Tetracot

Tetracot

Subgroups	2.3.5, 2.3.5.11, 2.3.5.11.13
Comma basis	20000/19683 (2.3.5); 100/99, 243/242 (2.3.5.11) 100/99, 144/143, 243/242 (2.3.5.11.13)
Reduced mapping	⟨1; 4 9 10 -2]
ET join	7 & 27e
Generators (CWE)	~10/9 = 176.1 ¢
MOS scales	6L 1s, 7L 6s, 7L 13s
Ploidacot	tetracot
Pergen	(P8, P5/4)
Color name	Saquadyo
Minimax error	5-odd-limit: 3.07 ¢; 2.3.5.11.13 15-odd-limit: 10.9 ¢
Target scale size	5-odd-limit: 13 notes; 2.3.5.11.13 15-odd-limit: 20 notes

This page is about the regular temperament. For the ploidacot signature, see Ploidacot/Tetracot.

Tetracot, in this article, is the rank-2 temperament in the 2.3.5.11.13 subgroup generated by a submajor second of about 174–178 ¢ which represents both 10/9 and 11/10. It is so named because the generator is a quarter of fifth: four such generators make a perfect fifth which approximates 3/2, which cannot occur in 12edo, resulting in 100/99, 144/143, and 243/242 being tempered out. This is in contrast to meantone, where 10/9 is tuned sharper than or equal to just in order to be equated with 9/8.

Equal temperaments that support tetracot include 27, 34, and 41.

Tetracot has four strong extensions for the 7-, 11-, and 13-limit, which use the same methods of obtaining the 11th and 13th harmonics (10 generators up and 2 generators down, respectively) but differ in their methods of obtaining the 7th harmonic:

• Monkey (34 & 41) obtains the 7th harmonic at 15 generators down, tempering out 875/864 and thereby equating 7/4 with (6/5)³;
• Bunya (34d & 41) obtains the 7th harmonic at 26 generators up, tempering out 225/224 and thereby equating 7/2 with (15/8)²;
• Modus (27e & 34d) obtains the 7th harmonic at 8 generators down, tempering out 64/63 and thereby equating 7/4 with 16/9;
• Wollemia (27e & 34) obtains the 7th harmonic at 19 generators up, tempering out 126/125 and thereby equating 7/1 with (5/3)³(3/2).

See Tetracot family for technical data.

Intervals

Interval chain

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

#	Cents*	Approximate ratios
0	0.0	1/1
1	175.8	11/10, 10/9
2	350.6	11/9, 16/13
3	527.4	15/11
4	703.3	3/2
5	879.1	5/3
6	1054.9	11/6, 24/13
7	30.7	55/54, 45/44, 40/39
8	206.5	9/8
9	382.3	5/4
10	558.2	11/8, 18/13
11	734.0	20/13
12	909.8	22/13
13	1085.6	15/8
14	61.4	33/32, 27/26, 25/24
15	237.2	15/13

* In 2.3.5.11.13 subgroup CTE tuning

As a detemperament of 7et

Tetracot as a 34-tone 7et detempering

Tetracot is considered as a cluster temperament with 7 clusters of notes in an octave, so it is naturally a detemperament of the 7 equal temperament. The diagram on the right shows a 34-tone detempered scale, with a generator range of −16 to +17, which covers all the intervals in the no-7 13-odd-limit. Each category is divided into four or five qualities separated by 7 generator steps, which represent 40/39, 45/44, 55/54, 65/64, 66/65, 81/80, and 121/120 all at once.

Scales

• Tetracot7 – 6L 1s scale
• Tetracot13 – improper 7L 6s
• Tetracot20 – improper 7L 13s

Tunings

5-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 176.0283 ¢	CWE: ~10/9 = 176.0965 ¢	POTE: ~10/9 = 176.1598 ¢

2.3.5.11-subgroup norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 175.7765 ¢	CWE: ~10/9 = 175.8847 ¢	POTE: ~10/9 = 175.9849 ¢

2.3.5.11.13-subgroup norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 175.8150 ¢	CWE: ~10/9 = 176.0854 ¢	POTE: ~10/9 = 176.1965 ¢

Tuning spectrum

Edo generator	Eigenmonzo (unchanged-interval)*	Generator (¢)	Comments
	11/10	165.004
	243/200	168.574	1/2-comma
1\7		171.429	Lower bound of 2.3.5.11 subgroup 11-odd-limit, 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	27/20	173.184	1/3-comma
	11/9	173.704
	81/80	174.501	2/7-comma
	11/6	174.894
7\48		175.000
	11/8	175.132	2.3.5.11-subgroup 11-odd-limit minimax
	3/2	175.489	1/4-comma
6\41		175.610
	13/11	175.899	2.3.5.11.13-subgroup 13- and 15-odd-limit minimax
	15/8	176.021
	5/4	176.257	5-odd-limit and 5-limit 9-odd-limit minimax, 2/9-comma
	13/9	176.338
5\34		176.471
	15/13	176.516
	5/3	176.872	1/5-comma
	13/10	176.890
	13/12	176.905
4\27		177.778	27e val, upper bound of 2.3.5.11.13 subgroup 13- and 15-odd-limit diamond monotone
	27/25	177.794	1/6-comma
	243/125	178.452	1/7-comma
	15/11	178.984
	13/8	179.736
3\20		180.000	20ce val, upper bound of 2.3.5.11-subgroup 11-odd-limit diamond monotone
	9/5	182.404

* Besides the octave

Music

Flora Canou

• "October Dieting Plan" from TOTMC Suite (2023–2025) – in modus, 34edo tuning

Zhea Erose

• Modal Studies in Tetracot (2021) – in 34edo tuning

Dustin Schallert

• Tetracot Perc-Sitar
• Tetracot Jam
• Tetracot Pump – all in modus, 27edo tuning

Xotla

• "Electrostat" from Lesser Groove (2020) – Spotify | Bandcamp | YouTube – ambient electro in Tetracot[13], 34edo tuning