24edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2011-06-~~10 18~~:30:31 UTC</tt>.

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-25 17:26:22 UTC</tt>.

: The original revision id was <tt>~~235859654~~</tt>.

: The original revision id was <tt>242793765</tt>.

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

The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 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.

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.

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

|| Degree || Cents ||= Approximate Ratios* ||

|| 0 || 0 ||= 1/1 ||

|| 1 || 50 ||= 33/32, 34/33 ||

|| 2 || 100 ||= 17/16, 18/17 ||

|| 3 || 150 ||= 12/11 ||

|| 4 || 200 ||= 9/8 ||

|| 5 || 250 ||= 22/19 ||

|| 6 || 300 ||= 19/16 ||

|| 7 || 350 ||= 11/9 ||

|| 8 || 400 ||= 24/19 ||

|| 9 || 450 ||= 22/17 ||

|| 10 || 500 ||= 4/3 ||

|| 11 || 550 ||= 11/8 ||

|| 12 || 600 ||= 17/12 ||

|| 13 || 650 ||= 16/11 ||

|| 14 || 700 ||= 3/2 ||

|| 15 || 750 ||= 17/11 ||

|| 16 || 800 ||= 19/12 ||

|| 17 || 850 ||= 18/11 ||

|| 18 || 900 ||= 32/19 ||

|| 19 || 950 ||= 19/11 ||

|| 20 || 1000 ||= 16/9 ||

|| 21 || 1050 ||= 11/6 ||

|| 22 || 1100 ||= 17/9, 32/17 ||

|| 23 || 1150 ||= 33/17, 64/33 ||

|| 24 || 1200 ||= 2/1 ||

*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)

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For example, the scale 0-5-10-15-20 is written as C-D[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] (or E[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]]) F G[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] (or A[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]]) Bb.

