Porcupine notation: Difference between revisions

Line 1:

<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:TallKite|TallKite]] and made on <tt>2016-12-28 06:00:12 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-28 06:52:54 UTC</tt>.

: The original revision id was <tt>~~602867018~~</tt>.

: The original revision id was <tt>602867982</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.

Line 122:

=Kite Giedraitis's approach =

[[xenharmonic/Ups and Downs Notation|Ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] 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 3 major 2nds. 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 as sharp-3, sharp-6, etc. For sharp-3 edos (15, 22, 29, etc.), the generator is

[[xenharmonic/Ups and Downs Notation|Ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] 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

generator = P4 / 3 = (A4 - A1) / 3 = A4 / 3 - A1 / 3 = M2 - ^31 / 3 = M2 - ^1 = vM2. For sharp-6 edos (~~30, 37, .~~.. 72), ~~we have P4 / 3 =~~ vvM2. Sharp-9 edos are rarely used, but it would be ~~vvvM2~~.

generator = P4 / 3 = (A4 - A1) / 3 = A4 / 3 - A1 / 3 = M2 - ^31 / 3 = M2 - ^1 = vM2

For sharp-6 edos (e.g. 30 or 72), the generator is vvM2. Sharp-9 edos are rarely used, but the generator would be v3M2.

For sharp-3 edos, the genchain is

For sharp-3 edos, the genchain is ^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - __**P1**__ - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8.

~~P1 -~~- ~~vM2~~ -- ^m3 -- P4 -- v5 -- ^m6 -- m7 -- v8

In C, this would be C^ - D - Ev - F^ - G - Av - Bb^ - __**C**__ - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv.

For sharp-6 edos, simply double all ups and downs: P1 - vvM2 - ^^m3 - P4 - vv5... and C - Dvv - Eb^^ - F - Gvv...

</pre></div>

This assumes a mapping of 5/4 that results in 250/243 mapping to zero edosteps. 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. (Arguably, the generated scale is still porcupine-like.)

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.</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>Porcupine Notation</title></head><body><a class="wiki_link" href="/Mike%20Battaglia">Mike Battaglia</a> posted the following description of a <a class="wiki_link" href="/Porcupine">Porcupine</a> notation to the Xenharmonic Alliance Facebook Group:

Line 248:

Line 251:

<h1 id="toc2"><a name="Kite Giedraitis's approach"></a>Kite Giedraitis's approach </h1>

<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">Ups and downs </a><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a> 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 3 major 2nds. 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 as sharp-3, sharp-6, etc. For sharp-3 edos (15, 22, 29, etc.), the generator is

<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">Ups and downs </a><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a> 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

generator = P4 / 3 = (A4 - A1) / 3 = A4 / 3 - A1 / 3 = M2 - ^31 / 3 = M2 - ^1 = vM2. For sharp-6 edos (~~30, 37, .~~.. 72), ~~we have P4 / 3 =~~ vvM2. Sharp-9 edos are rarely used, but it would be ~~vvvM2~~.

generator = P4 / 3 = (A4 - A1) / 3 = A4 / 3 - A1 / 3 = M2 - ^31 / 3 = M2 - ^1 = vM2

For sharp-6 edos (e.g. 30 or 72), the generator is vvM2. Sharp-9 edos are rarely used, but the generator would be v3M2.

For sharp-3 edos, the genchain is ^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - P1 - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8.

In C, this would be C^ - D - Ev - F^ - G - Av - Bb^ - C - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv.

For sharp-6 edos, simply double all ups and downs: P1 - vvM2 - ^^m3 - P4 - vv5... and C - Dvv - Eb^^ - F - Gvv...

~~For sharp-3~~ edos, the ~~genchain is <~~/~~span>~~ 

This assumes a mapping of 5/4 that results in 250/243 mapping to zero edosteps. 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. (Arguably, the generated scale is still porcupine-like.)

~~<~~span ~~class="commentBody"~~>~~P1 -- vM2 -- ^m3 -- P4 -- v5 -- ^m6 -~~- ~~m7 -- v8~~

~~ ~~

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

~~ ~~

</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>2016-12-28 06:00:12 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-28 06:52:54 UTC</tt>.<br>
-: The original revision id was <tt>602867018</tt>.<br>
+: The original revision id was <tt>602867982</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 122: / Line 122: @@
 =&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 3 major 2nds. 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 as sharp-3, sharp-6, etc. For sharp-3 edos (15, 22, 29, etc.), the generator is &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; 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. For sharp-6 edos (30, 37, ... 72), we have P4 / 3 = vvM2. Sharp-9 edos are rarely used, but it would be vvvM2.&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;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;For sharp-3 edos, the genchain is &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;P1 -- vM2 -- ^m3 -- P4 -- v5 -- ^m6 -- m7 -- v8&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; &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;
-&lt;span class="commentBody"&gt;
-&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. 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>
 <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;
@@ Line 248: / Line 251: @@
 &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;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 3 major 2nds. 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 as sharp-3, sharp-6, etc. For sharp-3 edos (15, 22, 29, etc.), the generator is &lt;/span&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; 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. For sharp-6 edos (30, 37, ... 72), we have P4 / 3 = vvM2. Sharp-9 edos are rarely used, but it would be vvvM2.&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;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;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;span class="commentBody"&gt;For sharp-3 edos, the genchain is &lt;/span&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. 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;P1 -- vM2 -- ^m3 -- P4 -- v5 -- ^m6 -- m7 -- v8&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;span class="commentBody"&gt; &lt;/span&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>
-&lt;span class="commentBody"&gt;&lt;br /&gt;
-&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>