80edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 79edo80edo81edo →
Prime factorization 24 × 5
Step size 15¢ 
Fifth 47\80 (705¢)
Semitones (A1:m2) 9:5 (135¢ : 75¢)
Consistency limit 19
Distinct consistency limit 11

80 equal divisions of the octave (abbreviated 80edo or 80ed2), also called 80-tone equal temperament (80tet) or 80 equal temperament (80et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 80 equal parts of exactly 15 ¢ each. Each step represents a frequency ratio of 21/80, or the 80th root of 2.

Theory

80edo is the first edo that represents the 19-odd-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. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just, with inconsistencies usually arising through 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. 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. It can also model larger primes if one is willing to accept their sharpness and for this purpose it does well for its size at the no-31's 41-limit, or even the 43-limit if you are fine with 43/32 being slightly flat causing more inconsistencies. If one wants higher precision as one goes to higher primes to try to convey the subtle harmonic qualities of those primes, 80et arguably fails in general, although many specific cases may be convincing. A promising alternative is using 80et as a model in which to fit higher-limit JI by way of approximating as much of the harmonic series as possible, for which it can model the 125-odd-limit quite well (corresponding to the 113-prime-limit), leading to an excellent Ringer scale described in the Ringer 80 section of this article.

80et tempers out 2048/2025 in the 5-limit; 1728/1715, 3136/3125, 4000/3969, and 4375/4374 in the 7-limit; 176/175, 540/539 and 4000/3993 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 and 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; 208/207, 253/252, 323/322 and 460/459 in the 23-limit; and 320/319 in the 29-limit. The last comma is notable as it equates a sharp 29/16 with a near-perfect 20/11, although this equivalence begins to make more sense when you consider the error cancellations with other sharp harmonics and as a way to give more reasonable interpretations to otherwise questionably mapped intervals. It 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.

Prime harmonics

Approximation of prime harmonics in 80edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43
Error Absolute (¢) +0.00 +3.04 +3.69 +6.17 +3.68 -0.53 +0.04 +2.49 +1.73 +5.42 -5.04 +3.66 +5.94 -1.52
Relative (%) +0.0 +20.3 +24.6 +41.2 +24.5 -3.5 +0.3 +16.6 +11.5 +36.2 -33.6 +24.4 +39.6 -10.1
Steps
(reduced)
80
(0)
127
(47)
186
(26)
225
(65)
277
(37)
296
(56)
327
(7)
340
(20)
362
(42)
389
(69)
396
(76)
417
(17)
429
(29)
434
(34)

Subsets and supersets

Since 80 factors into 24 × 5, 80edo has subset edos 2, 4, 5, 8, 10, 16, 20, and 40.

80edo is notable in not only it is consistent in the 19-odd-limit, but a large number of its supersets are also consistent in at least 19-odd-limit, if not larger. These are 320, 400, 1600, 1920, 2000, 2320, 3920, 4320. Temperament mergers of these produce various 80th-octave temperaments.

Intervals

Degree Cents Approximate Ratios*
0 0 1/1
1 15 64/63, 85/84, 91/90, 92/91, 96/95, 99/98, 100/99, 115/114, 120/119, 121/120, 126/125, 144/143
2 30 46/45, 50/49, 51/50, 52/51, 55/54, 56/55, 57/56, 58/57, 65/64, 66/65, 69/68, 70/69, 81/80
3 45 34/33, 36/35, 37/36, 38/37, 39/38, 40/39, 45/44, 49/48
4 60 26/25, 28/27, 29/28, 30/29, 33/32, 35/34
5 75 22/21, 23/22, 24/23, 25/24, 27/26
6 90 19/18, 20/19, 21/20, 39/37
7 105 16/15, 17/16, 18/17
8 120 15/14
9 135 13/12, 14/13, 27/25, 40/37
10 150 12/11
11 165 11/10
12 180 10/9, 21/19, 51/46
13 195 19/17, 28/25
14 210 9/8, 17/15, 26/23, 44/39
15 225 8/7, 25/22, 33/29
16 240 23/20, 85/74
17 255 15/13, 22/19, 29/25, 37/32, 81/70
18 270 7/6, 76/65
19 285 13/11, 20/17, 33/28
20 300 19/16, 25/21, 44/37
21 315 6/5
22 330 17/14, 23/19, 29/24, 40/33, 121/100
23 345 11/9
24 360 16/13
25 375 21/17, 36/29, 46/37, 57/46
26 390 5/4, 64/51
27 405 19/15, 24/19, 29/23
28 420 14/11, 23/18, 32/25, 37/29, 51/40
29 435 9/7
30 450 13/10, 22/17, 35/27, 48/37
31 465 17/13, 55/42, 72/55
32 480 21/16, 25/19, 29/22, 33/25, 37/28
33 495 4/3, 85/64
34 510 39/29, 51/38
35 525 19/14, 23/17, 65/48
36 540 15/11, 26/19
37 555 11/8, 29/21, 40/29, 51/37
38 570 18/13, 25/18, 32/23
39 585 7/5, 38/27, 52/37
40 600 17/12, 24/17, 45/32, 64/45, 99/70, 140/99

* based on treating 80edo as a no-31's 37-limit temperament; other approaches are possible. Inconsistent interpretations in italic.

