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 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 make 17/16. Equivalently, the fifth in leapday is ~2.3 cents sharp of 3/2 (approximately 704 ¢), so that 23/16 is represented by an augmented fourth (C–F♯), 13/8 is represented by an augmented fifth (C–G♯), 11/8 is represented by an augmented third (C–E♯), the harmonic seventh is represented by a doubly augmented fifth (C–G𝄪), the classical major third is represented by a triply augmented unison (C–C𝄪♯), and 17/16 is represented by an octave-reduced triply augmented sixth (C–A𝄪♯).
Like superpyth, Leapday goes in a completely different direction than meantone despite being based on the circle of fifths: Meantone (including 12edo) has 3/2 tuned flat so that the 5th harmonic's intervals are simple while the 7th, 11th, and 13th harmonics' intervals are complex, while Leapday has 3/2 tuned sharp so that the 7th, 11th, and 13th harmonics' intervals are (relatively) simple while the 5th harmonic's intervals are complex.
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 5/4 is complex and the canonical mapping for prime 19 is fairly off.
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 | No-19 23-limit extension | ||
| 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 | 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 |
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 | ||
| 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