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{{ED intro}}
{{ED intro}}


43EDT is related to [[27edo|27 EDO]], but with the 3/1 rather than the 2/1 being just. It has octaves compressed by about 5.7492{{c}} compressed and is consistent to the [[9-odd-limit|10-integer-limit]].
== Theory ==
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.


== Properties ==
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}}.
This tuning is related to 27EDO having 5.7{{c}} octave compression, 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|13EDT]] is not. The {{sl|4L 5s}} 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)}}


== Harmonics ==
=== Subsets and supersets ===
{{Harmonics in equal
43edt is the 14th [[prime equal division|prime edt]], following [[41edt]] and coming before [[47edt]].
| steps = 43
| num = 3
| denom = 1
| intervals = prime
}}
{{Harmonics in equal
| steps = 43
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = prime
}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2 right-3"
|-
|-
! Degrees
! #
! Cents
! Cents
! [[Hekt]]s
! [[Hekt]]s
! Corresponding<br />JI intervals
! Approximate ratios
|-
|-
| 1
| 1
| 44.232
| 44.2
| 30.233
| 30.2
| 40/39, 39/38
| 39/38, 40/39
|-
|-
| 2
| 2
| 88.463
| 88.5
| 60.465
| 60.5
| [[20/19]]
| [[20/19]]
|-
|-
| 3
| 3
| 132.695
| 132.7
| 90.698
| 90.7
| [[27/25]]
| [[27/25]]
|-
|-
| 4
| 4
| 176.926
| 176.9
| 120.93
| 120.9
| [[10/9]]
| [[10/9]]
|-
|-
| 5
| 5
| 221.158
| 221.2
| 151.163
| 151.2
| [[25/22]]
| [[25/22]]
|-
|-
| 6
| 6
| 265.389
| 265.4
| 181.395
| 181.4
| ([[7/6]])
| [[7/6]]
|-
|-
| 7
| 7
| 309.621
| 309.6
| 211.628
| 211.6
| [[6/5]]
| [[6/5]]
|-
|-
| 8
| 8
| 353.852
| 353.9
| 241.8605
| 241.9
| [[27/22]]
| [[27/22]]
|-
|-
| 9
| 9
| 398.084
| 398.1
| 272.093
| 272.1
| 24/19
| [[24/19]]
|-
|-
| 10
| 10
| 442.315
| 442.3
| 302.326
| 302.3
| 9/7
| [[9/7]]
|-
|-
| 11
| 11
| 486.547
| 486.5
| 332.558
| 332.6
| (45/34)
| [[45/34]]
|-
|-
| 12
| 12
| 530.778
| 530.8
| 362.791
| 362.8
| (34/25)
| [[34/25]]
|-
|-
| 13
| 13
| 575.01
| 575.0
| 393.023
| 393.0
| (39/28)
| [[39/28]]
|-
|-
| 14
| 14
| 619.241
| 619.2
| 423.256
| 423.3
| [[10/7]]
| [[10/7]]
|-
|-
| 15
| 15
| 663.473
| 663.5
| 453.488
| 453.5
| [[22/15]]
| [[22/15]]
|-
|-
| 16
| 16
| 707.704
| 707.7
| 483.721
| 483.7
| [[3/2]]
| [[3/2]]
|-
|-
| 17
| 17
| 751.936
| 751.9
| 513.9535
| 514.0
| 105/68, [[20/13]]
