Kite's ups and downs notation: 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:<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 03:56:09 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 04:32:13 UTC</tt>.<br>

: The original revision id was <tt>~~584786725~~</tt>.<br>

: The original revision id was <tt>584787143</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>

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For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot ~~EDO~~ chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with ~~15edo~~, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with ~~12edo~~ (3 or 4 ~~not~~ coprime with 12), but compatible with ~~24edo~~ (7 coprime with 24).

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).

All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:

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For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.

||= ||= Keyspan of # ||= value of i ||= genspan of ^ ||= ||= stepspan &

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||= 22-tone ||= 3 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 27-tone ||= 4 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 29-tone ||= 3 ||= 2, -1 ||= -17, +12 ||= C-Ebbb, C-B# ||= ~~dbl~~-dim 3rd, desc dim 2nd ||

||= 29-tone ||= 3 ||= 2, -1 ||= -17, +12 ||= C-Ebbb, C-B# ||= double-dim 3rd, desc dim 2nd ||

||= 31-tone ||= 2 ||= 1 ||= -12 ||= C-Dbb ||= dim 2nd ||

||= 32-tone ||= 5 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 37-tone ||= 6 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th ||

||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th, desc double-dim 3rd ||

||= 41-tone ||= 4 ||= 3, -1 ||= -29, +12 ||= C-Fbbbb, C-B# ||= ||

||= 41-tone ||= 4 ||= 3, -1 ||= -29, +12 ||= C-Fbbbb, C-B# ||= quadruple-dim 4th, desc dim 2nd ||

||= 42-tone ||= 7 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 43-tone ||= 3 ||= 1 ||= -12 ||= C-Dbb ||= dim 2nd ||

||= 45-tone ||= 2 ||= 1 ||= -19 || <span style="display: block; text-align: center;">C-Dbbb

</span> ||= double-dim 2nd ||

||= 49-tone ||= 7 ||= 0 ||= -7 ||= C-Cb ||= ~~desc~~ aug ~~unison~~ ||

||= 49-tone ||= 7 ||= 0, 1, 2, -1 ||= 1, -6, -13, 8 ||= C-G, C-Gb, C-Gbb, C-G# ||= perf 5th, dim 5th, double-dim 5th, aug 5th ||

||= 50-tone ||= 3 ||= ||= ||= ||= ||

||= 50-tone ||= 3 ||= 2, -1 ||= -31, +19 ||= C-Ebbbbb, C-Bx ||= quintuple-dim 3rd, desc double-dim 2nd ||

||= 53-tone ||= 5 ||= ||= ||= ||= ||</pre></div>

||= 53-tone ||= 5 ||= 4, -1 ||= -41, +12 ||= C-Gb6, C-B# ||= sixfold-dim 5th, desc dim 2nd ||

The value of i equals the stepspan of the up, except in the case of 49-tone.

For most frameworks, the genchain will be similar to that of 22-tone. But when i isn't 1, things are different.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a>&quot;Ups and Downs&quot; Notation</h1>

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<br />

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot ~~EDO~~ chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with ~~15edo~~, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with ~~12edo~~ (3 or 4 ~~not~~ coprime with 12), but compatible with ~~24edo~~ (7 coprime with 24).<br />

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).<br />

<br />

All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:<br />

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<br />

For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.<br />

<br />

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Line 4,146:

</td>

<td style="text-align: center;">double-dim 3rd, desc dim 2nd<br />

</td>

</tr>

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</td>

<td style="text-align: center;">triple-dim 4th<br />

<td style="text-align: center;">triple-dim 4th, desc double-dim 3rd<br />

</td>

</tr>

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<td style="text-align: center;">C-Fbbbb, C-B#<br />

</td>

<td style="text-align: center;">quadruple-dim 4th, desc dim 2nd<br />

</td>

</tr>

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</td>

</td>

</td>

</td>

<td style="text-align: center;">~~desc~~ aug ~~unison~~<br />

<td style="text-align: center;">perf 5th, dim 5th, double-dim 5th, aug 5th<br />

</td>

</tr>

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</td>

</td>

</td>

<td style="text-align: center;">C-Ebbbbb, C-Bx<br />

</td>

<td style="text-align: center;">quintuple-dim 3rd, desc double-dim 2nd<br />

</td>

</tr>

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</td>

</td>

</td>

</td>

<td style="text-align: center;">sixfold-dim 5th, desc dim 2nd<br />

</td>

</tr>

</table>

</body></html></pre></div>

The value of i equals the stepspan of the up, except in the case of 49-tone.<br />

<br />

For most frameworks, the genchain will be similar to that of 22-tone. But when i isn't 1, things are different.</body></html></pre></div>

