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 no-19 23-limit. It is based on the chain of fifths, but here, the fifth is tuned slightly sharp of just (approximately 704 ¢) so that 6 fifths give 23/16, 8 fifths give 13/8, 11 fifths give 11/8, 15 fifths give 7/4, 21 fifths give 5/4, and 24 fifths give 17/16.

Equivalently:

• 5/4, the classical major third, is represented by a triply augmented unison (C–C𝄪♯),
• 7/4, the harmonic seventh, is represented by a doubly augmented fifth (C–G𝄪),
• 11/8 is represented by an augmented third (C–E♯),
• 13/8 is represented by an augmented fifth (C–G♯),
• 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯), and
• 23/16 is represented by an augmented fourth (C–F♯).

As a result, leapday is very much the "opposite" of meantone in many respects, similar to superpyth: meantone (including 12edo) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of harmonics 7, 11, and 13 are complex, while leapday has the fifth tuned sharp so that intervals of 7, 11, and 13 are relatively simple while intervals of 5 are complex.

If ratios of 5 are omitted, the 2.3.7.11.13 subgroup version of leapday is known as leapfrog, notable as tempering parapyth (a rank-3 temperament of the 2.3.7.11.13 subgroup) to rank 2 by finding ~13/8 at (~9/8)⁴, that is, by tempering out the tetris comma, and is a good combination of simplicity and accuracy, as 5/4 is complex and the canonical mapping for prime 19 is fairly inaccurate.

Leapday was named by Herman Miller in 2004.

See Hemifamity temperaments #Leapday or No-fives subgroup temperaments #Leapfrog for more technical data.

Interval chain

In the following table, odd harmonics 1–23 are in bold.

#	Cents*	Approximate ratios
		13-limit	Additional ratios of 17 and 23
0	0.0	1/1
1	704.2	3/2
2	208.5	9/8	17/15, 26/23
3	912.7	22/13, 27/16	17/10
4	416.9	14/11, 33/26	23/18
5	1121.2	21/11, 40/21	23/12, 44/23
6	625.4	10/7, 13/9	23/16
7	129.6	13/12, 14/13, 15/14
8	833.9	13/8, 21/13	34/21
9	338.1	11/9, 39/32, 40/33	17/14, 28/23
10	1042.3	11/6, 20/11	42/23
11	546.6	11/8, 15/11
12	50.8	28/27, 33/32, 40/39, 45/44	34/33, 35/34
13	755.1	14/9, 20/13	17/11
14	259.3	7/6, 15/13
15	963.5	7/4	40/23
16	467.8	21/16	17/13, 30/23
17	1172.0	63/32, 160/81	45/23, 51/26
18	676.2	40/27	34/23
19	180.5	10/9
20	884.7	5/3
21	388.9	5/4
22	1093.2	15/8	17/9
23	597.4	45/32	17/12

* In 13-limit CWE tuning, octave reduced

Tunings

Norm-based tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 704.2257 ¢	CWE: ~3/2 = 704.2504 ¢	POTE: ~3/2 = 704.2634 ¢

13-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 704.2924 ¢	CWE: ~3/2 = 704.2346 ¢	POTE: ~3/2 = 704.2138 ¢

No-19 23-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 704.3142 ¢	CWE: ~3/2 = 704.2450 ¢	POTE: ~3/2 = 704.2246 ¢

Tuning spectrum

Edo generator	Unchanged interval (eigenmonzo)*	Generator (¢)	Comments
	3/2	701.955	Pythagorean tuning
24\41		702.439	41cc… val, lower bound of 5-odd-limit diamond monotone
	15/14	702.778
	7/5	702.915
	21/20	703.107
	15/11	703.359
	15/13	703.410
17\29		703.448	29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
	11/10	703.500
	13/10	703.522
	13/11	703.597
	23/15	703.750
	21/13	703.782
	23/20	703.869
	21/11	703.893
44\75		704.000	75dfg val
	15/8	704.012
	17/14	704.014
	17/13	704.027
	13/7	704.043
	5/4	704.110	5-odd-limit minimax
	17/11	704.126
71\121		704.132	121defg val
	5/3	704.218	7-, 15- and 17-odd-limit minimax
	23/21	704.251
	23/17	704.260
	21/17	704.272
	9/5	704.337	9-, 11- and 13-odd-limit minimax
27\46		704.348	17-odd-limit, no-19 21- and 23-odd-limit diamond monotone (singleton)
	17/16	704.373
	11/7	704.377
	21/16	704.424
	17/12	704.478
	23/14	704.506
	7/4	704.588
	17/9	704.593
	23/22	704.609
	11/8	704.665
	23/16	704.712
37\63		704.762	63c val
	7/6	704.776
	11/6	704.936
	9/7	704.994
	13/8	705.066
	23/12	705.264
	11/9	705.268
	13/12	705.510
10\17		705.882	17cg val, upper bound of 5-, 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone
	23/18	706.091
	13/9	706.103
	23/13	706.127
	17/10	706.214
	17/15	708.343

* Besides the octave

Music

Claudi Meneghin

• Lytel Twyelyghte Musicke (Little Twilight Music), for Brass, Winds, Strings, and Timpani, in 80-equal division of the octave, as the linear temperament generated by its fifth (2025)