Musical cells: Difference between revisions

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

'''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]''.

== Musical cells ==

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

* 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 ==

~~'''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.

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>

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== See also ==

* [[Octave]]

* [[Stretched tuning]]

* [[Wikipedia:Stretched octave]]

[[Category:~~Todo:add definition]]~~

[[Category:Well temperament]]

~~[[Category:Todo:expand~~]]

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-'''Musical cells''' is a system of [[Mario Pizarro]] for constructing [[circulating temperament]]s. Recently he has connected this to a system of stretched octaves he calls "'''toctaves'''". This is discussed on ''[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100427.html The corrected progression of musical cells]''.
+'''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]''.
+== 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>.
+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 ==
-'''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.
+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>
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 == See also ==
 * [[Octave]]
+* [[Stretched tuning]]
+* [[Wikipedia:Stretched octave]]
-[[Category:Todo:add definition]]
+[[Category:Well temperament]]
-[[Category:Todo:expand]]