Xenharmonic Wiki

Leapday: Difference between revisions

← Older edit
VisualWikitext
→Tunings: simplify the ratios; mark the vals
m Text replacement - "Eigenmonzo<br>(unchanged-interval)" to "Unchanged interval<br>(eigenmonzo)"
Tags: Mobile edit Mobile web edit
 
(17 intermediate revisions by 3 users not shown)
Line 1: Line 1:
: ''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 19-limit. 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{{cent}}), so that 13/8 is represented by an augmented fifth (e.g.&nbsp;C&ndash;G&#x266F;), 11/8 is represented by an augmented third (e.g.&nbsp;C&ndash;E&#x266F;), the harmonic seventh is represented by a doubly augmented fifth (e.g.&nbsp;C&ndash;G&#x1D12A;), the classical major third is represented by a triply augmented unison (e.g.&nbsp;C&ndash;C&#x1D12A;&#x266F;), and 17/16 is represented by an octave-reduced triply augmented sixth (e.g.&nbsp;C&ndash;A&#x1D12A;&#x266F;).
'''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]].  


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]])<sup>4</sup>, that is, by tempering out the [[tetris comma]], and is a good combination of simplicity and accuracy, as prime 5 is complex and the canonical mapping for prime 19 is fairly off.
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>.  
Line 10: Line 20:


== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 are in '''bold'''.  
In the following table, odd harmonics 1–23 are in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! #
|-
! Cents*
! rowspan="2" | #
! Approximate Ratios
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
|-
! 13-limit
! Additional ratios<br />of 17 and 23
|-
|-
| 0
| 0
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|-
|-
| 1
| 1
| 704.3
| 704.3
| '''3/2'''
| '''3/2'''
|
|-
|-
| 2
| 2
| 208.6
| 208.6
| '''9/8'''
| '''9/8'''
| 17/15, 26/23
|-
|-
| 3
| 3
| 912.9
| 912.9
| 22/13, 27/16
| 22/13, 27/16
| 17/10
|-
|-
| 4
| 4
| 417.2
| 417.2
| 14/11
| 14/11, 33/26
| 23/18
|-
|-
| 5
| 5
| 1121.5
| 1121.5
| 21/11, 40/21
| 21/11, 40/21
| 23/12, 44/23
|-
|-
| 6
| 6
| 625.8
| 625.8
| 10/7, 13/9
| 10/7, 13/9
| '''23/16'''
|-
|-
| 7
| 7
| 130.0
| 130.0
| 13/12, 14/13, 15/14
| 13/12, 14/13, 15/14
|
|-
|-
| 8
| 8
| 834.3
| 834.3
| '''13/8''', 21/13
| '''13/8''', 21/13
| 34/21
|-
|-
| 9
| 9
| 338.6
| 338.6
| 11/9, 39/32, 40/33
| 11/9, 39/32, 40/33
| 17/14, 28/23
|-
|-
| 10
| 10
| 1042.9
| 1042.9
| 11/6, 20/11
| 11/6, 20/11
| 42/23
|-
|-
| 11
| 11
| 547.2
| 547.2
| '''11/8''', 15/11
| '''11/8''', 15/11
|
|-
|-
| 12
| 12
| 51.5
| 51.5
| 28/27, 33/32, 40/39, 45/44
| 28/27, 33/32, 40/39, 45/44
| 34/33, 35/34
|-
|-
| 13
| 13
| 755.8
| 755.8
| 14/9, 20/13
| 14/9, 20/13
| 17/11
|-
|-
| 14
| 14
| 260.1
| 260.1
| 7/6, 15/13
| 7/6, 15/13
|
|-
|-
| 15
| 15
| 964.4
| 964.4
| '''7/4'''
| '''7/4'''
| 40/23
|-
|-
| 16
| 16
| 468.7
| 468.7
| '''21/16'''
| '''21/16'''
| 17/13, 30/23
|-
|-
| 17
| 17
| 1173.0
| 1173.0
| 63/32, 160/81
| 63/32, 160/81
| 45/23, 51/26
|-
|-
| 18
| 18
| 677.3
| 677.3
| 40/27
| 40/27
| 34/23
|-
|-
| 19
| 19
| 181.6
| 181.6
| 10/9
| 10/9
|
|-
|-
| 20
| 20
| 885.8
| 885.8
| 5/3
| 5/3
|
|-
|-
| 21
| 21
| 390.1
| 390.1
| '''5/4'''
| '''5/4'''
|
|-
|-
| 22
| 22
| 1094.4
| 1094.4
| '''15/8'''
| '''15/8'''
| 17/9
|-
|-
| 23
| 23
| 598.7
| 598.7
| 45/32
| 45/32
| 17/12
|}
|}
<nowiki />* In 13-limit CTE tuning
<nowiki />* In 13-limit CTE tuning
Line 117: Line 155:
== Tunings ==
== Tunings ==
=== Tuning spectrum ===
=== Tuning spectrum ===
This spectrum assumes 19-limit leapday.


{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 291: Line 329:
|-
|-
|  
|  
| 10/9
| 9/5
| 704.337
| 704.337
| 9-, 11- and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
Line 390: Line 428:
|  
|  
|}
|}
<nowiki>*</nowiki> Besides the octave
<nowiki/>* Besides the octave


== Notes ==
== References and external links ==
<references/>


[[Category:Leapday| ]] <!-- main article -->
[[Category:Leapday| ]] <!-- Main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Sengic temperaments]]
[[Category:Sengic temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]
Retrieved from "https://en.xen.wiki/w/Leapday"