24edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Wikispaces>TallKite
**Imported revision 599955646 - Original comment: **
Wikispaces>JosephRuhf
**Imported revision 601447262 - Original comment: **
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-21 06:16:48 UTC</tt>.<br>
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-05 14:50:46 UTC</tt>.<br>
: The original revision id was <tt>599955646</tt>.<br>
: The original revision id was <tt>601447262</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 84: Line 84:


===Commas===  
===Commas===  
24 EDO tempers out the following commas. (Note: This assumes val &lt; [[tel/24 38 56 67 83 89|24 38 56 67 83 89]] |.)
24 EDO tempers out the following commas. (Note: This assumes val &lt; 24 38 56 67 83 89/1 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||=  || | -19 12 &gt; ||&gt; 23.46 ||= Pythagorean Comma ||=  ||  ||
||=  || | -19 12 &gt; ||&gt; 23.46 ||= Pythagorean Comma ||=  ||  ||
Line 822: Line 822:
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x24edo as a temperament--Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Commas&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x24edo as a temperament--Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Commas&lt;/h3&gt;
  24 EDO tempers out the following commas. (Note: This assumes val &amp;lt; &lt;a class="wiki_link" href="http://tel.wikispaces.com/24%2038%2056%2067%2083%2089"&gt;24 38 56 67 83 89&lt;/a&gt; |.)&lt;br /&gt;
  24 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 24 38 56 67 83 89/1 |.)&lt;br /&gt;




Revision as of 14:50, 5 December 2016

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author JosephRuhf and made on 2016-12-05 14:50:46 UTC.
The original revision id was 601447262.
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:

[[toc]]
<span style="display: block; text-align: right;">[[xenharmonie/24edo|Deutsch]] - [[24平均律|日本語]]
</span>
=Basics= 
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 also because of its use in theory and occasionally in practice in [[Arabic, Turkish, Persian|Arabic]] music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - [[DIY Quartertone Composition with 12 equal tools|see this page]].

=24edo 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 almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 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 17-limit [[k*N subgroups|3*24 subgroup]] 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. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.

|| Degree || Cents coarse/fine
DMS ||= Approximate Ratios* ||||= [[Ups and Downs Notation|ups and downs notation]] ||
|| 0 || 0 ||= 1/1 || unison || P1 ||
|| 1 || 50, 60
15° ||= 33/32, 34/33 || up-unison,
downminor 2nd || ^P1, vm2 ||
|| 2 || 100, 120
30° ||= 17/16, 18/17 || aug unison,
minor 2nd || A1, m2 ||
|| 3 || 150, 180
45° ||= 12/11 || mid 2nd || ~2 ||
|| 4 || 200, 240
60° ||= 9/8 || major 2nd || M2 ||
|| 5 || 250, 300
75° ||= 22/19 || upmajor 2nd,
downminor 3rd || ^M2, vm3 ||
|| 6 || 300, 360
90° ||= 19/16 || minor 3rd || m3 ||
|| 7 || 350, 420
105° ||= 11/9 || mid 3rd || ~3 ||
|| 8 || 400, 480
120° ||= 24/19 || major 3rd || M3 ||
|| 9 || 450, 540
135° ||= 22/17 || upmajor 3rd,
down-fourth || ^M3, vP4 ||
|| 10 || 500, 600
150° ||= 4/3 || fourth || P4 ||
|| 11 || 550, 660
165° ||= 11/8 || up-fourth || ^P4 ||
|| 12 || 600, 720
180° ||= 17/12 || augmented 4th,
diminished 5th || A4, d5 ||
|| 13 || 650, 780
195° ||= 16/11 || down-fifth || vP5 ||
|| 14 || 700, 840
210° ||= 3/2 || fifth || P5 ||
|| 15 || 750, 900
225° ||= 17/11 || up-fifth,
downminor 6th || ^P5, vm6 ||
|| 16 || 800, 960
240* ||= 19/12 || minor 6th || m6 ||
|| 17 || 850, 1020
255° ||= 18/11 || mid 6th || ~6 ||
|| 18 || 900, 1080
270° ||= 32/19 || major 6th || M6 ||
|| 19 || 950, 1140
285° ||= 19/11 || upmajor 6th,
downminor 7th || ^M6, vm7 ||
|| 20 || 1000, 1200
300° ||= 16/9 || minor 7th || m7 ||
|| 21 || 1050, 1260
315° ||= 11/6 || mid 7th || ~7 ||
|| 22 || 1100, 1320
330° ||= 17/9, 32/17 || major 7th || M7 ||
|| 23 || 1150, 1380
345° ||= 33/17, 64/33 || upmajor 7th,
down-octave || ^M7, vP8 ||
*based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.

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 24-tone interval of 550 cents 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 the 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 ([[godzilla]]), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 ([[mohajira]]), a heptatonic scale close to several Arabic scales.)

===Commas=== 
24 EDO tempers out the following commas. (Note: This assumes val < 24 38 56 67 83 89/1 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||=   || | -19 12 > ||> 23.46 ||= Pythagorean Comma ||=   ||   ||
||= 648/625 || | 3 4 -4 > ||> 62.57 ||= Major Diesis ||= Diminished Comma ||   ||
||= 128/125 || | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||   ||
||= 81/80 || | -4 4 -1 > ||> 21.51 ||= Syntonic Comma ||= Didymos Comma || Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 > ||> 19.55 ||= Diaschisma ||=   ||   ||
||=   || | 26 -12 -3 > ||> 17.60 ||= Misty Comma ||=   ||   ||
||= 32805/32768 || | -15 8 1 > ||> 1.95 ||= Schisma ||=   ||   ||
||=   || | 161 -84 -12 > ||> 0.02 ||= Atom ||=   ||   ||
||= 49/48 || | -4 -1 0 2 > ||> 35.70 ||= Slendro Diesis ||=   ||   ||
||= 245/243 || | 0 -5 1 2 > ||> 14.19 ||= Sensamagic ||=   ||   ||
||= 19683/19600 || | -4 9 -2 -2 > ||> 7.32 ||= Cataharry ||=   ||   ||
||= 6144/6125 || | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||=   ||   ||
||= 121/120 || | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||=   ||   ||
||= 176/175 || | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||   ||
||= 896/891 || | 7 -4 0 1 -1 > ||> 9.69 ||= Pentacircle ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 > ||> 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||=   ||   ||
||= 9801/9800 || | -3 4 -2 -2 2 > ||> 0.18 ||= Kalisma ||= Gauss' Comma ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 > ||> 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 > ||> 2.56 ||= Parizeksma ||=   ||   ||
=Intervals= 

24 EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for
fourths, fifths, unison, and octave. For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.

