Misty: Difference between revisions

Revision as of 07:21, 12 February 2026

Misty is the regular temperament tempering out the misty comma. It equates the Pythagorean comma with the diesis, and splits this interval into three equal parts, one representing the schisma~diaschisma, and two representing the syntonic comma. Consequently, the octave is also split into three parts of 512/405 each. This temperament, supported by 12et, is notably in the schismic–Pythagorean equivalence continuum, with n = 3.

In the 7-limit, the canonical extension tempers out 3136/3125 and 5120/5103. Possible tunings include 87edo, 99edo and 111edo.

In the following table, odd harmonics 1–21 are in bold.

#	Period 0		Period 1		Period 2
#	Cents*	Approx. ratios	Cents*	Approx. ratios	Cents*	Approx. ratios
0	0.0	1/1	400.0	24/19, 34/27	800.0	19/12, 27/17
1	703.1	3/2	1103.1	17/9, 36/19	303.1	19/16, 25/21
2	206.1	9/8	606.1	17/12	1006.1	25/14, 34/19
3	909.2	27/16	109.2	16/15, 17/16	509.2	51/38, 75/56
4	412.2	19/15	812.2	8/5	12.2	126/125, 225/224
5	1115.3	19/10, 40/21	315.3	6/5	715.3	68/45
6	618.3	10/7	1018.3	9/5	218.3	17/15
7	121.4	15/14	521.4	27/20	921.4	17/10
8	824.4	45/28	24.4	64/63, 81/80	424.4	32/25
9	327.5	76/63, 135/112	727.5	32/21	1127.5	48/25
10	1030.5	38/21	230.5	8/7	630.5	36/25
11	533.6	19/14	933.6	12/7	133.6	27/25
12	36.7	50/49, 57/56	436.7	9/7	836.7	34/21

* In 7-limit CWE tuning, octave reduced

Edo Generator	Eigenmonzo (rnchanged-interval)	Generator (¢)	Comments
7\12		700.000	Lower bound of 9-odd-limit diamond monotone
	3/2	701.955
	81/80	702.688
65\111		702.703
	15/14	702.778
	7/5	702.915
	9/7	702.924
	9/5	702.933	9-odd-limit minimax (error = 1.955¢)
	7/6	703.012
58\99		703.030
	35/18	703.048
	49/48	703.062
	21/20	703.107
	7/4	703.117	7-odd-limit minimax (error = 1.217¢)
	5/3	703.128	5-odd-limit minimax (error = 1.173¢)
	21/16	703.247
	25/24	703.259
	63/32	703.408
	5/4	703.422
51\87		703.448
	15/8	703.910
44\75		704.000
37\63		704.762	Upper bound of 9-odd-limit diamond monotone

@@ Line 1: / Line 1: @@
+{{Infobox regtemp
+| Title = Misty
+| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.17.19
+| Comma basis = [[67108864/66430125]] (5-limit); <br>[[3136/3125]], [[5120/5103]] (7-limit); <br>[[256/255]], [[324/323]], [[400/399]], [[476/475]]<br>(2.3.5.7.17.19)
+| Edo join 1 = 12 | Edo join 2 = 99
+| Mapping = 3; 1 -4 -10 3 1
+| Generators = 3/2
+| Generators tuning = 703.1
+| Optimization method = CWE
+| MOS scales = [[3L 9s]], [[12L 3s]]
+| Odd limit 1 = 9 | Mistuning 1 = 1.96 | Complexity 1 = 39
+}}
 '''Misty''' is the [[regular temperament]] [[tempering out]] the [[misty comma]]. It equates the [[Pythagorean comma]] with the [[diesis]], and splits this interval into three equal parts, one representing the [[schisma]]~[[diaschisma]], and two representing the [[syntonic comma]]. Consequently, the octave is also split into three parts of [[512/405]] each. This temperament, supported by [[12edo|12et]], is notably in the [[schismic–Pythagorean equivalence continuum]], with {{nowrap|''n'' {{=}} 3}}.
 In the 7-limit, the canonical extension tempers out [[3136/3125]] and [[5120/5103]]. Possible tunings include [[87edo]], [[99edo]] and [[111edo]].
-See [[Misty family]] for technical data.
+See [[Misty family #Misty]] and [[Misty family #Septimal misty|#Septimal misty]] for technical data.
 == Interval chain ==
@@ Line 60: / Line 72: @@
 | '''8/5'''
 | 12.2
-| 126/125, 225/224, …
+| 126/125, 225/224
 |-
 | 5