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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>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-10 14:08:08 UTC</tt>.<br>
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| : The original revision id was <tt>245276651</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <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
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| //[[http://launch.groups.yahoo.com/group/tuning/message/100427|The corrected progression of musical cells]]//.
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| = Toctave = | | == Musical cells == |
| **Toctave** (short for "True Octave") is a frequency ratio of 27 * 2^(1/4) / 16 (ca. 1205.865 cents) which Pizarro cam 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>. |
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| [[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. |
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| | <math>Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots</math> |
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| This is 1/4 of a Pythagorean comma sharp. | | This is 1/4 of a Pythagorean comma sharp. |
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| Threads/Folders on the tuning list: | | Threads/Folders on the tuning list: |
| * http://launch.groups.yahoo.com/group/tuning/message/100919 - "The True Octave"
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| * http://launch.groups.yahoo.com/group/tuning/message/101004 - "A new equal tempered scale?"
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| * http://launch.groups.yahoo.com/group/tuning/files/MarioPizarro/
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| == 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>Musical cells 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;toctaves&quot;. This is discussed on<br />
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| <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 />
| | == See also == |
| <br />
| | * [[Octave]] |
| <!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Toctave"></a><!-- ws:end:WikiTextHeadingRule:1 --> Toctave </h1>
| | * [[Stretched tuning]] |
| <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 cam up with in connection with his theory of musical cells.<br />
| | * [[Wikipedia:Stretched octave]] |
| <br />
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| <!-- ws:start:WikiTextMathRule:0:
| | [[Category:Well temperament]] |
| [[math]]&lt;br/&gt;
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| Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots&lt;br/&gt;[[math]]
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| --><script type="math/tex">Toctave=\frac{27}{16}\sqrt[4]{2} \approx 2.0067870065670918\dots</script><!-- ws:end:WikiTextMathRule:0 --><br />
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| <br />
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| This is 1/4 of a Pythagorean comma sharp.<br />
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| <br />
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| Threads/Folders on the tuning list:<br />
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| <ul><li><!-- ws:start:WikiTextUrlRule:32:http://launch.groups.yahoo.com/group/tuning/message/100919 --><a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/100919" rel="nofollow">http://launch.groups.yahoo.com/group/tuning/message/100919</a><!-- ws:end:WikiTextUrlRule:32 --> - &quot;The True Octave&quot;</li><li><!-- ws:start:WikiTextUrlRule:33:http://launch.groups.yahoo.com/group/tuning/message/101004 --><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:33 --> - &quot;A new equal tempered scale?&quot;</li><li><!-- ws:start:WikiTextUrlRule:34:http://launch.groups.yahoo.com/group/tuning/files/MarioPizarro/ --><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:34 --></li></ul><br />
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| <!-- ws:start:WikiTextHeadingRule:3:&lt;h2&gt; --><h2 id="toc1"><a name="Toctave-see also"></a><!-- ws:end:WikiTextHeadingRule:3 --> see also </h2>
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| <ul><li><a class="wiki_link" href="/Octave">Octave</a></li></ul></body></html></pre></div>
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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 × 21/4)/(313 × 52);
- U: (212 × 52 × 31/2)/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:
See also