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:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-~~29 10~~:35:06 UTC</tt>.

: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-31 09:37:08 UTC</tt>.

: The original revision id was <tt>~~597443722~~</tt>.

: The original revision id was <tt>597550782</tt>.

: The revision comment was: <tt></tt>

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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

|| Degree || Cents coarse/fine

DMS ||= Approximate Ratios* ||

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

|| 1 || 50

|| 1 || 50, 60

15° ||= 33/32, 34/33 ||

|| 2 || 100

|| 2 || 100, 120

30° ||= 17/16, 18/17 ||

|| 3 || 150

|| 3 || 150, 180

45° ||= 12/11 ||

|| 4 || 200

|| 4 || 200, 240

60° ||= 9/8 ||

|| 5 || 250

|| 5 || 250, 300

75° ||= 22/19 ||

|| 6 || 300

|| 6 || 300, 360

90° ||= 19/16 ||

|| 7 || 350

|| 7 || 350, 420

105° ||= 11/9 ||

|| 8 || 400

|| 8 || 400, 480

120° ||= 24/19 ||

|| 9 || 450

|| 9 || 450, 540

135° ||= 22/17 ||

|| 10 || 500

|| 10 || 500, 600

150° ||= 4/3 ||

|| 11 || 550

|| 11 || 550, 660

165° ||= 11/8 ||

|| 12 || 600

|| 12 || 600, 720

180° ||= 17/12 ||

|| 13 || 650

|| 13 || 650, 780

195° ||= 16/11 ||

|| 14 || 700

|| 14 || 700, 840

210° ||= 3/2 ||

|| 15 || 750

|| 15 || 750, 900

225° ||= 17/11 ||

|| 16 || 800

|| 16 || 800, 960

240* ||= 19/12 ||

|| 17 || 850

|| 17 || 850, 1020

255° ||= 18/11 ||

|| 18 || 900

|| 18 || 900, 1080

270° ||= 32/19 ||

|| 19 || 950

|| 19 || 950, 1140

285° ||= 19/11 ||

|| 20 || 1000

|| 20 || 1000, 1200

300° ||= 16/9 ||

|| 21 || 1050

|| 21 || 1050, 1260

315° ||= 11/6 ||

|| 22 || 1100

|| 22 || 1100, 1320

330° ||= 17/9, 32/17 ||

|| 23 || 1150

|| 23 || 1150, 1380

345° ||= 33/17, 64/33 ||

*based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.

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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 ~~šlêndros~~. 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.)

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 < [[tel/24 38 56 67 83 89|24 38 56 67 83 89]] |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||

||= ~~[[tel:531441/524288|531441/524288]]~~ || | -19 12 > ||> 23.46 ||= Pythagorean Comma ||= || ||

||= || | -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 ||= || ||

||= ~~[[tel:5201701/5149091|5201701/5149091]]~~ || | 26 -12 -3 > ||> 17.60 ||= Misty Comma ||= || ||

||= || | 26 -12 -3 > ||> 17.60 ||= Misty Comma ||= || ||

||= 32805/32768 || | -15 8 1 > ||> 1.95 ||= Schisma ||= || ||

||= ~~[[tel:1465155/1465142|1465155/1465142]]~~ || | 161 -84 -12 > ||> 0.02 ||= Atom ||= || ||

||= || | 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 ||= || ||

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||= 676/675 || | 2 -3 -2 0 0 2 > ||> 2.56 ||= Parizeksma ||= || ||

=Intervals=

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

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.

