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, [27 & 34] or [34 & 41], is a 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. Note that this favors a sharp 3/2 (optimally around 3.2c sharp) and a sharp 13/8 (optimally around 6.9c sharp).
• Prime 11 is often added by equating 10/9 with 11/10 (thus placing 11/8 at +10 generators), 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). An alternate extension (27p & 34), associated with 7-limit Wollemia, places 11/8 at -24 generators.
• 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 (27 & 34d), and Wollemia (27 & 34).
  • The weak extension Octacot (27 & 41) is more elegant; it splits the Tetracot generator into two semitones (about 88.1c) representing 21/20, thus equating three Octacot generators with 7/6 (and 11 of them with 7/4). Octacot can be extended to have prime 19 (at 17 generators) by equating 21/20 to 20/19 (equivalently, 10/9 to 21/19 or 27/20 to 19/14).