16edo: Difference between revisions

Wikispaces>k9assassin
**Imported revision 599093816 - Original comment: **
Wikispaces>TallKite
**Imported revision 599358390 - Original comment: **
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:k9assassin|k9assassin]] and made on <tt>2016-11-10 09:24:58 UTC</tt>.<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-14 02:36:35 UTC</tt>.<br>
: The original revision id was <tt>599093816</tt>.<br>
: The original revision id was <tt>599358390</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 17: Line 17:


=Intervals=  
=Intervals=  
|| Degree || Cents (coarse, fine)
|| Degree || Cents  
(coarse, fine)
DMS ||= Approximate
DMS ||= Approximate
Ratios* || Interval Names
Ratios* || Interval Names
Just || Interval Names
Just || Interval Names
Simplified ||
Simplified || Interval names,
|| 0 || 0 ||= 1/1 || Unison || Unison ||
# &gt; b, major &gt; minor ||
|| 0 || 0 ||= 1/1 || Unison || Unison ||  ||
|| 1 || 75, 90
|| 1 || 75, 90
22º30' ||= 28/27, 27/26 || Subminor 2nd || Minor Second ||
22º30' ||= 28/27, 27/26 || Subminor 2nd || Minor Second || aug 1, dim 2nd ||
|| 2 || 150, 180
|| 2 || 150, 180
45° ||= 12/11, 35/32 || Neutral 2nd || Major Second ||
45° ||= 12/11, 35/32 || Neutral 2nd || Major Second || min 2nd ||
|| 3 || 225, 270
|| 3 || 225, 270
67°30' ||= 8/7 || Supermajor 2nd,
67°30' ||= 8/7 || Supermajor 2nd,
Septimal Whole-Tone || Perfect Second ||
Septimal Whole-Tone || Perfect Second || maj 2nd ||
|| 4 || 300, 360
|| 4 || 300, 360
90° ||= 19/16, 32/27 || Minor 3rd || Minor Third ||
90° ||= 19/16, 32/27 || Minor 3rd || Minor Third || min 3rd ||
|| 5 || 375, 450
|| 5 || 375, 450
112°30' ||= 5/4, 26/21 || Major 3rd || Major Third ||
112°30' ||= 5/4, 26/21 || Major 3rd || Major Third || maj 3rd ||
|| 6 || 450, 540
|| 6 || 450, 540
135° ||= 13/10, 35/27 || Sub-4th,
135° ||= 13/10, 35/27 || Sub-4th,
Supermajor 3rd || Minor Fourth ||
Supermajor 3rd || Minor Fourth || aug 3rd, dim 4th ||
|| 7 || 525, 630
|| 7 || 525, 630
157°30' ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth ||
157°30' ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth || perfect 4th ||
|| 8 || 600, 720
|| 8 || 600, 720
180° ||= 7/5, 10/7 || Tritone || Augmented fourth/
180° ||= 7/5, 10/7 || Tritone || Augmented fourth/
Diminished fifth ||
Diminished fifth || aug 4th, dim 5th ||
|| 9 || 675, 810
|| 9 || 675, 810
202°30' ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth ||
202°30' ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth || perfect 5th ||
|| 10 || 750, 900
|| 10 || 750, 900
225° ||= 20/13, 54/35 || Super-5th,
225° ||= 20/13, 54/35 || Super-5th,
Subminor 6th || Major Fifth ||
Subminor 6th || Major Fifth || aug 5th, dim 6th ||
|| 11 || 825, 990
|| 11 || 825, 990
247°30' ||= 8/5, 21/13 || Minor 6th || Minor Sixth ||
247°30' ||= 8/5, 21/13 || Minor 6th || Minor Sixth || min 6th ||
|| 12 || 900, 1080
|| 12 || 900, 1080
270° ||= 27/16, 32/19 || Major 6th || Major Sixth ||
270° ||= 27/16, 32/19 || Major 6th || Major Sixth || maj 6th ||
|| 13 || 975, 1170
|| 13 || 975, 1170
292°30' ||= 7/4 || Subminor 7th,
292°30' ||= 7/4 || Subminor 7th,
Septimal Minor 7th || Perfect Seventh ||
Septimal Minor 7th || Perfect Seventh || min 7th ||
|| 14 || 1050, 1260
|| 14 || 1050, 1260
315° ||= 11/6, 64/35 || Neutral 7th || Minor Seventh ||
315° ||= 11/6, 64/35 || Neutral 7th || Minor Seventh || maj 7th ||
|| 15 || 1125, 1350
|| 15 || 1125, 1350
337°30' ||= 27/14, 52/27 || Supermajor 7th || Major Seventh ||
337°30' ||= 27/14, 52/27 || Supermajor 7th || Major Seventh || aug 7th, dim 8ve ||
|| 16 || 1200, 1440
|| 16 || 1200, 1440
360° ||= 2/1 || Octave || Octave ||
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.
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.


