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Porcupine notation: Difference between revisions

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**Imported revision 602868058 - Original comment: **
Wikispaces>TallKite
**Imported revision 630868519 - Original comment: **
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<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:TallKite|TallKite]] and made on <tt>2016-12-28 06:58:31 UTC</tt>.<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-07-11 19:21:42 UTC</tt>.<br>
: The original revision id was <tt>602868058</tt>.<br>
: The original revision id was <tt>630868519</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>
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=&lt;span class="commentBody"&gt;Kite Giedraitis's approach &lt;/span&gt;=  
=&lt;span class="commentBody"&gt;Kite Giedraitis's approach &lt;/span&gt;=  


[[xenharmonic/Ups and Downs Notation|Ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]]&lt;span class="commentBody"&gt; can be used even though we don't know which edo we are in. We know that porcupine divides the perfect 4th into 3 equal steps. Also the aug 4th is always 3 major 2nds by definition. And from P4 to A4 is A1. So any edo which divides the aug1 into 3 equal steps will temper out porcupine. These are the edos marked on the scale tree on the linked page as sharp-3, sharp-6, etc. For sharp-3 edos (15, 22, 29, etc.), A1 = triple-up unison, and the generator is &lt;/span&gt;
&lt;span class="commentBody"&gt; Porcupine divides the perfect 4th into 3 equal steps, thus its [[pergen]] is (P8, P4/3). &lt;/span&gt;[[xenharmonic/Ups and Downs Notation|Ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]]&lt;span class="commentBody"&gt; can be used. The generator is vM2, and the enharmonic is v&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;A1 (C^&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt; = C#). The genchain is &lt;/span&gt;
&lt;span class="commentBody"&gt; generator = P4 / 3 = (A4 - A1) / 3 = A4 / 3 - A1 / 3 = M2 - ^&lt;/span&gt;&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;1 / 3 = M2 - ^1 = vM2 &lt;/span&gt;
&lt;span class="commentBody"&gt;...^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - __**P1**__ - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8...&lt;/span&gt;
&lt;span class="commentBody"&gt;For sharp-6 edos (e.g. 30 or 72), the generator is vvM2. Sharp-9 edos are rarely used, but the generator would be v&lt;/span&gt;&lt;span style="font-size: 14.4px; vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;M2.&lt;/span&gt;
&lt;span class="commentBody"&gt;In C, this would be &lt;/span&gt;
&lt;span class="commentBody"&gt;...C^ - D - Ev - F^ - G - Av - Bb^ - __**C**__ - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv...&lt;/span&gt;


&lt;span class="commentBody"&gt;For sharp-3 edos, the genchain is ^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - __**P1**__ - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8.&lt;/span&gt;
&lt;span class="commentBody"&gt;Example comma pump, with brackets indicating an enharmonic equivalence:&lt;/span&gt;
&lt;span class="commentBody"&gt;In C, this would be C^ - D - Ev - F^ - G - Av - Bb^ - __**C**__ - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv.&lt;/span&gt;
&lt;span class="commentBody"&gt;C.v -- Av.^m -- Dv.v -- [Bvv=Bb^].^m -- Eb^.v -- Ab^.v -- C.v&lt;/span&gt;
&lt;span class="commentBody"&gt;See the ups and downs page for an explanation of the chord names.&lt;/span&gt;


&lt;span class="commentBody"&gt;For sharp-6 edos, simply double all ups and downs: P1 - vvM2 - ^^m3 - P4 - vv5... and C - Dvv - Eb^^ - F - Gvv...&lt;/span&gt;
This is the rank-2 notation,&lt;span class="commentBody"&gt; for a generator of indeterminate cents&lt;/span&gt;. The edo notation uses ups and downs to represent one EDOstep. If an edo tempers out porcupine, it must be sharp-3, sharp-6, sharp-9, etc. (See the scale tree on the ups and downs page.) The notation is identical for sharp-3 edos&lt;span class="commentBody"&gt; (15, 22, 29, etc.). For sharp-6 edos (e.g. 30 or 72), simply double all ups and downs. The generator is vvM2. The genchain is P1 - vvM2 - ^^m3 - P4 - vv5... or C - Dvv - Eb^^ - F - Gvv... Sharp-9 edos are rarely used, but the generator would be v&lt;/span&gt;&lt;span style="font-size: 14.4px; vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;M2. &lt;/span&gt;</pre></div>
 
&lt;span class="commentBody"&gt;This assumes a mapping of 5/4 that results in 250/243 mapping to zero edosteps. Since 10/9 should be vM2, 5/4 should be 10/9 + 9/8 = vM2 +M2 = vM3. The obvious mapping often suffices, but sometimes the mapping needs tweaking, as with edos 36c, 43c, 57cc, 58c, 64ccc, 65c and 72cc. If the edo isn't tweaked, the generator won't map to 10/9, and two generators won't map to 6/5. (Arguabl&lt;/span&gt;y, the generated scale is still porcupine-like.)
 