These are the intervals of 24 EDO that do not exist in 12 EDO: 2
[[24 EDO Interval names and Harmonies|See full article on 24 Edo intervals.]]
||||~ The twelve new intervals in 24edo ||||||~ some nearby JI intervals ||
||~ cents ||~ common names ||~ frequency ratio ||~ cents ||~ common name ||
||= 50 ||= quartertone
infra second, wide unison ||= [[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 ||= ultra second
<span style="line-height: 1.5;">infra third</span> ||= [[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, ultra third, narrow fourth ||= [[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 ||= wide fourth ||= [[11_8|11/8]] ||= 551.318 ||= undecimal superfourth, harmonic 11th ||
||= 650 ||= narrow fifth ||= [[16_11|16/11]] ||= 648.682 ||= undecimal subfifth, 11th subharmonic ||
||= 750 ||= wide fifth, infra sixth ||= [[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 ||= ultra sixth , infra 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 ||= ultra seventh, narrow octave ||= [[31_16|31/16]]
[[33_17|33/17]]
[[35_18|35/18]] ||= 1145.036
1148.318
1151.230 ||= 31st harmonic
..
.. ||

=Notation= 
There have been disputes about disadvantages of various systems for notating quarter tones. Here are some of the few systems along with pros and cons.

==Mainstream Quartertone Notation== 
[[image:http://www.wfg.woodwind.org/n/s14_blue.gif]] or ^ = quarter-tone sharp or "Jump" or "up"
[[image:http://www.wfg.woodwind.org/n/s34_blue.gif]] or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
[[image:http://www.wfg.woodwind.org/n/f14_blue.gif]] or v = quarter-tone flat or "Drop" or "down"
[[image:http://www.wfg.woodwind.org/n/f34_blue.gif]] or bv or vb = three-quarter-tone flat or "Drop-Flat" or "downflat"

Pros: Familiar, fairly easy to learn
Cons: Clutters a score easily, can get confusing when sight read at faster paces

==Alternate Quartertone Accidentals== 
[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] = quarter-tone sharp or Jump
††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp
[[image:http://www.dolmetsch.com/onequarterflat.gif]] = quarter-tone flat or Drop
[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]] = three quarter-tones flat or drop-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.

Pros: Very easy to distinguish accidentals from one another
Cons: Not practical, tends to clutter a score

==<span style="line-height: 1.5;">And in Persian music</span>== 
[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Koron_sign.svg/200px-Koron_sign.svg.png width="12" height="25"]]Koron ([[http://en.wikipedia.org/wiki/Koron_%28music%29|en]] | [[http://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86|fa]]) = quarter-tone flat or Jump
[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Sori_sign.svg/200px-Sori_sign.svg.png width="15" height="25"]]Sori ([[http://fa.wikipedia.org/wiki/%D8%B3%D8%B1%DB%8C_%28%D9%85%D9%88%D8%B3%DB%8C%D9%82%DB%8C%29|fa]]) = quarter-tone sharp or Drop

Pros: Easy to read
Cons: Hard to write on a computer, doesn't fit with standard notation well

==Sagittal Notation== 
[[Sagittal notation]] works extremely well for 24 Edo notation as well as other systems.
It's easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat.
A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them
with Sagittal signs for sharp and flat.

[[image:sagittal 24.PNG]]

=Interval Alterations= 
The special alterations of the intervals and chords of 12 equal can be notated like this:

Supermajor or "**Tendo**" is a major interval raised a quarter tone
Subminor or "**Arto**" is a minor interval lowered a quarter tone
Neutral are intervals that exist between the major and minor version of an interval
The prefix **under** indicates a perfect interval lowered by one quarter tone
The prefix **over** indicates a perfect interval raised by a quarter tone

The Latin words "tendo" (meaning "expand") and "arto" (meaning "contract") can be used to replace the words "supermajor" and "subminor" in order to shorten the names of the intervals.

=Chord Structures= 
24edo features a rich variety on not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.

As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as
0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.
These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.
For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.

William Lynch considers these as some possible good tetrads:

[[image:Three chords.PNG]]

||~ Chord name ||~ Degrees of 24edo ||~ Chord spelling ||~ Audio example ||
|| neutral || 0 7 14 21 || 1 v3 5 v7 || [[media type="file" key="Neutral Tetrad on C.mp3" width="200" height="30"]] ||
|| arto || 0 5 14 20 || 1 bv3 5 b7 || [[media type="file" key="arto tetrad on C.mp3" width="200" height="30"]] ||
|| tendo || 0 9 14 19 || 1 ^3 5 bv7 || ... ||

Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.

=Naming Chords in 24edo= 
Naming chords in 24edo can be achieved by adding a few things to the already [[#|existing]] set of terms that are used to name 12edo chords.
They are:
Super + perfect interval such as "perfect fifth" means to raise it by a quarter tone
Sub + perfect interval means to lower a quarter tone
Sharp is to raise by one half tone
Flat is to raise by a half tone
Neutral, arto and tendo refer to triads or tetrads
Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively

Examples:
Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top
Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11

Alternatively, [[ups and downs notation]] can be used:
0-5-14 = D Fv A = D.vm = "D downminor"
0-6-14 = D F A = Dm = "D minor"
0-7-14 = D F^ A = D~ = "D mid"
0-8-14 = D F# A = D = "D" or "D major"
0-9-14 = D F#^ A = D.^ = "D dot up" or "D upmajor"


=Rank two temperaments= 
[[List of 24et rank two temperaments by badness]]
[[List of edo-distinct 24et rank two temperaments]]

||~ Periods per octave ||~ Generator ||~ Name ||
|| 1 || 1\24 ||   ||
|| 1 || 5\24 || [[Semaphore and Godzilla|Semaphore/godzilla]] / [[Chromatic pairs#Bridgetown|Bridgetown]] ||
|| 1 || 7\24 || [[Mohajira]] (or [[maqamic]] with 24d val) ||
|| 1 || 11\24 ||   ||
|| 2 || 1\24 || [[Shrutar]] ||
|| 2 || 5\24 || Similar to [[decimal]] ||
|| 3 || 1\24 || [[Semiaug]] ||
|| 3 || 3\24 || [[Triforce]] ||
|| 4 || 1\24 ||   ||
|| 6 || 1\24 ||   ||
|| 8 || 1\24 ||   ||
|| 12 || 1\24 || [[Catler]] ||

