Musical cells: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

'''Musical cells'''{{idiosyncratic}} are a system of [[Mario Pizarro]] for constructing [[well temperament]]s. Recently he has connected this to a system of stretched octaves he calls "'''toctaves'''"{{idiosyncratic}}. This is discussed on ''[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100427.html The corrected progression of musical cells]''.

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-08-21 12:15:36 UTC</tt>.<br>~~

~~: The original revision id was <tt>247394293</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>~~

~~<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">**Musical cells** is a system of [[Mario Pizarro]] for constructing [[~~circulating temperaments~~]]. Recently he has connected this to a system of stretched octaves he calls "**toctaves**". This is discussed on ~~//[~~[~~http~~://~~launch.groups~~.~~yahoo~~.~~com/group~~/tuning/~~message/100427|~~The corrected progression of musical cells]~~]//~~.

= ~~Toctave~~ =

== Musical cells ==

*~~*Toctave**~~ (~~short for "True Octave"~~) ~~is a frequency ratio of 27~~ * 2^(1/4) / 16 (~~ca. 1205.865 cents~~) ~~which Pizarro came up with in connection with his theory of musical cells~~.

Pizarro's musical cells are based on a set of three [[comma]]s:

* M: (3<sup>8</sup> × 5)/2<sup>15</sup> ([[schisma]]);

* J: (2<sup>25</sup> × 2<sup>1/4</sup>)/(3<sup>13</sup> × 5<sup>2</sup>);

* U: (2<sup>12</sup> × 5<sup>2</sup> × 3<sup>1/2</sup>)/3<sup>11</sup>.

[[~~math~~]]

Using combinations of these commas, he defined two "semitone factors" K and P, which he then put together to from the Piagui scale. This well-tempered scale follows an [[8L 4s]] pattern, with large steps K and small steps P. There are three variants (rotations) of this scale: Piagui I (KKP KKP KKP KKP), Piagui II (KPK KPK KPK KPK) and Piagui III (PKK PKK PKK PKK).

Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots

[[math]]

== Toctave ==

The toctave (short for "True Octave") is a frequency ratio of 27 * 2^(1/4) / 16 (ca. 1205.865 cents) which Pizarro came up with in connection with his theory of musical cells.

<math>Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots</math>

This is 1/4 of a Pythagorean comma sharp.

Threads/Folders on the tuning list:

* http://launch.groups.yahoo.com/group/tuning/message/100919 - "The True Octave"

* http://launch.groups.yahoo.com/group/tuning/message/101004 - "A new equal tempered scale?"

* http://launch.groups.yahoo.com/group/tuning/files/MarioPizarro/

~~== see also ==~~

* [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100894.html#100919 http://launch.groups.yahoo.com/group/tuning/message/100919] - "The True Octave"

* [~~[Octave]]</pre></div>~~

* [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101004.html http://launch.groups.yahoo.com/group/tuning/message/101004] - "A new equal tempered scale?"

~~<h4>Original HTML content:</h4>~~

* [https://drive.google.com/drive/folders/1Zu20johQ1YcmVyFGv0V0_QtMDsCeRp7H?usp=sharing MarioPizarro - Google Drive] - Collection of materials on the topic

~~<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>musical cells</title><~~/~~head><body><strong>Musical cells<~~/strong> is a system of <a class="wiki_link" href="/Mario%20Pizarro">Mario Pizarro</a> for constructing <a class="wiki_link" href="/circulating%20temperaments">circulating temperaments</a>. Recently he has connected this to a system of stretched octaves he calls &quot;<strong>toctaves</strong>&quot;. This is discussed on <em><a class="wiki_link_ext" href="http://launch.~~groups~~.~~yahoo.com/group~~/tuning/~~message/100427" rel="nofollow">The corrected progression of musical cells</a></em>~~.~~<br />~~

~~<br />~~

== See also ==

~~<h1 id="toc0"><a name="Toctave"></a> Toctave </h1>~~

* [[Octave]]