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<tr>

<td>~~[[#|~~Degree]]

<td>Degree

</td>

<td>Cents

<td>Cents coarse/fine

DMS

</td>

Line 485:

<td>1

</td>

<td>50

<td>50, 60

15°

</td>

Line 494:

<td>2

</td>

<td>100

<td>100, 120

30°

</td>

Line 503:

<td>3

</td>

<td>150

<td>150, 180

45°

</td>

Line 512:

<td>4

</td>

<td>200

<td>200, 240

60°

</td>

Line 521:

<td>5

</td>

<td>250

<td>250, 300

75°

</td>

Line 530:

<td>6

</td>

<td>300

<td>300, 360

90°

</td>

Line 539:

<td>7

</td>

<td>350

<td>350, 420

105°

</td>

Line 548:

<td>8

</td>

<td>400

<td>400, 480

120°

</td>

Line 557:

<td>9

</td>

<td>450

<td>450, 540

135°

</td>

Line 566:

<td>10

</td>

<td>500

<td>500, 600

150°

</td>

Line 575:

<td>11

</td>

<td>550

<td>550, 660

165°

</td>

Line 584:

<td>12

</td>

<td>600

<td>600, 720

180°

</td>

Line 593:

<td>13

</td>

<td>650

<td>650, 780

195°

</td>

Line 602:

<td>14

</td>

<td>700

<td>700, 840

210°

</td>

Line 611:

<td>15

</td>

<td>750

<td>750, 900

225°

</td>

Line 620:

<td>16

</td>

<td>800

<td>800, 960

240*

</td>

Line 629:

<td>17

</td>

<td>850

<td>850, 1020

255°

</td>

Line 638:

<td>18

</td>

<td>900

<td>900, 1080

270°

</td>

Line 647:

<td>19

</td>

<td>950

<td>950, 1140

285°

</td>

Line 656:

<td>20

</td>

<td>1000

<td>1000, 1200

300°

</td>

Line 665:

<td>21

</td>

<td>1050

<td>1050, 1260

315°

</td>

Line 674:

<td>22

</td>

<td>1100

<td>1100, 1320

330°

</td>

Line 683:

<td>23

</td>

<td>1150

<td>1150, 1380

345°

</td>

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Its inversion, 0-3-6-10-14 (&quot;minor&quot;)

0-7-14 (&quot;neutral&quot;)

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 ~~šlêndros~~. 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.)

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.)

<h3 id="toc2"><a name="x24edo as a temperament--Commas"></a>Commas</h3>

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</tr>

<tr>

<td style="text-align: center;">~~[[tel:531441/524288|531441/524288]]~~

</td>

<td>| -19 12 &gt;

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</tr>

<tr>

<td style="text-align: center;">~~[[tel:5201701/5149091|5201701/5149091]]~~

</td>

<td>| 26 -12 -3 &gt;

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</tr>

<tr>

<td style="text-align: center;">~~[[tel:1465155/1465142|1465155/1465142]]~~

</td>

<td>| 161 -84 -12 &gt;

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

~~<table class="wiki_table">~~

~~<tr>~~

~~<td>24 ~~

~~</td>~~

~~<td>1200~~

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~</table>~~

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.

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<h2 id="toc5"><a name="Notation-Mainstream Quartertone Notation"></a>Mainstream Quartertone Notation</h2>

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

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

<img src="http://www.wfg.woodwind.org/n/s34_blue.gif" alt="external image s34_blue.gif" title="external image s34_blue.gif" /> or #^ = three-quarter-tone sharp or &quot;Jump-Sharp&quot;

<img src="http://www.wfg.woodwind.org/n/s34_blue.gif" alt="external image s34_blue.gif" title="external image s34_blue.gif" /> or #^ = three-quarter-tone sharp or &quot;Jump-Sharp&quot;

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

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

<img src="http://www.wfg.woodwind.org/n/f34_blue.gif" alt="external image f34_blue.gif" title="external image f34_blue.gif" /> or bv = three-quarter-tone flat or &quot;Drop-Flat&quot;

<img src="http://www.wfg.woodwind.org/n/f34_blue.gif" alt="external image f34_blue.gif" title="external image f34_blue.gif" /> or bv = three-quarter-tone flat or &quot;Drop-Flat&quot;

