343edo: Difference between revisions

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343edo is only [[consistent]] to the [[3-odd-limit]] since its errors of [[harmonic]]s [[3/1|3]] and [[5/1|5]] are quite large. To start with, consider the 2.9.15.7 [[subgroup]], where it [[tempering out|tempers out]] 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649.

For the full 7-limit, the 343c [[val]] tempers out [[4375/4374]] and [[5120/5103]], [[support]]ing [[amity]]. The 343cdd val tempers out [[16875/16807]] and 65536/64827. The [[patent val]] tempers out [[10976/10935]] and 390625/387072.

For the full 7-limit, the 343c [[val]] tempers out [[4375/4374]] and [[5120/5103]], [[support]]ing [[amity]] (gen. 97\343, per. 343\343). The 343cdd val tempers out [[16875/16807]] and 65536/64827. The [[patent val]] tempers out [[10976/10935]] and 390625/387072.

=== Odd harmonics ===

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== Use as a NEJI ==

Of all n-[[afdo]]s where n is between 343 and 800, and where n is a multiple of a simple prime by any number of 2s, or a simple semiprime by any number of 2s, [[476afdo]] (''7x17x2x2'') approximates 343edo ~~best~~. (''See [[User:BudjarnLambeth/Approximating ~~434edo~~ in afdos|Approximating ~~434edo~~ in afdos]].)''

Of all n-[[afdo]]s where n is between 343 and 800, and where n is a multiple of a simple prime by any number of 2s, or a simple semiprime by any number of 2s, [[476afdo]] (''7x17x2x2'') approximates 343edo with the least [[relative error]]. (''See [[User:BudjarnLambeth/Approximating 343edo in afdos|Approximating 343edo in afdos]].)''

343edo could be approximated into 476afdo as a [[neji]] scale. Doing so would make it an over-17-by-7 scale (when viewed through a [[primodal]] lens).

343edo could be approximated into 476afdo as a [[neji]] scale. Doing so would make it an over-17-by-7 scale (when viewed through a [[primodal]] lens). (''[[User:BudjarnLambeth/Approximating 343edo in afdos#Scala file|Scala file]].)''

It would make sense to use smaller over-17, over-7, or over-17-by-7 JI scales as subsets of this neji.

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343edo is on the [[optimal ET sequence]] of [[gammy]] temperament (343be, 10\343 generator, 2/1 period), [[protolangwidge]] temperament (343, 200\343 g, 2/1 p) and [[anthoine]] temperament (343dd, 110\343 g, 2/1 p).

343edo might potentially be useful for [[49th-octave temperaments]] ''(see [[~~fractional~~-octave temperaments]])'', this is something which hasn't been explored yet.

343edo might potentially be useful for [[49th-octave temperaments]] ''(see [[Fractional-octave temperaments]])'', this is something which hasn't been explored yet.

{| class="wikitable center-4 center-5 center-6"

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** Octave size: 1199.761{{c}}

** TE error: 0.382{{c}}/octave

* 343 (patent val)

** Octave size: 1199.950{{c}}

** TE error: 0.395{{c}}/octave

* 343e

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Line 97:

** Octave size: 1199.888{{c}}

** TE error: 0.461{{c}}/octave

{{Harmonics in cet|3.498|intervals=odd|title=Odd harmonics in TE-tuned 343cf}}

== Scales ==

343edo includes every 49edo scale (see [[49edo#Scales]]).

==== Lucite[23] ====

'''Lucite[23]''' is a 23-tone [[MOS scale]] discovered by [[Gordon Wery]] in October 2025:

* 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13

; Properties

It is very similar to [[23edo]] and can be used as a [[well temperament]] of 23edo.

In his post on Discord describing it, Wery said of the scale:

"''Basically a more complicated version of 23edo, centered around a more minor (less neutral) anti-diatonic scale.''

''This scale has a sort of glassy quality, ample neutral seconds, and two sets of dual fifths--a true dual dual fifth scale.''"

