16edo: 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>2016-12-04 12:13:32 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-22 04:13:52 UTC</tt>.

: The original revision id was <tt>~~601336582~~</tt>.

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=Intervals=

|| Degree || Cents ~~(coarse,~~

|| Degree || Cents ||= Approximate

~~fine) DMS~~ ||= Approximate

Ratios* || Interval Names

Just || Interval Names

Simplified || Interval ~~Names~~ with

Simplified || Interval names

major ~~wider~~ than minor ||

with major wider

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

than minor || Interval names,

|| 1 || 75~~, 90~~

major narrower

~~22º30'~~ ||= 28/27, 27/26 || Subminor 2nd || Minor Second || aug 1, dim 2nd ||

than minor ||

|| 2 || 150~~, 180~~

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

~~45°~~ ||= 12/11, 35/32 || Neutral 2nd || Major Second || min 2nd ||

|| 1 || 75 ||= 28/27, 27/26 || Subminor 2nd || Minor Second || aug 1, dim 2nd || dim 1, aug 2nd ||

|| 3 || 225~~, 270~~

|| 2 || 150 ||= 12/11, 35/32 || Neutral 2nd || Major Second || min 2nd || maj 2nd ||

~~67°30'~~ ||= 8/7 || Supermajor 2nd,

|| 3 || 225 ||= 8/7 || Supermajor 2nd,

Septimal Whole-Tone || Perfect Second || maj 2nd ||

Septimal Whole-Tone || Perfect Second || maj 2nd || min 2nd ||

|| 4 || 300~~, 360~~

|| 4 || 300 ||= 19/16, 32/27 || Minor 3rd || Minor Third || min 3rd || maj 3rd ||

~~90°~~ ||= 19/16, 32/27 || Minor 3rd || Minor Third || min 3rd ||

|| 5 || 375 ||= 5/4, 26/21 || Major 3rd || Major Third || maj 3rd || min 3rd ||

|| 5 || 375~~, 450~~

|| 6 || 450 ||= 13/10, 35/27 || Sub-4th,

~~112°30'~~ ||= 5/4, 26/21 || Major 3rd || Major Third || maj 3rd ||

Supermajor 3rd || Minor Fourth || aug 3rd, dim 4th || dim 3rd, aug 4th ||

|| 6 || 450~~, 540~~

|| 7 || 525 ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth || perfect 4th || perfect 4th ||

~~135°~~ ||= 13/10, 35/27 || Sub-4th,

|| 8 || 600 ||= 7/5, 10/7 || Tritone || Augmented fourth/

Supermajor 3rd || Minor Fourth || aug 3rd, dim 4th ||

Diminished fifth || aug 4th, dim 5th || dim 4th, aug 5th ||

|| 7 || 525~~, 630~~

|| 9 || 675 ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth || perfect 5th || perfect 5th ||

~~157°30'~~ ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth || perfect 4th ||

|| 10 || 750 ||= 20/13, 54/35 || Super-5th,

|| 8 || 600~~, 720~~

Subminor 6th || Major Fifth || aug 5th, dim 6th || dim 5th, aug 6th ||

~~180°~~ ||= 7/5, 10/7 || Tritone || Augmented fourth/

|| 11 || 825 ||= 8/5, 21/13 || Minor 6th || Minor Sixth || min 6th || maj 6th ||

Diminished fifth || aug 4th, dim 5th ||

|| 12 || 900 ||= 27/16, 32/19 || Major 6th || Major Sixth || maj 6th || min 6th ||

|| 9 || 675~~, 810~~

|| 13 || 975 ||= 7/4 || Subminor 7th,

~~202°30'~~ ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth || perfect 5th ||

Septimal Minor 7th || Perfect Seventh || min 7th || maj 7th ||

|| 10 || 750~~, 900~~

|| 14 || 1050 ||= 11/6, 64/35 || Neutral 7th || Minor Seventh || maj 7th || min 7th ||

~~225°~~ ||= 20/13, 54/35 || Super-5th,

|| 15 || 1125 ||= 27/14, 52/27 || Supermajor 7th || Major Seventh || aug 7th, dim 8ve || dim 7th, aug 8ve ||

