Mothra

Mothra is a temperament in the 7-limit that is a strong extension to slendric, which is defined by splitting the interval of 3/2 into three 8/7s and tempering out 1029/1024. The fifth of mothra is flattened to a meantone fifth, so that it reaches 5/4 when stacked four times and 81/80 is tempered out, unlike that of the other slendric extension rodan, which is sharpened from just. This has the effect of bringing the generator 8/7 considerably closer to just, and also allowing MOS scales of mothra to be more melodically usable than those of other forms of slendric, as the structurally-pervasive small step known as the quark is larger here. EDOs that support mothra include 26edo, 31edo, and 36edo, and 31 is a particularly good tuning.

In the 11-limit, two extensions are of note: undecimal mothra (26 & 31), which tempers out 99/98, 385/384 and 441/440 to find the 11th harmonic at 8 generators down, and mosura (31 & 36), which tempers out 176/175 to find the 11th harmonic at 23 generators up. These two mappings merge at 31edo, which is therefore a uniquely suitable tuning for 11-limit mothra.

For technical data, see Gamelismic clan #Mothra.

Interval chains

In the following tables, odd harmonics and subharmonics 1–21 are labeled in bold.

#	Cents*	Approximate ratios
		7-limit intervals	Intervals of undecimal extensions
			Undecimal mothra	Mosura
0	0.0	1/1
1	232.3	8/7	55/48, 63/55	25/22
2	464.5	21/16, 35/27, 64/49	55/42, 72/55	33/25
3	696.8	3/2	49/33
4	929.0	12/7	55/32, 56/33
5	1161.3	35/18, 63/32, 96/49	55/28, 64/33, 108/55	88/45
6	193.5	9/8, 10/9	49/44, 55/49
7	425.8	9/7	14/11
8	658.0	35/24, 72/49	16/11	22/15
9	890.3	5/3, 27/16
10	1122.5	40/21, 27/14	21/11
11	154.8	35/32, 54/49	12/11	11/10
12	387.0	5/4		44/35
13	619.3	10/7	63/44
14	851.5	80/49	18/11	44/27, 33/20
15	1083.8	15/8, 50/27		66/35
16	116.0	15/14	35/33
17	348.3	60/49	27/22, 40/33	11/9
18	580.5	25/18, 45/32		88/63
19	812.8	45/28, 100/63	35/22
20	1045.0	90/49	20/11	11/6
21	77.3	25/24		22/21
22	309.5	25/21
23	541.8		15/11	11/8
24	774.0	25/16		11/7
25	1006.3	25/14		88/49
26	38.5	50/49	45/44	33/32, 55/54

* In 7-limit CWE tuning

Tuning spectrum

Vals refer to the appropriate undecimal extension in the EDO's range.

Edo generator	Eigenmonzo (unchanged interval)*	Generator (¢)	Extension	Comments
4\21		228.571		21c val, lower bound of 5-odd-limit diamond monotone
	10/9	230.401		1/2-comma meantone fifth
5\26		230.769		Lower bound of 7- and 9-odd-limit diamond monotone
	8/7	231.174		Untempered tuning
16\83		231.325		83bc val
	40/21	231.553
11\57		231.579
	5/3	231.595		1/3-comma meantone fifth
17\88		231.818
23\119		231.933		119be val
	25/24	231.937		2/7-comma meantone fifth
29\150		232.000		150be val
	10/7	232.114
	5/4	232.193		1/4-comma meantone fifth
6\31		232.258	↑ Undecimal mothra (99/98) ↓ Mosura (176/175)	Lower bound of (7-limit) 15- and 21-odd-limit diamond monotone
	15/14	232.465
31\160		232.500		160be val
	15/8	232.551		1/5-comma meantone fifth
25\129		232.558
19\98		232.653
32\165		232.727		165bc val
13\67		232.836
	96/49	232.861		1/5-comma slendric
20\103		233.010		103ce val
	12/7	233.282		1/4-comma slendric
7\36		233.333
	3/2	233.985		1/3-comma slendric
1\5		240.000		5e val, upper bound of (7-limit) 5- to 21-odd-limit diamond monotone

* Besides the octave