Tetracot (temperament)

Tetracot

Subgroups	2.3.5
Reduced mapping	⟨1; 4 9]
ET join	27 & 34
Generators (CWE)	~10/9 = 176.1¢
MOS scales	6L 1s, 7L 6s, 7L 13s
Ploidacot	tetracot
Comma basis	20000/19683 (2.3.5)
Minimax error	5-odd-limit: 3.07¢
Target scale size	5-odd-limit: 13 notes

Tetracot, 2.3.5[27 & 34], is a 2.3.5 temperament that splits 3/2 into four flattened 10/9's, so that:

• 2 generators = a neutral third
• 3 generators = 27/20
• 4 generators = 3/2
• 5 generators = 5/3
• 6 generators = a neutral seventh
• 7 generators = 81/80
• 8 generators = 9/8
• 9 generators = 5/4

Extensions

Tetracot has a number of strong extensions, but most of them are problematic in some way. This is because the Tetracot generator is, optimally, approximately 31/28 — not easily interpretable as LCJI.

• Prime 13 can be added by equating (10/9)^2 (the neutral third) with 16/13.
• Prime 11 is often added by equating 10/9 with 11/10, but this is questionable because it produces either a very sharp 11/8 (as in 27edo and 34edo) or a flat 5/4 (as in 41edo and 48edo).
• There isn't a canonical way to add prime 7. This is because 27edo and 41edo have good 7 approximations but 34edo does not. There are no less than 4 strong extensions to 2.3.5.7: Bunya (34d & 41), Monkey (34 & 41), Modus (27e & 34d), and Wollemia (27e & 34).
  • The weak extension Octacot (27 & 41) is more elegant; it splits the Tetracot generator into two semitones representing 21/20, thus equating three Octacot generators with 7/6 (and 11 of them with 7/4).