=Scales / Modes= 

|||| ====Pentatonic:==== ||
|| {{**2 8 3 6 5**}} || Anchihoye: Ethiopia ||
|| {{**5 5 4 5 5**}} || Quasi-equal Pentatonic - [[MOSScales|MOS]] of type [[4L 1s|4L 1s (bug)]] ||
|| {{**5 5 5 5 4**}} || Hába's Pentatonic - [[MOSScales|MOS]] of type [[4L 1s|4L 1s (bug)]] ||

|||| ====Hexatonic:==== ||
|| {{**1 1 8 4 2 8**}} || Spondeiakos ||

|||| ====Heptatonic:==== ||
|| {{**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 - [[MODMOS Scales|MODMOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 4 3 3 4 4 3**}} || Dastgah-e Sehgah, Neutral Diatonic Lydian - [[xenharmonic/MODMOS Scales|MODMOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 3 3 4 4 3 3**}} || Arabic Diatonic, Maqam Rast, Quasi-equal Heptatonic, Neutral Diatonic Phrygian - [[xenharmonic/MODMOS Scales|MODMOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 3 4 4 3 3 4**}} || Maqam Hussaini, Ushaq, Neutral Diatonic Dorian - [[xenharmonic/MODMOS Scales|MODMOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**3 4 4 3 3 4 3**}} || Maqam Sikah (Segah), Neutral Diatonic Hypolydian - [[xenharmonic/MODMOS Scales|MODMOS]] of type [[3L 4s|3L 4s (mosh)]] ||
|| {{**4 4 3 3 4 3 3**}} || Neutral Diatonic Hypophrygian - [[xenharmonic/MODMOS Scales|MODMOS]] 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 - [[xenharmonic/MODMOS Scales|MODMOS]] 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 ||
|| {{**3 3 4 4 2 4 4**}} || Maqam 'Ushshaq Turki, Urfa, Isfahan, Dastgah-e Shur ||
|| {{**3 3 4 4 4 2 4**}} || Maqam Nahfat ||
|| {{**3 3 2 6 2 4 4**}} || Maqam Saba ||
|| {{**3 3 2 6 2 6 2**}} || Maqam Sabr Jadid ||
|| {{**4 3 3 4 2 6 2**}} || Maqam Suznak (Soznak) ||
|| {{**4 3 3 4 4 4 2**}} || Maqam Mahur ||
|| {{**3 3 4 2 6 2 4**}} || Maqam Qarjighar, Bayati Shuri ||
|| {{**3 4 2 6 2 4 3**}} || Maqam Hizam (Huzam, El Houzam), Rahat al Arouah ||
|| {{**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 ||
|| {{**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 ||

|||| ====Octatonic:==== ||
|| {{**3 3 3 3 3 3 3 3**}} || 8-equal, Wyschnegradsky's octatonic ||
|| {{**3 3 4 4 2 1 3 4**}} || Maqam Bayati ||
|| {{**3 3 2 6 2 4 2 2**}} || Maqam Saba ||
|| {{**4 3 1 2 4 4 2 4**}} || Maqam Suzidil 'ara ||
|| {{**3 3 2 2 4 3 3 4**}} || Maqam Mansuri ||
|| {{**4 3 3 4 4 2 1 3**}} || Maqam Rast, Dilkashidah, Dilnishin ||
|| {{**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 ||
|| {{**3 1 2 4 4 2 4 4**}} || Maqam Jabburi ||
|| {{**1 4 4 2 4 4 4 1**}} || Giancarlo Dalmonte's new quarter-tone scale (see [[http://www.ottavanota.info/|http://www.ottavanota.info]]) ||

|||| ====Enneatonic:==== ||
|| {{**1 2 3 4 4 1 2 3 4**}} || Progressive Enneatonic ||
|| {{**4 1 4 1 4 1 4 1 4**}} || de Vries 9-tone - [[MOSScales|MOS]] of type [[5L 4s|5L 4s (unfair bug)]] ||
|| {{**3 4 2 2 2 2 2 4 3**}} || Maqam Huzam ||
|| {{**4 4 2 1 3 2 2 4 2**}} || Maqam Shawq Afza ||

|||| ====Decatonic:==== ||
|| {{**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 1 3 2 2 4 2 4 2 2**}} || Maqam Shawq Tarab ||
|| {{**4 2 1 3 2 2 4 2 1 3**}} || Maqam Basandida ||
|| {{**4 3 1 2 1 3 4 2 1 3**}} || Maqam Yakah ||

|||| ====Hendecatonic:==== ||
|| {{**4 2 1 3 2 2 2 2 3 1 2**}} || Maqam Hayyan ||

|||| ====Tridecatonic:==== ||
|| {{**1 2 2 2 2 2 2 1 2 2 2 2 2**}} || de Vries 13-tone - [[MOSScales|MOS]] of type [[11L 2s]] ||
|| {{**2 2 2 2 2 1 2 2 2 2 2 2 1**}} || Agmon Diatonic DS5, Ivan Wyschnegradsky's diatonicized chromatic scale - [[MOSScales|MOS]] of type [[11L 2s]] ||

|||| ====Tetradecatonic:==== ||
|| **<span style="font-family: monospace;">2 2 1 2 2 2 1 2 2 1 2 2 2 1</span>** || Young Half-Octave Diatonic - [[MOSScales|MOS]] of type [[10L 4s]] ||

=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

Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface Tuning|Iceface tuning]].

=Compositions= 
==Music== 
<span class="ymp-btn-page-play ymp-media-eb56a917c62000140296c2e1a6d89911">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex3.mp3|Microhex3]]</span> <span class="ymp-btn-page-play ymp-media-750620955011dabb8bc98c7c8830cbb8">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex4.mp3|Microhex4]]</span> and <span class="ymp-btn-page-play ymp-media-4aeb7c299c23983ba8081a591562e704">[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex5.mp3|Microhex5]]</span> by [[http://www.96edo.com/About_me.html|Shaahin Mohajeri]]
<span class="ymp-btn-page-play ymp-media-30832be5610668b4b42451c8ce82dc80">[[http://micro.soonlabel.com/24et/quarterpicnic.mp3|Quarterpicnic]]</span> by [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/BeDell/Quarter%20Tone%20Prelude%20for%20two%20Harps.mp3|Quarter Tone Prelude For Two Harps]] by [[http://soundcloud.com/cerbeus-1/quarter-tone-prelude-for-two|Nathan BeDell]]
<span class="ymp-btn-page-play ymp-media-2738730931ba168a7364a84b5417488d">[[http://micro.soonlabel.com/24et/daily20111021-fretless-1.mp3|Fretless Chrome 1]]</span> and <span class="ymp-btn-page-play ymp-media-d35cb71128e927701930058ebd16b778">[[http://micro.soonlabel.com/24et/daily20111021-fretless-2.mp3|Fretless Chrome 2]]</span> by [[@http://chrisvaisvil.com/?p=1477|Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lament.mp3|Lament]] by [[http://soundcloud.com/jdfreivald/lament|Jake Freivald]]. In the [[freivaldneutral24]] scale.
[[https://soundcloud.com/jdfreivald/mo-happy-happy|Mo - Happy - Happy]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/mo-happy-happy.mp3|play]] by Jake Freivald in Neuter[7] (2.3.11 mohajira), 24et tuning
[[http://www.youtube.com/watch?v=q1Lp8AtKK9o|Autumn Winds]], [[http://www.youtube.com/watch?v=Igxe3DwbFJ4&feature=c4-overview&list=UUvq5bg-LvOS6adpB5efPTyQ|Easter Time at Nine]], [[http://www.youtube.com/watch?v=z_3uhA9Cq08&list=UUvq5bg-LvOS6adpB5efPTyQ|Waters of Persia]] by William Lynch in mohajira, 24et tuning.
[[https://soundcloud.com/mason-l-green/serena|Serena]], by Mason Green (intro and coda in 24edo, the rest is in 12edo)
[[https://www.youtube.com/watch?v=yzvXEMYgHCY|Autumn Girl]], by Mason Green

