 3</strong></tt><br />
</td>
        <td>Maqam Huzam<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 2 6 2 4 3</strong></tt><br />
</td>
        <td>Maqam Hizam (Huzam, El Houzam), Rahat al Arouah<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 2 6 2 4 2 1</strong></tt><br />
</td>
        <td>Maqam Rahat al-Arwah<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 4 4 2 1</strong></tt><br />
</td>
        <td>Maqam Iraq<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 6 2 4 2 1 3 4</strong></tt><br />
</td>
        <td>Maqam Hijaz<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 4 2 1 3 4 3</strong></tt><br />
</td>
        <td>Maqam Musta'ar<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 4 4 2 4 2 1</strong></tt><br />
</td>
        <td>Maqam Farahnak<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 2 6 2 1</strong></tt><br />
</td>
        <td>Maqam Bastanikar, Tarz Nuin<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 6 2 4 2 1 3</strong></tt><br />
</td>
        <td>Maqam Farah Faza, Maqam Nakriz<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 1 3 2 2 2 2 3 1 2</strong></tt><br />
</td>
        <td>Maqam Hayyan<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 1 3 2 2 4 2 1 3</strong></tt><br />
</td>
        <td>Maqam Basandida<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 2 1 3 2 2 4 2</strong></tt><br />
</td>
        <td>Maqam Shawq Afza<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 1 3 2 2 4 2 4 2 2</strong></tt><br />
</td>
        <td>Maqam Shawq Tarab<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 1 2 4 4 2 4 4</strong></tt><br />
</td>
        <td>Maqam Jabburi<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 4 4 4 3 3 4</strong></tt><br />
</td>
        <td>Maqam Nawa<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 5 3 4 2 5 3</strong></tt><br />
</td>
        <td>Maqam Higaz-kar<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 4 2 4 4 3</strong></tt><br />
</td>
        <td>Maqam Su'ar, Naghmeh Abuata, Naghmeh Afshari<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 2 4 4 3 3</strong></tt><br />
</td>
        <td>Maqam Jahargah (Jiharkah), Naghmeh Bayat-e Tork, Naghmeh Dashti<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 2 4 2 4 4</strong></tt><br />
</td>
        <td>Dastgah-e Homayun<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 4 4 3 5 2</strong></tt><br />
</td>
        <td>Naghmeh Esfahan<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 6 1 5 2 6 1</strong></tt><br />
</td>
        <td>Maqam 'Awg 'ara (Aug-ara)<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 5 4 2 6 2</strong></tt><br />
</td>
        <td>Maqam Buselik<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 6 2 2 5 3</strong></tt><br />
</td>
        <td>Maqam Neuter<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 8 3 6 5</strong></tt><br />
</td>
        <td>Anchihoye: Ethiopia<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 3 4 4 2 4</strong></tt><br />
</td>
        <td>Dance scale of Yi people: China<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 2 3 1 4 6</strong></tt><br />
</td>
        <td>Daniel-mode of Spanish-Arab Jews<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 2 2 2 2 2 2 1 2 2 2 2 2</strong></tt><br />
</td>
        <td>de Vries 13-tone - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/11L%202s">11L 2s</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 4 1 4 1 4 1 4</strong></tt><br />
</td>
        <td>de Vries 9-tone - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/5L%204s">5L 4s (unfair bug)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>9 2 2 9 2</strong></tt><br />
</td>
        <td>de Vries 5-tone - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/2L%203s">2L 3s (classic pentatonic)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 7 4 3 7</strong></tt><br />
</td>
        <td>Spondeion<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 1 8 4 2 8</strong></tt><br />
</td>
        <td>Spondeiakos<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 2 3 4 4 1 2 3 4</strong></tt><br />
</td>
        <td>Progressive Enneatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 2 3 2 2 3 2 3 2</strong></tt><br />
</td>
        <td>Young Decatonic - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%205s">4L 5s (fair bug)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 1 2 2 2 1 2 2 1 2 2 2 1</strong></tt><br />
</td>
        <td>Young Half-Octave Diatonic - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/10L%204s">10L 4s</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 1 3 3 3 1 3 3 1 3</strong></tt><br />
</td>
        <td>Breed Decatonic - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/7L%203s">7L 3s (unfair mosh)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 2 2 3 2 3 2 2 3</strong></tt><br />
</td>
        <td>Oljare Decatonic - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%206s">4L 6s (fair bicycle)</a><br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 2 2 2 1 2 2 2 2 2 2 1</strong></tt><br />
</td>
        <td>Agmon Diatonic DS5 - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/11L%202s">11L 2s</a><br />
</td>
    </tr>
    <tr>
        <td><strong><span style="font-family: 'Courier New',Courier,monospace;">4 4 2 4 1 3 4 2</span></strong><br />
</td>
        <td>Giancarlo Dalmonte's octatonic scale (see <a class="wiki_link_ext" href="http://www.ottavanota.info/" rel="nofollow">http://www.ottavanota.info</a>)<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:12 -->Instruments<!-- ws:start:WikiTextAnchorRule:32:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@instruments&quot; title=&quot;Anchor: instruments&quot;/&gt; --><a name="instruments"></a><!-- ws:end:WikiTextAnchorRule:32 --></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 />
<!-- ws:start:WikiTextRemoteImageRule:799:&lt;img src=&quot;http://www.swordguitars.com/Sword_quartertone_stratsm.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 195px; width: 406px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextRemoteImageRule:799 -->24-tone &quot;1/4-tone&quot; Guitar by Ron Sword / Sword guitars<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:14 -->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 />
<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/List_of_quarter_tone_pieces" rel="nofollow">List of quartertone pieces on Wikipedia</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:16 -->Practical Theory / Books</h1>
 <br />
<!-- ws:start:WikiTextRemoteImageRule:800:&lt;img src=&quot;http://ronsword.com/images/24_tet_Coversm.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 236px; width: 179px;&quot; /&gt; --><img src="http://ronsword.com/images/24_tet_Coversm.jpg" alt="external image 24_tet_Coversm.jpg" title="external image 24_tet_Coversm.jpg" style="height: 236px; width: 179px;" /><!-- ws:end:WikiTextRemoteImageRule:800 --><strong>&quot;Icosakaiteraphonic Scales for Guitar&quot;</strong> - A Book for Twenty-Four Equal Divisions of the Octave on guitar, or 'Quarter-tones'. Features a practical approach to understanding the tuning, and over 550 Scale Examples on Nine-String finger board charts, which allows for both symmetrical tuning visualization and standard guitar tuning- helpful for bassists and large range guitarists as well. Includes MOS, DE, and *all* the Scales / Modes from the list above.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:18 -->External links</h1>
 <a class="wiki_link_ext" href="http://tonalsoft.com/enc/q/quarter-tone.aspx" rel="nofollow">quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory</a><br />
<a class="wiki_link_ext" href="http://www.96edo.com/24_EDO.html" rel="nofollow">About 24-EDO</a></body></html>
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