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46edo: Difference between revisions

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== Theory ==
== Theory ==


46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank_two_temperaments|Rank two temperaments]] it supports include sensi, valentine, shrutar, rodan, leapday and unidec. The [[11-limit|11-limit]] [[Target_tunings|minimax]] tuning for [[Starling_family|valentine temperament]], (11/7)^(1/10), is only 0.01 cents flat of 3/46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit|13-limit]], though others award that distinction to [[41edo|41edo]]. In fact, while 41 is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] but not a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral.
46et tempers out 507/500, 91/90, 686/675, 2048/2025, 121/120, 245/243, 126/125, 169/168, 176/175, 896/891, 196/195, 1029/1024, 5120/5103, 385/384, and 441/440 among other intervals, with various consequences. [[Rank two temperaments]] it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[Target_tunings|minimax]] tuning for valentine temperament, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]]. In fact, while 41 is a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]] but not a [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta integral.


The fifth of 46 equal is 2.39 cents sharp, which some people (eg, [https://en.xen.wiki/w/Margo_Schulter Margo Schulter]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.
The fifth of 46 equal is 2.39 cents sharp, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.


46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.
46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.


[[Magic22_as_srutis#shrutar22assrutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.
[[Magic22_as_srutis #shrutar22assrutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.


== Intervals ==
== Intervals ==
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| C upmajor or C up
| C upmajor or C up
|}
|}
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].


== Just approximation ==
== Just approximation ==
=== Selected just intervals ===
=== Selected just intervals ===
{| class="wikitable center-all"
{| class="wikitable center-all"
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| +6
| +6
|}
|}
The following table shows how [[15-odd-limit intervals]] are represented in 46edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
The following table shows how [[15-odd-limit intervals]] are represented in 46edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  


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|-
|-
! rowspan="2" |Error
! rowspan="2" |Error
! absolute (¢)
! [[TE error|absolute]] (¢)
| 0.75
| 0.75
| 0.90
| 0.90
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| 1.23
| 1.23
|-
|-
! relative (%)
! [[TE simple badness|relative]] (%)
| 2.89
| 2.89
| 3.45
| 3.45
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| [[1L_8s|1L 8s]] (9-tone)
| [[1L_8s|1L 8s]] (9-tone)


[[9L_1s]] (10-tone)
[[9L_1s|9L 1s]] (10-tone)


9L 10s (19-tone)
9L 10s (19-tone)
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| 13\46
| 13\46
| 339.130
| 339.130
| [[Amity]]/[[Hitchcock|hitchcock]]
| [[Amity]]/[[hitchcock]]
| [[4L 3s]] (7-tone)
| [[4L 3s]] (7-tone)


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| 4\46
| 4\46
| 104.348
| 104.348
| [[Srutal]]/[[Diaschismic|diaschismic]]
| [[Srutal]]/[[diaschismic]]
| 2L 2s (4-tone)
| 2L 2s (4-tone)


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46edo represents [[Overtone series|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.
46edo represents [[Overtone series|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.


8\46edo (208.696¢) stands in for frequency ratio [[9/8|9:8]] (203.910¢).
* 8\46edo (208.696¢) stands in for frequency ratio [[9/8|9:8]] (203.910¢).
 
* 7\46edo (182.609¢) stands in for [[10/9|10:9]] (182.404¢).
7\46edo (182.609¢) stands in for [[10/9|10:9]] (182.404¢).
* 6\46edo (156.522¢) stands in for [[11/10|11:10]] (165.004¢) and [[12/11|12:11]] (150.637¢).
 
* 5\46edo (130.435¢) stands in for [[13/12|13:12]] (138.573¢), [[14/13|14:13]] (128.298¢) and [[15/14|15:14]] (119.443¢).
6\46edo (156.522¢) stands in for [[11/10|11:10]] (165.004¢) and [[12/11|12:11]] (150.637¢).
* 4\46edo (104.348¢) stands in for [[16/15|16:15]] (111.731¢).
 
5\46edo (130.435¢) stands in for [[13/12|13:12]] (138.573¢), [[14/13|14:13]] (128.298¢) and [[15/14|15:14]] (119.443¢).
 
4\46edo (104.348¢) stands in for [[16/15|16:15]] (111.731¢).


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/46edo"