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 {{mos scalesig|4L 5s<3/1>|link=1}} [[mos]] has {{nowrap|L {{=}} 7|s {{=}} 3}}.

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Comma list: {{monzo| 0 63 -43 }}

~~Mapping:~~ {{~~mapping~~| 1 0 0 | 0 43 63 }}

Optimal tuning (POTE): ~{{monzo| 0 -41 28 }} = 44.2294

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Comma list: 4375/4374, 40353607/40000000

~~Mapping:~~ {{~~mapping~~| 1 0 0 1 | 0 43 63 49 }}

Optimal tuning (POTE): ~1029/1000 = 44.2288

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Comma list: 134217728/133984375, 512557306947/512000000000

~~Mapping:~~ {{~~mapping~~| 1 0 0 9 | 0 43 63 -168 }}

Optimal tuning (POTE): ~525/512 = 44.2320

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Comma list: 46656/46585, 131072/130977, 234375/234256

~~Mapping:~~ {{~~mapping~~| 1 0 0 9 -1 | 0 43 63 -168 121 }}

Optimal tuning (POTE): ~525/512 = 44.2312

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Comma list: 2080/2079, 4096/4095, 39366/39325, 109512/109375

~~Mapping:~~ {{~~mapping~~| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}

Optimal tuning (POTE): ~40/39 = 44.2312

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== See also ==

* [[16edf]] – relative edf

* [[27edo]] – relative edo

* [[70ed6]] – relative ed6

* [[90ed10]] – relative ed10

* [[97ed12]] – relative ed12

[[Category:27edo]]

@@ Line 3: / Line 3: @@
 == 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 {{mos scalesig|4L 5s<3/1>|link=1}} [[mos]] has {{nowrap|L {{=}} 7|s {{=}} 3}}.
@@ Line 247: / Line 247: @@
 Comma list: {{monzo| 0 63 -43 }}
-Mapping: {{mapping| 1 0 0 | 0 43 63 }}
+{{Mapping|legend=0| 1 0 0 | 0 43 63 }}
 Optimal tuning (POTE): ~{{monzo| 0 -41 28 }} = 44.2294
@@ Line 258: / Line 258: @@
 Comma list: 4375/4374, 40353607/40000000
-Mapping: {{mapping| 1 0 0 1 | 0 43 63 49 }}
+{{Mapping|legend=0| 1 0 0 1 | 0 43 63 49 }}
 Optimal tuning (POTE): ~1029/1000 = 44.2288
@@ Line 272: / Line 272: @@
 Comma list: 134217728/133984375, 512557306947/512000000000
-Mapping: {{mapping| 1 0 0 9 | 0 43 63 -168 }}
+{{Mapping|legend=0| 1 0 0 9 | 0 43 63 -168 }}
 Optimal tuning (POTE): ~525/512 = 44.2320
@@ Line 285: / Line 285: @@
 Comma list: 46656/46585, 131072/130977, 234375/234256
-Mapping: {{mapping| 1 0 0 9 -1 | 0 43 63 -168 121 }}
+{{Mapping|legend=0| 1 0 0 9 -1 | 0 43 63 -168 121 }}
 Optimal tuning (POTE): ~525/512 = 44.2312
@@ Line 298: / Line 298: @@
 Comma list: 2080/2079, 4096/4095, 39366/39325, 109512/109375
-Mapping: {{mapping| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}
+{{Mapping|legend=0| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}
 Optimal tuning (POTE): ~40/39 = 44.2312
@@ Line 307: / Line 307: @@
 == See also ==
+* [[16edf]] – relative edf
 * [[27edo]] – relative edo
 * [[70ed6]] – relative ed6
+* [[90ed10]] – relative ed10
 * [[97ed12]] – relative ed12
 [[Category:27edo]]