Misty

Misty, 12 & 87, is a 7-limit temperament with

• generator 3/2 (period-reduced: 25/21)
• period 1/3-octave which represents 63/50, the difference between 56/25 and 16/9.

In Misty, the diesis 128/125 is split into three 126/125s (which also represent 225/224). As a result, the octave is also split into three, because by definition a 5/4 and a third of 128/125 reach 1\3. The generator is 135/128, which stacks four times to 5/4 and raises by a 400c period to 4/3. Because it is a weak extension of Didacus, 5/4 is split into two parts that stack 5 times to 7/4.

Misty results from tempering out the following 2 commas:

• 5120/5103, the aberschisma, which equates 64/63 and 81/80
• 3136/3125, the Didacus comma

Notable Misty edos include 87edo, 99edo, and 111edo.

Theory

Intervals

Misty has the structural property of dividing the comma 81/80~64/63 into two equal kleismas 126/125~225/224.

# periods	-1 (mod 3)		0 (mod 3)		+1 (mod 3)
# gens	Cents*	JI	Cents*	JI	Cents*	JI
-1	703.1	3/2	1103.1	189/100	303.1	25/21
0	800	100/63	0	1/1	400	63/50
1	896.9	42/25	96.9	135/128, 200/189	496.9	4/3
2	993.8	16/9	193.8	28/25	593.8	45/32
3	1090.6	15/8	290.6	32/27	690.6
4	1187.5	125/126, 224/225	387.5	5/4	787.5	63/40, 128/81
5	84.4	21/20	484.4		884.4	5/3
6	181.3	10/9	581.3	7/5	981.3
7	278.2	75/64	678.2	40/27	1078.2	28/15
8	375.1	56/45	775.1	25/16	1175.1	63/64, 80/81
9	471.9	21/16	871.9		71.9	25/24
10	568.8	50/36	968.8	7/4	168.8
11	665.7		1065.7	50/27	265.7	7/6
12	762.6	14/9	1162.6	49/50, 125/128	362.6	100/81
13	859.5		59.5	28/27	459.5
14	956.4		156.4	35/32	556.4

(* exact-2/1, exact-7/4 tuning; octave-reduced)

Derivation of 1/3-octave period

1. 128/125 = 126/125 * 64/63 is equated to 126/125 * 81/80 by the aberschisma
2. 81/80 itself = 126/125 * 225/224
3. Didacus equates 225/224 to 126/125
4. So we have 128/125 ~= 126/125 * 81/80 = 126/125 * 126/125 * 225/224 ~= (126/125)³
5. Hence, 2/1 = 5/4 * 5/4 * 5/4 * 128/125 ~= (5/4 * 126/125)³ = (63/50)³

Patent vals

The following patent vals support 5-limit Misty, which tempers out [26 -12 -3⟩. Vals that are contorted in the 5-limit are not included.

Edo	7-limit extension	Fifth
12	12 & 87	700.000
123	12 & 123	702.439
111	12 & 87	702.703
210	12 & 87	702.857
99	12 & 87	703.030
384	12 & 87	703.125
285	12 & 87	703.158
471	12 & 87	703.185
186	12 & 87	703.226
273	12 & 87	703.297
360	87 & 360	703.333
87	12 & 87, 87 & 75	703.448
336	87 & 75	703.571
249	87 & 75	703.614
162	87 & 75	703.704
237	237 & 312	703.797
312	237 & 312	703.846
75	12 & 51, 87 & 75	704.000
138		704.348
63	12 & 51	704.762
51	12 & 51	705.882