Didacus
| Didacus |
Didacus is a temperament of the 2.5.7 subgroup, tempering out 3136/3125, 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 size and generator steps.
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.
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 | 23/15, 75/49 |
| 11 | 938.7 | 55/32, 112/65 | 128/75 | 12/7 | |
| 12 | 1133.1 | 25/13, 77/40 | 40/21 | 23/12, 48/25 | |
| 13 | 127.6 | 14/13 | 16/15 | 15/14 | |
| 14 | 322.0 | 77/64, 110/91 | 25/21 | 6/5 | |
| 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