And in persian music:

~~~~[[image:http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg caption="Sori,Koron" link="24edo"]]~~~~

[[image:http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg align="center" caption="Sori,Koron" link="24edo"]]

=Scales / Modes=

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

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.

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.

<tr>

<td>Degree

</td>

<td>Cents

</td>

<td style="text-align: center;">Approximate Ratios*

</td>

</tr>

<tr>

<td>0

</td>

<td>0

</td>

<td style="text-align: center;">1/1

</td>

</tr>

<tr>

<td>1

</td>

<td>50

</td>

<td style="text-align: center;">33/32, 34/33

</td>

</tr>

<tr>

<td>2

</td>

<td>100

</td>

<td style="text-align: center;">17/16, 18/17

</td>

</tr>

<tr>

<td>3

</td>

<td>150

</td>

<td style="text-align: center;">12/11

</td>

</tr>

<tr>

<td>4

</td>

<td>200

</td>

<td style="text-align: center;">9/8

</td>

</tr>

<tr>

<td>5

</td>

<td>250

</td>

<td style="text-align: center;">22/19

</td>

</tr>

<tr>

<td>6

</td>

<td>300

</td>

<td style="text-align: center;">19/16

</td>

</tr>

<tr>

<td>7

</td>

<td>350

</td>

<td style="text-align: center;">11/9

</td>

</tr>

<tr>

<td>8

</td>

<td>400

</td>

<td style="text-align: center;">24/19

</td>

</tr>

<tr>

<td>9

</td>

<td>450

</td>

<td style="text-align: center;">22/17

</td>

</tr>

<tr>

<td>10

</td>

<td>500

</td>

<td style="text-align: center;">4/3

</td>

</tr>

<tr>

<td>11

</td>

<td>550

</td>

<td style="text-align: center;">11/8

</td>

</tr>

<tr>

<td>12

</td>

<td>600

</td>

<td style="text-align: center;">17/12

</td>

</tr>

<tr>

<td>13

</td>

<td>650

</td>

<td style="text-align: center;">16/11

</td>

</tr>

<tr>

<td>14

</td>

<td>700

</td>

<td style="text-align: center;">3/2

</td>

</tr>

<tr>

<td>15

</td>

<td>750

</td>

<td style="text-align: center;">17/11

</td>

</tr>

<tr>

<td>16

</td>

<td>800

</td>

<td style="text-align: center;">19/12

</td>

</tr>

<tr>

<td>17

</td>

<td>850

</td>

<td style="text-align: center;">18/11

</td>

</tr>

<tr>

<td>18

</td>

<td>900

</td>

<td style="text-align: center;">32/19

</td>

</tr>

<tr>

<td>19

</td>

<td>950

</td>

<td style="text-align: center;">19/11

</td>

</tr>

<tr>

<td>20

</td>

<td>1000

</td>

<td style="text-align: center;">16/9

</td>

</tr>

<tr>

<td>21

</td>

<td>1050

</td>

<td style="text-align: center;">11/6

</td>

</tr>

<tr>

<td>22

</td>

<td>1100

</td>

<td style="text-align: center;">17/9, 32/17

</td>

</tr>

<tr>

<td>23

</td>

<td>1150

</td>

<td style="text-align: center;">33/17, 64/33

</td>

</tr>

<tr>

<td>24

</td>

<td>1200

</td>

<td style="text-align: center;">2/1

</td>

</tr>

</table>

*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 ~~major fourth in~~ 24-~~tET~~ is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):

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 (&quot;major&quot; chord with a 9:8 and a 11:8 above the root)

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<h1 id="toc4"><a name="Notation"></a>Notation</h1>

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

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

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

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

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

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

<img src="http://www.wfg.woodwind.org/n/f34_blue.gif" alt="external image f34_blue.gif" title="external image f34_blue.gif" /> = three-quarter-tone flat

<img src="http://www.wfg.woodwind.org/n/f34_blue.gif" alt="external image f34_blue.gif" title="external image f34_blue.gif" /> = three-quarter-tone flat

Alternately, in another notation system,

<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> can be used as a quarter-tone sharp,

<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> can be used as a quarter-tone sharp,

††† (the horizontal line should connect all three vertical lines) can be used as a three-quartertone sharp,

<img src="http://www.dolmetsch.com/onequarterflat.gif" alt="external image onequarterflat.gif" title="external image onequarterflat.gif" /> = quartertone flat,

<img src="http://www.dolmetsch.com/onequarterflat.gif" alt="external image onequarterflat.gif" title="external image onequarterflat.gif" /> = quartertone flat,

<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" /> can be used as a three-quarter-tone flat.

<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" /> can be used as a three-quarter-tone flat.

For example, the scale 0-5-10-15-20 is written as C-D<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> (or E<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" />) F G<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> (or A<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" />) Bb.

For example, the scale 0-5-10-15-20 is written as C-D<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> (or E<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" />) F G<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> (or A<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" />) Bb.

And in persian music:

<~~span~~ style=~~"display: block~~; text-align: center;">~~<table class="captionBox"><tr><td class="captionedImage"><a href="/24edo"><img src="http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg" alt="Sori,Koron" title="Sori,Koron" /></a></td></tr><tr><td class="imageCaption">Sori,Koron</td></tr></table><!-- ws:end:WikiTextRemoteImageRule:~~1104~~ --~~></span>

<table class="captionBox" style="margin-left:auto; margin-right:auto;"><tr><td class="captionedImage"><a href="/24edo"><img src="http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg" alt="Sori,Koron" title="Sori,Koron" /></a></td></tr><tr><td class="imageCaption">Sori,Koron</td></tr></table>

<h1 id="toc5"><a name="Scales / Modes"></a>Scales / Modes</h1>

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<h1 id="toc7"><a name="Instruments"></a>Instruments<a name="instruments"></a></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.

