80edo: Difference between revisions

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The ''80 equal temperament'', often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step represents a frequency ratio of exactly 15 [[cent|cent]]s. 80et is the first equal temperament that represents the [[19-limit|19-limit]] [[Tonality_diamond|tonality diamond]] [[consistent|consistent]]ly (it barely manages to do so).
The '''80 equal temperament''', often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step represents a frequency ratio of exactly 15 [[cent|cent]]s. 80et is the first equal temperament that represents the [[19-limit]] [[tonality diamond]] [[consistent|consistently]] (it barely manages to do so).


80 et [[tempering_out|tempers out]] 136/135, 169/168, 176/175, 190/189, 221/220, 256/255, 286/285, 289/288, 325/324, 351/350, 352/351, 361/360, 364/363, 400/399, 456/455, 476/475, 540/539, 561/560, 595/594, 715/714, 936/935, 969/968, 1001/1000, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.
80et [[Tempering_out|tempers out]] 136/135, 169/168, 176/175, 190/189, 221/220, 256/255, 286/285, 289/288, 325/324, 351/350, 352/351, 361/360, 364/363, 400/399, 456/455, 476/475, 540/539, 561/560, 595/594, 715/714, 936/935, 969/968, 1001/1000, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.


80 supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention:
80 supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention:
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In each case, the numbers joined by an ampersand represent 19-limit [[Patent_val|patent vals]] (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.
In each case, the numbers joined by an ampersand represent 19-limit [[Patent_val|patent vals]] (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.


=Intervals of 80edo=
== Intervals ==


{| class="wikitable"
{| class="wikitable center-all right-2 left-3"
|-
|-
! | degrees
! Degree
! | cents
! Cents
! | ratios*
! Approximate Ratios*
|-
|-
| 0
| 0
|0
| 0
| | 1/1
| 1/1
|-
|-
| | 1
| 1
| | 15
| 15
| | 64/63
| 64/63
|-
|-
| | 2
| 2
| | 30
| 30
| | 81/80
| 81/80
|-
|-
| | 3
| 3
| | 45
| 45
| | 34/33, 36/35
| 34/33, 36/35
|-
|-
| | 4
| 4
| | 60
| 60
| | 26/25, 28/27, 33/32, 35/34
| 26/25, 28/27, 33/32, 35/34
|-
|-
| | 5
| 5
| | 75
| 75
| | 22/21, 25/24, 27/26
| 22/21, 25/24, 27/26
|-
|-
| | 6
| 6
| | 90
| 90
| | 19/18, 20/19, 21/20
| 19/18, 20/19, 21/20
|-
|-
| | 7
| 7
| | 105
| 105
| | 16/15, 17/16, 18/17
| 16/15, 17/16, 18/17
|-
|-
| | 8
| 8
| | 120
| 120
| | 14/13, 15/14
| 14/13, 15/14
|-
|-
| | 9
| 9
| | 135
| 135
| | 13/12
| 13/12
|-
|-
| | 10
| 10
| | 150
| 150
| | 12/11
| 12/11
|-
|-
| | 11
| 11
| | 165
| 165
| | 11/10
| 11/10
|-
|-
| | 12
| 12
| | 180
| 180
| | 10/9, 21/19
| 10/9, 21/19
|-
|-
| | 13
| 13
| | 195
| 195
| | 19/17
| 19/17
|-
|-
| | 14
| 14
| | 210
| 210
| | 9/8, 17/15
| 9/8, 17/15
|-
|-
| | 15
| 15
| | 225
| 225
| | 8/7
| 8/7
|-
|-
| | 16
| 16
| | 240
| 240
| |  
|  
|-
|-
| | 17
| 17
| | 255
| 255
| | 15/13, 22/19
| 15/13, 22/19
|-
|-
| | 18
| 18
| | 270
| 270
| | 7/6
| 7/6
|-
|-
| | 19
| 19
| | 285
| 285
| | 13/11, 20/17
| 13/11, 20/17
|-
|-
| | 20
| 20
| | 300
| 300
| | 19/16, 25/21
| 19/16, 25/21
|-
|-
| | 21
| 21
| | 315
| 315
| | 6/5
| 6/5
|-
|-
| | 22
| 22
| | 330
| 330
| | 17/14
| 17/14
|-
|-
| | 23
| 23
| | 345
| 345
| | 11/9
| 11/9
|-
|-
| | 24
| 24
| | 360
| 360
| | 16/13, 21/17
| 16/13, 21/17
|-
|-
| | 25
| 25
| | 375
| 375
| |  
|  
|-
|-
| | 26
| 26
| | 390
| 390
| | 5/4
| 5/4
|-
|-
| | 27
| 27
| | 405
| 405
| | 19/15, 24/19
| 19/15, 24/19
|-
|-
| | 28
| 28
| | 420
| 420
| | 14/11
| 14/11
|-
|-
| | 29
| 29
| | 435
| 435
| | 9/7
| 9/7
|-
|-
| | 30
| 30
| | 450
| 450
| | 13/10, 22/17
| 13/10, 22/17
|-
|-
| | 31
| 31
| | 465
| 465
| | 17/13
| 17/13
|-
|-
| | 32
| 32
| | 480
| 480
| | 21/16, 25/19
| 21/16, 25/19
|-
|-
| | 33
| 33
| | 495
| 495
| | 4/3
| 4/3
|-
|-
| | 34
| 34
| | 510
| 510
| |  
|  
|-
|-
| | 35
| 35
| | 525
| 525
| | 19/14
| 19/14
|-
|-
| | 36
| 36
| | 540
| 540
| | 26/19
| 26/19
|-
|-
| | 37
| 37
| | 555
| 555
| | 11/8
| 11/8
|-
|-
| | 38
| 38
| | 570
| 570
| | 18/13
| 18/13
|-
|-
| | 39
| 39
| | 585
| 585
| | 7/5
| 7/5
|-
|-
| | 40
| 40
| | 600
| 600
| | 17/12, 24/17
| 17/12, 24/17
|}
|}
*based on treating 80edo as a [[19-limit|19-limit]] temperament; other approaches are possible.
<nowiki>*</nowiki> based on treating 80edo as a [[19-limit]] temperament; other approaches are possible.
 
