4edt: Difference between revisions

Line 1:

~~4 Equal Divisions~~ of the ~~Tritave~~

'''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 [[~~8edt|~~8edt]].

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: 1/1

{| 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

|}

~~1: 475~~.~~489 cents "4~~/3"

==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.

2: ~~950.978 cents "5~~/3"

===19-limit 15g&43&53===

Commas: 120/119, 171/170, 176/175, 325/323, 363/361

3: ~~1426~~.~~466 cents "7/3"~~

POTE generator: ~17/13 = 475.455

4: ~~1901~~.~~955 tritave~~ 3/1

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:~~edt~~]]

[[Category:~~tritave~~]]

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]]

@@ Line 1: / Line 1: @@
-Equal Divisions of the Tritave
+'''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 [[8edt|8edt]].
+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]].
-: 1/1
+{| 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
+|}
-: 475.489 cents "4/3"
+==Related regular temperaments==
+EDT 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.
-: 950.978 cents "5/3"
+===19-limit 15g&amp;43&amp;53===
+Commas: 120/119, 171/170, 176/175, 325/323, 363/361
-: 1426.466 cents "7/3"
+POTE generator: ~17/13 = 475.455
-: 1901.955 tritave 3/1
+Map: [&lt;1 0 0 4 0 -1 -1 0|, &lt;0 4 0 -3 -3 6 7 -1|, &lt;0 0 1 0 2 1 1 2|]
-[[Category:edt]]
-[[Category:tritave]]
+EDOs: 5egh, 10e, 15g, 38f, 43, 53
+===19-limit interpental (43&amp;53)===
+Commas: 99/98, 120/119, 169/168, 171/170, 176/175, 325/323
+POTE generator: ~16/13 = 362.440
+Map: [&lt;1 4 -1 1 -5 4 5 -3|, &lt;0 -8 11 6 28 -1 -3 24|]
+EDOs: 10e, 43, 53
+[[Category:Edt]]
+[[Category:Edonoi]]
+[[Category:Macrotonal]]