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

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

<h1 id="toc8"><a name="Compositions"></a>Compositions</h1>

Line 1,421:

Line 1,665:

<h1 id="toc11"><a name="Practical Theory / Books"></a>Practical Theory / Books</h1>

<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;" />&quot;Icosakaiteraphonic Scales for Guitar&quot; - 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.

<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;" />&quot;Icosakaiteraphonic Scales for Guitar&quot; - 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.

<h1 id="toc12"><a name="External links"></a>External links</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-10 18:30:31 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-25 17:26:22 UTC</tt>.<br>
-: The original revision id was <tt>235859654</tt>.<br>
+: The original revision id was <tt>242793765</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 14: / Line 14: @@
 The [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.
-The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 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.
+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.
-The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):
+|| Degree || Cents ||= Approximate Ratios* ||
+|| 0 || 0 ||= 1/1 ||
+|| 1 || 50 ||= 33/32, 34/33 ||
+|| 2 || 100 ||= 17/16, 18/17 ||
+|| 3 || 150 ||= 12/11 ||
+|| 4 || 200 ||= 9/8 ||
+|| 5 || 250 ||= 22/19 ||
+|| 6 || 300 ||= 19/16 ||
+|| 7 || 350 ||= 11/9 ||
+|| 8 || 400 ||= 24/19 ||
+|| 9 || 450 ||= 22/17 ||
+|| 10 || 500 ||= 4/3 ||
+|| 11 || 550 ||= 11/8 ||
+|| 12 || 600 ||= 17/12 ||
+|| 13 || 650 ||= 16/11 ||
+|| 14 || 700 ||= 3/2 ||
+|| 15 || 750 ||= 17/11 ||
+|| 16 || 800 ||= 19/12 ||
+|| 17 || 850 ||= 18/11 ||
+|| 18 || 900 ||= 32/19 ||
+|| 19 || 950 ||= 19/11 ||
+|| 20 || 1000 ||= 16/9 ||
+|| 21 || 1050 ||= 11/6 ||
+|| 22 || 1100 ||= 17/9, 32/17 ||
+|| 23 || 1150 ||= 33/17, 64/33 ||
+|| 24 || 1200 ||= 2/1 ||
+*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):
 -4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
@@ Line 128: / Line 156: @@
 For example, the scale 0-5-10-15-20 is written as C-D[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] (or E[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]]) F G[[image:http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif]] (or A[[image:http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png]]) Bb.
 And in persian music:
-&lt;span style="display: block; text-align: center;"&gt;[[image:http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg caption="Sori,Koron" link="24edo"]]&lt;/span&gt;
+[[image:http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg align="center" caption="Sori,Koron" link="24edo"]]
 =Scales / Modes=
@@ Line 264: / Line 293: @@
   The &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit&lt;/a&gt; approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.&lt;br /&gt;
 &lt;br /&gt;
-The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;3*24 subgroup&lt;/a&gt; 2.3.125.35.11.325.17 &lt;a class="wiki_link" href="/just%20intonation%20subgroup"&gt;just intonation subgroup&lt;/a&gt;, making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17.&lt;br /&gt;
+The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit &lt;a class="wiki_link" href="/k%2AN%20subgroups"&gt;3*24 subgroup&lt;/a&gt; 2.3.125.35.11.325.17 &lt;a class="wiki_link" href="/just%20intonation%20subgroup"&gt;just intonation subgroup&lt;/a&gt;, making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Degree&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Cents&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Approximate Ratios*&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;50&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;33/32, 34/33&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;100&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17/16, 18/17&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;150&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;12/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;200&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;9/8&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;250&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;22/19&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;300&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;19/16&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;350&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;11/9&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;400&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;24/19&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;450&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;22/17&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;500&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;4/3&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;550&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;11/8&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;600&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17/12&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;650&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;16/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;700&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;3/2&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;750&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;16&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;800&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;19/12&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;850&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;18/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;900&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;32/19&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;950&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;19/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;20&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1000&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;16/9&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;21&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1050&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;11/6&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;22&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1100&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17/9, 32/17&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1150&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;33/17, 64/33&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;24&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;1200&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+*based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.&lt;br /&gt;
