Leapday

Not to be confused with calendar-based scales such as those in 293edo, 400edo, 353edo or Irvian mode.

Leapday is a regular temperament for the 7-, 11-, 13-, 17-, and 19-limit JI. The fifth is tuned slightly sharp of just so that 8 fifths give a 13/8, 11 fifths make an 11/8, 15 fifths give 7/4, twenty-one fifths give 5/4, and twenty-four of them makes ~17/16. Equivalently, the fifth in Leapday is ~2.3 cents sharp of 3/2 (approximately 704 ¢), so that 13/8 is represented by an augmented fifth (e.g. C–G♯), 11/8 is represented by an augmented third (e.g. C–E♯), the harmonic seventh is represented by a doubly-augmented fifth (e.g. C–G𝄪), the major third is represented by a triple-augmented unison (e.g. C–C𝄪♯), and 17/16 is represented by an inverted triply-diminished third (e.g. E𝄫𝄫–C).

The temperament was named by Herman Miller in 2004.

See Hemifamity temperaments #Leapday for more technical data.

Interval chain

In the following table, odd harmonics 1–13 are in bold.

#	Cents*	Approximate Ratios
0	0.0	1/1
1	704.2	3/2
2	208.5	9/8
3	912.7	22/13, 27/16
4	416.9	14/11
5	1121.2	21/11, 40/21
6	625.4	10/7
7	129.6	13/12, 14/13, 15/14
8	833.9	13/8, 21/13
9	338.1	39/32, 40/33
10	1042.3	11/6, 20/11
11	546.6	11/8, 15/11
12	50.8	28/27, 33/32, 40/39, 45/44
13	755.0	14/9, 20/13
14	259.3	7/6, 15/13
15	963.5	7/4
16	467.8	21/16
17	1172.0	63/32, 160/81
18	676.2	40/27
19	180.5	10/9
20	884.7	5/3
21	388.9	5/4
22	1093.2	15/8
23	597.4	45/32

* in 13-limit CTE tuning

Tuning spectrum

Gencom: [2 3/2; 91/90 121/120 133/132 136/135 154/153 169/168]

Gencom mapping: [⟨1 1 -10 -6 -3 -1 -10 6], ⟨0 1 21 15 11 8 24 -3]]

EDO generator	eigenmonzo (unchanged-interval)	generator (¢)	comments
	19/16	700.829
	24/19	701.110
	19/18	701.279
	4/3	701.955
24\41		702.439
	15/14	702.778
	7/5	702.915
	21/20	703.107
	15/11	703.359
	15/13	703.410
17\29		703.448
	11/10	703.500
	13/10	703.522
	13/11	703.597
	19/15	703.630
	20/19	703.700
	26/21	703.782
	22/19	703.843
	21/19	703.856
	22/21	703.893
	26/19	703.910
	19/14	703.962
	19/17	703.979	19- and 21-odd-limit minimax
44\75		704.000
	16/15	704.012
	17/14	704.014
	17/13	704.027
	14/13	704.043
	5/4	704.110	5-odd-limit minimax
	22/17	704.126
71\121		704.132
	6/5	704.218	7-, 15- and 17-odd-limit minimax
	21/17	704.272
	10/9	704.337	9-, 11- and 13-odd-limit minimax
27\46		704.348
	17/16	704.373
	14/11	704.377
	21/16	704.424
	24/17	704.478
	8/7	704.588
	18/17	704.593
	11/8	704.665
37\63		704.762
	7/6	704.776
	12/11	704.936
	9/7	704.994
	16/13	705.066
	11/9	705.268
	13/12	705.510
10\17		705.882
	18/13	706.103
	20/17	706.214
	17/15	708.343