Octoid

Octoid

Subgroups	2.3.5.7, 2.3.5.7.11
Comma basis	4375/4374, 16875/16807 (7-limit); 540/539, 1375/1372, 4000/3993 (11-limit)
Reduced mapping	⟨8; 3 4 5 3]
ET join	72 & 80
Generators (CWE)	~7/5 = 583.948 ¢
MOS scales	8L 64s, 72L 8s
Ploidacot	octaploid alpha-tricot
Pergen	(P8/8, P4/3)
Minimax error	11-odd-limit: 1.286 ¢; 11-limit 15-odd-limit: 1.473 ¢
Target scale size	11-odd-limit: 56 notes; 11-limit 15-odd-limit: 64 notes

Octoid is a regular temperament which takes a period of 1/8 octave, which represents 12/11, and adds a single generator which represents 6/5, 7/5, 9/7 or 11/10. It tempers out 4375/4374 and 16875/16807 in the 7-limit, and 540/539, 1375/1372, and 4000/3993 in the 11-limit.

There are some extensions for the 13-limit including tridecimal octoid (72 & 224) and octopus (72 & 80).

See Ragismic microtemperaments #Octoid for technical details.

Interval chain

Generator	Period 1		Period 2		Period 3		Period 4		Period 5		Period 6		Period 7		Period 8
	Cents	Approx. ratios	Cents	Approx. ratios	Cents	Approx. ratios	Cents	Approx. ratios	Cents	Approx. ratios	Cents	Approx. ratios	Cents	Approx. ratios	Cents	Approx. ratios
0	150.000	12/11	300.000	25/21	450.000	35/27	600.000	99/70, 140/99	750.000	54/35	900.000	42/25	1050.000	11/6	1200.000	2/1
1	133.948	27/25	283.948	33/28	433.948	9/7	583.948	7/5	733.948		883.948	5/3	1033.948	20/11	1183.948
2	117.895	15/14	267.895	7/6	417.895	14/11	567.895	25/18	717.895	50/33	867.895	33/20	1017.895	9/5	1167.895	49/25, 55/28
3	101.843	35/33	251.843		401.843		551.843	11/8	701.843	3/2	851.843	18/11	1001.843	25/14	1151.843	35/18
4	85.791	21/20	235.791		385.791	5/4	535.791	15/11	685.791	49/33	835.791		985.791		1135.791	27/14
5	69.739	25/24	219.739	25/22	369.739		519.739	27/20	669.739		819.739	45/28	969.739	7/4	1119.739	21/11
6	53.686	33/32	203.686	9/8	353.686	27/22	503.686		653.686	35/24	803.686	35/22	953.686		1103.686
7	37.634	45/44, 49/48	187.634	49/44	337.634		487.634		637.634		787.634		937.634		1087.634	15/8
8	21.582	81/80	171.582		321.582		471.582	21/16	621.582		771.582		921.582		1071.582

* in 11-limit CWE tuning

Scales

• Octoid72
• Octoid80

Tunings

Norm-based tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~7/5 = 583.9418 ¢	CWE: ~7/5 = 583.9411 ¢	POTE: ~7/5 = 583.9404 ¢

11-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~7/5 = 583.9297 ¢	CWE: ~7/5 = 583.9477 ¢	POTE: ~7/5 = 583.9622 ¢

Tuning spectrum

Edo generator	Unchanged interval (eigenmonzo)*	Generator (¢)	Comments
1\8		150.000	8d val, lower bound of 7-odd-limit diamond monotone
12\88		163.636	88bcde val, lower bound of 9- and 11-odd-limit diamond monotone
	9/7	164.916
11\80		165.000
	11/10	165.004
32\232		165.517	232d val
	5/3	165.641
21\152		165.789
	11/9	165.803
	5/4	165.922	5-odd-limit minimax
52\376		165.957
	3/2	166.015	11-limit 15-odd-limit minimax
31\224		166.071
	9/5	166.202	9- and 11-odd-limit minimax
41\296		166.216
	11/8	166.227
	7/4	166.235	7-odd-limit minimax
	11/7	166.246
	7/6	166.565
10\72		166.667
	7/5	167.488
9\64		168.750	64cd val, upper bound of 9- and 11-odd-limit diamond monotone
8\56		171.429	56bccdde val, upper bound of 7-odd-limit diamond monotone

* Besides the octave

Music

• Dreyfus (archived 2010) by Gene Ward Smith – SoundCloud | details | play – Octoid[72] in 224edo tuning