 &lt;br /&gt;
-The 11th harmonic, and intervals derived from it, (11:10, 11:9, 11:8, 11:6, 12:11, 15:11, 16:11, 18:11, 20:11) are very well approximated in 24-tone equal temperament. The major fourth in 24-tET is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half a perfect fifth. Some good chords in 24-tET are (the numbers are degree numbers, e.g. 4 is a major second, 8 is a major third):&lt;br /&gt;
+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):&lt;br /&gt;
 &lt;br /&gt;
 -4-8-11-14 (&amp;quot;major&amp;quot; chord with a 9:8 and a 11:8 above the root)&lt;br /&gt;
@@ Line 797: / Line 1,041: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Notation&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextRemoteImageRule:1092:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/s14_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/s14_blue.gif" alt="external image s14_blue.gif" title="external image s14_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1092 --&gt; = quarter-tone sharp&lt;br /&gt;
+  &lt;!-- ws:start:WikiTextRemoteImageRule:1302:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/s14_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/s14_blue.gif" alt="external image s14_blue.gif" title="external image s14_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1302 --&gt; = quarter-tone sharp&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1093:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/s34_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/s34_blue.gif" alt="external image s34_blue.gif" title="external image s34_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1093 --&gt; = three-quarter-tone sharp&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1303:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/s34_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/s34_blue.gif" alt="external image s34_blue.gif" title="external image s34_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1303 --&gt; = three-quarter-tone sharp&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1094:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/f14_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/f14_blue.gif" alt="external image f14_blue.gif" title="external image f14_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1094 --&gt; = quarter-tone flat&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1304:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/f14_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/f14_blue.gif" alt="external image f14_blue.gif" title="external image f14_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1304 --&gt; = quarter-tone flat&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1095:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/f34_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/f34_blue.gif" alt="external image f34_blue.gif" title="external image f34_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1095 --&gt; = three-quarter-tone flat&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1305:&amp;lt;img src=&amp;quot;http://www.wfg.woodwind.org/n/f34_blue.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.wfg.woodwind.org/n/f34_blue.gif" alt="external image f34_blue.gif" title="external image f34_blue.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1305 --&gt; = three-quarter-tone flat&lt;br /&gt;
 &lt;br /&gt;
 Alternately, in another notation system,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1096:&amp;lt;img src=&amp;quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1096 --&gt; can be used as a quarter-tone sharp,&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1306:&amp;lt;img src=&amp;quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1306 --&gt; can be used as a quarter-tone sharp,&lt;br /&gt;
 ††† (the horizontal line should connect all three vertical lines) can be used as a three-quartertone sharp,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1097:&amp;lt;img src=&amp;quot;http://www.dolmetsch.com/onequarterflat.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.dolmetsch.com/onequarterflat.gif" alt="external image onequarterflat.gif" title="external image onequarterflat.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1097 --&gt; = quartertone flat,&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1307:&amp;lt;img src=&amp;quot;http://www.dolmetsch.com/onequarterflat.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.dolmetsch.com/onequarterflat.gif" alt="external image onequarterflat.gif" title="external image onequarterflat.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1307 --&gt; = quartertone flat,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1098:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;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" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1098 --&gt; can be used as a three-quarter-tone flat.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1308:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;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" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1308 --&gt; can be used as a three-quarter-tone flat.&lt;br /&gt;