&lt;span class="commentBody"&gt; If we're not in an edo at all, but in a rank-2 tuning with a generator of indeterminate cents, use the sharp-3 notation.&lt;/span&gt;</pre></div>
<h4>Original HTML content:</h4>
<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;Porcupine Notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;a class="wiki_link" href="/Mike%20Battaglia"&gt;Mike Battaglia&lt;/a&gt; posted the following description of a &lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt; notation to the Xenharmonic Alliance Facebook Group:&lt;br /&gt;
<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;Porcupine Notation&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;a class="wiki_link" href="/Mike%20Battaglia"&gt;Mike Battaglia&lt;/a&gt; posted the following description of a &lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt; notation to the Xenharmonic Alliance Facebook Group:&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Kite Giedraitis's approach"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;span class="commentBody"&gt;Kite Giedraitis's approach &lt;/span&gt;&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Kite Giedraitis's approach"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;span class="commentBody"&gt;Kite Giedraitis's approach &lt;/span&gt;&lt;/h1&gt;
  &lt;br /&gt;
  &lt;br /&gt;
&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;Ups and downs &lt;/a&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;notation&lt;/a&gt;&lt;span class="commentBody"&gt; can be used even though we don't know which edo we are in. We know that porcupine divides the perfect 4th into 3 equal steps. Also the aug 4th is always 3 major 2nds by definition. And from P4 to A4 is A1. So any edo which divides the aug1 into 3 equal steps will temper out porcupine. These are the edos marked on the scale tree on the linked page as sharp-3, sharp-6, etc. For sharp-3 edos (15, 22, 29, etc.), A1 = triple-up unison, and the generator is &lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt; Porcupine divides the perfect 4th into 3 equal steps, thus its &lt;a class="wiki_link" href="/pergen"&gt;pergen&lt;/a&gt; is (P8, P4/3). &lt;/span&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;Ups and downs &lt;/a&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation"&gt;notation&lt;/a&gt;&lt;span class="commentBody"&gt; can be used. The generator is vM2, and the enharmonic is v&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;A1 (C^&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt; = C#). The genchain is &lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt; generator = P4 / 3 = (A4 - A1) / 3 = A4 / 3 - A1 / 3 = M2 - ^&lt;/span&gt;&lt;span style="font-size: 90%; vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;1 / 3 = M2 - ^1 = vM2 &lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt;...^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - &lt;u&gt;&lt;strong&gt;P1&lt;/strong&gt;&lt;/u&gt; - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8...&lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt;For sharp-6 edos (e.g. 30 or 72), the generator is vvM2. Sharp-9 edos are rarely used, but the generator would be v&lt;/span&gt;&lt;span style="font-size: 14.4px; vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;M2.&lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt;In C, this would be &lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;span class="commentBody"&gt;...C^ - D - Ev - F^ - G - Av - Bb^ - &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv...&lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt;For sharp-3 edos, the genchain is ^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - &lt;u&gt;&lt;strong&gt;P1&lt;/strong&gt;&lt;/u&gt; - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8.&lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt;In C, this would be C^ - D - Ev - F^ - G - Av - Bb^ - &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv.&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;span class="commentBody"&gt;For sharp-6 edos, simply double all ups and downs: P1 - vvM2 - ^^m3 - P4 - vv5... and C - Dvv - Eb^^ - F - Gvv...&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;span class="commentBody"&gt;This assumes a mapping of 5/4 that results in 250/243 mapping to zero edosteps. Since 10/9 should be vM2, 5/4 should be 10/9 + 9/8 = vM2 +M2 = vM3. The obvious mapping often suffices, but sometimes the mapping needs tweaking, as with edos 36c, 43c, 57cc, 58c, 64ccc, 65c and 72cc. If the edo isn't tweaked, the generator won't map to 10/9, and two generators won't map to 6/5. (Arguabl&lt;/span&gt;y, the generated scale is still porcupine-like.)&lt;br /&gt;
&lt;span class="commentBody"&gt;Example comma pump, with brackets indicating an enharmonic equivalence:&lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt;C.v -- Av.^m -- Dv.v -- [Bvv=Bb^].^m -- Eb^.v -- Ab^.v -- C.v&lt;/span&gt;&lt;br /&gt;
&lt;span class="commentBody"&gt;See the ups and downs page for an explanation of the chord names.&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;span class="commentBody"&gt; If we're not in an edo at all, but in a rank-2 tuning with a generator of indeterminate cents, use the sharp-3 notation.&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
This is the rank-2 notation,&lt;span class="commentBody"&gt; for a generator of indeterminate cents&lt;/span&gt;. The edo notation uses ups and downs to represent one EDOstep. If an edo tempers out porcupine, it must be sharp-3, sharp-6, sharp-9, etc. (See the scale tree on the ups and downs page.) The notation is identical for sharp-3 edos&lt;span class="commentBody"&gt; (15, 22, 29, etc.). For sharp-6 edos (e.g. 30 or 72), simply double all ups and downs. The generator is vvM2. The genchain is P1 - vvM2 - ^^m3 - P4 - vv5... or C - Dvv - Eb^^ - F - Gvv... Sharp-9 edos are rarely used, but the generator would be v&lt;/span&gt;&lt;span style="font-size: 14.4px; vertical-align: super;"&gt;3&lt;/span&gt;&lt;span class="commentBody"&gt;M2. &lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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