Subminor 6th || Major Fifth || aug 5th, dim 6th ||

|| 16 || 1200 ||= 2/1 || Octave || Octave || perfect 8ve || perfect 8ve ||

|| 11 || 825~~, 990~~

~~247°30'~~ ||= 8/5, 21/13 || Minor 6th || Minor Sixth || min 6th ||

|| 12 || 900~~, 1080~~

~~270°~~ ||= 27/16, 32/19 || Major 6th || Major Sixth || maj 6th ||

|| 13 || 975~~, 1170~~

~~292°30'~~ ||= 7/4 || Subminor 7th,

Septimal Minor 7th || Perfect Seventh || min 7th ||

|| 14 || 1050~~, 1260~~

~~315°~~ ||= 11/6, 64/35 || Neutral 7th || Minor Seventh || maj 7th ||

|| 15 || 1125~~, 1350~~

~~337°30'~~ ||= 27/14, 52/27 || Supermajor 7th || Major Seventh || aug 7th, dim 8ve ||

|| 16 || 1200~~, 1440~~

~~360°~~ ||= 2/1 || Octave || Octave || perfect 8ve ||

*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

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=Commas=

16 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] < 16 25 37 45 55 59/1 |.)

16 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] < 16 25 37 45 55 59 |.)

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

||= 135/128 ||< | -7 3 1 > ||> 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||

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<h1 id="toc0"><a name="Images"></a>Images</h1>

