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:igliashon|igliashon]] and made on <tt>2011-07-21 00:47:25 UTC</tt>.

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-21 00:52:03 UTC</tt>.

: The original revision id was <tt>~~242219447~~</tt>.

: The original revision id was <tt>242219813</tt>.

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

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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**16edo** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|5/4]] which is eleven cents flat. Four steps of it gives the 300 cent minor third interval identical to that of 12-edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.

|| Degree || Cents || Approximate

|| Degree || Cents ||= Approximate

Ratios* || Interval Name ||

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

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

|| 1 || 75 || 28/27, 27/26 || ||

|| 1 || 75 ||= 28/27, 27/26 || Subminor 2nd ||

|| 2 || 150 || 35/32 || ||

|| 2 || 150 ||= 35/32 || Neutral 2nd ||

|| 3 || 225 || 8/7 || ||

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

|| 4 || 300 || 19/16, 32/27 || ||

Septimal Whole-Tone ||

|| 5 || 375 || 5/4, 16/13 || ||

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

|| 6 || 450 || 13/10 || ||

|| 5 || 375 ||= 5/4, 16/13 || Major 3rd ||

|| 7 || 525 || 27/20, 70/52, 26/19 || ||

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

|| 8 || 600 || 7/5, 10/7 || ||

Supermajor 3rd ||

|| 9 || 675 || 40/27, 52/35, 19/13 || ||

|| 7 || 525 ||= 27/20, 70/52, 26/19 || Wide 4th ||

|| 10 || 750 || 20/13 || ||

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

|| 11 || 825 || 8/5, 13/8 || ||

|| 9 || 675 ||= 40/27, 52/35, 19/13 || Narrow 5th ||

|| 12 || 900 || 27/16, 32/19 || ||

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

|| 13 || 975 || 7/4 || ||

Subminor 6th ||

|| 14 || 1050 || 64/35 || ||

|| 11 || 825 ||= 8/5, 13/8 || Minor 6th ||

|| 15 || 1125 || 27/14, 52/27 || ||

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

|| 16 || 1200 || 2/1 || ||

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

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

Septimal Minor 7th ||

|| 14 || 1050 ||= 64/35 || Neutral 7th ||

|| 15 || 1125 ||= 27/14, 52/27 || Supermajor 7th ||

|| 16 || 1200 ||= 2/1 || Octave ||

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

=Hexadecaphonic Octave Theory=

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<td>Cents

</td>

<td>Approximate

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

Ratios*

</td>

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<td>0

</td>

<td>1/1

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

</td>

<td>

<td>Unison

</td>

</tr>

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<td>75

</td>

<td>28/27, 27/26

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

</td>

<td>

<td>Subminor 2nd

</td>

</tr>

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<td>150

</td>

<td>35/32

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

</td>

<td>

<td>Neutral 2nd

</td>

</tr>

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<td>225

</td>

<td>8/7

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

</td>

<td>

<td>Supermajor 2nd,

Septimal Whole-Tone

</td>

</tr>

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<td>300

</td>

<td>19/16, 32/27

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

</td>

<td>

<td>Minor 3rd

</td>

</tr>

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<td>375

</td>

<td>5/4, 16/13

<td style="text-align: center;">5/4, 16/13

</td>

<td>

<td>Major 3rd

</td>

</tr>

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<td>450

</td>

<td>13/10

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

</td>

<td>

<td>Sub-4th,

Supermajor 3rd

</td>

</tr>

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<td>525

</td>

<td>27/20, 70/52, 26/19

<td style="text-align: center;">27/20, 70/52, 26/19

</td>

<td>

<td>Wide 4th

</td>

</tr>

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<td>600

</td>

<td>7/5, 10/7

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

</td>

<td>

<td>Tritone

</td>

</tr>

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<td>675

</td>

<td>40/27, 52/35, 19/13

<td style="text-align: center;">40/27, 52/35, 19/13

</td>

<td>

<td>Narrow 5th

</td>

</tr>

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<td>750

</td>

<td>20/13

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

</td>

<td>

<td>Super-5th,

Subminor 6th

</td>

</tr>

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<td>825

</td>

<td>8/5, 13/8

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

</td>

<td>

<td>Minor 6th

</td>

</tr>

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<td>900

</td>

<td>27/16, 32/19

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

</td>

<td>

<td>Major 6th

</td>

</tr>

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<td>975

</td>

<td>7/4

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

</td>

<td>

<td>Subminor 7th,

Septimal Minor 7th

</td>

</tr>

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<td>1050

</td>

<td>64/35

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

</td>

<td>

<td>Neutral 7th

</td>

</tr>

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<td>1125

</td>

<td>27/14, 52/27

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

</td>

<td>

<td>Supermajor 7th

</td>

</tr>

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<td>1200

</td>

<td>2/1

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

</td>

<td>

<td>Octave

</td>

</tr>

</table>

*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.

