43edt: Difference between revisions

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== Theory ==

43edt is related to [[27edo]], but with the 3/1 rather than the 2/1 being just. ~~It has octaves compressed by about 5.7492{{c}} and~~ is consistent to the [[9-odd-limit|10-integer-limit]]. ~~The octave compression is~~ a small but significant deviation. This is particularly relevant because the harmonics 27edo approximates well—3, 5, 7, and 13—are all tuned sharp, so 43edt improves those approximations.

43edt is related to [[27edo]], but with the 3/1 rather than the 2/1 being just. Like 27edo, it is consistent to the [[9-odd-limit|10-integer-limit]]. It has octaves compressed by about 5.7492{{c}}, a small but significant deviation. This is particularly relevant because the harmonics 27edo approximates well—3, 5, 7, and 13—are all tuned sharp, so 43edt improves those approximations.

However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, [[Bohlen–Pierce]] harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that [[13edt]] is not. The {{sl|4L 5s}} [[mos]] has {{nowrap|L {{=}} 7~~}}, {{nowrap~~|s {{=}} 3}}.

However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, [[Bohlen–Pierce]] harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that [[13edt]] is not. The {{mos scalesig|4L 5s<3/1>|link=1}} [[mos]] has {{nowrap|L {{=}} 7|s {{=}} 3}}.

=== Harmonics ===

{{Harmonics in equal|43|3|1|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 43edt (continued)}}

=== Subsets and supersets ===

43edt is the 14th [[prime equal division|prime edt]], following [[41edt]] and coming before [[47edt]].

== Intervals ==

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|}

== ~~As a~~ regular ~~temperament~~ ==

== Related regular temperaments ==

43edt tempers out the no-twos comma of {{monzo| 0 63 -43 }}, leading to the regular temperament [[support]]ed by [[27edo|27-]], [[190edo|190-]], and [[217edo]].

=== 27 & 190 temperament ===

==== 5-limit ====

Subgroup: 2.3.5

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Badness: 0.1659

=== 217 & 407 temperament ===

==== 7-limit ====

Subgroup: 2.3.5.7

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Badness: 0.0503

== See also ==

* [[16edf]] – relative edf

* [[27edo]] – relative edo

* [[70ed6]] – relative ed6

* [[90ed10]] – relative ed10

* [[97ed12]] – relative ed12

[[Category:27edo]]

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 == Theory ==
-edt is related to [[27edo]], but with the 3/1 rather than the 2/1 being just. It has octaves compressed by about 5.7492{{c}} and is consistent to the [[9-odd-limit|10-integer-limit]]. The octave compression is a small but significant deviation. This is particularly relevant because the harmonics 27edo approximates well—3, 5, 7, and 13—are all tuned sharp, so 43edt improves those approximations.
+edt is related to [[27edo]], but with the 3/1 rather than the 2/1 being just. Like 27edo, it is consistent to the [[9-odd-limit|10-integer-limit]]. It has octaves compressed by about 5.7492{{c}}, a small but significant deviation. This is particularly relevant because the harmonics 27edo approximates well—3, 5, 7, and 13—are all tuned sharp, so 43edt improves those approximations.
-However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, [[Bohlen–Pierce]] harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that [[13edt]] is not. The {{sl|4L 5s}} [[mos]] has {{nowrap|L {{=}} 7}}, {{nowrap|s {{=}} 3}}.
+However, in addition to its rich octave-based harmony, the 43edt is also a fine tritave-based tuning: with a 7/3 of 1460 cents and such a near perfect 5/3, [[Bohlen–Pierce]] harmony is very clear and hearty, as well as capable of extended enharmonic distinctions that [[13edt]] is not. The {{mos scalesig|4L 5s<3/1>|link=1}} [[mos]] has {{nowrap|L {{=}} 7|s {{=}} 3}}.
 === Harmonics ===
 {{Harmonics in equal|43|3|1}}
 {{Harmonics in equal|43|3|1|start=12|columns=12|collapsed=true|title=Approximation of harmonics in 43edt (continued)}}
+=== Subsets and supersets ===
+edt is the 14th [[prime equal division|prime edt]], following [[41edt]] and coming before [[47edt]].
 == Intervals ==
@@ Line 235: / Line 238: @@
 |}
-== As a regular temperament ==
+== Related regular temperaments ==
 edt tempers out the no-twos comma of {{monzo| 0 63 -43 }}, leading to the regular temperament [[support]]ed by [[27edo|27-]], [[190edo|190-]], and [[217edo]].
-=== 27 & 190 temperament ===
+=== 27 &amp; 190 temperament ===
 ==== 5-limit ====
 Subgroup: 2.3.5
@@ Line 263: / Line 266: @@
 Badness: 0.1659
-=== 217 & 407 temperament ===
+=== 217 &amp; 407 temperament ===
 ==== 7-limit ====
 Subgroup: 2.3.5.7
@@ Line 302: / Line 305: @@
 Badness: 0.0503
+== See also ==
+* [[16edf]] – relative edf
+* [[27edo]] – relative edo
+* [[70ed6]] – relative ed6
+* [[90ed10]] – relative ed10
+* [[97ed12]] – relative ed12
+[[Category:27edo]]