==About== 
"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://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.webcitation.org/5xeFMH6cd|Permalink]]
[[http://www.96edo.com/24_EDO.html|About 24-EDO]] by Shaahin Mohajeri [[http://www.webcitation.org/5xeFBNdQW|Permalink]]
[[https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit|Notation and Chord Names for 24-EDO]] by William Lynch
[[http://sonic-arts.org/darreg/contents.htm|The place of QUARTERTONES in Today's Xenharmonics]] by [[Ivor Darreg]] [[http://sonic-arts.org/darreg/dar8.htm|Permalink]]

Original HTML content:

<html><head><title>24edo</title></head><body><!-- ws:start:WikiTextTocRule:60:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 1em;"><a href="#Basics">Basics</a></div>
<!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><div style="margin-left: 1em;"><a href="#x24edo as a temperament">24edo as a temperament</a></div>
<!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 3em;"><a href="#x24edo as a temperament--Commas">Commas</a></div>
<!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div>
<!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 1em;"><a href="#Notation">Notation</a></div>
<!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 2em;"><a href="#Notation-Mainstream Quartertone Notation">Mainstream Quartertone Notation</a></div>
<!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 2em;"><a href="#Notation-Alternate Quartertone Accidentals">Alternate Quartertone Accidentals</a></div>
<!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Notation-And in Persian music">And in Persian music</a></div>
<!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 2em;"><a href="#Notation-Sagittal Notation">Sagittal Notation</a></div>
<!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 1em;"><a href="#Interval Alterations">Interval Alterations</a></div>
<!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 1em;"><a href="#Chord Structures">Chord Structures</a></div>
<!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 1em;"><a href="#Naming Chords in 24edo">Naming Chords in 24edo</a></div>
<!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div>
<!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 1em;"><a href="#Scales / Modes">Scales / Modes</a></div>
<!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Pentatonic:">Pentatonic:</a></div>
<!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Hexatonic:">Hexatonic:</a></div>
<!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Heptatonic:">Heptatonic:</a></div>
<!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Octatonic:">Octatonic:</a></div>
<!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Enneatonic:">Enneatonic:</a></div>
<!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Decatonic:">Decatonic:</a></div>
<!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Hendecatonic:">Hendecatonic:</a></div>
<!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Tridecatonic:">Tridecatonic:</a></div>
<!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 4em;"><a href="#Scales / Modes---Tetradecatonic:">Tetradecatonic:</a></div>
<!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 1em;"><a href="#Instruments">Instruments</a></div>
<!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div>
<!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 2em;"><a href="#Compositions-Music">Music</a></div>
<!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 2em;"><a href="#Compositions-About">About</a></div>
<!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 1em;"><a href="#Practical Theory / Books">Practical Theory / Books</a></div>
<!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 1em;"><a href="#External links">External links</a></div>
<!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --></div>
<!-- ws:end:WikiTextTocRule:90 --><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/24edo">Deutsch</a> - <a class="wiki_link" href="/24%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br />
</span><br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="Basics"></a><!-- ws:end:WikiTextHeadingRule:2 -->Basics</h1>
 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 also because of its use in theory and occasionally in practice in <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">Arabic</a> music. It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments - <a class="wiki_link" href="/DIY%20Quartertone%20Composition%20with%2012%20equal%20tools">see this page</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc1"><a name="x24edo as a temperament"></a><!-- ws:end:WikiTextHeadingRule:4 -->24edo as a temperament</h1>
 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 almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 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 17-limit <a class="wiki_link" href="/k%2AN%20subgroups">3*24 subgroup</a> 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. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents coarse/fine<br />
DMS<br />
</td>
        <td style="text-align: center;">Approximate Ratios*<br />
</td>
        <td colspan="2" style="text-align: center;"><a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs notation</a><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
        <td>unison<br />
</td>
        <td>P1<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>50, 60<br />
15°<br />
</td>
        <td style="text-align: center;">33/32, 34/33<br />
</td>
        <td>up-unison,<br />
downminor 2nd<br />
</td>
        <td>^P1, vm2<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>100, 120<br />
30°<br />
</td>
        <td style="text-align: center;">17/16, 18/17<br />
</td>
        <td>aug unison,<br />
minor 2nd<br />
</td>
        <td>A1, m2<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>150, 180<br />
45°<br />
</td>
        <td style="text-align: center;">12/11<br />
</td>
        <td>mid 2nd<br />
</td>
        <td>~2<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>200, 240<br />
60°<br />
</td>
        <td style="text-align: center;">9/8<br />
</td>
        <td>major 2nd<br />
</td>
        <td>M2<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>250, 300<br />
75°<br />
</td>
        <td style="text-align: center;">22/19<br />
</td>
        <td>upmajor 2nd,<br />
downminor 3rd<br />
</td>
        <td>^M2, vm3<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>300, 360<br />
90°<br />
</td>
        <td style="text-align: center;">19/16<br />
</td>
        <td>minor 3rd<br />
</td>
        <td>m3<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>350, 420<br />
105°<br />
</td>
        <td style="text-align: center;">11/9<br />
</td>
        <td>mid 3rd<br />
</td>
        <td>~3<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>400, 480<br />
120°<br />
</td>
        <td style="text-align: center;">24/19<br />
</td>
        <td>major 3rd<br />
</td>
        <td>M3<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>450, 540<br />
135°<br />
</td>
        <td style="text-align: center;">22/17<br />
</td>
        <td>upmajor 3rd,<br />
down-fourth<br />
</td>
        <td>^M3, vP4<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>500, 600<br />
150°<br />
</td>
        <td style="text-align: center;">4/3<br />
</td>
        <td>fourth<br />
</td>
        <td>P4<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>550, 660<br />
165°<br />
</td>
        <td style="text-align: center;">11/8<br />
</td>
        <td>up-fourth<br />
</td>
        <td>^P4<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>600, 720<br />
180°<br />
</td>
        <td style="text-align: center;">17/12<br />
</td>
        <td>augmented 4th,<br />
diminished 5th<br />
</td>
        <td>A4, d5<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>650, 780<br />
195°<br />
</td>
        <td style="text-align: center;">16/11<br />
</td>
        <td>down-fifth<br />
</td>
        <td>vP5<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>700, 840<br />
210°<br />
</td>
        <td style="text-align: center;">3/2<br />
</td>
        <td>fifth<br />
</td>
        <td>P5<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>750, 900<br />
225°<br />
</td>
        <td style="text-align: center;">17/11<br />
</td>
        <td>up-fifth,<br />
downminor 6th<br />
</td>
        <td>^P5, vm6<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>800, 960<br />
240*<br />
</td>
        <td style="text-align: center;">19/12<br />
</td>
        <td>minor 6th<br />
</td>
        <td>m6<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>850, 1020<br />
255°<br />
</td>
        <td style="text-align: center;">18/11<br />
</td>
        <td>mid 6th<br />
</td>
        <td>~6<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>900, 1080<br />
270°<br />
</td>
        <td style="text-align: center;">32/19<br />
</td>
        <td>major 6th<br />
</td>
        <td>M6<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>950, 1140<br />
285°<br />
</td>
        <td style="text-align: center;">19/11<br />
</td>
        <td>upmajor 6th,<br />
downminor 7th<br />
</td>
        <td>^M6, vm7<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>1000, 1200<br />
300°<br />
</td>
        <td style="text-align: center;">16/9<br />
</td>
        <td>minor 7th<br />
</td>
        <td>m7<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>1050, 1260<br />
315°<br />
</td>
        <td style="text-align: center;">11/6<br />
</td>
        <td>mid 7th<br />
</td>
        <td>~7<br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>1100, 1320<br />
330°<br />
</td>
        <td style="text-align: center;">17/9, 32/17<br />
</td>
        <td>major 7th<br />
</td>
        <td>M7<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>1150, 1380<br />
345°<br />
</td>
        <td style="text-align: center;">33/17, 64/33<br />
</td>
        <td>upmajor 7th,<br />
down-octave<br />
</td>
        <td>^M7, vP8<br />
</td>
    </tr>
</table>