-For example, the scale 0-5-10-15-20 is written as C-D&lt;!-- ws:start:WikiTextRemoteImageRule:1099:&amp;lt;img src=&amp;quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1099 --&gt; (or E&lt;!-- ws:start:WikiTextRemoteImageRule:1100:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;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" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1100 --&gt;) F G&lt;!-- ws:start:WikiTextRemoteImageRule:1101:&amp;lt;img src=&amp;quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1101 --&gt; (or A&lt;!-- ws:start:WikiTextRemoteImageRule:1102:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;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" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1102 --&gt;) Bb.&lt;br /&gt;
+For example, the scale 0-5-10-15-20 is written as C-D&lt;!-- ws:start:WikiTextRemoteImageRule:1309:&amp;lt;img src=&amp;quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1309 --&gt; (or E&lt;!-- ws:start:WikiTextRemoteImageRule:1310:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;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" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1310 --&gt;) F G&lt;!-- ws:start:WikiTextRemoteImageRule:1311:&amp;lt;img src=&amp;quot;http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1311 --&gt; (or A&lt;!-- ws:start:WikiTextRemoteImageRule:1312:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Arabic_music_notation_half_flat.svg/9px-Arabic_music_notation_half_flat.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;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" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1312 --&gt;) Bb.&lt;br /&gt;
 And in persian music:&lt;br /&gt;
-&lt;span style="display: block; text-align: center;"&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:1104:&amp;lt;a href=&amp;quot;/24edo&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg&amp;quot; alt=&amp;quot;Sori,Koron&amp;quot; title=&amp;quot;Sori,Koron&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt; --&gt;&lt;table class="captionBox"&gt;&lt;tr&gt;&lt;td class="captionedImage"&gt;&lt;a href="/24edo"&gt;&lt;img src="http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg" alt="Sori,Koron" title="Sori,Koron" /&gt;&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td class="imageCaption"&gt;Sori,Koron&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1104 --&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1314:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;a href=&amp;quot;/24edo&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg&amp;quot; alt=&amp;quot;Sori,Koron&amp;quot; title=&amp;quot;Sori,Koron&amp;quot; /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;table class="captionBox" style="margin-left:auto; margin-right:auto;"&gt;&lt;tr&gt;&lt;td class="captionedImage"&gt;&lt;a href="/24edo"&gt;&lt;img src="http://240edo.googlepages.com/2.jpg/2-custom;size:430,120.jpg" alt="Sori,Koron" title="Sori,Koron" /&gt;&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td class="imageCaption"&gt;Sori,Koron&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1314 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Scales / Modes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Scales / Modes&lt;/h1&gt;
@@ Line 1,407: / Line 1,651: @@
 &lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Instruments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Instruments&lt;!-- ws:start:WikiTextAnchorRule:42:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@instruments&amp;quot; title=&amp;quot;Anchor: instruments&amp;quot;/&amp;gt; --&gt;&lt;a name="instruments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:42 --&gt;&lt;/h1&gt;
   The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This &lt;a class="wiki_link" href="/Microtonal%20Keyboards#twelvenoteoctavescales"&gt;&amp;quot;12 note octave scales&amp;quot;&lt;/a&gt; approach is used in a wide part of the existing literature - see below.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1105:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/Sword_quartertone_stratsm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 195px; width: 406px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/Sword_quartertone_stratsm.jpg" alt="external image Sword_quartertone_stratsm.jpg" title="external image Sword_quartertone_stratsm.jpg" style="height: 195px; width: 406px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1105 --&gt;24-tone &amp;quot;1/4-tone&amp;quot; Guitar by Ron Sword / Sword guitars&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1315:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/Sword_quartertone_stratsm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 195px; width: 406px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/Sword_quartertone_stratsm.jpg" alt="external image Sword_quartertone_stratsm.jpg" title="external image Sword_quartertone_stratsm.jpg" style="height: 195px; width: 406px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1315 --&gt;24-tone &amp;quot;1/4-tone&amp;quot; Guitar by Ron Sword / Sword guitars&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;Compositions&lt;/h1&gt;
@@ Line 1,421: / Line 1,665: @@
 &lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&gt;Practical Theory / Books&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1106:&amp;lt;img src=&amp;quot;http://ronsword.com/images/24_tet_Coversm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 236px; width: 179px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1106 --&gt;&lt;strong&gt;&amp;quot;Icosakaiteraphonic Scales for Guitar&amp;quot;&lt;/strong&gt; - 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.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1316:&amp;lt;img src=&amp;quot;http://ronsword.com/images/24_tet_Coversm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 236px; width: 179px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1316 --&gt;&lt;strong&gt;&amp;quot;Icosakaiteraphonic Scales for Guitar&amp;quot;&lt;/strong&gt; - 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.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="External links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;External links&lt;/h1&gt;