| [[20/13]], 105/68
|-
|-
| 18
| 18
| 796.167
| 796.2
| 544.186
| 544.2
| [[19/12]]
| [[19/12]]
|-
|-
| 19
| 19
| 840.399
| 840.4
| 574.419
| 574.4
| [[13/8]]
| [[13/8]]
|-
|-
| 20
| 20
| 884.63
| 884.6
| 604.651
| 604.7
| [[5/3]]
| [[5/3]]
|-
|-
| 21
| 21
| 928.862
| 928.9
| 634.883
| 634.9
| [[12/7]]
| [[12/7]]
|-
|-
| 22
| 22
| 973.093
| 973.1
| 665.116
| 665.1
| 7/4
| [[7/4]]
|-
|-
| 23
| 23
| 1017.325
| 1017.3
| 695.349
| 695.3
| [[9/5]]
| [[9/5]]
|-
|-
| 24
| 24
| 1061.556
| 1061.6
| 725.581
| 725.6
| [[24/13]]
| [[24/13]]
|-
|-
| 25
| 25
| 1105.788
| 1105.8
| 755.814
| 755.8
| [[36/19]]
| [[36/19]]
|-
|-
| 26
| 26
| 1150.019
| 1150.0
| 786.0465
| 786.0
| 68/35, 39/20
| [[39/20]], [[68/35]]
|-
|-
| 27
| 27
| 1194.251
| 1194.3
| 816.279
| 816.3
| [[2/1]]
| [[2/1]]
|-
|-
| 28
| 28
| 1238.482
| 1238.5
| 846.511
| 846.5
| [[45/44|45/22]]
| [[45/22]]
|-
|-
| 29
| 29
| 1282.713
| 1282.7
| 876.744
| 876.7
| ([[21/20|21/10]])
| [[21/10]]
|-
|-
| 30
| 30
| 1326.946
| 1326.9
| 906.977
| 907.0
| ([[14/13|28/13]])
| [[28/13]]
|-
|-
| 31
| 31
| 1371.177
| 1371.2
| 937.209
| 937.2
| (75/34)
| 75/34
|-
|-
| 32
| 32
| 1415.408
| 1415.4
| 967.442
| 967.4
| ([[17/15|34/15]])
| [[34/15]]
|-
|-
| 33
| 33
| 1459.640
| 1459.6
| 997.674
| 997.7
| 7/3
| [[7/3]]
|-
|-
| 34
| 34
| 1503.871
| 1503.9
| 1027.907
| 1027.9
| 19/8
| [[19/8]]
|-
|-
| 35
| 35
| 1548.193
| 1548.1
| 1058.1395
| 1058.1
| [[11/9|22/9]]
| [[22/9]]
|-
|-
| 36
| 36
| 1592.334
| 1592.3
| 1088.372
| 1088.3
| 5/2
| [[5/2]]
|-
|-
| 37
| 37
| 1636.566
| 1636.6
| 1118.605
| 1118.6
| ([[9/7|18/7]])
| [[18/7]]
|-
|-
| 38
| 38
| 1680.797
| 1680.8
| 1148.837
| 1148.8
| [[33/25|66/25]]
| [[66/25]]
|-
|-
| 39
| 39
| 1725.029
| 1725.0
| 1179.069
| 1179.1
| 27/10
| [[27/10]]
|-
|-
| 40
| 40
| 1769.261
| 1769.3
| 1209.302
| 1209.3
| [[25/18|25/9]]
| [[25/9]]
|-
|-
| 41
| 41
| 1813.492
| 1813.5
| 1239.5345
| 1239.5
| 57/20
| 57/20
|-
|-
| 42
| 42
| 1857.724
| 1857.7
| 1269.767
| 1269.8
| 117/40, [[19/13|38/13]]
| [[38/13]], 117/40
|-
|-
| 43
| 43
| 1901.955
| 1902.0
| 1300
| 1300.0
| '''exact [[3/1]]'''
| [[3/1]]
|}
|}


= 43EDT as a regular temperament =
== Related regular temperaments ==
43EDT tempers out a no-twos comma of {{vector|0 63 -43}}, leading the regular temperament supported by [[27edo|27]], [[190edo|190]], and [[217edo|217]] EDOs.
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]].


