24edo: Difference between revisions

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=24edo as a temperament=

The [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are ~~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 [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 72-tET, 84-tET or 156-tET.

The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.

|| Degree || Cents coarse/fine

DMS ||= Approximate Ratios* ||

DMS ||= Approximate Ratios* ||||= [[Ups and Downs Notation|ups and downs notation]] ||

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

|| 0 || 0 ||= 1/1 || unison || P1 ||

|| 1 || 50, 60

15° ||= 33/32, 34/33 ||

15° ||= 33/32, 34/33 || up-unison,

downminor 2nd || ^P1, vm2 ||

|| 2 || 100, 120

30° ||= 17/16, 18/17 ||

30° ||= 17/16, 18/17 || aug unison,

minor 2nd || A1, m2 ||

|| 3 || 150, 180

45° ||= 12/11 ||

45° ||= 12/11 || mid 2nd || ~2 ||

|| 4 || 200, 240

60° ||= 9/8 ||

60° ||= 9/8 || major 2nd || M2 ||

|| 5 || 250, 300

75° ||= 22/19 ||

75° ||= 22/19 || upmajor 2nd,

downminor 3rd || ^M2, vm3 ||

|| 6 || 300, 360

90° ||= 19/16 ||

90° ||= 19/16 || minor 3rd || m3 ||

|| 7 || 350, 420

105° ||= 11/9 ||

105° ||= 11/9 || mid 3rd || ~3 ||

|| 8 || 400, 480

120° ||= 24/19 ||

120° ||= 24/19 || major 3rd || M3 ||

|| 9 || 450, 540

135° ||= 22/17 ||

135° ||= 22/17 || upmajor 3rd,

down-fourth || ^M3, vP4 ||

|| 10 || 500, 600

150° ||= 4/3 ||

150° ||= 4/3 || fourth || P4 ||

|| 11 || 550, 660

165° ||= 11/8 ||

165° ||= 11/8 || up-fourth || ^P4 ||

|| 12 || 600, 720

180° ||= 17/12 ||

180° ||= 17/12 || augmented 4th,

diminished 5th || A4, d5 ||

|| 13 || 650, 780

195° ||= 16/11 ||

195° ||= 16/11 || down-fifth || vP5 ||

|| 14 || 700, 840

210° ||= 3/2 ||

210° ||= 3/2 || fifth || P5 ||

|| 15 || 750, 900

225° ||= 17/11 ||

225° ||= 17/11 || up-fifth,

downminor 6th || ^P5, vm6 ||

|| 16 || 800, 960

240* ||= 19/12 ||

240* ||= 19/12 || minor 6th || m6 ||

|| 17 || 850, 1020

255° ||= 18/11 ||

255° ||= 18/11 || mid 6th || ~6 ||

|| 18 || 900, 1080

270° ||= 32/19 ||

270° ||= 32/19 || major 6th || M6 ||

|| 19 || 950, 1140

285° ||= 19/11 ||

285° ||= 19/11 || upmajor 6th,

downminor 7th || ^M6, vm7 ||

|| 20 || 1000, 1200

300° ||= 16/9 ||

300° ||= 16/9 || minor 7th || m7 ||

|| 21 || 1050, 1260

315° ||= 11/6 ||

315° ||= 11/6 || mid 7th || ~7 ||

|| 22 || 1100, 1320

330° ||= 17/9, 32/17 ||

330° ||= 17/9, 32/17 || major 7th || M7 ||

|| 23 || 1150, 1380

345° ||= 33/17, 64/33 ||

345° ||= 33/17, 64/33 || upmajor 7th,

down-octave || ^M7, vP8 ||

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

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==Mainstream Quartertone Notation==

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

[[image:http://www.wfg.woodwind.org/n/s14_blue.gif]] or ^ = quarter-tone sharp or "Jump" or "up"

[[image:http://www.wfg.woodwind.org/n/s34_blue.gif]] or #^ = three-quarter-tone sharp or "Jump-Sharp"

[[image:http://www.wfg.woodwind.org/n/s34_blue.gif]] or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"

[[image:http://www.wfg.woodwind.org/n/f14_blue.gif]] or v = quarter-tone flat or "Drop"

[[image:http://www.wfg.woodwind.org/n/f14_blue.gif]] or v = quarter-tone flat or "Drop" or "down"

[[image:http://www.wfg.woodwind.org/n/f34_blue.gif]] or bv = three-quarter-tone flat or "Drop-Flat"

[[image:http://www.wfg.woodwind.org/n/f34_blue.gif]] or bv or vb = three-quarter-tone flat or "Drop-Flat" or "downflat"

Pros: Familiar, fairly easy to learn

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

As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral triad 0-7-14 however there are other options such as

As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as

0-9-14 (Ultra Triad) and 0-5-14 (Infra Triad), the chord names being based on what kind of third is in the chord.

