Musical cells

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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).


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:

See also