43edt

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Division of the third harmonic into 43 equal parts (43EDT) is related to 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 10-integer-limit.

Properties

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 4L+5s MOS has L=7 s=3.

degrees cents value hekts corresponding
JI intervals
1 44.232 30.233 40/39, 39/38
2 88.463 60.465 20/19
3 132.695 90.698 27/25
4 176.926 120.93 10/9
5 221.158 151.163 25/22
6 265.389 181.395 (7/6)
7 309.621 211.628 6/5
8 353.852 241.8605 27/22
9 398.084 272.093 24/19
10 442.315 302.326 9/7
11 486.547 332.558 (45/34)
12 530.778 362.791 (34/25)
13 575.01 393.023 (39/28)
14 619.241 423.256 10/7
15 663.473 453.488 22/15
16 707.704 483.721 3/2
17 751.936 513.9535 105/68, 20/13
18 796.167 544.186 19/12
19 840.399 574.419 13/8
20 884.63 604.651 5/3
21 928.862 634.883 12/7
22 973.093 665.116 7/4
23 1017.325 695.349 9/5
24 1061.556 725.581 24/13
25 1105.788 755.814 36/19
26 1150.019 786.0465 68/35, 39/20
27 1194.251 816.279 2/1
28 1238.482 846.511 45/22
29 1282.713 876.744 (21/10)
30 1326.946 906.977 (28/13)
31 1371.177 937.209 (75/34)
32 1415.408 967.442 (34/15)
33 1459.640 997.674 7/3
34 1503.871 1027.907 19/8
35 1548.193 1058.1395 22/9
36 1592.334 1088.372 5/2
37 1636.566 1118.605 (18/7)
38 1680.797 1148.837 66/25
39 1725.029 1179.069 27/10
40 1769.261 1209.302 25/9
41 1813.492 1239.5345 57/20
42 1857.724 1269.767 117/40, 38/13
43 1901.955 1300 exact 3/1

43EDT as a regular temperament

43EDT tempers out a no-twos comma of |0 63 -43>, leading the regular temperament supported by 27, 190, and 217 EDOs.

27&190 temperament

5-limit

Comma: |0 63 -43>

POTE generator: ~|0 -41 28> = 44.2294

Map: [<1 0 0|, <0 43 63|]

EDOs: 27, 190, 217, 407, 597, 624, 841

7-limit

Commas: 4375/4374, 40353607/40000000

POTE generator: ~1029/1000 = 44.2288

Map: [<1 0 0 1|, <0 43 63 49|]

EDOs: 27, 190, 217

Badness: 0.1659

217&407 temperament

7-limit

Commas: 134217728/133984375, 512557306947/512000000000

POTE generator: ~525/512 = 44.2320

Map: [<1 0 0 9|, <0 43 63 -168|]

EDOs: 217, 407, 624, 841, 1058, 1465

Badness: 0.3544

11-limit

Commas: 46656/46585, 131072/130977, 234375/234256

POTE generator: ~525/512 = 44.2312

Map: [<1 0 0 9 -1|, <0 43 63 -168 121|]

EDOs: 217, 407, 624

Badness: 0.1129

13-limit

Commas: 2080/2079, 4096/4095, 39366/39325, 109512/109375

POTE generator: ~40/39 = 44.2312

Map: [<1 0 0 9 -1 3|, <0 43 63 -168 121 19|]

EDOs: 217, 407, 624

Badness: 0.0503