Musical cells: Difference between revisions
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'''Musical cells''' | '''Musical cells''' 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'''". 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 == | ||
Revision as of 19:58, 27 March 2022
Musical cells are a system of Mario Pizarro for constructing well temperaments. Recently he has connected this to a system of stretched octaves he calls "toctaves". 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
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