Pros: Familiar, fairly easy to learn

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<h2 id="toc6"><a name="Notation-Alternate Quartertone Accidentals"></a>Alternate Quartertone Accidentals</h2>

<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> = quarter-tone sharp or Jump

<img src="http://faculty.mansfield.edu/jmurphy/oldsite/quartertonesharp.gif" alt="external image quartertonesharp.gif" title="external image quartertonesharp.gif" /> = quarter-tone sharp or Jump

††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp

<img src="http://www.dolmetsch.com/onequarterflat.gif" alt="external image onequarterflat.gif" title="external image onequarterflat.gif" /> = quarter-tone flat or Drop

<img src="http://www.dolmetsch.com/onequarterflat.gif" alt="external image onequarterflat.gif" title="external image onequarterflat.gif" /> = quarter-tone flat or Drop

<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" /> = three quarter-tones flat or drop-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" /> = three quarter-tones flat or drop-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.

Pros: Very easy to distinguish accidentals from one another

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<h2 id="toc7"><a name="Notation-And in Persian music"></a>And in Persian music</h2>

<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;" />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

<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;" />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

<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;" />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

<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;" />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

Pros: Easy to read

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with Sagittal signs for sharp and flat.

<img src="/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG" alt="sagittal 24.PNG" title="sagittal 24.PNG" />

<img src="/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG" alt="sagittal 24.PNG" title="sagittal 24.PNG" />

<h1 id="toc9"><a name="Interval Alterations"></a>Interval Alterations</h1>

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William Lynch considers these as some possible good tetrads:

<img src="/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG" alt="Three chords.PNG" title="Three chords.PNG" />

<img src="/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG" alt="Three chords.PNG" title="Three chords.PNG" />

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<h1 id="toc23"><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

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>.

