Musical cells

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]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

See also

• Octave
• Stretched tuning
• Wikipedia:Stretched octave