It is [[generator|generated]] by 30\343 (104.956{{c}}).

Lucite[23] can be generalised into a 17-limit [[regular temperament]] called '''[[User:BudjarnLambeth/Regular temperament interpretation of lucite23|lucite temperament]]'''.

; Naming

Lucite is another name for acrylic glass.

Wery named the temperament "lucite" because musically, it sounds like frosted glass (perhaps to do with the timbre/partials of struck glass).

Some coincidences that make the name "lucite" particularly fitting

* Lucite is often installed in double layers in building, and lucite temperament has two sizes of perfect fifth-like interval.

* Lucite[23] is close to ripple[23], but turned inside out; and lucite is reflective and clear like water, but solid instead of liquid

* Lucite is an especially lightweight material, and lucite temperament is lightweight in the way it only needs 18 generators to reach every 17-limit prime.

; Subsets

* [[Modmos]] of lucite[6]: 60 60 30 40 93 60

=== Other MOS scales ===

* Amity[7]: 52 52 45 52 45 52 45

* Amity[11]: 45 7 45 45 7 45 7 45 45 7 42

* Amity[18]: 7 38 7 38 7 7 38 7 7 38 7 38 7 7 38 7 38 7

* Amity[25]: 7 31 7 7 7 31 7 7 31 7 7 7 31 7 7 7 31 7 7 31 7 7 7 31 7

* Amity[32]: 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7

* Amity[39]: 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7

* Amity[53]: 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7

** ''Try approximating scales fron 53edo ([[53edo#Scales]]) within the amity[53] scale''

* Amity[99]: 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3

** ''Try approximating scales fron 99edo ([[99edo#Scales]]) within the amity[99] scale''

* Lucite[23]: 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13

** ''Try approximating scales fron 23edo ([[23edo#Scales]]) within the lucite[23] scale''

* Lucite[34]: 13 4 13 13 4 13 13 4 13 13 4 13 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13

** ''Try approximating scales fron 34edo ([[34edo#Scales]]) within the lucite[34] scale''

=== 343ed16 ===

'''343ed16''' is contained within 343edo (it is every 4th step of 343edo). It is like [[86edo]] with the [[octave stretching|octave stretched]] by 3.5 [[cents]].

It is quite similar to [[136edt]].

Compared to 86edo it improves harmonics 3, 5, 7 and 11. Its mappings of multiple-of-2 harmonics are very inconsistent, though some composers may enjoy this due to the potential to play tricks on the listener by having [[octave equivalence]] fall on a scale step one might not expect.

Many temperaments and scales from 86edo can be used here in 343ed16 too.

{{Harmonics in equal|86|2|1|intervals=integer|columns=12|collapsed=1|title=86edo for comparison}}

=== 34.3edo ===

'''34.3edo''' is contained within 343edo (it is every 10th step of 343edo). It is like [[34edo]] with the [[octave shrinking|octave compressed]] by 11.51 [[cents]].

It has a step size of 34.985{{c}}.

It was discovered by chaseofspades513 and [[YoVariable]] on the [[Xenharmonic Alliance]] Discord server and further described by [[Gordon Wery]].

Compared to 34edo it improves harmonics 7, 11 and 13, at the expense of 2, 3 and 5. Its mappings of multiple-of-2 and multiple-of-3 harmonics are very inconsistent, though some composers may enjoy this due to the potential to play tricks on the listener by having [[octave equivalence]] fall on a scale step one might not expect.

Many temperaments and [[34edo#Scales|scales from 34edo]] can be used here in 34.3edo too.