@@ 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:TallKite|TallKite]] and made on <tt>2016-06-04 03:56:09 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 04:32:13 UTC</tt>.<br>
-: The original revision id was <tt>584786725</tt>.<br>
+: The original revision id was <tt>584787143</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 819: / Line 819: @@
-For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot EDO chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).
+For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).
 All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:
@@ Line 931: / Line 931: @@
 For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.
 ||=   ||= Keyspan of # ||= value of i ||= genspan of ^ ||=   ||= stepspan &amp;
@@ Line 937: / Line 939: @@
 ||= 22-tone ||= 3 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||
 ||= 27-tone ||= 4 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||
-||= 29-tone ||= 3 ||= 2, -1 ||= -17, +12 ||= C-Ebbb, C-B# ||= dbl-dim 3rd, desc dim 2nd ||
+||= 29-tone ||= 3 ||= 2, -1 ||= -17, +12 ||= C-Ebbb, C-B# ||= double-dim 3rd, desc dim 2nd ||
 ||= 31-tone ||= 2 ||= 1 ||= -12 ||= C-Dbb ||= dim 2nd ||
 ||= 32-tone ||= 5 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||
 ||= 37-tone ||= 6 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||
-||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th ||
+||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th, desc double-dim 3rd ||
-||= 41-tone ||= 4 ||= 3, -1 ||= -29, +12 ||= C-Fbbbb, C-B# ||=   ||
+||= 41-tone ||= 4 ||= 3, -1 ||= -29, +12 ||= C-Fbbbb, C-B# ||= quadruple-dim 4th, desc dim 2nd ||
 ||= 42-tone ||= 7 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||
 ||= 43-tone ||= 3 ||= 1 ||= -12 ||= C-Dbb ||= dim 2nd ||
 ||= 45-tone ||= 2 ||= 1 ||= -19 || &lt;span style="display: block; text-align: center;"&gt;C-Dbbb
 &lt;/span&gt; ||= double-dim 2nd ||
-||= 49-tone ||= 7 ||= 0 ||= -7 ||= C-Cb ||= desc aug unison ||
+||= 49-tone ||= 7 ||= 0, 1, 2, -1 ||= 1, -6, -13, 8 ||= C-G, C-Gb, C-Gbb, C-G# ||= perf 5th, dim 5th, double-dim 5th, aug 5th ||
-||= 50-tone ||= 3 ||=   ||=   ||=   ||=   ||
+||= 50-tone ||= 3 ||= 2, -1 ||= -31, +19 ||= C-Ebbbbb, C-Bx ||= quintuple-dim 3rd, desc double-dim 2nd ||
-||= 53-tone ||= 5 ||=   ||=   ||=   ||=   ||</pre></div>
+||= 53-tone ||= 5 ||= 4, -1 ||= -41, +12 ||= C-Gb6, C-B# ||= sixfold-dim 5th, desc dim 2nd ||
+The value of i equals the stepspan of the up, except in the case of 49-tone.
+For most frameworks, the genchain will be similar to that of 22-tone. But when i isn't 1, things are different.</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Ups and Downs Notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x&amp;quot;Ups and Downs&amp;quot; Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&amp;quot;Ups and Downs&amp;quot; Notation&lt;/h1&gt;
@@ Line 3,065: / Line 3,070: @@
 &lt;br /&gt;
 &lt;br /&gt;
-For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. Each node in the Stern-Brocot EDO chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15edo, because the fifth's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).&lt;br /&gt;
+For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).&lt;br /&gt;
 &lt;br /&gt;
 All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:&lt;br /&gt;
@@ Line 4,067: / Line 4,072: @@
 &lt;br /&gt;
 For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
 &lt;br /&gt;
@@ Line 4,139: / Line 4,146: @@
          &lt;td style="text-align: center;"&gt;C-Ebbb, C-B#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;dbl-dim 3rd, desc dim 2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double-dim 3rd, desc dim 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,195: / Line 4,202: @@
          &lt;td style="text-align: center;"&gt;C-Fbbb, C-Ax&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;triple-dim 4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;triple-dim 4th, desc double-dim 3rd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,209: / Line 4,216: @@
          &lt;td style="text-align: center;"&gt;C-Fbbbb, C-B#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;quadruple-dim 4th, desc dim 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,260: / Line 4,267: @@
          &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;0, 1, 2, -1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;-7&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1, -6, -13, 8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C-Cb&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C-G, C-Gb, C-Gbb, C-G#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;desc aug unison&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;perf 5th, dim 5th, double-dim 5th, aug 5th&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,274: / Line 4,281: @@
          &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;2, -1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;-31, +19&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C-Ebbbbb, C-Bx&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;quintuple-dim 3rd, desc double-dim 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,288: / Line 4,295: @@
          &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;4, -1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;-41, +12&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C-Gb6, C-B#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;sixfold-dim 5th, desc dim 2nd&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
 &lt;/table&gt;
-&lt;/body&gt;&lt;/html&gt;</pre></div>
+The value of i equals the stepspan of the up, except in the case of 49-tone.&lt;br /&gt;
+&lt;br /&gt;
+For most frameworks, the genchain will be similar to that of 22-tone. But when i isn't 1, things are different.&lt;/body&gt;&lt;/html&gt;</pre></div>