<img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" />

<img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" />

<h1 id="toc1"><a name="Intervals"></a>Intervals</h1>

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<td>Degree

</td>

<td>Cents ~~(coarse, ~~

<td>Cents

~~fine) DMS~~

</td>

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

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Simplified

</td>

<td>Interval ~~Names~~ with

<td>Interval names

major ~~wider~~ than minor

with major wider

than minor

</td>

<td>Interval names,

major narrower

than minor

</td>

</tr>

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<td>Unison

</td>

<td>

<td>Unison

</td>

<td>Unison

</td>

</tr>

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<td>1

</td>

<td>75~~, 90 ~~

<td>75

~~22º30'~~

</td>

<td style="text-align: center;">28/27, 27/26

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</td>

<td>aug 1, dim 2nd

</td>

<td>dim 1, aug 2nd

</td>

</tr>

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<td>2

</td>

<td>150~~, 180 ~~

<td>150

~~45°~~

</td>

<td style="text-align: center;">12/11, 35/32

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</td>

<td>min 2nd

</td>

<td>maj 2nd

</td>

</tr>

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<td>3

</td>

<td>225~~, 270 ~~

<td>225

~~67°30'~~

</td>

<td style="text-align: center;">8/7

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</td>

<td>maj 2nd

</td>

<td>min 2nd

</td>

</tr>

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<td>4

</td>

<td>300~~, 360 ~~

<td>300

~~90°~~

</td>

<td style="text-align: center;">19/16, 32/27

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</td>

<td>min 3rd

</td>

<td>maj 3rd

</td>

</tr>

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<td>5

</td>

<td>375~~, 450 ~~

<td>375

~~112°30'~~

</td>

<td style="text-align: center;">5/4, 26/21

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</td>

<td>maj 3rd

</td>

<td>min 3rd

</td>

</tr>

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<td>6

</td>

<td>450~~, 540 ~~

<td>450

~~135°~~

</td>

<td style="text-align: center;">13/10, 35/27

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</td>

<td>aug 3rd, dim 4th

</td>

<td>dim 3rd, aug 4th

</td>

</tr>

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<td>7

</td>

<td>525~~, 630 ~~

<td>525

~~157°30'~~

</td>

<td style="text-align: center;">27/20, 52/35, 256/189

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</td>

<td>Major Fourth

</td>

<td>perfect 4th

</td>

<td>perfect 4th

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<td>8

</td>

<td>600~~, 720 ~~

<td>600

~~180°~~

</td>

<td style="text-align: center;">7/5, 10/7

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</td>

<td>aug 4th, dim 5th

</td>

<td>dim 4th, aug 5th

</td>

</tr>

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<td>9

</td>

<td>675~~, 810 ~~

<td>675

~~202°30'~~

</td>

<td style="text-align: center;">40/27, 35/26, 189/128

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</td>

<td>Minor Fifth

</td>

<td>perfect 5th

</td>

<td>perfect 5th

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<td>10

</td>

<td>750~~, 900 ~~

<td>750

~~225°~~

</td>

<td style="text-align: center;">20/13, 54/35

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</td>

<td>aug 5th, dim 6th

</td>

<td>dim 5th, aug 6th

</td>

</tr>

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<td>11

</td>

<td>825~~, 990 ~~

<td>825

~~247°30'~~

</td>

<td style="text-align: center;">8/5, 21/13

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</td>

<td>min 6th

</td>

<td>maj 6th

</td>

</tr>

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<td>12

</td>

<td>900~~, 1080 ~~

<td>900

~~270°~~

</td>

<td style="text-align: center;">27/16, 32/19

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</td>

<td>maj 6th

</td>

<td>min 6th

</td>

</tr>

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<td>13

</td>

<td>975~~, 1170 ~~

<td>975

~~292°30'~~

</td>

<td style="text-align: center;">7/4

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</td>

<td>min 7th

</td>

<td>maj 7th

</td>

</tr>

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<td>14

</td>

<td>1050~~, 1260 ~~

<td>1050

~~315°~~

</td>

<td style="text-align: center;">11/6, 64/35

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</td>

<td>maj 7th

</td>

<td>min 7th

</td>

</tr>

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<td>15

</td>

<td>1125~~, 1350 ~~

<td>1125

~~337°30'~~

</td>

<td style="text-align: center;">27/14, 52/27

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</td>

<td>aug 7th, dim 8ve

</td>

<td>dim 7th, aug 8ve

</td>

</tr>

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<td>16

</td>

<td>1200~~, 1440 ~~

<td>1200

~~360°~~

</td>

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

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</td>

<td>Octave

</td>

<td>perfect 8ve

</td>

<td>perfect 8ve

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<object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/16ed2-001.svg">alt : Your browser has no SVG support.</object>

<div class="objectEmbed"><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');"><img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="16ed2-001.svg" /></a><div><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');" class="filename" title="16ed2-001.svg">16ed2-001.svg</a> <ul><li><a href="/file/detail/16ed2-001.svg">Details</a></li><li><a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg">Download</a></li><li style="color: #666">18 KB</li></ul></div></div>

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<h1 id="toc3"><a name="Hexadecaphonic Octave Theory"></a>Hexadecaphonic Octave Theory</h1>

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Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's Mavila as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).

<img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" />

<img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" />

The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.

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16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.

<img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="external image DSgoldsmith_piece.jpg" title="external image DSgoldsmith_piece.jpg" style="height: 342px; width: 1008px;" />

<img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="external image DSgoldsmith_piece.jpg" title="external image DSgoldsmith_piece.jpg" style="height: 342px; width: 1008px;" />

<hr />

In 16-EDO diatonic scales are dissonant and &quot;shimmery&quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.

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<h2 id="toc5"><a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"></a>16 Tone Piano Layout Based on the Anti-diatonic Scale</h2>

<img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" />

<img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" />

This Layout places Mavila[7] on the black keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.

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<h1 id="toc9"><a name="Commas"></a>Commas</h1>

16 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes <a class="wiki_link" href="/val">val</a> &lt; 16 25 37 45 55 59/1 |.)

16 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes <a class="wiki_link" href="/val">val</a> &lt; 16 25 37 45 55 59 |.)

