Leapday: Difference between revisions

: ''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|temperament]] based on the [[chain of fifths]], but the fifth is tuned slightly sharp of just (approximately 704{{cent}}) so that 15 fifths give [[7/4]] and 21 fifths give [[5/4]]. In other words, the classical major third (5/4) is represented by a triple-augmented unison (C–C𝄪♯), and the harmonic seventh (7/4) is represented by a double-augmented fifth (C–G𝄪).

Leapday can be easily extended to the [[13-limit]] by identifying [[14/11]] with the major third and [[13/11]] with the minor third. This implies 11/8 is represented by an augmented third (C–E♯) and 13/8 is represented by an augmented fifth (C–G♯).  

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.

 

Since ratios of 5 are complex, they can be omitted, and the [[2.3.7.11.13 subgroup|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.

 

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>.

 

See [[Hemifamity temperaments #Leapday]] for technical data.

In the following table, odd harmonics 1–23 are in '''bold'''.  

! rowspan="2" | #

! rowspan="2" | Cents*

! colspan="2" | Approximate ratios

| 704.2

| 208.5

| 912.7

| 416.9

| 14/11, 33/26

| 1121.2

| 625.4

| 129.6

| 833.9

| 338.1

| 1042.3

| 546.6

| 50.8

| 755.1

| 259.3

| 963.5

| 467.8

| 1172.0

| 676.2

| 180.5

| 884.7

| 388.9

| 1093.2

| 597.4

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings

! rowspan="2" |

! colspan="3" | Euclidean

| CTE: ~3/2 = 704.2924{{c}}

| CWE: ~3/2 = 704.2346{{c}}

| POTE: ~3/2 = 704.2138{{c}}

|+ style="font-size: 105%; white-space: nowrap;" | No-19 23-limit norm-based tunings

! colspan="3" | Euclidean

|-

|-

| POTE: ~3/2 = 704.2246{{c}}

|}

! Edo<br>generator

! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*

| 3/2

| Pythagorean tuning

| 41cc… val, lower bound of 5-odd-limit diamond monotone

| 29g val, lower bound of 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone

| 23/15

| 703.750

| 21/13

| 23/20

| 703.869

| 21/11

| 75dfg val

| 15/8

| 13/7

| 17/11

| 121defg val

| 5/3

| 9/5

| 17-odd-limit, no-19 21- and 23-odd-limit diamond monotone (singleton)

| 11/7

| 17/12

| 23/14

| 7/4

| 17/9

| 63c val

| 11/6

| 13/8

| 17cg val, upper bound of 5-, 7-, 9-, 11-, 13-, and 15-odd-limit diamond monotone

| 13/9

| 23/13

| 17/10

<nowiki/>* Besides the octave

 

== References and external links ==

[[Category:Leapday| ]] <!-- Main article -->

[[Category:Rank-2 temperaments]]

[[Category:Aberschismic temperaments]]