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<h1 id="toc27"><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="toc28"><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:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-29 10:35:06 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-31 09:37:08 UTC</tt>.<br>
-: The original revision id was <tt>597443722</tt>.<br>
+: The original revision id was <tt>597550782</tt>.<br>
 : The revision comment was: <tt></tt><br>
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@@ Line 17: / Line 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.
-|| [[#|Degree]] || Cents
+|| Degree || Cents coarse/fine
 DMS ||= Approximate Ratios* ||
 || 0 || 0 ||= 1/1 ||
-|| 1 || 50
+|| 1 || 50, 60
 ° ||= 33/32, 34/33 ||
-|| 2 || 100
+|| 2 || 100, 120
 ° ||= 17/16, 18/17 ||
-|| 3 || 150
+|| 3 || 150, 180
 ° ||= 12/11 ||
-|| 4 || 200
+|| 4 || 200, 240
 ° ||= 9/8 ||
-|| 5 || 250
+|| 5 || 250, 300
 ° ||= 22/19 ||
-|| 6 || 300
+|| 6 || 300, 360
 ° ||= 19/16 ||
-|| 7 || 350
+|| 7 || 350, 420
 ° ||= 11/9 ||
-|| 8 || 400
+|| 8 || 400, 480
 ° ||= 24/19 ||
-|| 9 || 450
+|| 9 || 450, 540
 ° ||= 22/17 ||
-|| 10 || 500
+|| 10 || 500, 600
 ° ||= 4/3 ||
-|| 11 || 550
+|| 11 || 550, 660
 ° ||= 11/8 ||
-|| 12 || 600
+|| 12 || 600, 720
 ° ||= 17/12 ||
-|| 13 || 650
+|| 13 || 650, 780
 ° ||= 16/11 ||
-|| 14 || 700
+|| 14 || 700, 840
 ° ||= 3/2 ||
-|| 15 || 750
+|| 15 || 750, 900
 ° ||= 17/11 ||
-|| 16 || 800
+|| 16 || 800, 960
 * ||= 19/12 ||
-|| 17 || 850
+|| 17 || 850, 1020
 ° ||= 18/11 ||
-|| 18 || 900
+|| 18 || 900, 1080
 ° ||= 32/19 ||
-|| 19 || 950
+|| 19 || 950, 1140
 ° ||= 19/11 ||
-|| 20 || 1000
+|| 20 || 1000, 1200
 ° ||= 16/9 ||
-|| 21 || 1050
+|| 21 || 1050, 1260
 ° ||= 11/6 ||
-|| 22 || 1100
+|| 22 || 1100, 1320
 ° ||= 17/9, 32/17 ||
-|| 23 || 1150
+|| 23 || 1150, 1380
 ° ||= 33/17, 64/33 ||
 *based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.
@@ Line 73: / Line 73: @@
 Its inversion, 0-3-6-10-14 ("minor")
 -7-14 ("neutral")
--5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 ([[godzilla]]), that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 ([[mohajira]]), a heptatonic scale close to several Arabic scales.)
+-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===
 EDO tempers out the following commas. (Note: This assumes val &lt; [[tel/24 38 56 67 83 89|24 38 56 67 83 89]] |.)
 ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
-||= [[tel:531441/524288|531441/524288]] || | -19 12 &gt; ||&gt; 23.46 ||= Pythagorean Comma ||=   ||   ||
+||=   || | -19 12 &gt; ||&gt; 23.46 ||= Pythagorean Comma ||=   ||   ||
 ||= 648/625 || | 3 4 -4 &gt; ||&gt; 62.57 ||= Major Diesis ||= Diminished Comma ||   ||
 ||= 128/125 || | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||   ||
 ||= 81/80 || | -4 4 -1 &gt; ||&gt; 21.51 ||= Syntonic Comma ||= Didymos Comma || Meantone Comma ||
 ||= 2048/2025 || | 11 -4 -2 &gt; ||&gt; 19.55 ||= Diaschisma ||=   ||   ||
-||= [[tel:5201701/5149091|5201701/5149091]] || | 26 -12 -3 &gt; ||&gt; 17.60 ||= Misty Comma ||=   ||   ||
+||=   || | 26 -12 -3 &gt; ||&gt; 17.60 ||= Misty Comma ||=   ||   ||
 ||= 32805/32768 || | -15 8 1 &gt; ||&gt; 1.95 ||= Schisma ||=   ||   ||
-||= [[tel:1465155/1465142|1465155/1465142]] || | 161 -84 -12 &gt; ||&gt; 0.02 ||= Atom ||=   ||   ||
+||=   || | 161 -84 -12 &gt; ||&gt; 0.02 ||= Atom ||=   ||   ||
 ||= 49/48 || | -4 -1 0 2 &gt; ||&gt; 35.70 ||= Slendro Diesis ||=   ||   ||
 ||= 245/243 || | 0 -5 1 2 &gt; ||&gt; 14.19 ||= Sensamagic ||=   ||   ||
@@ Line 99: / Line 99: @@
 ||= 676/675 || | 2 -3 -2 0 0 2 &gt; ||&gt; 2.56 ||= Parizeksma ||=   ||   ||
 =Intervals=
-|| 24 || 1200 ||= 2/1 ||
 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.
@@ Line 466: / Line 466: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td&gt;[[#|Degree]]&lt;br /&gt;
+         &lt;td&gt;Degree&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Cents&lt;br /&gt;
+         &lt;td&gt;Cents coarse/fine&lt;br /&gt;
 DMS&lt;br /&gt;
 &lt;/td&gt;
@@ Line 485: / Line 485: @@
          &lt;td&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;50&lt;br /&gt;
+         &lt;td&gt;50, 60&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 494: / Line 494: @@
          &lt;td&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;100&lt;br /&gt;
+         &lt;td&gt;100, 120&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 503: / Line 503: @@
          &lt;td&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;150&lt;br /&gt;
+         &lt;td&gt;150, 180&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 512: / Line 512: @@
          &lt;td&gt;4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;200&lt;br /&gt;
+         &lt;td&gt;200, 240&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 521: / Line 521: @@
          &lt;td&gt;5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;250&lt;br /&gt;
+         &lt;td&gt;250, 300&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 530: / Line 530: @@
