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.

Interval chain

In the following table, odd harmonics and subharmonics 1–15 are in bold.

#	Cents*	Approximate ratios
0	0.0	1/1
1	176.3	10/9
2	352.5
3	528.8	27/20
4	705.0	3/2
5	881.3	5/3
6	1057.5
7	33.8	81/80
8	210.1	9/8
9	386.3	5/4
10	562.6
11	738.8
12	915.1	27/16
13	1091.3	15/8

* in exact-5/2 tuning

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 or 41 & 48), Modus (27 & 34d), and Wollemia (27 & 34).
  • In terms of commas:
    • Bunya tempers out 225/224, the Marvel comma.
    • Monkey is the extension tempering out 5120/5103, the aberschisma.
    • Modus tempers out 64/63.
    • Wollemia tempers out 126/125.
  • 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). The elegance of the extension can also be seen in the fact that it tempers out S49. 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).

List of patent vals

The following patent vals support 2.3.5 Tetracot. Contorted vals are not included.

Edo	Generator tuning	Fifth tuning
7	171.429	685.714
48	175.000	700.000
41	175.610	702.439
116	175.862	703.448
191	175.916	703.665
75	176.000	704.000
259	176.062	704.247
184	176.087	704.348
109	176.147	704.587
143	176.224	704.895
177	176.271	705.085
34	176.471	705.882
95	176.842	707.368
61	177.049	708.197
27	177.778	711.111