Didacus
| Didacus |
176/175, 1375/1372 (2.5.7.11)
(11-odd limit) 4.13c
(11-odd limit) 19 notes
Didacus is an 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, notable for being one of the most efficient traversals of the no-threes subgroup. 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. Further extensions to primes 13, 17, and 19, known as roulette and mediantone, are also possible.
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 spans 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 | ||
|---|---|---|---|---|
| 2.5.7.11 intervals | Intervals of extensions | |||
| Hemithirds | Hemiwürschmidt | |||
| 0 | 0.0 | 1/1 | ||
| 1 | 194.4 | 28/25, 49/44, 55/49 | ||
| 2 | 388.9 | 5/4, 44/35 | 144/115 | |
| 3 | 583.3 | 7/5 | ||
| 4 | 777.7 | 11/7, 25/16 | 36/23 | |
| 5 | 972.1 | 7/4, 44/25 | 184/105 | |
| 6 | 1166.6 | 49/25, 55/28 | 96/49, 45/23 | |
| 7 | 161.0 | 11/10, 35/32 | 23/21, 126/115 | |
| 8 | 355.4 | 49/40, 121/98 | 128/105 | 60/49, 92/75 |
| 9 | 549.9 | 11/8 | 48/35, 63/46, 115/84 | |
| 10 | 744.3 | 49/32, 77/50 | 32/21 | 75/49, 23/15 |
| 11 | 938.7 | 55/32, 121/70 | 128/75 | 12/7 |
| 12 | 1133.1 | 77/40 | 40/21 | 48/25, 23/12 |
| 13 | 127.6 | 121/112 | 16/15 | 15/14 |
| 14 | 322.0 | 77/64, 121/100 | 25/21 | 6/5, 115/96 |
| 15 | 516.4 | 4/3 | 75/56 | |
| 16 | 710.8 | 121/80 | 112/75 | 3/2 |
| 17 | 905.3 | 5/3 | 42/25 | |
| 18 | 1099.7 | 121/64 | 28/15 | 15/8 |
| 19 | 94.1 | 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
| 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 | ||
| 9\56 | 192.857 | 56e val | |
| 14\87 | 193.103 | 87e val | |
| 5/4 | 193.157 | 1/2-comma | |
| 19\118 | 193.220 | 118ee val | |
| 5\31 | 193.548 | ||
| 7/4 | 193.765 | 2/5-comma | |
| 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 | |
| 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 | ||
| 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 = 1\1, ~28/25 = 194.061
Images
A chart of the tuning spectrum of didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled.