46edo: Difference between revisions

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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 temperament]]s it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[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.

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 temperament]]s it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[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 [[41-EDO]]. 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 (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.

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

[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of ~~46edo~~ for [[Indian]] music.

[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46-EDO for [[Indian]] music.

== Intervals ==

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<nowiki>*</nowiki> Based on treating 46-~~edo~~ as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46-~~edo~~ has the 15th harmony poorly approximated in general, because, while both the 3rd and 5th harmonies are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\~~46edo~~ is closer to 15/13 by a hair; 10\~~46edo~~ represents the difference between, for instance, ~~46edo~~'s 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.

<nowiki>*</nowiki> Based on treating 46-EDO as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46-EDO has the 15th harmony poorly approximated in general, because, while both the 3rd and 5th harmonies are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\46-EDO is closer to 15/13 by a hair; 10\46-EDO represents the difference between, for instance, 46-EDO's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.

Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:

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| 9/7, 12/7

|}

All ~~46edo~~ chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:

All 46-EDO chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:

{| class="wikitable center-all"

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! color of the 3rd

! JI chord

! notes as ~~edosteps~~

! notes as EDO steps

! notes of C chord

! written name

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== JI approximation ==

=== 15-odd-limit interval mappings ===

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 46-EDO. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

{| class="wikitable center-all"

|+Direct mapping (even if inconsistent)

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=== Harmonic scales ===

~~46edo~~ represents [[harmonic 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.

46-EDO represents [[harmonic 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]] (203.910¢).

* 8\46 (208.696¢) stands in for frequency ratio [[9/8]] (203.910¢).

* 7\~~46edo~~ (182.609¢) stands in for [[10/9]] (182.404¢).

* 7\46 (182.609¢) stands in for [[10/9]] (182.404¢).

* 6\~~46edo~~ (156.522¢) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).

* 6\46 (156.522¢) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).

* 5\~~46edo~~ (130.435¢) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).

* 5\46 (130.435¢) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).

* 4\~~46edo~~ (104.348¢) stands in for [[16/15]] (111.731¢).

* 4\46 (104.348¢) stands in for [[16/15]] (111.731¢).

== Music ==

* [http://aaronkristerjohnson.bandcamp.com/track/satiesque Satiesque] by [[Aaron Krister Johnson]].

* [http://www.archive.org/details/Chromosounds Chromosounds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3 play] by [[Gene Ward Smith]].

* [http://www.archive.org/details/MusicForYourEars Music For Your Ears] [http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] by [[Gene Ward Smith]]. The central portion is in [[~~27edo~~]], the rest is in ~~46edo~~.

* [http://www.archive.org/details/MusicForYourEars Music For Your Ears] [http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] by [[Gene Ward Smith]]. The central portion is in [[27-EDO]], the rest is in 46-EDO.

* [http://andrewheathwaite.bandcamp.com/track/rats Rats] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2001%20Rats.mp3 play] by [[Andrew Heathwaite]].

* [http://andrewheathwaite.bandcamp.com/track/tumbledown-stew Tumbledown Stew] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2012%20Tumbledown%20Stew.mp3 play] by [[Andrew Heathwaite]].

@@ Line 69: / Line 69: @@
 |}
-et 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 temperament]]s it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[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.
+et 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 temperament]]s it supports include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-limit]] [[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 [[41-EDO]]. 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 (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.
-edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.
+-EDO can be treated as two [[23-EDO]]'s separated by an interval of 26.087 cents.
-[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.
+[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46-EDO for [[Indian]] music.
 == Intervals ==
@@ Line 463: / Line 463: @@
 | do
 |}
-<nowiki>*</nowiki> Based on treating 46-edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46-edo has the 15th harmony poorly approximated in general, because, while both the 3rd and 5th harmonies are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\46edo is closer to 15/13 by a hair; 10\46edo represents the difference between, for instance, 46edo's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.
+<nowiki>*</nowiki> Based on treating 46-EDO as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46-EDO has the 15th harmony poorly approximated in general, because, while both the 3rd and 5th harmonies are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\46-EDO is closer to 15/13 by a hair; 10\46-EDO represents the difference between, for instance, 46-EDO's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.
 Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
@@ Line 514: / Line 514: @@
 | 9/7, 12/7
 |}
-All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:
+All 46-EDO chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:
 {| class="wikitable center-all"
@@ Line 520: / Line 520: @@
 ! color of the 3rd
 ! JI chord
-! notes as edosteps
+! notes as EDO steps
 ! notes of C chord
 ! written name
@@ Line 571: / Line 571: @@
 == JI approximation ==
 === 15-odd-limit interval mappings ===
-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 46-EDO. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
 {| class="wikitable center-all"
 |+Direct mapping (even if inconsistent)
@@ Line 895: / Line 895: @@
 === Harmonic scales ===
-edo represents [[harmonic 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.
+-EDO represents [[harmonic 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]] (203.910¢).
+* 8\46 (208.696¢) stands in for frequency ratio [[9/8]] (203.910¢).
-* 7\46edo (182.609¢) stands in for [[10/9]] (182.404¢).
+* 7\46 (182.609¢) stands in for [[10/9]] (182.404¢).
-* 6\46edo (156.522¢) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).
+* 6\46 (156.522¢) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).
-* 5\46edo (130.435¢) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).
+* 5\46 (130.435¢) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).
-* 4\46edo (104.348¢) stands in for [[16/15]] (111.731¢).
+* 4\46 (104.348¢) stands in for [[16/15]] (111.731¢).
 == Music ==
 * [http://aaronkristerjohnson.bandcamp.com/track/satiesque Satiesque] by [[Aaron Krister Johnson]].
 * [http://www.archive.org/details/Chromosounds Chromosounds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3 play] by [[Gene Ward Smith]].
-* [http://www.archive.org/details/MusicForYourEars Music For Your Ears] [http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] by [[Gene Ward Smith]]. The central portion is in [[27edo]], the rest is in 46edo.
+* [http://www.archive.org/details/MusicForYourEars Music For Your Ears] [http://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] by [[Gene Ward Smith]]. The central portion is in [[27-EDO]], the rest is in 46-EDO.
 * [http://andrewheathwaite.bandcamp.com/track/rats Rats] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2001%20Rats.mp3 play] by [[Andrew Heathwaite]].
 * [http://andrewheathwaite.bandcamp.com/track/tumbledown-stew Tumbledown Stew] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2012%20Tumbledown%20Stew.mp3 play] by [[Andrew Heathwaite]].