*based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.<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 24-tone interval of 550 cents 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 the 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 (<a class="wiki_link" href="/godzilla">godzilla</a>), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 (<a class="wiki_link" href="/mohajira">mohajira</a>), a heptatonic scale close to several Arabic scales.)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc2"><a name="x24edo as a temperament--Commas"></a><!-- ws:end:WikiTextHeadingRule:6 -->Commas</h3>
 24 EDO tempers out the following commas. (Note: This assumes val &lt; 24 38 56 67 83 89/1 |.)<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;"><br />
</td>
        <td>| -19 12 &gt;<br />
</td>
        <td style="text-align: right;">23.46<br />
</td>
        <td style="text-align: center;">Pythagorean Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">648/625<br />
</td>
        <td>| 3 4 -4 &gt;<br />
</td>
        <td style="text-align: right;">62.57<br />
</td>
        <td style="text-align: center;">Major Diesis<br />
</td>
        <td style="text-align: center;">Diminished Comma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">128/125<br />
</td>
        <td>| 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><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">81/80<br />
</td>
        <td>| -4 4 -1 &gt;<br />
</td>
        <td style="text-align: right;">21.51<br />
</td>
        <td style="text-align: center;">Syntonic Comma<br />
</td>
        <td style="text-align: center;">Didymos Comma<br />
</td>
        <td>Meantone Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2048/2025<br />
</td>
        <td>| 11 -4 -2 &gt;<br />
</td>
        <td style="text-align: right;">19.55<br />
</td>
        <td style="text-align: center;">Diaschisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td>| 26 -12 -3 &gt;<br />
</td>
        <td style="text-align: right;">17.60<br />
</td>
        <td style="text-align: center;">Misty Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32805/32768<br />
</td>
        <td>| -15 8 1 &gt;<br />
</td>
        <td style="text-align: right;">1.95<br />
</td>
        <td style="text-align: center;">Schisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td>| 161 -84 -12 &gt;<br />
</td>
        <td style="text-align: right;">0.02<br />
</td>
        <td style="text-align: center;">Atom<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">49/48<br />
</td>
        <td>| -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><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">245/243<br />
</td>
        <td>| 0 -5 1 2 &gt;<br />
</td>
        <td style="text-align: right;">14.19<br />
</td>
        <td style="text-align: center;">Sensamagic<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19683/19600<br />
</td>
        <td>| -4 9 -2 -2 &gt;<br />
</td>
        <td style="text-align: right;">7.32<br />
</td>
        <td style="text-align: center;">Cataharry<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6144/6125<br />
</td>
        <td>| 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><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">121/120<br />
</td>
        <td>| -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><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td>| 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><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">896/891<br />
</td>
        <td>| 7 -4 0 1 -1 &gt;<br />
</td>
        <td style="text-align: right;">9.69<br />
</td>
        <td style="text-align: center;">Pentacircle<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">243/242<br />
</td>
        <td>| -1 5 0 0 -2 &gt;<br />
</td>
        <td style="text-align: right;">7.14<br />
</td>
        <td style="text-align: center;">Rastma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td>| -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><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9801/9800<br />
</td>
        <td>| -3 4 -2 -2 2 &gt;<br />
</td>
        <td style="text-align: right;">0.18<br />
</td>
        <td style="text-align: center;">Kalisma<br />
</td>
        <td style="text-align: center;">Gauss' Comma<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">91/90<br />
</td>
        <td>| -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><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">676/675<br />
</td>
        <td>| 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><br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:8 -->Intervals</h1>
 <br />
24 EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for<br />
fourths, fifths, unison, and octave. For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.<br />
<br />
These are the intervals of 24 EDO that do not exist in 12 EDO: 2<br />
<a class="wiki_link" href="/24%20EDO%20Interval%20names%20and%20Harmonies">See full article on 24 Edo intervals.</a><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 names<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;">quartertone<br />
infra second, wide unison<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;">ultra second<br />
<span style="line-height: 1.5;">infra third</span><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 />
..<br />
..<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, ultra third, narrow fourth<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;">wide fourth<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;">narrow fifth<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;">wide fifth, infra sixth<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;">ultra sixth , infra 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 />
..<br />
..<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;">ultra seventh, narrow octave<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 />
..<br />
..<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc4"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:10 -->Notation</h1>
 There have been disputes about disadvantages of various systems for notating quarter tones. Here are some of the few systems along with pros and cons.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc5"><a name="Notation-Mainstream Quartertone Notation"></a><!-- ws:end:WikiTextHeadingRule:12 -->Mainstream Quartertone Notation</h2>
 <!-- ws:start:WikiTextRemoteImageRule:1632:&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:1632 --> or ^ = quarter-tone sharp or &quot;Jump&quot; or &quot;up&quot;<br />
<!-- ws:start:WikiTextRemoteImageRule:1633:&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:1633 --> or #^ or ^# = three-quarter-tone sharp or &quot;Jump-Sharp&quot; or &quot;upsharp&quot;<br />
<!-- ws:start:WikiTextRemoteImageRule:1634:&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:1634 --> or v = quarter-tone flat or &quot;Drop&quot; or &quot;down&quot;<br />
<!-- ws:start:WikiTextRemoteImageRule:1635:&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:1635 --> or bv or vb = three-quarter-tone flat or &quot;Drop-Flat&quot; or &quot;downflat&quot;<br />
<br />
Pros: Familiar, fairly easy to learn<br />
Cons: Clutters a score easily, can get confusing when sight read at faster paces<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc6"><a name="Notation-Alternate Quartertone Accidentals"></a><!-- ws:end:WikiTextHeadingRule:14 -->Alternate Quartertone Accidentals</h2>
 <!-- ws:start:WikiTextRemoteImageRule:1636:&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:1636 --> = quarter-tone sharp or Jump<br />
††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp<br />
<!-- ws:start:WikiTextRemoteImageRule:1637:&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:1637 --> = quarter-tone flat or Drop<br />
<!-- ws:start:WikiTextRemoteImageRule:1638:&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:1638 --> = three quarter-tones flat or drop-flat<br />
For example, the scale 0-5-10-15-20 is written as C-D<!-- ws:start:WikiTextRemoteImageRule:1639:&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:1639 --> (or E<!-- ws:start:WikiTextRemoteImageRule:1640:&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:1640 -->) F G<!-- ws:start:WikiTextRemoteImageRule:1641:&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:1641 --> (or A<!-- ws:start:WikiTextRemoteImageRule:1642:&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:1642 -->) Bb.<br />
<br />
Pros: Very easy to distinguish accidentals from one another<br />
Cons: Not practical, tends to clutter a score<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc7"><a name="Notation-And in Persian music"></a><!-- ws:end:WikiTextHeadingRule:16 --><span style="line-height: 1.5;">And in Persian music</span></h2>
 <!-- ws:start:WikiTextRemoteImageRule:1643:&lt;img src=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Koron_sign.svg/200px-Koron_sign.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 25px; width: 12px;&quot; /&gt; --><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Koron_sign.svg/200px-Koron_sign.svg.png" alt="external image 200px-Koron_sign.svg.png" title="external image 200px-Koron_sign.svg.png" style="height: 25px; width: 12px;" /><!-- ws:end:WikiTextRemoteImageRule:1643 -->Koron (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Koron_%28music%29" rel="nofollow">en</a> | <a class="wiki_link_ext" href="http://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86" rel="nofollow">fa</a>) = quarter-tone flat or Jump<br />
<!-- ws:start:WikiTextRemoteImageRule:1644:&lt;img src=&quot;http://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Sori_sign.svg/200px-Sori_sign.svg.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 25px; width: 15px;&quot; /&gt; --><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Sori_sign.svg/200px-Sori_sign.svg.png" alt="external image 200px-Sori_sign.svg.png" title="external image 200px-Sori_sign.svg.png" style="height: 25px; width: 15px;" /><!-- ws:end:WikiTextRemoteImageRule:1644 -->Sori (<a class="wiki_link_ext" href="http://fa.wikipedia.org/wiki/%D8%B3%D8%B1%DB%8C_%28%D9%85%D9%88%D8%B3%DB%8C%D9%82%DB%8C%29" rel="nofollow">fa</a>) = quarter-tone sharp or Drop<br />
<br />
Pros: Easy to read<br />
Cons: Hard to write on a computer, doesn't fit with standard notation well<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc8"><a name="Notation-Sagittal Notation"></a><!-- ws:end:WikiTextHeadingRule:18 -->Sagittal Notation</h2>
 <a class="wiki_link" href="/Sagittal%20notation">Sagittal notation</a> works extremely well for 24 Edo notation as well as other systems.<br />
It's easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat.<br />
A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them<br />
with Sagittal signs for sharp and flat.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:1630:&lt;img src=&quot;/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG" alt="sagittal 24.PNG" title="sagittal 24.PNG" /><!-- ws:end:WikiTextLocalImageRule:1630 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc9"><a name="Interval Alterations"></a><!-- ws:end:WikiTextHeadingRule:20 -->Interval Alterations</h1>
 The special alterations of the intervals and chords of 12 equal can be notated like this:<br />
<br />
Supermajor or &quot;<strong>Tendo</strong>&quot; is a major interval raised a quarter tone<br />
Subminor or &quot;<strong>Arto</strong>&quot; is a minor interval lowered a quarter tone<br />
Neutral are intervals that exist between the major and minor version of an interval<br />
The prefix <strong>under</strong> indicates a perfect interval lowered by one quarter tone<br />
The prefix <strong>over</strong> indicates a perfect interval raised by a quarter tone<br />
<br />
The Latin words &quot;tendo&quot; (meaning &quot;expand&quot;) and &quot;arto&quot; (meaning &quot;contract&quot;) can be used to replace the words &quot;supermajor&quot; and &quot;subminor&quot; in order to shorten the names of the intervals.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc10"><a name="Chord Structures"></a><!-- ws:end:WikiTextHeadingRule:22 -->Chord Structures</h1>
 24edo features a rich variety on not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.<br />
<br />
As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as<br />
0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.<br />
These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.<br />
For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 <a class="wiki_link" href="/William%20Lynch">William Lynch</a> considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.<br />
<br />
William Lynch considers these as some possible good tetrads:<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:1631:&lt;img src=&quot;/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG" alt="Three chords.PNG" title="Three chords.PNG" /><!-- ws:end:WikiTextLocalImageRule:1631 --><br />
<br />


