Parapyth

Parapyth is the rank-3 temperament tempering out 352/351 and 364/363 in the 2.3.7.11.13 subgroup.

Inspired by George Secor's 29-tone high tolerance temperament, parapyth was found by Margo Schulter in 2002, and it continued to be developed as part of her neoclassical tuning theory (NTT), although a regular temperament perspective is as viable.

In the early prototype, there was only a single chain of fifths, tuned a little sharp such that:

• the major sixth (+3 fifths) hits 22/13, tempering out 352/351;
• the major third (+4 fifths) hits 14/11, tempering out 896/891;
• the augmented unison (+7 fifths) hits 14/13, tempering out 28672/28431.

This is now known as pepperoni. Parapyth encapsulates pepperoni, and adds a spacer representing 28/27~33/32. Prime harmonics 7, 11 and 13 are all made available simply using two chains of fifths.

See Pentacircle clan #Parapyth for technical data.

Parapyth edos

Define a nondegenerate parapyth edo as an edo whose patent val in the 2.3.7.11.13 group tempers out 352/351 and 364/363 but not 243/242 (thereby not equating 16/13 and 11/9). Then the noncontorted nondegenerate parapyth edos below 311 are 22, 29, 46, 63, 80, 87, 104, 109, 121, 128, 133, 145, 150, 167, 172, 184, 191, 196, 213, 230, 232, 237, 254, 259, 271, 278, 283, and 295.

Interval lattice

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• 

These diagrams differ by lattice bases and tunings. The first diagram is generated by {~2, ~3, ~7/4}, corresponding to the octave-reduced form of the mapping, and tuned to the 2.3.7.11.13 subgroup CTE tuning. The second diagram shows the preferred settings in Margo Schulter's neoclassical tuning theory, where it is generated by {~2, ~3, ~33/32}, and tuned to MET-24.

Scales

• Parapyth12 – 12-tone Fokker block in 2.3.7.11.13 TOP tuning
• Pepperoni7 – 7-tone single chain of fifths in 271edo tuning
• Pepperoni12 – 12-tone single chain of fifths in 271edo tuning

See also

• Peppermint-24

External links

• Xenharmonikon | MET-24: A Milder Extended Temperament by Margo Schulter
• A Friendly Introduction to "Rank-3" Temperaments: Designing a System with Three Degrees of Freedom by Margo Schulter
• The MET-24 temperament for Maqam music: Partitions or divisions of the apotome in context by Margo Schulter