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'''[[Edt|Division of the third harmonic]] into 43 equal parts''' (43EDT) is related to [[27edo|27 EDO]], but with the 3/1 rather than the 2/1 being just. The octave is about 5.7492 cents compressed and the step size is about 44.2315 cents. It is consistent to the [[9-odd-limit|10-integer-limit]].
{{Infobox ET}}
{{ED intro}}


=Properties=
== 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.


This tuning is related to 27EDO having ~5.7 cent octave compression, a small but significant deviation. This is particularly relevant because 27EDO is a "sharp tending" system, and flattening its octaves has been suggested before as an improvement (I think by no less than Ivor Darreg, but I'll have to check that).
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}}.


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 4L+5s MOS has 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)}}


{| class="wikitable"
=== Subsets and supersets ===
43edt is the 14th [[prime equal division|prime edt]], following [[41edt]] and coming before [[47edt]].
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
|-
! | degrees
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! [[Hekt]]s
! Approximate ratios
|-
|-
| | 1
| 1
| | 44.232
| 44.2
| | 40/39, 39/38
| 30.2
| 39/38, 40/39
|-
|-
| | 2
| 2
| | 88.463
| 88.5
| | [[20/19]]
| 60.5
| [[20/19]]
|-
|-
| | 3
| 3
| | 132.695
| 132.7
| | [[27/25]]
| 90.7
| [[27/25]]
|-
|-
| | 4
| 4
| | 176.926
| 176.9
| |  
| 120.9
| [[10/9]]
|-
|-
| | 5
| 5
| | 221.158
| 221.2
| | [[25/22]]
| 151.2
| [[25/22]]
|-
|-
| | 6
| 6
| | 265.389
| 265.4
| | ([[7/6]])
| 181.4
| [[7/6]]
|-
|-
| | 7
| 7
| | 309.621
| 309.6
| |  
| 211.6
| [[6/5]]
|-
|-
| | 8
| 8
| | 353.852
| 353.9
| | [[27/22]]
| 241.9
| [[27/22]]
|-
|-
| | 9
| 9
| | 398.084
| 398.1
| |  
| 272.1
| [[24/19]]
|-
|-
| | 10
| 10
| | 442.315
| 442.3
| |  
| 302.3
| [[9/7]]
|-
|-
| | 11
| 11
| | 486.547
| 486.5
| | (45/34)
| 332.6
| [[45/34]]
|-
|-
| | 12
| 12
| | 530.778
| 530.8
| | (34/25)
| 362.8
| [[34/25]]
|-
|-
| | 13
| 13
| | 575.010
| 575.0
| | (39/28)
| 393.0
| [[39/28]]
|-
|-
| | 14
| 14
| | 619.241
| 619.2
| | ([[10/7]])
| 423.3
| [[10/7]]
|-
|-
| | 15
| 15
| | 663.473
| 663.5
| | [[22/15]]
| 453.5
| [[22/15]]
|-
|-
| | 16
| 16
| | 707.704
| 707.7
| |  
| 483.7
| [[3/2]]
|-
|-
| | 17
| 17
| | 751.936
| 751.9
| |  
| 514.0
| [[20/13]], 105/68
|-
|-
| | 18
| 18
| | 796.167
| 796.2
| | [[19/12]]
| 544.2
| [[19/12]]
|-
|-
| | 19
| 19
| | 840.399
| 840.4
| | [[13/8]]
| 574.4
| [[13/8]]
|-
|-
| | 20
| 20
| | 884.630
| 884.6
| | [[5/3]]
| 604.7
| [[5/3]]
|-
|-
| | 21
| 21
| | 928.862
| 928.9
| |  
| 634.9
| [[12/7]]
|-
|-
| | 22
| 22
| | 973.093
| 973.1
| |  
| 665.1
| [[7/4]]
|-
|-
| | 23
| 23
| | 1017.325
| 1017.3
| | [[9/5]]
| 695.3
| [[9/5]]
|-
|-
| | 24
| 24
| | 1061.556
| 1061.6
| | [[24/13]]
| 725.6
| [[24/13]]
|-
|-
| | 25
| 25
| | 1105.788
| 1105.8
| | [[36/19]]
| 755.8
| [[36/19]]
|-
|-
| | 26
| 26
| | 1150.019
| 1150.0
| | 68/35
| 786.0
| [[39/20]], [[68/35]]
|-
|-
| | 27
| 27
| | 1194.251
| 1194.3
| |  
| 816.3
| [[2/1]]
|-
|-
| | 28
| 28
| | 1238.482
| 1238.5
| | [[45/44|45/22]]
| 846.5
| [[45/22]]
|-
|-
| | 29
| 29
| | 1282.713
| 1282.7
| | ([[21/20|21/10]])
| 876.7
| [[21/10]]
|-
|-
| | 30
| 30
| | 1326.946
| 1326.9
| | ([[14/13|28/13]])
| 907.0
| [[28/13]]
|-
|-
| | 31
| 31
| | 1371.177
| 1371.2
| |  
| 937.2
| 75/34
|-
|-
| | 32
| 32
| | 1415.408
| 1415.4
| | ([[17/15|34/15]])
| 967.4
| [[34/15]]
|-
|-
| | 33
| 33
| | 1459.640
| 1459.6
| |  
| 997.7
| [[7/3]]
|-
|-
| | 34
| 34
| | 1503.871
| 1503.9
| |  
| 1027.9
| [[19/8]]
|-
|-
| | 35
| 35
| | 1548.193
| 1548.1
| | [[11/9|22/9]]
| 1058.1
| [[22/9]]
|-
|-
| | 36
| 36
| | 1592.334
| 1592.3
| |  
| 1088.3
| [[5/2]]
|-
|-
| | 37
| 37
| | 1636.566
| 1636.6
| | ([[9/7|18/7]])
| 1118.6
| [[18/7]]
|-
|-
| | 38
| 38
| | 1680.797
| 1680.8
| | [[33/25|66/25]]
| 1148.8
| [[66/25]]
|-
|-
| | 39
| 39
| | 1725.029
| 1725.0
| |  
| 1179.1
| [[27/10]]
|-
|-
| | 40
| 40
| | 1769.261
| 1769.3
| | [[25/18|25/9]]
| 1209.3
| [[25/9]]
|-
|-
| | 41
| 41
| | 1813.492
| 1813.5
| | 57/20
| 1239.5
| 57/20
|-
|-
| | 42
| 42
| | 1857.724
| 1857.7
| | [[19/13|38/13]]
| 1269.8
| [[38/13]], 117/40
|-
|-
| | 43
| 43
| | 1901.955
| 1902.0
| | '''exact [[3/1]]'''
| 1300.0
| [[3/1]]
|}
|}