<table class="wiki_table">
    <tr>
        <th>Chord name<br />
</th>
        <th>Degrees of 24edo<br />
</th>
        <th>Chord spelling<br />
</th>
        <th>Audio example<br />
</th>
    </tr>
    <tr>
        <td>neutral<br />
</td>
        <td>0 7 14 21<br />
</td>
        <td>1 v3 5 v7<br />
</td>
        <td><!-- ws:start:WikiTextMediaRule:0:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/Neutral%20Tetrad%20on%20C.mp3?h=30&amp;w=200&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;Neutral Tetrad on C.mp3&amp;quot; width=&amp;quot;200&amp;quot; height=&amp;quot;30&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;30&quot; width=&quot;200&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="200" height="30" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FNeutral%252BTetrad%252Bon%252BC.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=200&height=30"></embed><!-- ws:end:WikiTextMediaRule:0 --><br />
</td>
    </tr>
    <tr>
        <td>arto<br />
</td>
        <td>0 5 14 20<br />
</td>
        <td>1 bv3 5 b7<br />
</td>
        <td><!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/arto%20tetrad%20on%20C.mp3?h=30&amp;w=200&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;arto tetrad on C.mp3&amp;quot; width=&amp;quot;200&amp;quot; height=&amp;quot;30&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;30&quot; width=&quot;200&quot;/&gt; --><embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="200" height="30" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Farto%252Btetrad%252Bon%252BC.mp3?file_extension=mp3&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=200&height=30"></embed><!-- ws:end:WikiTextMediaRule:1 --><br />
</td>
    </tr>
    <tr>
        <td>tendo<br />
</td>
        <td>0 9 14 19<br />
</td>
        <td>1 ^3 5 bv7<br />
</td>
        <td>...<br />
</td>
    </tr>
</table>