0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.

These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.

For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.

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Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top

Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11

Alternatively, [[ups and downs notation]] can be used:

0-5-14 = D Fv A = D.vm = "D downminor"

0-6-14 = D F A = Dm = "D minor"

0-7-14 = D F^ A = D~ = "D mid"

0-8-14 = D F# A = D = "D" or "D major"

0-9-14 = D F#^ A = D.^ = "D dot up" or "D upmajor"

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<h1 id="toc1"><a name="x24edo as a temperament"></a>24edo as a temperament</h1>

The <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are ~~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 <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a> approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 72-tET, 84-tET or 156-tET.

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.

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</td>

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

</td>

<td colspan="2" style="text-align: center;"><a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs notation</a>

</td>

</tr>

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</td>

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

</td>

<td>unison

</td>

<td>P1

</td>

</tr>

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Line 510:

</td>

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

</td>

<td>up-unison,

downminor 2nd

</td>

<td>^P1, vm2

</td>

</tr>

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</td>

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

</td>

<td>aug unison,

minor 2nd

</td>

<td>A1, m2

</td>

</tr>

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Line 538:

</td>

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

</td>

<td>mid 2nd

</td>

<td>~2

</td>

</tr>

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Line 551:

</td>

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

</td>

<td>major 2nd

</td>

<td>M2

</td>

</tr>

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Line 564:

</td>

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

</td>

<td>upmajor 2nd,

downminor 3rd

</td>

<td>^M2, vm3

</td>

</tr>

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Line 578:

</td>

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

</td>

<td>minor 3rd

</td>

<td>m3

</td>

</tr>

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Line 591:

</td>

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

</td>

<td>mid 3rd

</td>

<td>~3

</td>

</tr>

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Line 604:

</td>

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

</td>

<td>major 3rd

</td>

<td>M3

</td>

</tr>

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Line 617:

</td>

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

</td>

<td>upmajor 3rd,

down-fourth

</td>

<td>^M3, vP4

</td>

</tr>

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Line 631:

</td>

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

</td>

<td>fourth

</td>

<td>P4

</td>

</tr>

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Line 644:

</td>

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

</td>

<td>up-fourth

</td>

<td>^P4

</td>

</tr>

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Line 657:

</td>

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

</td>

<td>augmented 4th,

diminished 5th

</td>

<td>A4, d5

</td>

</tr>

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Line 671:

</td>

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

</td>

<td>down-fifth

</td>

<td>vP5

</td>

</tr>

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Line 684:

</td>

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

</td>

<td>fifth

</td>

<td>P5

</td>

</tr>

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</td>

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

</td>

<td>up-fifth,

downminor 6th

</td>

<td>^P5, vm6

</td>

</tr>

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</td>

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

</td>

<td>minor 6th

</td>

<td>m6

</td>

</tr>

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</td>

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

</td>

<td>mid 6th

</td>

<td>~6

</td>

</tr>

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Line 737:

</td>

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

</td>

<td>major 6th

</td>

<td>M6

</td>

</tr>

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</td>

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

</td>

<td>upmajor 6th,

downminor 7th

</td>

<td>^M6, vm7

</td>

</tr>

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</td>

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

</td>

<td>minor 7th

</td>

<td>m7

</td>

</tr>

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Line 777:

</td>

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

</td>

<td>mid 7th

</td>

<td>~7

</td>

</tr>

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</td>

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

</td>

<td>major 7th

</td>

<td>M7

</td>

</tr>

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</td>

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

</td>

<td>upmajor 7th,

down-octave

</td>

<td>^M7, vP8

</td>

</tr>

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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; or &quot;up&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 #^ or ^# = three-quarter-tone sharp or &quot;Jump-Sharp&quot; or &quot;upsharp&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; or &quot;down&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 or vb = three-quarter-tone flat or &quot;Drop-Flat&quot; or &quot;downflat&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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24edo features a rich variety on not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.

As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral triad 0-7-14 however there are other options such as

As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as

0-9-14 (Ultra Triad) and 0-5-14 (Infra Triad), the chord names being based on what kind of third is in the chord.

0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.

These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.

For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 <a class="wiki_link" href="/William%20Lynch">William Lynch</a> considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.