=43EDT as a regular temperament=
== Related regular temperaments ==
43EDT tempers out a no-twos comma of |0 63 -43&gt;, 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]].
 
=== 27 &amp; 190 temperament ===
==== 5-limit ====
Subgroup: 2.3.5
 
Comma list: {{monzo| 0 63 -43 }}
 
Mapping: {{mapping| 1 0 0 | 0 43 63 }}
 
Optimal tuning (POTE): ~{{monzo| 0 -41 28 }} = 44.2294
 
{{Optimal ET sequence|legend=0| 27, 190, 217, 407, 597, 624, 841 }}
 
==== 7-limit ====
Subgroup: 2.3.5.7
 
Comma list: 4375/4374, 40353607/40000000
 
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 }}


==27&amp;190 temperament==
Badness: 0.1659
===5-limit===
Comma: |0 63 -43&gt;


POTE generator: ~|0 -41 28&gt; = 44.2294
=== 217 &amp; 407 temperament ===
==== 7-limit ====
Subgroup: 2.3.5.7


Map: [&lt;1 0 0|, &lt;0 43 63|]
Comma list: 134217728/133984375, 512557306947/512000000000


EDOs: 27, 190, 217, 407, 597, 624, 841
Mapping: {{mapping| 1 0 0 9 | 0 43 63 -168 }}


===7-limit===
Optimal tuning (POTE): ~525/512 = 44.2320
Commas: 4375/4374, 40353607/40000000


POTE generator: ~1029/1000 = 44.2288
{{Optimal ET sequence|legend=0| 217, 407, 624, 841, 1058, 1465 }}


Map: [&lt;1 0 0 1|, &lt;0 43 63 49|]
Badness: 0.3544


EDOs: 27, 190, 217
==== 11-limit ====
Subgroup: 2.3.5.7.11


==217&amp;407 temperament==
Comma list: 46656/46585, 131072/130977, 234375/234256
===7-limit===
Commas: 134217728/133984375, 512557306947/512000000000


POTE generator: ~525/512 = 44.2320
Mapping: {{mapping| 1 0 0 9 -1 | 0 43 63 -168 121 }}


Map: [&lt;1 0 0 9|, &lt;0 43 63 -168|]
Optimal tuning (POTE): ~525/512 = 44.2312


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


===11-limit===
Badness: 0.1129
Commas: 46656/46585, 131072/130977, 234375/234256


POTE generator: ~525/512 = 44.2312
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 9 -1|, &lt;0 43 63 -168 121|]
Comma list: 2080/2079, 4096/4095, 39366/39325, 109512/109375


EDOs: 217, 407, 624
Mapping: {{mapping| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}


===13-limit===
Optimal tuning (POTE): ~40/39 = 44.2312
Commas: 2080/2079, 4096/4095, 39366/39325, 109512/109375


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


Map: [&lt;1 0 0 9 -1 3|, &lt;0 43 63 -168 121 19|]
Badness: 0.0503


EDOs: 217, 407, 624
== See also ==
* [[16edf]] – relative edf
* [[27edo]] – relative edo
* [[70ed6]] – relative ed6
* [[90ed10]] – relative ed10
* [[97ed12]] – relative ed12


[[Category:Edt]]
[[Category:27edo]]
[[Category:Edonoi]]
Retrieved from "https://en.xen.wiki/w/43edt"