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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
!hekts
! [[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 |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 }}


==27&amp;190 temperament==
Mapping: {{mapping| 1 0 0 | 0 43 63 }}
===5-limit===
Comma: |0 63 -43&gt;


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


Map: [&lt;1 0 0|, &lt;0 43 63|]
{{Optimal ET sequence|legend=0| 27, 190, 217, 407, 597, 624, 841 }}


EDOs: 27, 190, 217, 407, 597, 624, 841
==== 7-limit ====
Subgroup: 2.3.5.7


===7-limit===
Comma list: 4375/4374, 40353607/40000000
Commas: 4375/4374, 40353607/40000000


POTE generator: ~1029/1000 = 44.2288
Mapping: {{mapping| 1 0 0 1 | 0 43 63 49 }}


Map: [&lt;1 0 0 1|, &lt;0 43 63 49|]
Optimal tuning (POTE): ~1029/1000 = 44.2288


EDOs: 27, 190, 217
{{Optimal ET sequence|legend=0| 27, 190, 217 }}


Badness: 0.1659
Badness: 0.1659


==217&amp;407 temperament==
=== 217 &amp; 407 temperament ===
===7-limit===
==== 7-limit ====
Commas: 134217728/133984375, 512557306947/512000000000
Subgroup: 2.3.5.7


POTE generator: ~525/512 = 44.2320
Comma list: 134217728/133984375, 512557306947/512000000000


Map: [&lt;1 0 0 9|, &lt;0 43 63 -168|]
Mapping: {{mapping| 1 0 0 9 | 0 43 63 -168 }}


EDOs: 217, 407, 624, 841, 1058, 1465
Optimal tuning (POTE): ~525/512 = 44.2320
 
{{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


POTE generator: ~525/512 = 44.2312
Comma list: 46656/46585, 131072/130977, 234375/234256


Map: [&lt;1 0 0 9 -1|, &lt;0 43 63 -168 121|]
Mapping: {{mapping| 1 0 0 9 -1 | 0 43 63 -168 121 }}


EDOs: 217, 407, 624
Optimal tuning (POTE): ~525/512 = 44.2312
 
{{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


Map: [&lt;1 0 0 9 -1 3|, &lt;0 43 63 -168 121 19|]
Mapping: {{mapping| 1 0 0 9 -1 3 | 0 43 63 -168 121 19 }}


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]]

Latest revision as of 19:09, 25 June 2025

← 42edt 43edt 44edt →
Prime factorization 43 (prime)
Step size 44.2315 ¢ 
Octave 27\43edt (1194.25 ¢)
Consistency limit 10
Distinct consistency limit 8

43 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 43edt or 43ed3), is a nonoctave tuning system that divides the interval of 3/1 into 43 equal parts of about 44.2 ¢ each. Each step represents a frequency ratio of 31/43, or the 43rd root of 3.

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 10-integer-limit. It has octaves compressed by about 5.7492 ¢, 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 4L 5s⟨3/1⟩ mos has L = 7, s = 3.

Harmonics

Approximation of harmonics in 43edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.7 +0.0 -11.5 +0.3 -5.7 -7.2 -17.2 +0.0 -5.5 +6.4 -11.5
Relative (%) -13.0 +0.0 -26.0 +0.6 -13.0 -16.3 -39.0 +0.0 -12.4 +14.6 -26.0
Steps
(reduced) 		27
(27) 		43
(0) 		54
(11) 		63
(20) 		70
(27) 		76
(33) 		81
(38) 		86
(0) 		90
(4) 		94
(8) 		97
(11)
Approximation of harmonics in 43edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -17.4 -13.0 +0.3 +21.2 +4.7 -5.7 -10.9 -11.2 -7.2 +0.7 +12.2 -17.2
Relative (%) -39.3 -29.3 +0.6 +48.0 +10.7 -13.0 -24.6 -25.4 -16.3 +1.6 +27.6 -39.0
Steps
(reduced) 		100
(14) 		103
(17) 		106
(20) 		109
(23) 		111
(25) 		113
(27) 		115
(29) 		117
(31) 		119
(33) 		121
(35) 		123
(37) 		124
(38)

Subsets and supersets

43edt is the 14th prime edt, following 41edt and coming before 47edt.

Intervals

# Cents Hekts Approximate ratios
1 44.2 30.2 39/38, 40/39
2 88.5 60.5 20/19
3 132.7 90.7 27/25
4 176.9 120.9 10/9
5 221.2 151.2 25/22
6 265.4 181.4 7/6
7 309.6 211.6 6/5
8 353.9 241.9 27/22
9 398.1 272.1 24/19
10 442.3 302.3 9/7
11 486.5 332.6 45/34
12 530.8 362.8 34/25
13 575.0 393.0 39/28
14 619.2 423.3 10/7
15 663.5 453.5 22/15
16 707.7 483.7 3/2
17 751.9 514.0 20/13, 105/68
18 796.2 544.2 19/12
19 840.4 574.4 13/8
20 884.6 604.7 5/3
21 928.9 634.9 12/7
22 973.1 665.1 7/4
23 1017.3 695.3 9/5
24 1061.6 725.6 24/13
25 1105.8 755.8 36/19
26 1150.0 786.0 39/20, 68/35
27 1194.3 816.3 2/1
28 1238.5 846.5 45/22
29 1282.7 876.7 21/10
30 1326.9 907.0 28/13
31 1371.2 937.2 75/34
32 1415.4 967.4 34/15
33 1459.6 997.7 7/3
34 1503.9 1027.9 19/8
35 1548.1 1058.1 22/9
36 1592.3 1088.3 5/2
37 1636.6 1118.6 18/7
38 1680.8 1148.8 66/25
39 1725.0 1179.1 27/10
40 1769.3 1209.3 25/9
41 1813.5 1239.5 57/20
42 1857.7 1269.8 38/13, 117/40
43 1902.0 1300.0 3/1

Related regular temperaments

43edt tempers out the no-twos comma of [0 63 -43⟩, leading to the regular temperament supported by 27-, 190-, and 217edo.

27 & 190 temperament

5-limit

Subgroup: 2.3.5

Comma list: [0 63 -43⟩

Mapping: [⟨1 0 0], ⟨0 43 63]]

Optimal tuning (POTE): ~[0 -41 28⟩ = 44.2294

Optimal ET sequence: 27, 190, 217, 407, 597, 624, 841

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, 40353607/40000000

Mapping: [⟨1 0 0 1], ⟨0 43 63 49]]

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

Optimal ET sequence: 27, 190, 217

Badness: 0.1659

217 & 407 temperament

7-limit

Subgroup: 2.3.5.7

Comma list: 134217728/133984375, 512557306947/512000000000

Mapping: [⟨1 0 0 9], ⟨0 43 63 -168]]

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

Optimal ET sequence: 217, 407, 624, 841, 1058, 1465

Badness: 0.3544

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 0 9 -1], ⟨0 43 63 -168 121]]

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

Optimal ET sequence: 217, 407, 624

Badness: 0.1129

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 9 -1 3], ⟨0 43 63 -168 121 19]]

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

Optimal ET sequence: 217, 407, 624

Badness: 0.0503

See also

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