Approximation to JI

Consistent circles

80edo is home to a staggering amount of consistent circles, both ones closing after generating all 80 notes and ones closing after generating a subset edo like 2, 4, 5, 8, 10, 16 or 20.

80-note circles by gen. with related temperaments organized by period
Interval Closing
Error
Consistency 1\1 1\2 1\4 1\5 1\8 1\10 1\16 1\20
39/38 16.2% Strong Quartonic extension Biquartonic Quartiquart Quintiquart extension ? Decistearn, deca ? Degrees
17/16 23.8% Strong Septendesemi Srutal archagall Bidia Pentorwell 80 & 104 Linus retraction ? Degrees
11/10 2.3% Strong Trienparapyth Echidna, semisupermajor ? Trisey, dodgy Octopus Decistearn, deca Hexadecoid Degrees
9/7 44.9% Normal Supermajor Echidna, semisupermajor ? ? Octopus Decistearn, deca Hexadecoid Degrees
10-note circles by gen. with related 10th-octave temperaments
Interval Closing
Error
Consistency Associated
Edostep
Temperaments
15/14 37.1% Normal 1\10 Decistearn, deca, linus
16/13 35.2% Normal 3\10 Decistearn, deca, linus
69/56 93.6% Weak 3\10 Decistearn
85/69 69.3% Weak 3\10 Decistearn
8-note circles by gen. with related 8th-octave temperaments
Interval Closing
Error
Consistency Associated
Edostep
Temperaments
12/11 34.0% Normal 1\8 Octopus, 11-limit octoid
85/78 64.7% Weak 1\8 Octopus, octoid
13/10 224.7% Sub-weak 3\8 Octopus
35/27 38.7% Normal 3\8 Octopus, 7-limit octoid
22/17 194.0% Sub-weak 3\8 Octopus
48/37 32.6% Normal 3\8 Octoid/octopus extension
83/64 2.5% Super-strong 3\8 Octoid/octopus extension
5-note circles by gen. with related 5th-octave temperaments
Interval Closing
Error
Consistency Associated
Edostep
Temperaments
23/20 65.4% Weak 1\5 Trisey add-23, dodgy add-23
85/74 2.5% Super-strong 1\5 29-limit add-37 trisey extension
33/25 21.5% Strong 2\5 Countdown
37/28 83.9% Weak 2\5 29-limit add-37 trisey extension
95/72 2.8% Super-strong 2\5 29-limit add-37 trisey extension
4-note circles with related 4th-octave temperaments
Interval Closing Error Consistency Associated
Edostep
Temperaments
19/16 66.3% Weak 1\4 Bidia
25/21 49.2% Normal 1\4 Bidia
44/37 0.7% Super-strong 1\4 Berylic, bidia extension

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [127 -80 [80 127]] −0.961 0.960 6.40
2.3.5 2048/2025, 390625000/387420489 [80 127 186]] −1.169 0.837 5.59
2.3.5.7 1728/1715, 2048/2025, 3136/3125 [80 127 186 225]] −1.426 0.851 5.68
2.3.5.7.11 176/175, 540/539, 896/891, 1331/1323 [80 127 186 225 277]] −1.353 0.775 5.17
2.3.5.7.11.13 169/168, 176/175, 325/324, 364/363, 540/539 [80 127 186 225 277 296]] −1.105 0.901 6.01
2.3.5.7.11.13.17 136/135, 169/168, 176/175, 221/220, 364/363, 540/539 [80 127 186 225 277 296 327]] −0.949 0.917 6.12
2.3.5.7.11.13.17.19 136/135, 169/168, 176/175, 190/189, 221/220, 364/363, 400/399 [80 127 186 225 277 296 327 340]] −0.903 0.867 5.78

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 8ve
Generator* Cents* Associated
Ratio*
Temperament
1 1\80 15 121/120 Yarman I
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
Semisupermajor
2 33\80
(7\80)
495
(105)
4/3
(17/16)
Srutal
4 33\80
(7\80)
495
(105)
4/3
(17/16)
Bidia
5 15\80
(1\80)
225
(15)
8/7
(64/63)
Pentorwell
5 37\80
(5\80)
555
(75)
11/8
(25/24)
Trisedodge / countdown
8 39\80
(1\80)
585
(15)
7/5
(99/98~100/99)
Octopus
10 21\80
(3\80)
315
(45)
6/5
(40/39)
Deca
20 33\80
(1\80)
495
(15)
4/3
(99/98~100/99)
Degrees

