Musical cells: Difference between revisions

Latest revision as of 20:55, 12 December 2023

Musical cells^{[idiosyncratic term]} are a system of Mario Pizarro for constructing well temperaments. Recently he has connected this to a system of stretched octaves he calls "toctaves"^{[idiosyncratic term]}. This is discussed on The corrected progression of musical cells.

Musical cells

Pizarro's musical cells are based on a set of three commas:

M: (3⁸ × 5)/2¹⁵ (schisma);
J: (2²⁵ × 2^1/4)/(3¹³ × 5²);
U: (2¹² × 5² × 3^1/2)/3¹¹.

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

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]\displaystyle{ 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?"
MarioPizarro - Google Drive - Collection of materials on the topic

@@ 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>2011-08-10 04:33:01 UTC</tt>.<br>
+== Musical cells ==
-: The original revision id was <tt>245196657</tt>.<br>
+Pizarro's musical cells are based on a set of three [[comma]]s:
-: The revision comment was: <tt></tt><br>
+* M: (3<sup>8</sup> × 5)/2<sup>15</sup> ([[schisma]]);
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
+* J: (2<sup>25</sup> × 2<sup>1/4</sup>)/(3<sup>13</sup> × 5<sup>2</sup>);
-<h4>Original Wikitext content:</h4>
+* U: (2<sup>12</sup> × 5<sup>2</sup> × 3<sup>1/2</sup>)/3<sup>11</sup>.
-<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">//[[http://launch.groups.yahoo.com/group/tuning/message/100427|The corrected progression of musical cells]]// - Mario Pizarro puts "imperfection of the man ear" in relation to the detectability of differences from the pure 2:1-ratio
+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 ==
+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:
+* [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100894.html#100919 http://launch.groups.yahoo.com/group/tuning/message/100919] - "The True Octave"
+* [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101004.html http://launch.groups.yahoo.com/group/tuning/message/101004] - "A new equal tempered scale?"
+* [https://drive.google.com/drive/folders/1Zu20johQ1YcmVyFGv0V0_QtMDsCeRp7H?usp=sharing MarioPizarro - Google Drive] - Collection of materials on the topic
 == See also ==
-* [[Toctave]]</pre></div>
+* [[Octave]]
-<h4>Original HTML content:</h4>
+* [[Stretched tuning]]
-<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;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; - Mario Pizarro puts &amp;quot;imperfection of the man ear&amp;quot; in relation to the detectability of differences from the pure 2:1-ratio&lt;br /&gt;
+* [[Wikipedia:Stretched octave]]
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-See also"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt; See also &lt;/h2&gt;
+[[Category:Well temperament]]
-&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Toctave"&gt;Toctave&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Musical cells: Difference between revisions

Latest revision as of 20:55, 12 December 2023

Musical cells

Toctave

See also

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