Line 110: Line 112:


=Hexadecaphonic Notation=  
=Hexadecaphonic Notation=  
16edo can be notated with conventional notation, including the staff, note names, chord names, relative notation, etc. in two ways. The first preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is higher pitched than the latter:
D * E * * F * G * A * B * * C * D
D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8
||= D ||=  ||= E ||=  ||=  ||= F ||=  ||= G ||=  ||= A ||=  ||= B ||=  ||=  ||= C ||=  ||= D ||
|| __D__ || D#/Eb || __E__ || _E#_ || _Fb_ || __F__ || F#/Gb || __G__ || G#/Ab || __A__ || A#/Bb || __B__ || _B#_ || _Cb_ || __C__ || C#/Db || __D__ ||
||= P1 ||= A1/d2 ||= m2 ||= M2 ||= m3 ||= M3 ||= A3/d4 ||= P4 ||= A4/d5 ||= P5 ||= A5/d6 ||= m6 ||= M6 ||= m7 ||= M7 ||= A7/d8 ||= P8 ||
The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
The second approach is to preserve the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is further fifthwards on the chain of fifths than the latter:
||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||
||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||
||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||  ||


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.
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.
Line 245: Line 264:
&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;!-- 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;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:769:&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:769 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:1029:&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:1029 --&gt;&lt;br /&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;
&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 254: Line 273:
         &lt;td&gt;Degree&lt;br /&gt;
         &lt;td&gt;Degree&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Cents (coarse, fine)&lt;br /&gt;
         &lt;td&gt;Cents &lt;br /&gt;
(coarse, fine)&lt;br /&gt;
DMS&lt;br /&gt;
DMS&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 266: Line 286:
Simplified&lt;br /&gt;
Simplified&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
        &lt;td&gt;Interval names,&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;&amp;gt; b, major &amp;gt; minor&lt;/li&gt;&lt;/ol&gt;&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
Line 277: Line 299:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Unison&lt;br /&gt;
         &lt;td&gt;Unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 290: Line 314:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Minor Second&lt;br /&gt;
         &lt;td&gt;Minor Second&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;aug 1, dim 2nd&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 303: Line 329:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Major Second&lt;br /&gt;
         &lt;td&gt;Major Second&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;min 2nd&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 317: Line 345:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Perfect Second&lt;br /&gt;
         &lt;td&gt;Perfect Second&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;maj 2nd&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 330: Line 360:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Minor Third&lt;br /&gt;
         &lt;td&gt;Minor Third&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;min 3rd&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 343: Line 375:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Major Third&lt;br /&gt;
         &lt;td&gt;Major Third&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;maj 3rd&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 357: Line 391:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Minor Fourth&lt;br /&gt;
         &lt;td&gt;Minor Fourth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;aug 3rd, dim 4th&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 370: Line 406:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Major Fourth&lt;br /&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;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 384: Line 422:
         &lt;td&gt;Augmented fourth/&lt;br /&gt;
         &lt;td&gt;Augmented fourth/&lt;br /&gt;
Diminished fifth&lt;br /&gt;
Diminished fifth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;aug 4th, dim 5th&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 397: Line 437:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Minor Fifth&lt;br /&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;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 411: Line 453:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Major Fifth&lt;br /&gt;
         &lt;td&gt;Major Fifth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;aug 5th, dim 6th&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 424: Line 468:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Minor Sixth&lt;br /&gt;
         &lt;td&gt;Minor Sixth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;min 6th&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 437: Line 483:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Major Sixth&lt;br /&gt;
         &lt;td&gt;Major Sixth&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;maj 6th&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 451: Line 499:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Perfect Seventh&lt;br /&gt;
         &lt;td&gt;Perfect Seventh&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;min 7th&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 464: Line 514:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Minor Seventh&lt;br /&gt;
         &lt;td&gt;Minor Seventh&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;maj 7th&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 477: Line 529:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Major Seventh&lt;br /&gt;
         &lt;td&gt;Major Seventh&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;aug 7th, dim 8ve&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 490: Line 544:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td&gt;Octave&lt;br /&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;
     &lt;/tr&gt;
     &lt;/tr&gt;
Line 656: Line 712:
&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24810096?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;24810096&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/16ed2-001.svg"&gt;alt : Your browser has no SVG support.&lt;/object&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24810096?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;24810096&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/16ed2-001.svg"&gt;alt : Your browser has no SVG support.&lt;/object&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextFileRule:773:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/16ed2-001.svg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@16ed2-001.svg&amp;quot; title=&amp;quot;File: 16ed2-001.svg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="16ed2-001.svg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;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"&gt;16ed2-001.svg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/16ed2-001.svg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;18 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:773 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextFileRule:1033:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/16ed2-001.svg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@16ed2-001.svg&amp;quot; title=&amp;quot;File: 16ed2-001.svg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="16ed2-001.svg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;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"&gt;16ed2-001.svg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/16ed2-001.svg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;18 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:1033 --&gt;&lt;br /&gt;
&lt;br /&gt;
&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;
&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 668: Line 724:
&lt;br /&gt;
&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;
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;
&lt;!-- ws:start:WikiTextRemoteImageRule:771:&amp;lt;img src=&amp;quot;http://www.ronsword.com/161928%20copy.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 179px; width: 316px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:771 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:1031:&amp;lt;img src=&amp;quot;http://www.ronsword.com/161928%20copy.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 179px; width: 316px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1031 --&gt;&lt;br /&gt;
&lt;br /&gt;
&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;
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 675: Line 731:
&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Hexadecaphonic Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Hexadecaphonic Notation&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Hexadecaphonic Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Hexadecaphonic Notation&lt;/h1&gt;
  &lt;br /&gt;
  &lt;br /&gt;
16edo can be notated with conventional notation, including the staff, note names, chord names, relative notation, etc. in two ways. The first preserves the &lt;u&gt;melodic&lt;/u&gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is higher pitched than the latter:&lt;br /&gt;
&lt;br /&gt;
D * E * * F * G * A * B * * C * D&lt;br /&gt;
D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D&lt;br /&gt;
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;u&gt;D&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;D#/Eb&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;E&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;_E#_&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;_Fb_&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;F&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;F#/Gb&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;G&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;G#/Ab&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;A&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;A#/Bb&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;B&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;_B#_&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;_Cb_&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;C&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;C#/Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;u&gt;D&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A1/d2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M3&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A3/d4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P4&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A4/d5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A5/d6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M6&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;m7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;M7&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A7/d8&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&lt;br /&gt;
&lt;br /&gt;
The second approach is to preserve the &lt;u&gt;harmonic&lt;/u&gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is further fifthwards on the chain of fifths than the latter:&lt;br /&gt;
&lt;br /&gt;
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
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.&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:772:&amp;lt;img src=&amp;quot;http://ronsword.com/DSgoldsmith_piece.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 342px; width: 1008px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:772 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:1032:&amp;lt;img src=&amp;quot;http://ronsword.com/DSgoldsmith_piece.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 342px; width: 1008px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:1032 --&gt;&lt;br /&gt;
&lt;hr /&gt;
&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;
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 686: Line 979:
&lt;br /&gt;
&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: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:770:&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:770 --&gt;&lt;br /&gt;
  &lt;!-- ws:start:WikiTextLocalImageRule:1030:&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:1030 --&gt;&lt;br /&gt;
This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.&lt;br /&gt;
This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;