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Detemperaments

Ringer 80

80edo is a great essentially no-limit system for conceptualising and internalising harmonic series interval categories/structures through Ringer 80 which contains the entirety of the no-127's no-135's no-141's 145-odd-limit. An astounding ~84% of all intervals present are mapped consistently in Ringer 80. The Ringer 80 described below uses the best-performing val for 125-odd-limit consistency by a variety of metrics (squared error, sum of error, number of inconsistencies, number of inconsistencies if we require <25% error, etc.). The primes 31, 47, 53, 61, 67, 73, 79, 107, 109 are sharpened by one step compared to their flat patent val mapping (i.e. are mapped to their second-best mapping); all other primes are of the patent val. It is maybe worth noting that the least intuitive of these warts for prime 73 corresponds to getting the interval 73/63 to be mapped consistently, which is not insignificant because 80edo has an accurate enough approximation that it is practically a giant circle of 73/63's, among other such circles. Warting prime 73 also allows the introduction of the 145th harmonic which adds a lot of low-complexity and consistent intervals; specifically, all intervals made in ratio with the 145th harmonic that simplify are mapped consistently.

This scale has a few remarkable properties. Firstly, all the intervals that are inconsistent are mapped – at worst – to their second-best mapping, meaning you will never have a categorical/interval mapping exceeding 15 cents of error. Although this property is not as rare as it may sound it is still musically useful. Secondly, all of the "filler harmonics" beyond the 125-odd-limit fit in an obvious way; note how there are no warts beyond the 113-prime-limit (which the 125-odd-limit corresponds to due to the sizeable record prime gap from 113 to 127), meaning all composite harmonics were either already part of the 113-prime-limit or if prime the primes were patent val and sharp-tending. "In an obvious way" also means that every superparticular (n + 1)/n in the 125-odd-limit that was mapped to 2 steps is split into (2n + 2)/(2n + 1) and (2n + 1)/(2n), retaining the lowest possible complexity and most elegant possible structure for a ringer scale. Finally, note that while composite odd harmonics start going missing after the 125th harmonic, prime harmonics are very much not lacking. This scale exists inside the no-127's no-151's no-163's 179-prime-limit, meaning that all primes up to and including 179 are present excluding only those three, making it full of prime flavour on top of its capability for representing high compositeness due to the 125-odd-limit corresponding to a record prime gap. Note that prime 127 cannot be included because to match the increasing trend of sharpness it would need to be warted, leading to 128/127 being tempered out.

Mode 63 of the harmonic series (corresponding to 125-odd-limit) with added odds from mode 63 × 2 = 126 in square brackets:

63:64:[129]:65:[131]:66:[133]:67:68:[137]:69:[139]:70:71:[143]:72:[145]:73:74:[149]
75:76:[153]:77:78:[157]:79:80:[161]:81:82:83:[167]:84:85:86:[173]:87:88:89
[179]:90:91:92:[185]:93:94:95:96:97:[195]:98:99:100:101:102:103:104:[209]:105
106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125(:126)

The above is split into 20 harmonics per line a.k.a. ~300¢ worth of harmonic content per line.

In lowest terms as a /105 scale corresponding to a primodal /53 scale, among other possible interpretations:

105:106:107:108:109:110:111:112:113:114:115:116:117:118:119:120:121:122:123:124:125:126:128:129:130:131:132:133:134:136:137:138:139:140:142:143:144:145:146:148:149:150:152:153:154:156:157:158:160:161:162:164:166:167:168:170:172:173:174:176:178:179:180:182:184:185:186:188:190:192:194:195:196:198:200:202:204:206:208:209:210

This form is useful for copy-pasting into tools that accept colon-separated harmonic series chord enumerations as scales.

As reduced, rooted intervals (16 intervals per line):

129/128, 65/64, 131/128, 33/32, 133/128, 67/64, 17/16, 137/128, 69/64, 139/128, 35/32, 71/64, 143/128, 9/8, 145/128, 73/64,
37/32, 75/64, 19/16, 153/128, 77/64, 39/32, 157/128, 79/64, 5/4, 161/128, 81/64, 41/32, 83/64, 167/128, 21/16, 85/64,
43/32, 173/128, 87/64, 11/8, 89/64, 179/128, 45/32, 91/64, 23/16, 185/128, 93/64, 47/32, 95/64, 3/2, 97/64, 195/128,
49/32, 99/64, 25/16, 101/64, 51/32, 103/64, 13/8, 209/128, 105/64, 53/32, 107/64, 27/16, 109/64, 55/32, 111/64, 7/4,
113/64, 57/32, 115/64, 29/16, 117/64, 59/32, 119/64, 15/8, 121/64, 61/32, 123/64, 31/16, 125/64, 63/32, 2/1

Music

Godtone
Xotla