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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]]. | '''[[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]]. | ||
= | =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). | 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). | ||
| Line 171: | Line 171: | ||
| | 40 | | | 40 | ||
| | 1769.261 | | | 1769.261 | ||
| | [[25/9]] | | | [[25/18|25/9]] | ||
|- | |- | ||
| | 41 | | | 41 | ||
| Line 185: | Line 185: | ||
| | '''exact [[3/1]]''' | | | '''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 [[27edo|27]], [[190edo|190]], and [[217edo|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 | |||
==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 | |||
===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 | |||
===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 | |||
[[Category:Edt]] | [[Category:Edt]] | ||
[[Category:Edonoi]] | [[Category:Edonoi]] | ||
Revision as of 09:30, 7 March 2019
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 | corresponding JI intervals |
|---|---|---|
| 1 | 44.232 | 40/39, 39/38 |
| 2 | 88.463 | 20/19 |
| 3 | 132.695 | 27/25 |
| 4 | 176.926 | |
| 5 | 221.158 | 25/22 |
| 6 | 265.389 | (7/6) |
| 7 | 309.621 | |
| 8 | 353.852 | 27/22 |
| 9 | 398.084 | |
| 10 | 442.315 | |
| 11 | 486.547 | (45/34) |
| 12 | 530.778 | (34/25) |
| 13 | 575.010 | (39/28) |
| 14 | 619.241 | (10/7) |
| 15 | 663.473 | 22/15 |
| 16 | 707.704 | |
| 17 | 751.936 | |
| 18 | 796.167 | 19/12 |
| 19 | 840.399 | 13/8 |
| 20 | 884.630 | 5/3 |
| 21 | 928.862 | |
| 22 | 973.093 | |
| 23 | 1017.325 | 9/5 |
| 24 | 1061.556 | 24/13 |
| 25 | 1105.788 | 36/19 |
| 26 | 1150.019 | 68/35 |
| 27 | 1194.251 | |
| 28 | 1238.482 | 45/22 |
| 29 | 1282.713 | (21/10) |
| 30 | 1326.946 | (28/13) |
| 31 | 1371.177 | |
| 32 | 1415.408 | (34/15) |
| 33 | 1459.640 | |
| 34 | 1503.871 | |
| 35 | 1548.193 | 22/9 |
| 36 | 1592.334 | |
| 37 | 1636.566 | (18/7) |
| 38 | 1680.797 | 66/25 |
| 39 | 1725.029 | |
| 40 | 1769.261 | 25/9 |
| 41 | 1813.492 | 57/20 |
| 42 | 1857.724 | 38/13 |
| 43 | 1901.955 | 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
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
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
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