@@ 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>2016-12-04 12:13:32 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-22 04:13:52 UTC</tt>.<br>
-: The original revision id was <tt>601336582</tt>.<br>
+: The original revision id was <tt>602738502</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 17: / Line 17: @@
 =Intervals=
-|| Degree || Cents (coarse,
+|| Degree || Cents ||= Approximate
-fine) DMS ||= Approximate
 Ratios* || Interval Names
 Just || Interval Names
-Simplified || Interval Names with
+Simplified || Interval names
-major wider than minor ||
+with major wider
-|| 0 || 0 ||= 1/1 || Unison || Unison ||   ||
+than minor || Interval names,
-|| 1 || 75, 90
+major narrower
-º30' ||= 28/27, 27/26 || Subminor 2nd || Minor Second || aug 1, dim 2nd ||
+than minor ||
-|| 2 || 150, 180
+|| 0 || 0 ||= 1/1 || Unison || Unison || Unison || Unison ||
-° ||= 12/11, 35/32 || Neutral 2nd || Major Second || min 2nd ||
+|| 1 || 75 ||= 28/27, 27/26 || Subminor 2nd || Minor Second || aug 1, dim 2nd || dim 1, aug 2nd ||
-|| 3 || 225, 270
+|| 2 || 150 ||= 12/11, 35/32 || Neutral 2nd || Major Second || min 2nd || maj 2nd ||
-°30' ||= 8/7 || Supermajor 2nd,
+|| 3 || 225 ||= 8/7 || Supermajor 2nd,
-Septimal Whole-Tone || Perfect Second || maj 2nd ||
+Septimal Whole-Tone || Perfect Second || maj 2nd || min 2nd ||
-|| 4 || 300, 360
+|| 4 || 300 ||= 19/16, 32/27 || Minor 3rd || Minor Third || min 3rd || maj 3rd ||
-° ||= 19/16, 32/27 || Minor 3rd || Minor Third || min 3rd ||
+|| 5 || 375 ||= 5/4, 26/21 || Major 3rd || Major Third || maj 3rd || min 3rd ||
-|| 5 || 375, 450
+|| 6 || 450 ||= 13/10, 35/27 || Sub-4th,
-°30' ||= 5/4, 26/21 || Major 3rd || Major Third || maj 3rd ||
+Supermajor 3rd || Minor Fourth || aug 3rd, dim 4th || dim 3rd, aug 4th ||
-|| 6 || 450, 540
+|| 7 || 525 ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth || perfect 4th || perfect 4th ||
-° ||= 13/10, 35/27 || Sub-4th,
+|| 8 || 600 ||= 7/5, 10/7 || Tritone || Augmented fourth/
-Supermajor 3rd || Minor Fourth || aug 3rd, dim 4th ||
+Diminished fifth || aug 4th, dim 5th || dim 4th, aug 5th ||
-|| 7 || 525, 630
+|| 9 || 675 ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth || perfect 5th || perfect 5th ||
-°30' ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth || perfect 4th ||
+|| 10 || 750 ||= 20/13, 54/35 || Super-5th,
-|| 8 || 600, 720
+Subminor 6th || Major Fifth || aug 5th, dim 6th || dim 5th, aug 6th ||
-° ||= 7/5, 10/7 || Tritone || Augmented fourth/
+|| 11 || 825 ||= 8/5, 21/13 || Minor 6th || Minor Sixth || min 6th || maj 6th ||
-Diminished fifth || aug 4th, dim 5th ||
+|| 12 || 900 ||= 27/16, 32/19 || Major 6th || Major Sixth || maj 6th || min 6th ||
-|| 9 || 675, 810
+|| 13 || 975 ||= 7/4 || Subminor 7th,
-°30' ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth || perfect 5th ||
+Septimal Minor 7th || Perfect Seventh || min 7th || maj 7th ||
-|| 10 || 750, 900
+|| 14 || 1050 ||= 11/6, 64/35 || Neutral 7th || Minor Seventh || maj 7th || min 7th ||
-° ||= 20/13, 54/35 || Super-5th,
+|| 15 || 1125 ||= 27/14, 52/27 || Supermajor 7th || Major Seventh || aug 7th, dim 8ve || dim 7th, aug 8ve ||
-Subminor 6th || Major Fifth || aug 5th, dim 6th ||
+|| 16 || 1200 ||= 2/1 || Octave || Octave || perfect 8ve || perfect 8ve ||
-|| 11 || 825, 990
-°30' ||= 8/5, 21/13 || Minor 6th || Minor Sixth || min 6th ||
-|| 12 || 900, 1080
-° ||= 27/16, 32/19 || Major 6th || Major Sixth || maj 6th ||
-|| 13 || 975, 1170
-°30' ||= 7/4 || Subminor 7th,
-Septimal Minor 7th || Perfect Seventh || min 7th ||
-|| 14 || 1050, 1260
-° ||= 11/6, 64/35 || Neutral 7th || Minor Seventh || maj 7th ||
-|| 15 || 1125, 1350
-°30' ||= 27/14, 52/27 || Supermajor 7th || Major Seventh || aug 7th, dim 8ve ||
-|| 16 || 1200, 1440
-° ||= 2/1 || Octave || Octave || perfect 8ve ||
 *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
@@ Line 210: / Line 196: @@
 =Commas=
-EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] &lt; 16 25 37 45 55 59/1 |.)
+EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] &lt; 16 25 37 45 55 59 |.)
 ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
 ||= 135/128 ||&lt; | -7 3 1 &gt; ||&gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
@@ Line 276: / Line 262: @@
 &lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Images"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Images&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:1043:&amp;lt;img src=&amp;quot;/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 325px; width: 325px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1043 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:1079:&amp;lt;img src=&amp;quot;/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 325px; width: 325px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1079 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Intervals&lt;/h1&gt;
