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 JI. 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 temperament was named by Herman Miller in 2004[1][2].
See Hemifamity temperaments #Leapday 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
Tuning spectrum
Gencom: [2 3/2; 91/90 121/120 133/132 136/135 154/153 169/168]
Gencom mapping: [⟨1 1 -10 -6 -3 -1 -10 6], ⟨0 1 21 15 11 8 24 -3]]
| Edo Generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 19/16 | 700.829 | ||
| 24/19 | 701.110 | ||
| 19/18 | 701.279 | ||
| 4/3 | 701.955 | ||
| 24\41 | 702.439 | ||
| 15/14 | 702.778 | ||
| 7/5 | 702.915 | ||
| 21/20 | 703.107 | ||
| 15/11 | 703.359 | ||
| 15/13 | 703.410 | ||
| 17\29 | 703.448 | 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 | ||
| 20/19 | 703.700 | ||
| 26/21 | 703.782 | ||
| 22/19 | 703.843 | ||
| 21/19 | 703.856 | ||
| 22/21 | 703.893 | ||
| 26/19 | 703.910 | ||
| 19/14 | 703.962 | ||
| 19/17 | 703.979 | 19- and 21-odd-limit minimax | |
| 44\75 | 704.000 | ||
| 16/15 | 704.012 | ||
| 17/14 | 704.014 | ||
| 17/13 | 704.027 | ||
| 14/13 | 704.043 | ||
| 5/4 | 704.110 | 5-odd-limit minimax | |
| 22/17 | 704.126 | ||
| 71\121 | 704.132 | ||
| 6/5 | 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 | ||
| 14/11 | 704.377 | ||
| 21/16 | 704.424 | ||
| 24/17 | 704.478 | ||
| 8/7 | 704.588 | ||
| 18/17 | 704.593 | ||
| 11/8 | 704.665 | ||
| 37\63 | 704.762 | ||
| 7/6 | 704.776 | ||
| 12/11 | 704.936 | ||
| 9/7 | 704.994 | ||
| 16/13 | 705.066 | ||
| 11/9 | 705.268 | ||
| 13/12 | 705.510 | ||
| 10\17 | 705.882 | Upper bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone | |
| 18/13 | 706.103 | ||
| 20/17 | 706.214 | ||
| 17/15 | 708.343 |
* besides the octave