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'''4EDT''' is the [[Edt|equal division of the third harmonic]] into four parts of 475.4888 [[cent|cents]] each, corresponding to 2.5237 [[edo]]. | |||
The 4th root of 3, might be viewed alternately as a degenerate form or a fundamental building block of Bohlen-Pierce harmony, analogous to how 5edo relates to diatonic music. The situation is different however, as in this case both 5 and 7 are relatively well represented (opposed to just 3 in 5edo). While the approximations may seem excessively vague, and some might say impossible, they are nevertheless categorically important to the perception of the scale, and, may even be heard as those harmonies given the width of the "scale". It is doubtful however, that this scale could recieve much melodic treatment, and is more useful as a harmonic entity, either to demonstrate BP harmony, or as a component of scales like [[ | The 4th root of 3, might be viewed alternately as a degenerate form or a fundamental building block of Bohlen-Pierce harmony, analogous to how 5edo relates to diatonic music. The situation is different however, as in this case both 5 and 7 are relatively well represented (opposed to just 3 in 5edo). While the approximations may seem excessively vague, and some might say impossible, they are nevertheless categorically important to the perception of the scale, and, may even be heard as those harmonies given the width of the "scale". It is doubtful however, that this scale could recieve much melodic treatment, and is more useful as a harmonic entity, either to demonstrate BP harmony, or as a component of scales like [[8edt]]. | ||
0 | {| class="wikitable" | ||
|- | |||
! | degree | |||
! | cents value | |||
! | corresponding <br>JI intervals | |||
! | comments | |||
|- | |||
| | 0 | |||
| | 0.0000 | |||
| | '''exact [[1/1]]''' | |||
| | | |||
|- | |||
| | 1 | |||
| | 475.4888 | |||
| | [[17/13]], [[21/16]], 25/19, 33/25 | |||
| | | |||
|- | |||
| | 2 | |||
| | 950.9775 | |||
| | [[19/11]], 45/26, [[26/15]], (85/49), 33/19 | |||
| | | |||
|- | |||
| | 3 | |||
| | 1426.4663 | |||
| | 25/11, 57/25, [[16/7]], 39/17 | |||
| | | |||
|- | |||
| | 4 | |||
| | 1901.9550 | |||
| | '''exact [[3/1]]''' | |||
| | [[3/2|just perfect fifth]] plus an octave | |||
|} | |||
==Related regular temperaments== | |||
4EDT is a generator of the rank-three regular temperament which tempers out 120/119, 171/170, 176/175, 325/323, and 363/361 in the 19-limit, which is supported by [[15edo]] (15g val), [[43edo]], and [[53edo]]. Using two equal divisions of the interval which equals an octave minus the step interval of 4EDT as a generator, it leads the [[Marvel temperaments|interpental]] temperament, which tempers out 99/98, 120/119, 169/168, 171/170, 176/175, and 325/323 in the 19-limit. | |||
===19-limit 15g&43&53=== | |||
Commas: 120/119, 171/170, 176/175, 325/323, 363/361 | |||
POTE generator: ~17/13 = 475.455 | |||
4: | Map: [<1 0 0 4 0 -1 -1 0|, <0 4 0 -3 -3 6 7 -1|, <0 0 1 0 2 1 1 2|] | ||
[[Category: | |||
[[Category: | EDOs: 5egh, 10e, 15g, 38f, 43, 53 | ||
===19-limit interpental (43&53)=== | |||
Commas: 99/98, 120/119, 169/168, 171/170, 176/175, 325/323 | |||
POTE generator: ~16/13 = 362.440 | |||
Map: [<1 4 -1 1 -5 4 5 -3|, <0 -8 11 6 28 -1 -3 24|] | |||
EDOs: 10e, 43, 53 | |||
[[Category:Edt]] | |||
[[Category:Edonoi]] | |||
[[Category:Macrotonal]] | |||
Revision as of 10:15, 8 February 2019
4EDT is the equal division of the third harmonic into four parts of 475.4888 cents each, corresponding to 2.5237 edo.
The 4th root of 3, might be viewed alternately as a degenerate form or a fundamental building block of Bohlen-Pierce harmony, analogous to how 5edo relates to diatonic music. The situation is different however, as in this case both 5 and 7 are relatively well represented (opposed to just 3 in 5edo). While the approximations may seem excessively vague, and some might say impossible, they are nevertheless categorically important to the perception of the scale, and, may even be heard as those harmonies given the width of the "scale". It is doubtful however, that this scale could recieve much melodic treatment, and is more useful as a harmonic entity, either to demonstrate BP harmony, or as a component of scales like 8edt.
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 475.4888 | 17/13, 21/16, 25/19, 33/25 | |
| 2 | 950.9775 | 19/11, 45/26, 26/15, (85/49), 33/19 | |
| 3 | 1426.4663 | 25/11, 57/25, 16/7, 39/17 | |
| 4 | 1901.9550 | exact 3/1 | just perfect fifth plus an octave |
Related regular temperaments
4EDT is a generator of the rank-three regular temperament which tempers out 120/119, 171/170, 176/175, 325/323, and 363/361 in the 19-limit, which is supported by 15edo (15g val), 43edo, and 53edo. Using two equal divisions of the interval which equals an octave minus the step interval of 4EDT as a generator, it leads the interpental temperament, which tempers out 99/98, 120/119, 169/168, 171/170, 176/175, and 325/323 in the 19-limit.
19-limit 15g&43&53
Commas: 120/119, 171/170, 176/175, 325/323, 363/361
POTE generator: ~17/13 = 475.455
Map: [<1 0 0 4 0 -1 -1 0|, <0 4 0 -3 -3 6 7 -1|, <0 0 1 0 2 1 1 2|]
EDOs: 5egh, 10e, 15g, 38f, 43, 53
19-limit interpental (43&53)
Commas: 99/98, 120/119, 169/168, 171/170, 176/175, 325/323
POTE generator: ~16/13 = 362.440
Map: [<1 4 -1 1 -5 4 5 -3|, <0 -8 11 6 28 -1 -3 24|]
EDOs: 10e, 43, 53