<br />
Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc11"><a name="Naming Chords in 24edo"></a><!-- ws:end:WikiTextHeadingRule:24 -->Naming Chords in 24edo</h1>
 Naming chords in 24edo can be achieved by adding a few things to the already [[#|existing]] set of terms that are used to name 12edo chords.<br />
They are:<br />
Super + perfect interval such as &quot;perfect fifth&quot; means to raise it by a quarter tone<br />
Sub + perfect interval means to lower a quarter tone<br />
Sharp is to raise by one half tone<br />
Flat is to raise by a half tone<br />
Neutral, arto and tendo refer to triads or tetrads<br />
Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively<br />
<br />
Examples:<br />
Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top<br />
Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11<br />
<br />
Alternatively, <a class="wiki_link" href="/ups%20and%20downs%20notation">ups and downs notation</a> can be used:<br />
0-5-14 = D Fv A = D.vm = &quot;D downminor&quot;<br />
0-6-14 = D F A = Dm = &quot;D minor&quot;<br />
0-7-14 = D F^ A = D~ = &quot;D mid&quot;<br />
0-8-14 = D F# A = D = &quot;D&quot; or &quot;D major&quot;<br />
0-9-14 = D F#^ A = D.^ = &quot;D dot up&quot; or &quot;D upmajor&quot;<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc12"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:26 -->Rank two temperaments</h1>
 <a class="wiki_link" href="/List%20of%2024et%20rank%20two%20temperaments%20by%20badness">List of 24et rank two temperaments by badness</a><br />
<a class="wiki_link" href="/List%20of%20edo-distinct%2024et%20rank%20two%20temperaments">List of edo-distinct 24et rank two temperaments</a><br />
<br />


<table class="wiki_table">
    <tr>
        <th>Periods per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Name<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\24<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>5\24<br />
</td>
        <td><a class="wiki_link" href="/Semaphore%20and%20Godzilla">Semaphore/godzilla</a> / <a class="wiki_link" href="/Chromatic%20pairs#Bridgetown">Bridgetown</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>7\24<br />
</td>
        <td><a class="wiki_link" href="/Mohajira">Mohajira</a> (or <a class="wiki_link" href="/maqamic">maqamic</a> with 24d val)<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>11\24<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>1\24<br />
</td>
        <td><a class="wiki_link" href="/Shrutar">Shrutar</a><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>5\24<br />
</td>
        <td>Similar to <a class="wiki_link" href="/decimal">decimal</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>1\24<br />
</td>
        <td><a class="wiki_link" href="/Semiaug">Semiaug</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>3\24<br />
</td>
        <td><a class="wiki_link" href="/Triforce">Triforce</a><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>1\24<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>1\24<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>1\24<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>1\24<br />
</td>
        <td><a class="wiki_link" href="/Catler">Catler</a><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc13"><a name="Scales / Modes"></a><!-- ws:end:WikiTextHeadingRule:28 -->Scales / Modes</h1>
 <br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:30:&lt;h4&gt; --><h4 id="toc14"><a name="Scales / Modes---Pentatonic:"></a><!-- ws:end:WikiTextHeadingRule:30 -->Pentatonic:</h4>
</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>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>5 5 5 5 4</strong></tt><br />
</td>
        <td>Hába's 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>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:32:&lt;h4&gt; --><h4 id="toc15"><a name="Scales / Modes---Hexatonic:"></a><!-- ws:end:WikiTextHeadingRule:32 -->Hexatonic:</h4>
</td>
    </tr>
    <tr>
        <td><tt><strong>1 1 8 4 2 8</strong></tt><br />
</td>
        <td>Spondeiakos<br />
</td>
    </tr>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:34:&lt;h4&gt; --><h4 id="toc16"><a name="Scales / Modes---Heptatonic:"></a><!-- ws:end:WikiTextHeadingRule:34 -->Heptatonic:</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="/MODMOS%20Scales">MODMOS</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="http://xenharmonic.wikispaces.com/MODMOS%20Scales">MODMOS</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="http://xenharmonic.wikispaces.com/MODMOS%20Scales">MODMOS</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="http://xenharmonic.wikispaces.com/MODMOS%20Scales">MODMOS</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="http://xenharmonic.wikispaces.com/MODMOS%20Scales">MODMOS</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="http://xenharmonic.wikispaces.com/MODMOS%20Scales">MODMOS</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="http://xenharmonic.wikispaces.com/MODMOS%20Scales">MODMOS</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>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 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 6 2</strong></tt><br />
</td>
        <td>Maqam Sabr Jadid<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 4 2</strong></tt><br />
</td>
        <td>Maqam Mahur<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 6 2 4 3</strong></tt><br />
</td>
        <td>Maqam Hizam (Huzam, El Houzam), Rahat al Arouah<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>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>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:36:&lt;h4&gt; --><h4 id="toc17"><a name="Scales / Modes---Octatonic:"></a><!-- ws:end:WikiTextHeadingRule:36 -->Octatonic:</h4>
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 3 3 3 3 3 3</strong></tt><br />
</td>
        <td>8-equal, Wyschnegradsky's octatonic<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 2 6 2 4 2 2</strong></tt><br />
</td>
        <td>Maqam Saba<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 2 2 4 3 3 4</strong></tt><br />
</td>
        <td>Maqam Mansuri<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>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>3 1 2 4 4 2 4 4</strong></tt><br />
</td>
        <td>Maqam Jabburi<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 4 4 2 4 4 4 1</strong></tt><br />
</td>
        <td>Giancarlo Dalmonte's new quarter-tone scale (see <a class="wiki_link_ext" href="http://www.ottavanota.info/" rel="nofollow">http://www.ottavanota.info</a>)<br />
</td>
    </tr>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:38:&lt;h4&gt; --><h4 id="toc18"><a name="Scales / Modes---Enneatonic:"></a><!-- ws:end:WikiTextHeadingRule:38 -->Enneatonic:</h4>
</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>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>3 4 2 2 2 2 2 4 3</strong></tt><br />
</td>
        <td>Maqam Huzam<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>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:40:&lt;h4&gt; --><h4 id="toc19"><a name="Scales / Modes---Decatonic:"></a><!-- ws:end:WikiTextHeadingRule:40 -->Decatonic:</h4>
</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 1 3 2 2 4 2 4 2 2</strong></tt><br />
</td>
        <td>Maqam Shawq Tarab<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 3 1 2 1 3 4 2 1 3</strong></tt><br />
</td>
        <td>Maqam Yakah<br />
</td>
    </tr>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:42:&lt;h4&gt; --><h4 id="toc20"><a name="Scales / Modes---Hendecatonic:"></a><!-- ws:end:WikiTextHeadingRule:42 -->Hendecatonic:</h4>
</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>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:44:&lt;h4&gt; --><h4 id="toc21"><a name="Scales / Modes---Tridecatonic:"></a><!-- ws:end:WikiTextHeadingRule:44 -->Tridecatonic:</h4>
</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>2 2 2 2 2 1 2 2 2 2 2 2 1</strong></tt><br />
</td>
        <td>Agmon Diatonic DS5, Ivan Wyschnegradsky's diatonicized chromatic scale - <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/11L%202s">11L 2s</a><br />
</td>
    </tr>
</table>