<h1 id="toc0"><a name="Hexadecaphonic Octave Theory"></a>Hexadecaphonic Octave Theory</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:igliashon|igliashon]] and made on <tt>2011-07-21 00:47:25 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-21 00:52:03 UTC</tt>.<br>
-: The original revision id was <tt>242219447</tt>.<br>
+: The original revision id was <tt>242219813</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 11: / Line 11: @@
 **16edo** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|5/4]] which is eleven cents flat. Four steps of it gives the 300 cent minor third interval identical to that of 12-edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.
-|| Degree || Cents || Approximate
+|| Degree || Cents ||= Approximate
 Ratios* || Interval Name ||
-|| 0 || 0 || 1/1 ||   ||
+|| 0 || 0 ||= 1/1 || Unison ||
-|| 1 || 75 || 28/27, 27/26 ||   ||
+|| 1 || 75 ||= 28/27, 27/26 || Subminor 2nd ||
-|| 2 || 150 || 35/32 ||   ||
+|| 2 || 150 ||= 35/32 || Neutral 2nd ||
-|| 3 || 225 || 8/7 ||   ||
+|| 3 || 225 ||= 8/7 || Supermajor 2nd,
-|| 4 || 300 || 19/16, 32/27 ||   ||
+Septimal Whole-Tone ||
-|| 5 || 375 || 5/4, 16/13 ||   ||
+|| 4 || 300 ||= 19/16, 32/27 || Minor 3rd ||
-|| 6 || 450 || 13/10 ||   ||
+|| 5 || 375 ||= 5/4, 16/13 || Major 3rd ||
-|| 7 || 525 || 27/20, 70/52, 26/19 ||   ||
+|| 6 || 450 ||= 13/10, 35/27 || Sub-4th,
-|| 8 || 600 || 7/5, 10/7 ||   ||
+Supermajor 3rd ||
-|| 9 || 675 || 40/27, 52/35, 19/13 ||   ||
+|| 7 || 525 ||= 27/20, 70/52, 26/19 || Wide 4th ||
-|| 10 || 750 || 20/13 ||   ||
+|| 8 || 600 ||= 7/5, 10/7 || Tritone ||
-|| 11 || 825 || 8/5, 13/8 ||   ||
+|| 9 || 675 ||= 40/27, 52/35, 19/13 || Narrow 5th ||
-|| 12 || 900 || 27/16, 32/19 ||   ||
+|| 10 || 750 ||= 20/13, 54/35 || Super-5th,
-|| 13 || 975 || 7/4 ||   ||
+Subminor 6th ||
-|| 14 || 1050 || 64/35 ||   ||
+|| 11 || 825 ||= 8/5, 13/8 || Minor 6th ||
-|| 15 || 1125 || 27/14, 52/27 ||   ||
+|| 12 || 900 ||= 27/16, 32/19 || Major 6th ||
-|| 16 || 1200 || 2/1 ||   ||
+|| 13 || 975 ||= 7/4 || Subminor 7th,
-*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible
+Septimal Minor 7th ||
+|| 14 || 1050 ||= 64/35 || Neutral 7th ||
+|| 15 || 1125 ||= 27/14, 52/27 || Supermajor 7th ||
+|| 16 || 1200 ||= 2/1 || Octave ||
+*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
 =Hexadecaphonic Octave Theory=
@@ Line 144: / Line 148: @@
          &lt;td&gt;Cents&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Approximate&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Approximate&lt;br /&gt;
 Ratios*&lt;br /&gt;
 &lt;/td&gt;
@@ Line 155: / Line 159: @@
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1/1&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Unison&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 165: / Line 169: @@
          &lt;td&gt;75&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;28/27, 27/26&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;28/27, 27/26&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Subminor 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 175: / Line 179: @@
          &lt;td&gt;150&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;35/32&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;35/32&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Neutral 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 185: / Line 189: @@
          &lt;td&gt;225&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8/7&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Supermajor 2nd, &lt;br /&gt;
+Septimal Whole-Tone&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 195: / Line 200: @@
          &lt;td&gt;300&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;19/16, 32/27&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;19/16, 32/27&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Minor 3rd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 205: / Line 210: @@
          &lt;td&gt;375&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5/4, 16/13&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;5/4, 16/13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Major 3rd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 215: / Line 220: @@
          &lt;td&gt;450&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;13/10&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;13/10, 35/27&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Sub-4th, &lt;br /&gt;
+Supermajor 3rd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 225: / Line 231: @@
          &lt;td&gt;525&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;27/20, 70/52, 26/19&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;27/20, 70/52, 26/19&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Wide 4th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 235: / Line 241: @@
          &lt;td&gt;600&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7/5, 10/7&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;7/5, 10/7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Tritone&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 245: / Line 251: @@
          &lt;td&gt;675&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;40/27, 52/35, 19/13&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;40/27, 52/35, 19/13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Narrow 5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 255: / Line 261: @@
          &lt;td&gt;750&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;20/13&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;20/13, 54/35&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Super-5th, &lt;br /&gt;
+Subminor 6th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 265: / Line 272: @@
          &lt;td&gt;825&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8/5, 13/8&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;8/5, 13/8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Minor 6th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 275: / Line 282: @@
          &lt;td&gt;900&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;27/16, 32/19&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;27/16, 32/19&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Major 6th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 285: / Line 292: @@
          &lt;td&gt;975&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7/4&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Subminor 7th,&lt;br /&gt;
+Septimal Minor 7th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 295: / Line 303: @@
          &lt;td&gt;1050&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;64/35&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;64/35&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Neutral 7th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 305: / Line 313: @@
          &lt;td&gt;1125&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;27/14, 52/27&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;27/14, 52/27&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Supermajor 7th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 315: / Line 323: @@
          &lt;td&gt;1200&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2/1&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;Octave&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
 &lt;/table&gt;
-*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible&lt;br /&gt;
+*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Hexadecaphonic Octave Theory&lt;/h1&gt;