80edo: Difference between revisions

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80et provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[canou family]], especially in higher limits.  
80et provides the [[optimal patent val]] for 5-limit [[diaschismic]], for 13-limit [[srutal]], and for 7-, 11- and 13-limit [[bidia]]. It is a good tuning for various temperaments in [[canou family]], especially in higher limits.  
80et 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 val]]s (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.


== Intervals ==
== Intervals ==
Line 249: Line 227:
| +36.2
| +36.2
| -33.6
| -33.6
|}
== Rank-2 temperaments ==
80et 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 val]]s (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.
{| class="wikitable center-all left-5"
|+Rank-2 temperaments by generator
! Periods<br>per Octave
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated Ratio<br>(Reduced)
! Temperament
|-
| 1
| 1\80
| 15
| 121/120
| [[Yarman]]
|-
| 1
| 3\80
| 45
| 36/35~40/39
| [[Quartonic]]
|-
| 1
| 9\80
| 135
| 27/25
| [[Superlimmal]]
|-
| 1
| 21\80
| 315
| 6/5
| [[Parakleismic]] / parkleismic / paradigmic
|-
| 1
| 29\80
| 435
| 9/7
| [[Supermajor]]
|-
| 1
| 31\80
| 465
| 17/13
| [[Semisept]]
|-
| 1
| 39\80
| 585
| 7/5
| [[Pluto]]
|-
| 2
| 21\80<br>(19\80)
| 315<br>(285)
| 6/5<br>(33/28)
| [[Semiparakleismic]]
|-
| 2
| 29\80<br>(11\80)
| 435<br>(165)
| 9/7<br>(11/10)
| [[Echidna]]
|-
| 2
| 47\80<br>(7\80)
| 705<br>(105)
| 3/2<br>(17/16)
| [[Srutal]]
|-
| 4
| 47\80<br>(7\80)
| 705<br>(105)
| 3/2<br>(17/16)
| [[Bidia]]
|-
| 5
| 5\80
| 75
| 25/24
| [[Trisedodge]]
|-
| 5
| 61\80<br>(1\80)
| 975<br>(15)
| 7/4<br>(64/63)
| [[Pentorwell]]
|-
| 8
| 39\80<br>(1\80)
| 585<br>(15)
| 7/5<br>(99/98~100/99)
| [[Octoid]] / octopus
|-
| 10
| 21\80<br>(3\80)
| 315<br>(45)
| 6/5<br>(40/39)
| [[Deca]]
|-
| 20
| 47\80<br>(1\80)
| 705<br>(15)
| 3/2<br>(99/98~100/99)
| [[Degrees]]
|}
|}


Revision as of 04:18, 19 June 2021

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 is exactly 15 cents.

Theory

80et is the first equal temperament that represents the 19-limit tonality diamond consistently, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits in the 29-prime-limit are consistent, and its patent val generally does well at approximating (29-prime-limited) harmonic series segments, such as modes 16 through 30 but especially modes 8 through 15, with inconsistencies usually caused by not cancelling the over-sharpness of compound harmonics 21, 27, 35, 45 and 49 (and their octave-equivalents), which may be seen as an interesting limitation. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just. This means it can be used as a general-purpose approximate 29-prime-limit system with a relatively manageable number of tones, with some care taken around inconsistency.

80et tempers out 176/175 and 540/539 in the 11-limit, 169/168, 325/324, 351/350, 352/351, 364/363 and 1001/1000 in the 13-limit, 136/135, 221/220, 256/255, 289/288, 561/560, 595/594, 715/714, 936/935, 1275/1274 in the 17-limit, 190/189, 286/285, 361/360, 400/399, 456/455, 476/475, 969/968, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728 in the 19-limit, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.

80et provides the optimal patent val for 5-limit diaschismic, for 13-limit srutal, and for 7-, 11- and 13-limit bidia. It is a good tuning for various temperaments in canou family, especially in higher limits.

Intervals

Degree Cents Approximate Ratios*
0 0 1/1
1 15 64/63
2 30 81/80, 50/49
3 45 36/35, 49/48, 34/33
4 60 28/27, 33/32, 26/25, 35/34
5 75 25/24, 22/21, 27/26
6 90 21/20, 19/18, 20/19
7 105 16/15, 17/16, 18/17
8 120 15/14
9 135 13/12, 14/13
10 150 12/11
11 165 11/10
12 180 10/9, 21/19
13 195 19/17
14 210 9/8, 17/15, 26/23
15 225 8/7
16 240 23/20
17 255 81/70, 15/13, 22/19
18 270 7/6
19 285 13/11, 20/17
20 300 25/21, 19/16
21 315 6/5
22 330 17/14, 23/19, 29/24
23 345 11/9
24 360 16/13
25 375 21/17, 36/29
26 390 5/4
27 405 24/19, 19/15
28 420 14/11
29 435 9/7
30 450 35/27, 13/10, 22/17
31 465 17/13
32 480 21/16, 25/19, 29/22
33 495 4/3
34 510
35 525 19/14, 23/17
36 540 15/11, 26/19
37 555 11/8
38 570 18/13, 32/23
39 585 7/5
40 600 17/12, 24/17
… … …

* based on treating 80edo as a 29-limit temperament; other approaches are possible. Inconsistent interpretations in italic.

Just approximation

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23 prime 29 prime 31
Error absolute (¢) 0.00 +3.04 +3.69 +6.17 +3.68 -0.53 +0.04 +2.49 +1.73 +5.42 -5.04
relative (%) 0.0 +20.3 +24.6 +41.1 +24.5 -3.5 +0.3 +16.6 +11.5 +36.2 -33.6

Rank-2 temperaments

80et 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.

Rank-2 temperaments by generator
Periods
per Octave 		Generator
(Reduced) 		Cents
(Reduced) 		Associated Ratio
(Reduced) 		Temperament
1 1\80 15 121/120 Yarman
1 3\80 45 36/35~40/39 Quartonic
1 9\80 135 27/25 Superlimmal
1 21\80 315 6/5 Parakleismic / parkleismic / paradigmic
1 29\80 435 9/7 Supermajor
1 31\80 465 17/13 Semisept
1 39\80 585 7/5 Pluto
2 21\80
(19\80) 		315
(285) 		6/5
(33/28) 		Semiparakleismic
2 29\80
(11\80) 		435
(165) 		9/7
(11/10) 		Echidna
2 47\80
(7\80) 		705
(105) 		3/2
(17/16) 		Srutal
4 47\80
(7\80) 		705
(105) 		3/2
(17/16) 		Bidia
5 5\80 75 25/24 Trisedodge
5 61\80
(1\80) 		975
(15) 		7/4
(64/63) 		Pentorwell
8 39\80
(1\80) 		585
(15) 		7/5
(99/98~100/99) 		Octoid / octopus
10 21\80
(3\80) 		315
(45) 		6/5
(40/39) 		Deca
20 47\80
(1\80) 		705
(15) 		3/2
(99/98~100/99) 		Degrees
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