Line 1,287:

Line 1,408:

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" />

Line 1,350:

Line 1,471:

Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top

Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11

Alternatively, <a class="wiki_link" href="/ups%20and%20downs%20notation">ups and downs notation</a> can be used:

0-5-14 = D Fv A = D.vm = &quot;D downminor&quot;

0-6-14 = D F A = Dm = &quot;D minor&quot;

0-7-14 = D F^ A = D~ = &quot;D mid&quot;

0-8-14 = D F# A = D = &quot;D&quot; or &quot;D major&quot;

0-9-14 = D F#^ A = D.^ = &quot;D dot up&quot; or &quot;D upmajor&quot;

Line 2,136:

Line 2,264:

<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>.

Line 2,159:

Line 2,287:

<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>2016-10-31 09:37:08 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-21 06:16:48 UTC</tt>.<br>
-: The original revision id was <tt>597550782</tt>.<br>
+: The original revision id was <tt>599955646</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 13: / Line 13: @@
 =24edo as a temperament=
-The [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are 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 [[Harmonic Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 72-tET, 84-tET or 156-tET.
 The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.
 || Degree || Cents coarse/fine
-DMS ||= Approximate Ratios* ||
+DMS ||= Approximate Ratios* ||||= [[Ups and Downs Notation|ups and downs notation]] ||
-|| 0 || 0 ||= 1/1 ||
+|| 0 || 0 ||= 1/1 || unison || P1 ||
 || 1 || 50, 60
-° ||= 33/32, 34/33 ||
+° ||= 33/32, 34/33 || up-unison,
+downminor 2nd || ^P1, vm2 ||
 || 2 || 100, 120
-° ||= 17/16, 18/17 ||
+° ||= 17/16, 18/17 || aug unison,
+minor 2nd || A1, m2 ||
 || 3 || 150, 180
-° ||= 12/11 ||
+° ||= 12/11 || mid 2nd || ~2 ||
 || 4 || 200, 240
-° ||= 9/8 ||
+° ||= 9/8 || major 2nd || M2 ||
 || 5 || 250, 300
-° ||= 22/19 ||
+° ||= 22/19 || upmajor 2nd,
+downminor 3rd || ^M2, vm3 ||
 || 6 || 300, 360
-° ||= 19/16 ||
+° ||= 19/16 || minor 3rd || m3 ||
 || 7 || 350, 420
-° ||= 11/9 ||
+° ||= 11/9 || mid 3rd || ~3 ||
 || 8 || 400, 480
-° ||= 24/19 ||
+° ||= 24/19 || major 3rd || M3 ||
 || 9 || 450, 540
-° ||= 22/17 ||
+° ||= 22/17 || upmajor 3rd,
+down-fourth || ^M3, vP4 ||
 || 10 || 500, 600
-° ||= 4/3 ||
+° ||= 4/3 || fourth || P4 ||
 || 11 || 550, 660
-° ||= 11/8 ||
+° ||= 11/8 || up-fourth || ^P4 ||
 || 12 || 600, 720
-° ||= 17/12 ||
+° ||= 17/12 || augmented 4th,
+diminished 5th || A4, d5 ||
 || 13 || 650, 780
-° ||= 16/11 ||
+° ||= 16/11 || down-fifth || vP5 ||
 || 14 || 700, 840
-° ||= 3/2 ||
+° ||= 3/2 || fifth || P5 ||
 || 15 || 750, 900
-° ||= 17/11 ||
+° ||= 17/11 || up-fifth,
+downminor 6th || ^P5, vm6 ||
 || 16 || 800, 960
-* ||= 19/12 ||
+* ||= 19/12 || minor 6th || m6 ||
 || 17 || 850, 1020
-° ||= 18/11 ||
+° ||= 18/11 || mid 6th || ~6 ||
 || 18 || 900, 1080
-° ||= 32/19 ||
+° ||= 32/19 || major 6th || M6 ||
 || 19 || 950, 1140
-° ||= 19/11 ||
+° ||= 19/11 || upmajor 6th,
+downminor 7th || ^M6, vm7 ||
 || 20 || 1000, 1200
-° ||= 16/9 ||
+° ||= 16/9 || minor 7th || m7 ||
 || 21 || 1050, 1260
-° ||= 11/6 ||
+° ||= 11/6 || mid 7th || ~7 ||
 || 22 || 1100, 1320
-° ||= 17/9, 32/17 ||
+° ||= 17/9, 32/17 || major 7th || M7 ||
 || 23 || 1150, 1380
-° ||= 33/17, 64/33 ||
+° ||= 33/17, 64/33 || upmajor 7th,
+down-octave || ^M7, vP8 ||
 *based on treating 24-EDO as a 2.3.11.17.19 subgroup; other approaches are possible.
@@ Line 177: / Line 185: @@
 ==Mainstream Quartertone Notation==
-[[image:http://www.wfg.woodwind.org/n/s14_blue.gif]] or ^ = quarter-tone sharp or "Jump"
+[[image:http://www.wfg.woodwind.org/n/s14_blue.gif]] or ^ = quarter-tone sharp or "Jump" or "up"
-[[image:http://www.wfg.woodwind.org/n/s34_blue.gif]] or #^ = three-quarter-tone sharp or "Jump-Sharp"
+[[image:http://www.wfg.woodwind.org/n/s34_blue.gif]] or #^ or ^# = three-quarter-tone sharp or "Jump-Sharp" or "upsharp"
-[[image:http://www.wfg.woodwind.org/n/f14_blue.gif]] or v = quarter-tone flat or "Drop"
+[[image:http://www.wfg.woodwind.org/n/f14_blue.gif]] or v = quarter-tone flat or "Drop" or "down"
-[[image:http://www.wfg.woodwind.org/n/f34_blue.gif]] or bv = three-quarter-tone flat or "Drop-Flat"
+[[image:http://www.wfg.woodwind.org/n/f34_blue.gif]] or bv or vb = three-quarter-tone flat or "Drop-Flat" or "downflat"
 Pros: Familiar, fairly easy to learn
@@ Line 224: / Line 232: @@
 edo features a rich variety on not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.
-As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral triad 0-7-14 however there are other options such as
+As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as
--9-14 (Ultra Triad) and 0-5-14 (Infra Triad), the chord names being based on what kind of third is in the chord.
+-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.
 These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.
 For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.
@@ Line 253: / Line 261: @@
 Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top
 Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11
+Alternatively, [[ups and downs notation]] can be used:
+-5-14 = D Fv A = D.vm = "D downminor"
+-6-14 = D F A = Dm = "D minor"
+-7-14 = D F^ A = D~ = "D mid"
+-8-14 = D F# A = D = "D" or "D major"
+-9-14 = D F#^ A = D.^ = "D dot up" or "D upmajor"
@@ Line 458: / Line 473: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x24edo as a temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;24edo as a temperament&lt;/h1&gt;
-  The &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;5-limit&lt;/a&gt; approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals (7:4, 7:5 and 7:6) are just as bad in 24-tET as in 12-tET, although they are approximated by different intervals. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 72-tET, 84-tET or 156-tET.&lt;br /&gt;
+  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 almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like 36-tET, 72-tET, 84-tET or 156-tET.&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. 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;