          &lt;td&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;300&lt;br /&gt;
+         &lt;td&gt;300, 360&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 539: / Line 539: @@
          &lt;td&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;350&lt;br /&gt;
+         &lt;td&gt;350, 420&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 548: / Line 548: @@
          &lt;td&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;400&lt;br /&gt;
+         &lt;td&gt;400, 480&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 557: / Line 557: @@
          &lt;td&gt;9&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;450&lt;br /&gt;
+         &lt;td&gt;450, 540&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 566: / Line 566: @@
          &lt;td&gt;10&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;500&lt;br /&gt;
+         &lt;td&gt;500, 600&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 575: / Line 575: @@
          &lt;td&gt;11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;550&lt;br /&gt;
+         &lt;td&gt;550, 660&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 584: / Line 584: @@
          &lt;td&gt;12&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;600&lt;br /&gt;
+         &lt;td&gt;600, 720&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 593: / Line 593: @@
          &lt;td&gt;13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;650&lt;br /&gt;
+         &lt;td&gt;650, 780&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 602: / Line 602: @@
          &lt;td&gt;14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;700&lt;br /&gt;
+         &lt;td&gt;700, 840&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 611: / Line 611: @@
          &lt;td&gt;15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;750&lt;br /&gt;
+         &lt;td&gt;750, 900&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 620: / Line 620: @@
          &lt;td&gt;16&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;800&lt;br /&gt;
+         &lt;td&gt;800, 960&lt;br /&gt;
 *&lt;br /&gt;
 &lt;/td&gt;
@@ Line 629: / Line 629: @@
          &lt;td&gt;17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;850&lt;br /&gt;
+         &lt;td&gt;850, 1020&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 638: / Line 638: @@
          &lt;td&gt;18&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;900&lt;br /&gt;
+         &lt;td&gt;900, 1080&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 647: / Line 647: @@
          &lt;td&gt;19&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;950&lt;br /&gt;
+         &lt;td&gt;950, 1140&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 656: / Line 656: @@
          &lt;td&gt;20&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1000&lt;br /&gt;
+         &lt;td&gt;1000, 1200&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 665: / Line 665: @@
          &lt;td&gt;21&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1050&lt;br /&gt;
+         &lt;td&gt;1050, 1260&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 674: / Line 674: @@
          &lt;td&gt;22&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1100&lt;br /&gt;
+         &lt;td&gt;1100, 1320&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 683: / Line 683: @@
          &lt;td&gt;23&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1150&lt;br /&gt;
+         &lt;td&gt;1150, 1380&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 698: / Line 698: @@
 Its inversion, 0-3-6-10-14 (&amp;quot;minor&amp;quot;)&lt;br /&gt;
 -7-14 (&amp;quot;neutral&amp;quot;)&lt;br /&gt;
--5-10 (another kind of &amp;quot;neutral&amp;quot;, splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 (&lt;a class="wiki_link" href="/godzilla"&gt;godzilla&lt;/a&gt;), that is close to equi-pentatonic and also close to several Indonesian šlêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 (&lt;a class="wiki_link" href="/mohajira"&gt;mohajira&lt;/a&gt;), a heptatonic scale close to several Arabic scales.)&lt;br /&gt;
+-5-10 (another kind of &amp;quot;neutral&amp;quot;, splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 (&lt;a class="wiki_link" href="/godzilla"&gt;godzilla&lt;/a&gt;), 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 (&lt;a class="wiki_link" href="/mohajira"&gt;mohajira&lt;/a&gt;), a heptatonic scale close to several Arabic scales.)&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;
@@ Line 720: / Line 720: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;[[tel:531441/524288|531441/524288]]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;| -19 12 &amp;gt;&lt;br /&gt;
@@ Line 790: / Line 790: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;[[tel:5201701/5149091|5201701/5149091]]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;| 26 -12 -3 &amp;gt;&lt;br /&gt;