{{Harmonics in cet|34.985|intervals=integer|columns=12|title=Approximation of harmonics in 34.3edo}}

{{Harmonics in equal|34|2|1|intervals=integer|columns=12|collapsed=1|title=34edo for comparison}}

=== Scales approximated from JI ===

* [[4 of 7-17-19-21-51 pentany]]: 96 50 55 96 46 (sounds like minor pentatonic)

* [[4 of 7-17-19-21-51 by 3/2 tetrapentany]]: 3 9 46 4 34 21 29 8 47 3 9 46 38 46

* [[7-17-19-21 hexany]]: 50 46 50 55 87 55 (sounds like minor hexatonic)

* [[7-17-19-21 by 3/2 trihexany]]: 3 47 8 38 12 38 8 47 3 46 9 29 9 46

* [[9afdo]]: 40 37 34 32 30 28 52 47 43

* [[18afdo]]: 21 19 19 18 17 17 16 16 15 15 14 14 27 25 24 23 22 21

* [[36afdo]]: 11 10 9 10 9 10 9 9 8 9 8 9 8 8 8 8 7 8 7 8 7 7 7 7 14 13 13 12 12 12 12 11 11 11 11 10

* [[72afdo]]: 5 6 5 5 4 5 5 5 5 4 5 5 4 5 4 5 4 4 5 4 4 4 5 4 4 4 4 4 4 4 4 4 4 3 4 4 4 3 4 4 3 4 4 3 4 3 4 3 7 7 6 7 6 7 6 6 6 6 6 6 6 6 6 5 6 5 6 5 6 5 5 5

=== Other scales ===

* [[Equiheptatonic]] (as from [[7edo]]): 49 49 49 49 49 49 49

* [[49edo]]: 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

== Music ==

; [[Budjarn Lambeth]]

* [https://youtu.be/aWqdWHSk5J4 ''Odd Findings in the Caves''] (2025) - ''uses two copies of lucite[23]''