@@ Line 285: / Line 271: @@
          &lt;td&gt;Degree&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Cents (coarse,&lt;br /&gt;
+         &lt;td&gt;Cents&lt;br /&gt;
-fine) DMS&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Approximate&lt;br /&gt;
@@ Line 297: / Line 282: @@
 Simplified&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Interval Names with&lt;br /&gt;
+         &lt;td&gt;Interval names &lt;br /&gt;
-major wider than minor&lt;br /&gt;
+with major wider &lt;br /&gt;
+than minor&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Interval names,&lt;br /&gt;
+major narrower&lt;br /&gt;
+than minor&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 312: / Line 302: @@
          &lt;td&gt;Unison&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Unison&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Unison&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 318: / Line 310: @@
          &lt;td&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;75, 90&lt;br /&gt;
+         &lt;td&gt;75&lt;br /&gt;
-º30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;28/27, 27/26&lt;br /&gt;
@@ Line 328: / Line 319: @@
 &lt;/td&gt;
          &lt;td&gt;aug 1, dim 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim 1, aug 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 333: / Line 326: @@
          &lt;td&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;150, 180&lt;br /&gt;
+         &lt;td&gt;150&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;12/11, 35/32&lt;br /&gt;
@@ Line 343: / Line 335: @@
 &lt;/td&gt;
          &lt;td&gt;min 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;maj 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 348: / Line 342: @@
          &lt;td&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;225, 270&lt;br /&gt;
+         &lt;td&gt;225&lt;br /&gt;
-°30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
@@ Line 359: / Line 352: @@
 &lt;/td&gt;
          &lt;td&gt;maj 2nd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;min 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 364: / Line 359: @@
          &lt;td&gt;4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;300, 360&lt;br /&gt;
+         &lt;td&gt;300&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;19/16, 32/27&lt;br /&gt;
@@ Line 374: / Line 368: @@
 &lt;/td&gt;
          &lt;td&gt;min 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;maj 3rd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 379: / Line 375: @@
          &lt;td&gt;5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;375, 450&lt;br /&gt;
+         &lt;td&gt;375&lt;br /&gt;
-°30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5/4, 26/21&lt;br /&gt;
@@ Line 389: / Line 384: @@
 &lt;/td&gt;
          &lt;td&gt;maj 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;min 3rd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 394: / Line 391: @@
          &lt;td&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;450, 540&lt;br /&gt;
+         &lt;td&gt;450&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;13/10, 35/27&lt;br /&gt;
@@ Line 405: / Line 401: @@
 &lt;/td&gt;
          &lt;td&gt;aug 3rd, dim 4th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim 3rd, aug 4th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 410: / Line 408: @@
          &lt;td&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;525, 630&lt;br /&gt;
+         &lt;td&gt;525&lt;br /&gt;
-°30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/20, 52/35, 256/189&lt;br /&gt;
@@ Line 418: / Line 415: @@
 &lt;/td&gt;
          &lt;td&gt;Major Fourth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;perfect 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;perfect 4th&lt;br /&gt;
@@ Line 425: / Line 424: @@
          &lt;td&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;600, 720&lt;br /&gt;
+         &lt;td&gt;600&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7/5, 10/7&lt;br /&gt;
@@ Line 436: / Line 434: @@
 &lt;/td&gt;
          &lt;td&gt;aug 4th, dim 5th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim 4th, aug 5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 441: / Line 441: @@
          &lt;td&gt;9&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;675, 810&lt;br /&gt;
+         &lt;td&gt;675&lt;br /&gt;
-°30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;40/27, 35/26, 189/128&lt;br /&gt;
@@ Line 449: / Line 448: @@
 &lt;/td&gt;
          &lt;td&gt;Minor Fifth&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;perfect 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;perfect 5th&lt;br /&gt;
@@ Line 456: / Line 457: @@
          &lt;td&gt;10&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;750, 900&lt;br /&gt;
+         &lt;td&gt;750&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;20/13, 54/35&lt;br /&gt;