@@ Line 818: / Line 818: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;[[tel:1465155/1465142|1465155/1465142]]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;| 161 -84 -12 &amp;gt;&lt;br /&gt;
@@ Line 1,002: / Line 1,002: @@
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Intervals&lt;/h1&gt;
+  &lt;br /&gt;
-&lt;table class="wiki_table"&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;
 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&lt;br /&gt;
 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.&lt;br /&gt;
@@ Line 1,245: / Line 1,233: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Notation-Mainstream Quartertone Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Mainstream Quartertone Notation&lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextRemoteImageRule:1544:&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:1544 --&gt; or ^ = quarter-tone sharp or &amp;quot;Jump&amp;quot;&lt;br /&gt;
+  &lt;!-- ws:start:WikiTextRemoteImageRule:1534:&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:1534 --&gt; or ^ = quarter-tone sharp or &amp;quot;Jump&amp;quot;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1545:&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:1545 --&gt; or #^ = three-quarter-tone sharp or &amp;quot;Jump-Sharp&amp;quot;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1535:&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:1535 --&gt; or #^ = three-quarter-tone sharp or &amp;quot;Jump-Sharp&amp;quot;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1546:&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:1546 --&gt; or v = quarter-tone flat or &amp;quot;Drop&amp;quot;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1536:&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:1536 --&gt; or v = quarter-tone flat or &amp;quot;Drop&amp;quot;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1547:&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:1547 --&gt; or bv = three-quarter-tone flat or &amp;quot;Drop-Flat&amp;quot;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1537:&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:1537 --&gt; or bv = three-quarter-tone flat or &amp;quot;Drop-Flat&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
 Pros: Familiar, fairly easy to learn&lt;br /&gt;
@@ Line 1,254: / Line 1,242: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Notation-Alternate Quartertone Accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Alternate Quartertone Accidentals&lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextRemoteImageRule:1548:&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:1548 --&gt; = quarter-tone sharp or Jump&lt;br /&gt;
+  &lt;!-- ws:start:WikiTextRemoteImageRule:1538:&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:1538 --&gt; = quarter-tone sharp or Jump&lt;br /&gt;
 ††† (the horizontal line should connect all three vertical lines) = three quarter-tones sharp or Jump-Sharp&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1549:&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:1549 --&gt; = quarter-tone flat or Drop&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1539:&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:1539 --&gt; = quarter-tone flat or Drop&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1550:&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:1550 --&gt; = three quarter-tones flat or drop-flat&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1540:&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:1540 --&gt; = three quarter-tones flat or drop-flat&lt;br /&gt;
-For example, the scale 0-5-10-15-20 is written as C-D&lt;!-- ws:start:WikiTextRemoteImageRule:1551:&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:1551 --&gt; (or E&lt;!-- ws:start:WikiTextRemoteImageRule:1552:&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:1552 --&gt;) F G&lt;!-- ws:start:WikiTextRemoteImageRule:1553:&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:1553 --&gt; (or A&lt;!-- ws:start:WikiTextRemoteImageRule:1554:&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:1554 --&gt;) Bb.&lt;br /&gt;
+For example, the scale 0-5-10-15-20 is written as C-D&lt;!-- ws:start:WikiTextRemoteImageRule:1541:&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:1541 --&gt; (or E&lt;!-- ws:start:WikiTextRemoteImageRule:1542:&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:1542 --&gt;) F G&lt;!-- ws:start:WikiTextRemoteImageRule:1543:&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:1543 --&gt; (or A&lt;!-- ws:start:WikiTextRemoteImageRule:1544:&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:1544 --&gt;) Bb.&lt;br /&gt;
 &lt;br /&gt;
 Pros: Very easy to distinguish accidentals from one another&lt;br /&gt;