== {{nowrap|27 &amp; 190}} temperament ==
=== 27 &amp; 190 temperament ===
=== 5-limit ===
==== 5-limit ====
Comma: {{vector|0 63 -43}}
Subgroup: 2.3.5


POTE generator: ~{{vector|0 -41 28}} = 44.2294
Comma list: {{monzo| 0 63 -43 }}


Mapping: [{{map|1 0 0}}, {{map|0 43 63}}]
Mapping: {{mapping| 1 0 0 | 0 43 63 }}


EDOs: {{EDOs|27, 190, 217, 407, 597, 624, 841}}
Optimal tuning (POTE): ~{{monzo| 0 -41 28 }} = 44.2294


=== 7-limit ===
{{Optimal ET sequence|legend=0| 27, 190, 217, 407, 597, 624, 841 }}
Commas: 4375/4374, 40353607/40000000


POTE generator: ~1029/1000 = 44.2288
==== 7-limit ====
Subgroup: 2.3.5.7


Mapping: [{{map|1 0 0 1}}, {{map|0 43 63 49}}]
Comma list: 4375/4374, 40353607/40000000


EDOs: {{EDOs|27, 190, 217}}
Mapping: {{mapping| 1 0 0 1 | 0 43 63 49 }}
 
Optimal tuning (POTE): ~1029/1000 = 44.2288
 
{{Optimal ET sequence|legend=0| 27, 190, 217 }}


Badness: 0.1659
Badness: 0.1659


== {{nowrap|217 &amp; 407}} temperament ==
=== 217 &amp; 407 temperament ===
=== 7-limit ===
==== 7-limit ====
Commas: 134217728/133984375, 512557306947/512000000000
Subgroup: 2.3.5.7
 
Comma list: 134217728/133984375, 512557306947/512000000000


POTE generator: ~525/512 = 44.2320
Mapping: {{mapping| 1 0 0 9 | 0 43 63 -168 }}


Mapping: [{{map|1 0 0 9}}, {{map|0 43 63 -168}}]
Optimal tuning (POTE): ~525/512 = 44.2320


EDOs: {{EDOs|217, 407, 624, 841, 1058, 1465}}
{{Optimal ET sequence|legend=0| 217, 407, 624, 841, 1058, 1465 }}


Badness: 0.3544
Badness: 0.3544


=== 11-limit ===
==== 11-limit ====
Commas: 46656/46585, 131072/130977, 234375/234256
Subgroup: 2.3.5.7.11
 
Comma list: 46656/46585, 131072/130977, 234375/234256


POTE generator: ~525/512 = 44.2312
Mapping: {{mapping| 1 0 0 9 -1 | 0 43 63 -168 121 }}


Mapping: [{{map|1 0 0 9 -1}}, {{map|0 43 63 -168 121}}]
Optimal tuning (POTE): ~525/512 = 44.2312


EDOs: {{EDOs|217, 407, 624}}
{{Optimal ET sequence|legend=0| 217, 407, 624 }}


Badness: 0.1129
Badness: 0.1129


=== 13-limit ===
==== 13-limit ====
Commas: 2080/2079, 4096/4095, 39366/39325, 109512/109375
Subgroup: 2.3.5.7.11.13


POTE generator: ~40/39 = 44.2312
Comma list: 2080/2079, 4096/4095, 39366/39325, 109512/109375


Mapping: [{{map|1 0 0 9 -1 3}}, {{map|0 43 63 -168 121 19}}]
Mapping: {{mapping| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}


EDOs: {{EDOs|217, 407, 624}}
Optimal tuning (POTE): ~40/39 = 44.2312
 
{{Optimal ET sequence|legend=0| 217, 407, 624 }}


Badness: 0.0503
Badness: 0.0503


[[Category:Edt]]
== See also ==
[[Category:Edonoi]]
* [[16edf]] – relative edf
* [[27edo]] – relative edo
* [[70ed6]] – relative ed6
* [[90ed10]] – relative ed10
* [[97ed12]] – relative ed12
 
[[Category:27edo]]
Retrieved from "https://en.xen.wiki/w/43edt"