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. It is based on the chain of fifths, but here, 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 classical major third is represented by a triply augmented unison (e.g. C–C𝄪♯), and 17/16 is represented by an octave-reduced triply augmented sixth (e.g. C–A𝄪♯).

The no-5's 13-limit version of leapday, known as leapfrog, is notable as tempering parapythic (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 prime 5 is complex and the canonical mapping for prime 19 is fairly off.

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 1–21 are in bold.

#	Cents*	Approximate Ratios
0	0.0	1/1
1	704.3	3/2
2	208.6	9/8
3	912.9	22/13, 27/16
4	417.2	14/11
5	1121.5	21/11, 40/21
6	625.8	10/7, 13/9
7	130.0	13/12, 14/13, 15/14
8	834.3	13/8, 21/13
9	338.6	11/9, 39/32, 40/33
10	1042.9	11/6, 20/11
11	547.2	11/8, 15/11
12	51.5	28/27, 33/32, 40/39, 45/44
13	755.8	14/9, 20/13
14	260.1	7/6, 15/13
15	964.4	7/4
16	468.7	21/16
17	1173.0	63/32, 160/81
18	677.3	40/27
19	181.6	10/9
20	885.8	5/3
21	390.1	5/4
22	1094.4	15/8
23	598.7	45/32

* In 13-limit CTE tuning

Tunings

Tuning spectrum

Edo generator	Eigenmonzo (unchanged-interval)*	Generator (¢)	Comments
	19/16	700.829
	19/12	701.110
	19/18	701.279
	3/2	701.955
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
	19/15	703.630
	19/10	703.700
	21/13	703.782
	19/11	703.843
	21/19	703.856
	21/11	703.893
	19/13	703.910
	19/14	703.962
	19/17	703.979	19- and 21-odd-limit minimax
44\75		704.000	75dfgh 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	121defgh val
	5/3	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
	11/7	704.377
	21/16	704.424
	17/12	704.478
	7/4	704.588
	17/9	704.593
	11/8	704.665
37\63		704.762	63ch val
	7/6	704.776
	11/6	704.936
	9/7	704.994
	13/8	705.066
	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
	13/9	706.103
	17/10	706.214
	17/15	708.343

* Besides the octave