<strong>Toctave</strong> (short for &quot;True Octave&quot;) is a frequency ratio of 27 * 2^(1/4) / 16 (ca. 1205.865 cents) which Pizarro came up with in connection with his theory of musical cells.<br />

* [[Stretched tuning]]

~~<br />~~

* [[Wikipedia:Stretched octave]]

~~<!-- ws:start:WikiTextMathRule:0:~~

~~[[math]]&lt;br/&gt;~~

[[Category:Well temperament]]

~~Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots&lt;br/&gt;[[math]]~~

~~--><script type="math/tex">Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots</script><br />~~

~~<br />~~

~~This is 1/4 of a Pythagorean comma sharp.<br />~~

~~<br />~~

~~Threads/Folders on the tuning list:<br />~~

~~<ul><li> - &quot;~~The True Octave~~&quot;</li><li><a class="wiki_link_ext" href="http://launch~~.~~groups.yahoo.com/group/tuning/message/101004" rel="nofollow">~~http://launch.groups.yahoo.com/group/tuning/message/101004~~</a><!~~-~~- ws:end:WikiTextUrlRule:32 --> - &quot;~~A new equal tempered scale?~~&quot;</li><li><a class=~~"~~wiki_link_ext" href="http~~://~~launch.groups~~.~~yahoo~~.com/~~group~~/~~tuning~~/~~files/MarioPizarro/" rel~~=~~"nofollow">http://launch.groups.yahoo.com/group/tuning/files/~~MarioPizarro~~/</a><!~~-- ~~ws:end:WikiTextUrlRule:33 --></li></ul><br />~~

~~<h2 id~~=~~"toc1"><a name~~=~~"Toctave-see also"></a> see~~ also ~~</h2>~~

~~<ul><li><a class~~=~~"wiki_link" href~~="/Octave~~">Octave</a></li></ul></body></html></pre></div>~~