@@ Line 5: / Line 5: @@
 edo is only [[consistent]] to the [[3-odd-limit]] since its errors of [[harmonic]]s [[3/1|3]] and [[5/1|5]] are quite large. To start with, consider the 2.9.15.7 [[subgroup]], where it [[tempering out|tempers out]] 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649.
-For the full 7-limit, the 343c [[val]] tempers out [[4375/4374]] and [[5120/5103]], [[support]]ing [[amity]]. The 343cdd val tempers out [[16875/16807]] and 65536/64827. The [[patent val]] tempers out [[10976/10935]] and 390625/387072.
+For the full 7-limit, the 343c [[val]] tempers out [[4375/4374]] and [[5120/5103]], [[support]]ing [[amity]] (gen. 97\343, per. 343\343). The 343cdd val tempers out [[16875/16807]] and 65536/64827. The [[patent val]] tempers out [[10976/10935]] and 390625/387072.
 === Odd harmonics ===
@@ Line 14: / Line 14: @@
 == Use as a NEJI ==
-Of all n-[[afdo]]s where n is between 343 and 800, and where n is a multiple of a simple prime by any number of 2s, or a simple semiprime by any number of 2s, [[476afdo]] (''7x17x2x2'') approximates 343edo best. (''See [[User:BudjarnLambeth/Approximating 434edo in afdos|Approximating 434edo in afdos]].)''
+Of all n-[[afdo]]s where n is between 343 and 800, and where n is a multiple of a simple prime by any number of 2s, or a simple semiprime by any number of 2s, [[476afdo]] (''7x17x2x2'') approximates 343edo with the least [[relative error]]. (''See [[User:BudjarnLambeth/Approximating 343edo in afdos|Approximating 343edo in afdos]].)''
-edo could be approximated into 476afdo as a [[neji]] scale. Doing so would make it an over-17-by-7 scale (when viewed through a [[primodal]] lens).
+edo could be approximated into 476afdo as a [[neji]] scale. Doing so would make it an over-17-by-7 scale (when viewed through a [[primodal]] lens).  (''[[User:BudjarnLambeth/Approximating 343edo in afdos#Scala file|Scala file]].)''
 It would make sense to use smaller over-17, over-7, or over-17-by-7 JI scales as subsets of this neji.
@@ Line 23: / Line 23: @@
 edo is on the [[optimal ET sequence]] of [[gammy]] temperament (343be, 10\343 generator, 2/1 period), [[protolangwidge]] temperament (343, 200\343 g, 2/1 p) and [[anthoine]] temperament (343dd, 110\343 g, 2/1 p).
-edo might potentially be useful for [[49th-octave temperaments]] ''(see [[fractional-octave temperaments]])'', this is something which hasn't been explored yet.
+edo might potentially be useful for [[49th-octave temperaments]] ''(see [[Fractional-octave temperaments]])'', this is something which hasn't been explored yet.
 {| class="wikitable center-4 center-5 center-6"
@@ Line 81: / Line 81: @@
 ** Octave size: 1199.761{{c}}
 ** TE error: 0.382{{c}}/octave
+* 343 (patent val)
+** Octave size: 1199.950{{c}}
+** TE error: 0.395{{c}}/octave
 * 343e
@@ Line 93: / Line 97: @@
 ** Octave size: 1199.888{{c}}
 ** TE error: 0.461{{c}}/octave
+{{Harmonics in cet|3.498|intervals=odd|title=Odd harmonics in TE-tuned 343cf}}
+== Scales ==
+edo includes every 49edo scale (see [[49edo#Scales]]).
+==== Lucite[23] ====
+'''Lucite[23]''' is a 23-tone [[MOS scale]] discovered by [[Gordon Wery]] in October 2025:
+* 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13
+; Properties
+It is very similar to [[23edo]] and can be used as a [[well temperament]] of 23edo.
+In his post on Discord describing it, Wery said of the scale:
+"''Basically a more complicated version of 23edo, centered around a more minor (less neutral) anti-diatonic scale.''
+''This scale has a sort of glassy quality, ample neutral seconds, and two sets of dual fifths--a true dual dual fifth scale.''"
+It is [[generator|generated]] by 30\343 (104.956{{c}}).
+Lucite[23] can be generalised into a 17-limit [[regular temperament]] called '''[[User:BudjarnLambeth/Regular temperament interpretation of lucite23|lucite temperament]]'''.
+; Naming
+Lucite is another name for acrylic glass.
+Wery named the temperament "lucite" because musically, it sounds like frosted glass (perhaps to do with the timbre/partials of struck glass).
+Some coincidences that make the name "lucite" particularly fitting
+* Lucite is often installed in double layers in building, and lucite temperament has two sizes of perfect fifth-like interval.