@@ Line 472: / Line 487: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Approximate Ratios*&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td colspan="2" style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs notation&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 480: / Line 497: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;unison&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;P1&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 489: / Line 510: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;33/32, 34/33&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;up-unison,&lt;br /&gt;
+downminor 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;^P1, vm2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 498: / Line 524: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;17/16, 18/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;aug unison,&lt;br /&gt;
+minor 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;A1, m2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 507: / Line 538: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;12/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;mid 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;~2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 516: / Line 551: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;9/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;major 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;M2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 525: / Line 564: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;22/19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;upmajor 2nd,&lt;br /&gt;
+downminor 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;^M2, vm3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 534: / Line 578: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;19/16&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;minor 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;m3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 543: / Line 591: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11/9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;mid 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;~3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 552: / Line 604: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;24/19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;major 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;M3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 561: / Line 617: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;22/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;upmajor 3rd,&lt;br /&gt;
+down-fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;^M3, vP4&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 570: / Line 631: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;4/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;P4&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 579: / Line 644: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;up-fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;^P4&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 588: / Line 657: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;17/12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;augmented 4th,&lt;br /&gt;
+diminished 5th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;A4, d5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 597: / Line 671: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;16/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;down-fifth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;vP5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 606: / Line 684: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;3/2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;fifth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;P5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 615: / Line 697: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;17/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;up-fifth,&lt;br /&gt;
+downminor 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;^P5, vm6&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 624: / Line 711: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;19/12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;minor 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;m6&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 633: / Line 724: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;18/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;mid 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;~6&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 642: / Line 737: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;32/19&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;major 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;M6&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 651: / Line 750: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;19/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;upmajor 6th,&lt;br /&gt;
+downminor 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;^M6, vm7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 660: / Line 764: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;16/9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;minor 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;m7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 669: / Line 777: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11/6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;mid 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;~7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 678: / Line 790: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;17/9, 32/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;major 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;M7&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 687: / Line 803: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;33/17, 64/33&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;upmajor 7th,&lt;br /&gt;