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+'''Musical cells'''{{idiosyncratic}} are a system of [[Mario Pizarro]] for constructing [[well temperament]]s. Recently he has connected this to a system of stretched octaves he calls "'''toctaves'''"{{idiosyncratic}}. This is discussed on ''[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100427.html The corrected progression of musical cells]''.
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-08-21 12:15:36 UTC</tt>.<br>
-: The original revision id was <tt>247394293</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>
-<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">**Musical cells** is a system of [[Mario Pizarro]] for constructing [[circulating temperaments]]. Recently he has connected this to a system of stretched octaves he calls "**toctaves**". This is discussed on //[[http://launch.groups.yahoo.com/group/tuning/message/100427|The corrected progression of musical cells]]//.
-= Toctave =
+== Musical cells ==
-**Toctave** (short for "True Octave") is a frequency ratio of 27 * 2^(1/4) / 16 (ca. 1205.865 cents) which Pizarro came up with in connection with his theory of musical cells.
+Pizarro's musical cells are based on a set of three [[comma]]s:
+* M: (3<sup>8</sup> × 5)/2<sup>15</sup> ([[schisma]]);
+* J: (2<sup>25</sup> × 2<sup>1/4</sup>)/(3<sup>13</sup> × 5<sup>2</sup>);
+* U: (2<sup>12</sup> × 5<sup>2</sup> × 3<sup>1/2</sup>)/3<sup>11</sup>.
-[[math]]
+Using combinations of these commas, he defined two "semitone factors" K and P, which he then put together to from the Piagui scale. This well-tempered scale follows an [[8L 4s]] pattern, with large steps K and small steps P. There are three variants (rotations) of this scale: Piagui I (KKP KKP KKP KKP), Piagui II (KPK KPK KPK KPK) and Piagui III (PKK PKK PKK PKK).
-Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots
-[[math]]
+== Toctave ==
+The toctave (short for "True Octave") is a frequency ratio of 27 * 2^(1/4) / 16 (ca. 1205.865 cents) which Pizarro came up with in connection with his theory of musical cells.
+<math>Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots</math>
 This is 1/4 of a Pythagorean comma sharp.
 Threads/Folders on the tuning list:
-* http://launch.groups.yahoo.com/group/tuning/message/100919 - "The True Octave"
-* http://launch.groups.yahoo.com/group/tuning/message/101004 - "A new equal tempered scale?"
-* http://launch.groups.yahoo.com/group/tuning/files/MarioPizarro/
-== see also ==
+* [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100894.html#100919 http://launch.groups.yahoo.com/group/tuning/message/100919] - "The True Octave"
-* [[Octave]]</pre></div>
+* [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101004.html http://launch.groups.yahoo.com/group/tuning/message/101004] - "A new equal tempered scale?"
-<h4>Original HTML content:</h4>
+* [https://drive.google.com/drive/folders/1Zu20johQ1YcmVyFGv0V0_QtMDsCeRp7H?usp=sharing MarioPizarro - Google Drive] - Collection of materials on the topic
-<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;musical cells&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;Musical cells&lt;/strong&gt; is a system of &lt;a class="wiki_link" href="/Mario%20Pizarro"&gt;Mario Pizarro&lt;/a&gt; for constructing &lt;a class="wiki_link" href="/circulating%20temperaments"&gt;circulating temperaments&lt;/a&gt;. Recently he has connected this to a system of stretched octaves he calls &amp;quot;&lt;strong&gt;toctaves&lt;/strong&gt;&amp;quot;. This is discussed on &lt;em&gt;&lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/100427" rel="nofollow"&gt;The corrected progression of musical cells&lt;/a&gt;&lt;/em&gt;.&lt;br /&gt;
-&lt;br /&gt;
+== See also ==
-&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Toctave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt; Toctave &lt;/h1&gt;
+* [[Octave]]
-&lt;strong&gt;Toctave&lt;/strong&gt; (short for &amp;quot;True Octave&amp;quot;) is a frequency ratio of 27 * 2^(1/4) / 16 (ca. 1205.865 cents) which Pizarro came up with in connection with his theory of musical cells.&lt;br /&gt;
+* [[Stretched tuning]]
-&lt;br /&gt;
+* [[Wikipedia:Stretched octave]]
-&lt;!-- ws:start:WikiTextMathRule:0:
-[[math]]&amp;lt;br/&amp;gt;
+[[Category:Well temperament]]
-Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots&amp;lt;br/&amp;gt;[[math]]
- --&gt;&lt;script type="math/tex"&gt;Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
-&lt;br /&gt;
-This is 1/4 of a Pythagorean comma sharp.&lt;br /&gt;
-&lt;br /&gt;
-Threads/Folders on the tuning list:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;&lt;!-- ws:start:WikiTextUrlRule:31:http://launch.groups.yahoo.com/group/tuning/message/100919 --&gt;&lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/100919" rel="nofollow"&gt;http://launch.groups.yahoo.com/group/tuning/message/100919&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:31 --&gt; - &amp;quot;The True Octave&amp;quot;&lt;/li&gt;&lt;li&gt;&lt;!-- ws:start:WikiTextUrlRule:32:http://launch.groups.yahoo.com/group/tuning/message/101004 --&gt;&lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/101004" rel="nofollow"&gt;http://launch.groups.yahoo.com/group/tuning/message/101004&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:32 --&gt; - &amp;quot;A new equal tempered scale?&amp;quot;&lt;/li&gt;&lt;li&gt;&lt;!-- ws:start:WikiTextUrlRule:33:http://launch.groups.yahoo.com/group/tuning/files/MarioPizarro/ --&gt;&lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/files/MarioPizarro/" rel="nofollow"&gt;http://launch.groups.yahoo.com/group/tuning/files/MarioPizarro/&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:33 --&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Toctave-see also"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt; see also &lt;/h2&gt;
-&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Octave"&gt;Octave&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>