+* Lucite[23] is close to ripple[23], but turned inside out; and lucite is reflective and clear like water, but solid instead of liquid
+* Lucite is an especially lightweight material, and lucite temperament is lightweight in the way it only needs 18 generators to reach every 17-limit prime.
+; Subsets
+* [[Modmos]] of lucite[6]: 60 60 30 40 93 60
+=== Other MOS scales ===
+* Amity[7]: 52 52 45 52 45 52 45
+* Amity[11]: 45 7 45 45 7 45 7 45 45 7 42
+* Amity[18]: 7 38 7 38 7 7 38 7 7 38 7 38 7 7 38 7 38 7
+* Amity[25]: 7 31 7 7 7 31 7 7 31 7 7 7 31 7 7 7 31 7 7 31 7 7 7 31 7
+* Amity[32]: 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7 7 24 7 7 7 7 24 7 7
+* Amity[39]: 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 7 17 7 7 7 7 17 7 7 7 7 7 17 7 7
+* Amity[53]: 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7 7 7 7 3 7 7 7 7 7 7 7 3 7 7 7
+** ''Try approximating scales fron 53edo ([[53edo#Scales]]) within the amity[53] scale''
+* Amity[99]: 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3
+** ''Try approximating scales fron 99edo ([[99edo#Scales]]) within the amity[99] scale''
+* Lucite[23]: 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13 17 13
+** ''Try approximating scales fron 23edo ([[23edo#Scales]]) within the lucite[23] scale''
+* Lucite[34]: 13 4 13 13 4 13 13 4 13 13 4 13 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13 13 4 13
+** ''Try approximating scales fron 34edo ([[34edo#Scales]]) within the lucite[34] scale''
+=== 343ed16 ===
+'''343ed16''' is contained within 343edo (it is every 4th step of 343edo). It is like [[86edo]] with the [[octave stretching|octave stretched]] by 3.5 [[cents]].
+It is quite similar to [[136edt]].
+Compared to 86edo it improves harmonics 3, 5, 7 and 11. Its mappings of multiple-of-2 harmonics are very inconsistent, though some composers may enjoy this due to the potential to play tricks on the listener by having [[octave equivalence]] fall on a scale step one might not expect.
+Many temperaments and scales from 86edo can be used here in 343ed16 too.
+{{Harmonics in equal|343|16|1|intervals=integer|columns=12}}
+{{Harmonics in equal|86|2|1|intervals=integer|columns=12|collapsed=1|title=86edo for comparison}}
+=== 34.3edo ===
+'''34.3edo''' is contained within 343edo (it is every 10th step of 343edo). It is like [[34edo]] with the [[octave shrinking|octave compressed]] by 11.51 [[cents]].
+It has a step size of 34.985{{c}}.
+It was discovered by chaseofspades513 and [[YoVariable]] on the [[Xenharmonic Alliance]] Discord server and further described by [[Gordon Wery]].
+Compared to 34edo it improves harmonics 7, 11 and 13, at the expense of 2, 3 and 5. Its mappings of multiple-of-2 and multiple-of-3 harmonics are very inconsistent, though some composers may enjoy this due to the potential to play tricks on the listener by having [[octave equivalence]] fall on a scale step one might not expect.
+Many temperaments and [[34edo#Scales|scales from 34edo]] can be used here in 34.3edo too.
+{{Harmonics in cet|34.985|intervals=integer|columns=12|title=Approximation of harmonics in 34.3edo}}
+{{Harmonics in equal|34|2|1|intervals=integer|columns=12|collapsed=1|title=34edo for comparison}}
+=== Scales approximated from JI ===
+* [[4 of 7-17-19-21-51 pentany]]: 96 50 55 96 46 (sounds like minor pentatonic)
+* [[4 of 7-17-19-21-51 by 3/2 tetrapentany]]: 3 9 46 4 34 21 29 8 47 3 9 46 38 46
+* [[7-17-19-21 hexany]]: 50 46 50 55 87 55 (sounds like minor hexatonic)
+* [[7-17-19-21 by 3/2 trihexany]]: 3 47 8 38 12 38 8 47 3 46 9 29 9 46
+* [[9afdo]]: 40 37 34 32 30 28 52 47 43
+* [[18afdo]]: 21 19 19 18 17 17 16 16 15 15 14 14 27 25 24 23 22 21
+* [[36afdo]]: 11 10 9 10 9 10 9 9 8 9 8 9 8 8 8 8 7 8 7 8 7 7 7 7 14 13 13 12 12 12 12 11 11 11 11 10
+* [[72afdo]]: 5 6 5 5 4 5 5 5 5 4 5 5 4 5 4 5 4 4 5 4 4 4 5 4 4 4 4 4 4 4 4 4 4 3 4 4 4 3 4 4 3 4 4 3 4 3 4 3 7 7 6 7 6 7 6 6 6 6 6 6 6 6 6 5 6 5 6 5 6 5 5 5
+=== Other scales ===
+* [[Equiheptatonic]] (as from [[7edo]]): 49 49 49 49 49 49 49
+* [[49edo]]: 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
+== Music ==
+; [[Budjarn Lambeth]]
+* [https://youtu.be/aWqdWHSk5J4 ''Odd Findings in the Caves''] (2025) - ''uses two copies of lucite[23]''