Luna and hemithirds

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The 7-limit hemithirds temperament functions as a strong extension of didacus, the 2.5.7 subgroup temperament, in the range between 25edo and 31edo tuning, 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 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 118edo 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	175/128	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	175/96	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

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		192.000	25ef val, lower bound of 7- and 9-odd-limit diamond monotone
13\81		192.593	81bef val
9\56		192.857	Lower bound of 11- to 15-odd-limit diamond monotone
	32/21	192.922
23\143		193.007
	64/63	193.091
14\87		193.103
	5/4	193.157	1/2-didacus comma
33\205		193.171	205d val (hemithirds) ↑ Hemithirds ↓ Lunatic
	4/3	193.203	(2.3.7) 21- and 27-odd-limit minimax tuning (hemithirds)
19\118		193.220	↑ Lunatic ↓ Hemithirds
	28/27	193.259	2.3.7 CEE tuning
	14/9	193.283	{1, 3, 7, 9} minimax tuning
	7/6	193.344	{1, 3, 7} minimax tuning
	49/48	193.428
5\31		193.548	Upper bound of 9- to 15-odd-limit diamond monotone
6\37		194.595	37b val, upper bound of 7-odd-limit diamond monotone

* Besides the octave

Gencom: [2 28/25; 196/195 352/351 385/384 625/624]

Gencom mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]

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