[[Category:19-limit]]
[[Category:19-limit]]
[[Category:21-limit]]
[[Category:21-limit]]
[[Category:edo]]
[[Category:edo]]

Revision as of 03:36, 5 October 2020

The 80 equal temperament, often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step represents a frequency ratio of exactly 15 cents. 80et is the first equal temperament that represents the 19-limit tonality diamond consistently (it barely manages to do so).

80et tempers out 136/135, 169/168, 176/175, 190/189, 221/220, 256/255, 286/285, 289/288, 325/324, 351/350, 352/351, 361/360, 364/363, 400/399, 456/455, 476/475, 540/539, 561/560, 595/594, 715/714, 936/935, 969/968, 1001/1000, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.

80 supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention:

31&80 <<7 6 15 27 -24 -23 -20 ... ||

72&80 <<24 30 40 24 32 24 0 ... ||

34&80 <<2 -4 -50 22 16 2 -40 ... ||

46&80 <<2 -4 30 22 16 2 40 ... ||

29&80 <<3 34 45 33 24 -37 20 ... ||

12&80 <<4 -8 -20 -36 32 4 0 ... ||

22&80 <<6 -10 12 -14 -32 6 -40 ... ||

58&80 <<6 -10 12 -14 -32 6 40 ... ||

41&80 <<7 26 25 -3 -24 -33 20 ... ||

In each case, the numbers joined by an ampersand represent 19-limit patent vals (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.

Intervals

Degree Cents Approximate Ratios*
0 0 1/1
1 15 64/63
2 30 81/80
3 45 34/33, 36/35
4 60 26/25, 28/27, 33/32, 35/34
5 75 22/21, 25/24, 27/26
6 90 19/18, 20/19, 21/20
7 105 16/15, 17/16, 18/17
8 120 14/13, 15/14
9 135 13/12
10 150 12/11
11 165 11/10
12 180 10/9, 21/19
13 195 19/17
14 210 9/8, 17/15
15 225 8/7
16 240
17 255 15/13, 22/19
18 270 7/6
19 285 13/11, 20/17
20 300 19/16, 25/21
21 315 6/5
22 330 17/14
23 345 11/9
24 360 16/13, 21/17
25 375
26 390 5/4
27 405 19/15, 24/19
28 420 14/11
29 435 9/7
30 450 13/10, 22/17
31 465 17/13
32 480 21/16, 25/19
33 495 4/3
34 510
35 525 19/14
36 540 26/19
37 555 11/8
38 570 18/13
39 585 7/5
40 600 17/12, 24/17

* based on treating 80edo as a 19-limit temperament; other approaches are possible.

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