Didacus
| Didacus |
176/175, 1375/1372 (2.5.7.11)
(11-odd limit) 4.13 ¢
(11-odd limit) 19 notes
Didacus is a temperament of the 2.5.7 subgroup, tempering out 3136/3125, the hemimean comma, such that two intervals of 7/5 reach the same point as three intervals of 5/4; the generator is therefore (7/5)/(5/4) = 28/25, two of which stack to 5/4 and three of which stack to 7/5, meaning that the 4:5:7 chord is "locked" to (0 2 5) in terms of logarithmic ratios. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).
31edo is a very good tuning of didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but 25edo, 37edo, and 68edo among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, MOS scales of didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing 50/49~128/125), therefore bearing similar properties to those of slendric.
It also has a simple extension to prime 11 - undecimal didacus, by tempering out 176/175, the valinorsma, so that (5/4)2 is equated to 11/7 and 9 generators stack to 11/4; prime 13 can be found by tempering out 640/637, equating 16/13 to 49/40, and thereby putting the 13th harmonic 8 generators down. Beyond tridecimal didacus, further extensions to primes 17 and 19, known as roulette and mediantone, are also possible, sharing in common the interpretation of the generator as 19/17.
As for prime 3, while didacus has as a weak extension (among others) septimal meantone, strong extensions that include 3 are rather complex. Hemithirds (25 & 31) tempers out 1029/1024 to find the fifth at 3/2 ~ (8/7)3, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out 2401/2400 so that (5/4)8 is equated to 6/1, finding the 3rd harmonic 16 generators up (and as described for the page for 5-limit würschmidt, there is also a free extension to find 23/1 at 28 generators). These two mappings intersect in 31edo, though the latter hews closer to the optimal range for undecimal didacus specifically.
For technical data, see Hemimean clan #Didacus.
Interval chain
In the following table, odd harmonics and subharmonics 1–35 are labeled in bold.
| # | Cents* | Approximate ratios | |||
|---|---|---|---|---|---|
| 2.5.7 intervals | Intervals of extensions | ||||
| Tridecimal didacus | Hemithirds | Hemiwürschmidt | |||
| 0 | 0.0 | 1/1 | |||
| 1 | 194.4 | 28/25, 125/112 | 49/44, 55/49 | ||
| 2 | 388.9 | 5/4 | 44/35 | 144/115 | |
| 3 | 583.3 | 7/5 | 128/91 | ||
| 4 | 777.7 | 25/16 | 11/7 | 36/23 | |
| 5 | 972.1 | 7/4 | 44/25, 160/91 | 184/105 | |
| 6 | 1166.6 | 49/25, 125/64 | 55/28, 128/65 | 96/49, 45/23 | |
| 7 | 161.0 | 35/32 | 11/10, 100/91 | 23/21, 126/115 | |
| 8 | 355.4 | 49/40 | 16/13 | 128/105 | 60/49, 92/75 |
| 9 | 549.9 | 175/128 | 11/8 | 48/35, 63/46, 115/84 | |
| 10 | 744.3 | 49/32 | 20/13, 77/50 | 32/21 | 75/49, 23/15 |
| 11 | 938.7 | 55/32, 112/65 | 128/75 | 12/7 | |
| 12 | 1133.1 | 25/13, 77/40 | 40/21 | 48/25, 23/12 | |
| 13 | 127.6 | 14/13 | 16/15 | 15/14 | |
| 14 | 322.0 | 77/64, 110/91 | 25/21 | 6/5, 115/96 | |
| 15 | 516.4 | 35/26, 88/65 | 4/3 | 75/56 | |
| 16 | 710.8 | 98/65 | 112/75 | 3/2 | |
| 17 | 905.3 | 22/13 | 5/3 | 42/25 | |
| 18 | 1099.7 | 49/26 | 28/15 | 15/8 | |
| 19 | 94.1 | 55/52 | 25/24 | 21/20 | |
* In CWE undecimal didacus
Tunings
Optimized tunings
| Weight-skew\Order | Euclidean | |
|---|---|---|
| Constrained | Destretched | |
| Tenney | (2.5.7) CTE: ~28/25 = 193.650¢ | (2.5.7) POTE: ~28/25 = 193.772¢ |
| Equilateral | (2.5.7) CEE: ~28/25 = 193.681¢ (12/29-comma) | |
| Tenney | (2.5.7.11) CTE: ~28/25 = 194.246¢ | (2.5.7.11) POTE: ~28/25 = 194.556¢ |
| Optimized chord | Generator value | Polynomial | Further notes |
|---|---|---|---|
| 4:5:7 (+1 +2) | ~28/25 = 197.346 | g5 − 3g2 + 2 = 0 | Close to -1/5-comma |
| 5:7:8 (+2 +1) | ~28/25 = 193.829 | g5 − g2 − 4 = 0 | Close to 37/95-comma |
| 7:8:10 (+1 +2) | ~28/25 = 193.630 | g5 + g2 − 3 = 0 | Close to 33/80-comma |
| 8:11:14 (+1 +1) | ~28/25 = 195.043 | g9 − g5 − 1 = 0 | |
| 11:14:16 (+3 +2) | ~28/25 = 192.698 | g9 − 5g5 + 6 = 0 |
Tuning spectrum
The below tuning spectrum assumes undecimal didacus.
| EDO generator |
Eigenmonzo (unchanged interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 1\7 | 171.429 | 7dee val, lower bound of (2.5.7) 7-odd-limit diamond monotone | |
| 2\13 | 184.615 | 13e val, lower bound of (2.5.7.11) 11-odd-limit diamond monotone | |
| 3\19 | 189.474 | 19e val | |
| 125/112 | 190.115 | Full-comma | |
| 7\44 | 190.909 | 44dee val | |
| 4\25 | 192.000 | ||
| 13\81 | 192.593 | 81ee val | |
| 9\56 | 192.857 | 56e val | |
| 14\87 | 193.103 | 87e val | |
| 5/4 | 193.157 | 1/2-comma, lower bound of (2.5.7.11) 7- and 11-odd-limit diamond tradeoff | |
| 19\118 | 193.220 | 118ee val | |
| 5\31 | 193.548 | ||
| 35/32 | 193.591 | 3/7-comma | |
| 7/4 | 193.765 | 2/5-comma | |
| 26\161 | 193.789 | 161e val | |
| 21\130 | 193.846 | 130e val | |
| 16\99 | 193.939 | ||
| 27\167 | 194.012 | 167e val | |
| 11\68 | 194.118 | ||
| 7/5 | 194.171 | 1/3-comma, upper bound of (2.5.7) 7-odd-limit diamond tradeoff | |
| 17\105 | 194.286 | ||
| 23\142 | 194.366 | ||
| 11/8 | 194.591 | ||
| 6\37 | 194.595 | ||
| 196/125 | 194.678 | 1/4-comma | |
| 19\117 | 194.872 | 117d val | |
| 13\80 | 195.000 | ||
| 11/10 | 195.001 | ||
| 7\43 | 195.349 | ||
| 11/7 | 195.623 | upper bound of (2.5.7.11) 11-odd-limit diamond tradeoff | |
| 8\49 | 195.918 | ||
| 28/25 | 196.198 | Untempered tuning | |
| 9\55 | 196.364 | 55de val | |
| 1\6 | 200.000 | Upper bound of (no-threes) 7- and 11-odd-limit diamond monotone |
* Besides the octave
Other tunings
- DKW (2.5.7): ~2 = 1200.000, ~28/25 = 194.061