+down-octave&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;^M7, vP8&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,233: / Line 1,354: @@
 &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: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:1632:&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:1632 --&gt; or ^ = quarter-tone sharp or &amp;quot;Jump&amp;quot; or &amp;quot;up&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:1633:&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:1633 --&gt; or #^ or ^# = three-quarter-tone sharp or &amp;quot;Jump-Sharp&amp;quot; or &amp;quot;upsharp&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:1634:&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:1634 --&gt; or v = quarter-tone flat or &amp;quot;Drop&amp;quot; or &amp;quot;down&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;!-- ws:start:WikiTextRemoteImageRule:1635:&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:1635 --&gt; or bv or vb = three-quarter-tone flat or &amp;quot;Drop-Flat&amp;quot; or &amp;quot;downflat&amp;quot;&lt;br /&gt;
 &lt;br /&gt;
 Pros: Familiar, fairly easy to learn&lt;br /&gt;
@@ Line 1,242: / Line 1,363: @@
 &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: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;
+  &lt;!-- ws:start:WikiTextRemoteImageRule:1636:&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:1636 --&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: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:1637:&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:1637 --&gt; = quarter-tone flat or Drop&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;
+&lt;!-- ws:start:WikiTextRemoteImageRule:1638:&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:1638 --&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: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;
+For example, the scale 0-5-10-15-20 is written as C-D&lt;!-- ws:start:WikiTextRemoteImageRule:1639:&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:1639 --&gt; (or E&lt;!-- ws:start:WikiTextRemoteImageRule:1640:&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:1640 --&gt;) F G&lt;!-- ws:start:WikiTextRemoteImageRule:1641:&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:1641 --&gt; (or A&lt;!-- ws:start:WikiTextRemoteImageRule:1642:&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:1642 --&gt;) Bb.&lt;br /&gt;
 &lt;br /&gt;
 Pros: Very easy to distinguish accidentals from one another&lt;br /&gt;
@@ Line 1,252: / Line 1,373: @@
 &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: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:1643:&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:1643 --&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: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;!-- ws:start:WikiTextRemoteImageRule:1644:&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:1644 --&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 1,264: / Line 1,385: @@
 with Sagittal signs for sharp and flat.&lt;br /&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;!-- ws:start:WikiTextLocalImageRule:1630:&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:1630 --&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 1,280: / Line 1,401: @@
 edo features a rich variety on not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.&lt;br /&gt;
 &lt;br /&gt;
-As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral triad 0-7-14 however there are other options such as&lt;br /&gt;
+As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as&lt;br /&gt;
--9-14 (Ultra Triad) and 0-5-14 (Infra Triad), the chord names being based on what kind of third is in the chord.&lt;br /&gt;
+-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.&lt;br /&gt;
 These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities but can be resolved of that issue by using tetrads in place of triads.&lt;br /&gt;
 For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 &lt;a class="wiki_link" href="/William%20Lynch"&gt;William Lynch&lt;/a&gt; considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.&lt;br /&gt;
@@ Line 1,287: / Line 1,408: @@
 William Lynch considers these as some possible good tetrads:&lt;br /&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;!-- ws:start:WikiTextLocalImageRule:1631:&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:1631 --&gt;&lt;br /&gt;
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@@ Line 1,350: / Line 1,471: @@
 Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top&lt;br /&gt;
 Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11&lt;br /&gt;
+&lt;br /&gt;
+Alternatively, &lt;a class="wiki_link" href="/ups%20and%20downs%20notation"&gt;ups and downs notation&lt;/a&gt; can be used:&lt;br /&gt;
+-5-14 = D Fv A = D.vm = &amp;quot;D downminor&amp;quot;&lt;br /&gt;
+-6-14 = D F A = Dm = &amp;quot;D minor&amp;quot;&lt;br /&gt;
+-7-14 = D F^ A = D~ = &amp;quot;D mid&amp;quot;&lt;br /&gt;
+-8-14 = D F# A = D = &amp;quot;D&amp;quot; or &amp;quot;D major&amp;quot;&lt;br /&gt;
+-9-14 = D F#^ A = D.^ = &amp;quot;D dot up&amp;quot; or &amp;quot;D upmajor&amp;quot;&lt;br /&gt;
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@@ Line 2,136: / Line 2,264: @@
 &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: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;!-- ws:start:WikiTextRemoteImageRule:1645:&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:1645 --&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 2,159: / Line 2,287: @@
 &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: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;!-- ws:start:WikiTextRemoteImageRule:1646:&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:1646 --&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;