Leapday

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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[1][2].

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
1 704.3 3/2
2 208.6 9/8 17/15, 26/23
3 912.9 22/13, 27/16 17/10
4 417.2 14/11, 33/26 23/18
5 1121.5 21/11, 40/21 23/12, 44/23
6 625.8 10/7, 13/9 23/16
7 130.0 13/12, 14/13, 15/14
8 834.3 13/8, 21/13 34/21
9 338.6 11/9, 39/32, 40/33 17/14, 28/23
10 1042.9 11/6, 20/11 42/23
11 547.2 11/8, 15/11
12 51.5 28/27, 33/32, 40/39, 45/44 34/33, 35/34
13 755.8 14/9, 20/13 17/11
14 260.1 7/6, 15/13
15 964.4 7/4 40/23
16 468.7 21/16 17/13, 30/23
17 1173.0 63/32, 160/81 45/23, 51/26
18 677.3 40/27 34/23
19 181.6 10/9
20 885.8 5/3
21 390.1 5/4
22 1094.4 15/8 17/9
23 598.7 45/32 17/12

* In 13-limit CTE tuning

Tunings

Tuning spectrum

This spectrum assumes 19-limit leapday.

Edo
generator
Unchanged interval
(eigenmonzo)
*
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
9/5 704.337 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

References and external links