@@ Line 467: / Line 467: @@
 &lt;/td&gt;
          &lt;td&gt;aug 5th, dim 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim 5th, aug 6th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 472: / Line 474: @@
          &lt;td&gt;11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;825, 990&lt;br /&gt;
+         &lt;td&gt;825&lt;br /&gt;
-°30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;8/5, 21/13&lt;br /&gt;
@@ Line 482: / Line 483: @@
 &lt;/td&gt;
          &lt;td&gt;min 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;maj 6th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 487: / Line 490: @@
          &lt;td&gt;12&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;900, 1080&lt;br /&gt;
+         &lt;td&gt;900&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/16, 32/19&lt;br /&gt;
@@ Line 497: / Line 499: @@
 &lt;/td&gt;
          &lt;td&gt;maj 6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;min 6th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 502: / Line 506: @@
          &lt;td&gt;13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;975, 1170&lt;br /&gt;
+         &lt;td&gt;975&lt;br /&gt;
-°30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
@@ Line 513: / Line 516: @@
 &lt;/td&gt;
          &lt;td&gt;min 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;maj 7th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 518: / Line 523: @@
          &lt;td&gt;14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1050, 1260&lt;br /&gt;
+         &lt;td&gt;1050&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;11/6, 64/35&lt;br /&gt;
@@ Line 528: / Line 532: @@
 &lt;/td&gt;
          &lt;td&gt;maj 7th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;min 7th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 533: / Line 539: @@
          &lt;td&gt;15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1125, 1350&lt;br /&gt;
+         &lt;td&gt;1125&lt;br /&gt;
-°30'&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/14, 52/27&lt;br /&gt;
@@ Line 543: / Line 548: @@
 &lt;/td&gt;
          &lt;td&gt;aug 7th, dim 8ve&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;dim 7th, aug 8ve&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 548: / Line 555: @@
          &lt;td&gt;16&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1200, 1440&lt;br /&gt;
+         &lt;td&gt;1200&lt;br /&gt;
-°&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
@@ Line 556: / Line 562: @@
 &lt;/td&gt;
          &lt;td&gt;Octave&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;perfect 8ve&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;perfect 8ve&lt;br /&gt;
@@ Line 724: / Line 732: @@
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Hexadecaphonic Octave Theory&lt;/h1&gt;
@@ Line 736: / Line 744: @@
 &lt;br /&gt;
 Another interesting approach can include two interwoven &lt;a class="wiki_link" href="/8edo"&gt;8-EDO&lt;/a&gt; scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's &lt;strong&gt;Mavila&lt;/strong&gt; as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (&lt;a class="wiki_link" href="/19_16"&gt;19/16&lt;/a&gt;, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below).&lt;br /&gt;
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 &lt;br /&gt;
 The interval between the 28th &amp;amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &amp;quot;narrow fifth&amp;quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.&lt;br /&gt;
@@ Line 976: / Line 984: @@
 &lt;br /&gt;
 -EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.&lt;br /&gt;
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 &lt;hr /&gt;
 In 16-EDO diatonic scales are dissonant and &amp;quot;shimmery&amp;quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.&lt;br /&gt;
@@ Line 986: / Line 994: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;16 Tone Piano Layout Based on the Anti-diatonic Scale&lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextLocalImageRule:1044:&amp;lt;img src=&amp;quot;/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 293px; width: 748px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1044 --&gt;&lt;br /&gt;
+  &lt;!-- ws:start:WikiTextLocalImageRule:1080:&amp;lt;img src=&amp;quot;/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 293px; width: 748px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:1080 --&gt;&lt;br /&gt;
 This Layout places Mavila[7] on the black keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.&lt;br /&gt;
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@@ Line 1,137: / Line 1,145: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&gt;Commas&lt;/h1&gt;
-EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt; 16 25 37 45 55 59/1 |.)&lt;br /&gt;
+EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt; 16 25 37 45 55 59 |.)&lt;br /&gt;