Leapday

Leapday

Subgroups	2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis	686/675, 5120/5103 (7-limit); 121/120, 441/440, 686/675 (11-limit); 91/90, 121/120, 169/168, 352/351 (13-limit)
Reduced mapping	⟨1; 1 21 15 11 8]
ET join	29 & 46
Generators (CWE)	~3/2 = 704.2 ¢
MOS scales	2L 3s, 5L 2s, 5L 7s, 12L 5s
Ploidacot	monocot
Minimax error	9-odd-limit: 8.53 ¢; 13-limit 21-odd-limit: 10.6 ¢
Target scale size	9-odd-limit: 29 notes; 13-limit 21-odd-limit: 29 notes

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

Leapday is a temperament based on the chain of fifths, but the fifth is tuned slightly sharp of just (approximately 704 ¢) so that 15 fifths give 7/4 and 21 fifths give 5/4. In other words, the classical major third (5/4) is represented by a triple-augmented unison (C–C𝄪♯), and the harmonic seventh (7/4) is represented by a double-augmented fifth (C–G𝄪).

Leapday can be easily extended to the 13-limit by identifying 14/11 with the major third and 13/11 with the minor third. This implies 11/8 is represented by an augmented third (C–E♯) and 13/8 is represented by an augmented fifth (C–G♯).

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.

Further extensions for primes 17 and 23 are available, where 17/16 is represented by an octave-reduced triple-augmented sixth (C–A𝄪♯), and 23/16 is represented by an augmented fourth (C–F♯).

Since ratios of 5 are complex, they can be omitted, and 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.

Leapday was named by Herman Miller in 2004.

See Hemifamity temperaments #Leapday for 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)