@@ Line 1,264: / Line 1,252: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Notation-And in Persian music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;span style="line-height: 1.5;"&gt;And in Persian music&lt;/span&gt;&lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextRemoteImageRule:1555:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Koron_sign.svg/200px-Koron_sign.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 25px; width: 12px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1555 --&gt;Koron (&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Koron_%28music%29" rel="nofollow"&gt;en&lt;/a&gt; | &lt;a class="wiki_link_ext" href="http://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86" rel="nofollow"&gt;fa&lt;/a&gt;) = quarter-tone flat or Jump&lt;br /&gt;
+  &lt;!-- ws:start:WikiTextRemoteImageRule:1545:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/Koron_sign.svg/200px-Koron_sign.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 25px; width: 12px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1545 --&gt;Koron (&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Koron_%28music%29" rel="nofollow"&gt;en&lt;/a&gt; | &lt;a class="wiki_link_ext" href="http://fa.wikipedia.org/wiki/%DA%A9%D8%B1%D9%86" rel="nofollow"&gt;fa&lt;/a&gt;) = quarter-tone flat or Jump&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1556:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Sori_sign.svg/200px-Sori_sign.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 25px; width: 15px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1556 --&gt;Sori (&lt;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"&gt;fa&lt;/a&gt;) = quarter-tone sharp or Drop&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1546:&amp;lt;img src=&amp;quot;http://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Sori_sign.svg/200px-Sori_sign.svg.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 25px; width: 15px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1546 --&gt;Sori (&lt;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"&gt;fa&lt;/a&gt;) = quarter-tone sharp or Drop&lt;br /&gt;
 &lt;br /&gt;
 Pros: Easy to read&lt;br /&gt;
@@ Line 1,276: / Line 1,264: @@
 with Sagittal signs for sharp and flat.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1542:&amp;lt;img src=&amp;quot;/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG" alt="sagittal 24.PNG" title="sagittal 24.PNG" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1542 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:1532:&amp;lt;img src=&amp;quot;/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/sagittal%2024.PNG/500168246/sagittal%2024.PNG" alt="sagittal 24.PNG" title="sagittal 24.PNG" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1532 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Interval Alterations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Interval Alterations&lt;/h1&gt;
@@ Line 1,299: / Line 1,287: @@
 William Lynch considers these as some possible good tetrads:&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1543:&amp;lt;img src=&amp;quot;/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG" alt="Three chords.PNG" title="Three chords.PNG" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1543 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:1533:&amp;lt;img src=&amp;quot;/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Three%20chords.PNG/489521008/Three%20chords.PNG" alt="Three chords.PNG" title="Three chords.PNG" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1533 --&gt;&lt;br /&gt;
 &lt;br /&gt;
@@ Line 2,148: / Line 2,136: @@
 &lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc23"&gt;&lt;a name="Instruments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Instruments&lt;!-- ws:start:WikiTextAnchorRule:91:&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:91 --&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:1557:&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:1557 --&gt;24-tone &amp;quot;1/4-tone&amp;quot; Guitar by Ron Sword / Sword guitars&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1547:&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:1547 --&gt;24-tone &amp;quot;1/4-tone&amp;quot; Guitar by Ron Sword / Sword guitars&lt;br /&gt;
 &lt;br /&gt;
 Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called &lt;a class="wiki_link" href="/Iceface%20Tuning"&gt;Iceface tuning&lt;/a&gt;.&lt;br /&gt;
@@ Line 2,171: / Line 2,159: @@
 &lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc27"&gt;&lt;a name="Practical Theory / Books"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;Practical Theory / Books&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:1558:&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:1558 --&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:1548:&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:1548 --&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:58:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc28"&gt;&lt;a name="External links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;External links&lt;/h1&gt;