<br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:46:&lt;h4&gt; --><h4 id="toc22"><a name="Scales / Modes---Tetradecatonic:"></a><!-- ws:end:WikiTextHeadingRule:46 -->Tetradecatonic:</h4>
</td>
    </tr>
    <tr>
        <td><strong><span style="font-family: monospace;">2 2 1 2 2 2 1 2 2 1 2 2 2 1</span></strong><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>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc23"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:48 -->Instruments<!-- ws:start:WikiTextAnchorRule:91:&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:91 --></h1>
 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:1645:&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:1645 -->24-tone &quot;1/4-tone&quot; Guitar by Ron Sword / Sword guitars<br />
<br />
Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called <a class="wiki_link" href="/Iceface%20Tuning">Iceface tuning</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc24"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:50 -->Compositions</h1>
 <!-- ws:start:WikiTextHeadingRule:52:&lt;h2&gt; --><h2 id="toc25"><a name="Compositions-Music"></a><!-- ws:end:WikiTextHeadingRule:52 -->Music</h2>
 <span class="ymp-btn-page-play ymp-media-eb56a917c62000140296c2e1a6d89911"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex3.mp3" rel="nofollow">Microhex3</a></span> <span class="ymp-btn-page-play ymp-media-750620955011dabb8bc98c7c8830cbb8"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex4.mp3" rel="nofollow">Microhex4</a></span> and <span class="ymp-btn-page-play ymp-media-4aeb7c299c23983ba8081a591562e704"><a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex5.mp3" rel="nofollow">Microhex5</a></span> by <a class="wiki_link_ext" href="http://www.96edo.com/About_me.html" rel="nofollow">Shaahin Mohajeri</a><br />
<span class="ymp-btn-page-play ymp-media-30832be5610668b4b42451c8ce82dc80"><a class="wiki_link_ext" href="http://micro.soonlabel.com/24et/quarterpicnic.mp3" rel="nofollow">Quarterpicnic</a></span> by <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/BeDell/Quarter%20Tone%20Prelude%20for%20two%20Harps.mp3" rel="nofollow">Quarter Tone Prelude For Two Harps</a> by <a class="wiki_link_ext" href="http://soundcloud.com/cerbeus-1/quarter-tone-prelude-for-two" rel="nofollow">Nathan BeDell</a><br />
<span class="ymp-btn-page-play ymp-media-2738730931ba168a7364a84b5417488d"><a class="wiki_link_ext" href="http://micro.soonlabel.com/24et/daily20111021-fretless-1.mp3" rel="nofollow">Fretless Chrome 1</a></span> and <span class="ymp-btn-page-play ymp-media-d35cb71128e927701930058ebd16b778"><a class="wiki_link_ext" href="http://micro.soonlabel.com/24et/daily20111021-fretless-2.mp3" rel="nofollow">Fretless Chrome 2</a></span> by <a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1477" rel="nofollow" target="_blank">Chris Vaisvil</a><br />
<a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lament.mp3" rel="nofollow">Lament</a> by <a class="wiki_link_ext" href="http://soundcloud.com/jdfreivald/lament" rel="nofollow">Jake Freivald</a>. In the <a class="wiki_link" href="/freivaldneutral24">freivaldneutral24</a> scale.<br />
<a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/mo-happy-happy" rel="nofollow">Mo - Happy - Happy</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/mo-happy-happy.mp3" rel="nofollow">play</a> by Jake Freivald in Neuter[7] (2.3.11 mohajira), 24et tuning<br />
<a class="wiki_link_ext" href="http://www.youtube.com/watch?v=q1Lp8AtKK9o" rel="nofollow">Autumn Winds</a>, <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Igxe3DwbFJ4&amp;feature=c4-overview&amp;list=UUvq5bg-LvOS6adpB5efPTyQ" rel="nofollow">Easter Time at Nine</a>, <a class="wiki_link_ext" href="http://www.youtube.com/watch?v=z_3uhA9Cq08&amp;list=UUvq5bg-LvOS6adpB5efPTyQ" rel="nofollow">Waters of Persia</a> by William Lynch in mohajira, 24et tuning.<br />
<a class="wiki_link_ext" href="https://soundcloud.com/mason-l-green/serena" rel="nofollow">Serena</a>, by Mason Green (intro and coda in 24edo, the rest is in 12edo)<br />
<a class="wiki_link_ext" href="https://www.youtube.com/watch?v=yzvXEMYgHCY" rel="nofollow">Autumn Girl</a>, by Mason Green<br />
<br />
<!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc26"><a name="Compositions-About"></a><!-- ws:end:WikiTextHeadingRule:54 -->About</h2>
 &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://en.wikipedia.org/wiki/List_of_quarter_tone_pieces" rel="nofollow">List of quartertone pieces on Wikipedia</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:56:&lt;h1&gt; --><h1 id="toc27"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:56 -->Practical Theory / Books</h1>
 <br />
<!-- ws:start:WikiTextRemoteImageRule:1646:&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:1646 --><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:58:&lt;h1&gt; --><h1 id="toc28"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:58 -->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> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeFMH6cd" rel="nofollow">Permalink</a><br />
<a class="wiki_link_ext" href="http://www.96edo.com/24_EDO.html" rel="nofollow">About 24-EDO</a> by Shaahin Mohajeri <a class="wiki_link_ext" href="http://www.webcitation.org/5xeFBNdQW" rel="nofollow">Permalink</a><br />
<a class="wiki_link_ext" href="https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit" rel="nofollow">Notation and Chord Names for 24-EDO</a> by William Lynch<br />
<a class="wiki_link_ext" href="http://sonic-arts.org/darreg/contents.htm" rel="nofollow">The place of QUARTERTONES in Today's Xenharmonics</a> by <a class="wiki_link" href="/Ivor%20Darreg">Ivor Darreg</a> <a class="wiki_link_ext" href="http://sonic-arts.org/darreg/dar8.htm" rel="nofollow">Permalink</a></body></html>
Retrieved from "https://en.xen.wiki/index.php?title=24edo&oldid=7089"