Leapday: Difference between revisions
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: ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]] or [[Irvic scale|Irvian mode]].'' | : ''Not to be confused with calendar-based scales such as those in [[293edo]], [[400edo]], [[353edo]], or [[Irvic scale|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 | '''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{{cent}}) 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]])<sup>4</sup>, 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<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10589.html Yahoo! Tuning Group (Archive) | ''Some 13-limit temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10604.html Yahoo! Tuning Group (Archive) | ''24 13-limit temperaments supported by 46'']</ref>. | Leapday was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10589.html Yahoo! Tuning Group (Archive) | ''Some 13-limit temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10604.html Yahoo! Tuning Group (Archive) | ''24 13-limit temperaments supported by 46'']</ref>. | ||
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{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
|- | |- | ||
! rowspan="2" | | ! rowspan="2" | # | ||
! rowspan="2" | Cents* | ! rowspan="2" | Cents* | ||
! colspan="2" | Approximate ratios | ! colspan="2" | Approximate ratios | ||
|- | |- | ||
! 13-limit | ! 13-limit | ||
! | ! Additional ratios<br />of 17 and 23 | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 45: | Line 53: | ||
| 4 | | 4 | ||
| 417.2 | | 417.2 | ||
| 14/11 | | 14/11, 33/26 | ||
| 23/18 | | 23/18 | ||
|- | |- | ||
| Line 150: | Line 158: | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
! Edo<br | ! Edo<br>generator | ||
! [[Eigenmonzo| | ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
| Line 420: | Line 428: | ||
| | | | ||
|} | |} | ||
<nowiki />* Besides the octave | <nowiki/>* Besides the octave | ||
== References and external links == | == References and external links == | ||
<references /> | <references/> | ||
[[Category:Leapday| ]] <!-- Main article --> | [[Category:Leapday| ]] <!-- Main article --> | ||
[[Category: | [[Category:Rank-2 temperaments]] | ||
[[Category:Sengic temperaments]] | [[Category:Sengic temperaments]] | ||
[[Category:Hemifamity temperaments]] | [[Category:Hemifamity temperaments]] | ||
Latest revision as of 06:56, 21 June 2025
- 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