Luna and hemithirds

Hemithirds

Subgroups	2.3.5.7
Comma basis	1029/1024, 3136/3125 (2.3.5.7)
Reduced mapping	⟨1; -15 2 5]
Edo join	31 & 56
Generator (CWE)	~28/25 = 193.2 ¢
MOS scales	1L 5s, 6L 1s, 6L 7s, 6L 13s, 6L 19s, 25L 6s
Ploidacot	beta-pentadecacot
Pergen	(P8, ccP4/15)
Color name	Latrizo & Zozoquinguti
Minimax error	(7-odd limit) 2.5 ¢; ((7-limit) 25-odd limit) 3.6 ¢
Target scale size	(7-odd limit) 56 notes; ((7-limit) 25-odd limit) 87 notes

The 7-limit hemithirds temperament functions as a strong extension of didacus, the 2.5.7 subgroup temperament, which is defined by tempering out 3136/3125 such that two of its generators (hemithird, ~28/25, around 193.2 cents) reach ~5/4, three reach ~7/5, and therefore five reach ~7/4. Hemithirds extends didacus in the range between 25edo and 31edo tuning, by tempering out 1029/1024, such that three intervals of ~8/7 reach ~3/2, therefore finding ~4/3 after fifteen generators in total. The canonical extension to the 13-limit tempers out 385/384 and 441/440 to reach ~55/32 at four ~8/7s and therefore ~11/8 at 22 generators down, and then 196/195 (along with 352/351, 625/624, and 1001/1000) to interpret the generator as ~143/128 and find ~13/8 at 23 generators up.

Luna is a restriction of hemithirds to the 5-limit that is a microtemperament, supported by such high-precision tuning systems as 323edo and 441edo; another notable tuning of luna is 1000edo. It can further be re-extended to the 7-limit in the form of lunatic by adding 4375/4374 to the comma list, but that extension is extremely complex (finding the 7th harmonic at 113 generators down).

See Hemimean clan #Hemithirds and Luna family #Luna for more information.

Intervals

In the following table, odd harmonics and subharmonics 1–39 are labeled in bold.

#	Cents*	Approximate ratios
		7-limit hemithirds	13-limit extension
0	0.0	1/1
1	193.2	28/25, 125/112	39/35
2	386.5	5/4	96/77
3	579.7	7/5	39/28, 88/63
4	773.0	25/16	39/25, 120/77
5	966.2	7/4	96/55, 110/63
6	1159.4	49/25, 125/64	39/20, 88/45
7	152.7	35/32	12/11
8	345.9	49/40, 128/105	11/9, 39/32
9	539.2		15/11
10	732.4	32/21, 49/32	55/36, 84/55
11	925.6	128/75	77/45
12	1118.9	40/21	21/11
13	112.1	16/15	77/72
14	305.3	25/21
15	498.6	4/3
16	691.8	112/75	52/35
17	885.1	5/3	128/77
18	1078.3	28/15	13/7
19	71.5	25/24	26/25, 80/77
20	264.8	7/6	64/55
21	458.0	98/75, 125/96	13/10
22	651.3	35/24	16/11
23	844.5	49/30	13/8, 44/27
24	1037.7		20/11
25	31.0	64/63, 49/48	55/54, 56/55, 65/64

* In CWE 7-limit hemithirds tuning

Chords

Main article: Chords of hemithirds

Tunings

Tuning spectrum

Vals are for 13-limit hemithirds and 7-limit lunatic in their respective ranges.

EDO generator	Eigenmonzo (unchanged interval)*	Generator (¢)	Comments
4\25		192.0000	25ef val, lower bound of 7- and 9-odd-limit diamond monotone
13\81		192.5926	81bef val
9\56		192.8571	Lower bound of 11- to 15-odd-limit diamond monotone
	32/21	192.9219
	40/21	192.9601
23\143		193.0070
	64/63	193.0906
14\87		193.1034
	5/4	193.1569	1/2-comma didacus
33\205		193.1707	205d val (hemithirds) ↑ Hemithirds ↓ Lunatic
52\323		193.1889
	25/24	193.1933
71\441		193.1973
	5/3	193.1976	5-odd-limit minimax tuning
	10/9	193.2001
90\559		193.2021
	4/3	193.2030	(2.3.7) 21- and 27-odd-limit minimax tuning (hemithirds)
	16/15	193.2101
19\118		193.2203	↑ Lunatic ↓ Hemithirds
	28/27	193.2592	2.3.7 CEE tuning
	14/9	193.2833	9-odd-limit minimax tuning
24\149		193.2886	149f val
29\180		193.3333	180ef val
	7/6	193.3435	7-odd-limit minimax tuning
	28/15	193.3643
34\211		193.3649	211eff val
	49/48	193.4279
5\31		193.5484	Upper bound of 9- to 15-odd-limit diamond monotone
	7/4	193.7652	2/5-comma didacus
6\37		194.5946	37beff val, upper bound of 7-odd-limit diamond monotone
	28/25	196.1985	Untempered didacus

* Besides the octave

15-odd-limit eigenmonzos

15-odd-limit eigenmonzos of 13-limit hemithirds

Eigenmonzo (Unchanged-interval)	Generator (¢)	Comments
14/13	192.872
12/11	192.948
15/11	192.995
13/10	193.058
16/13	193.066
13/11	193.094
15/13	193.118
13/12	193.120
11/8	193.122
11/10	193.125
18/13	193.144
5/4	193.157
6/5	193.198	5-odd-limit minimax
10/9	193.200
4/3	193.203
16/15	193.210
14/11	193.241	11-odd-limit minimax
9/7	193.283	9-odd-limit minimax
7/6	193.344	7-odd-limit minimax
15/14	193.364
11/9	193.426
8/7	193.765
7/5	194.171