80edo

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.

Rank-2 temperaments

80et supports a profusion of 19-limit (and lower) rank-2 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