Didacus
| Didacus |
Didacus is a highly efficient 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.
Septimal Meantone is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension.
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 (2.3.5.7.11.13.23) | |||
| 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 | 11/9, 27/22, 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
List of patent vals
See Didacus/Patent vals.
| View • Talk • EditRegular temperaments | |
|---|---|
| Rank-2 | |
| Acot | Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave) |
| Monocot | Meantone • Schismic • Leapday • Archy |
| Complexity 2 | Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot) |
| Complexity 3 | Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot) |
| Complexity 4 | Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot) |
| Complexity 5-6 | Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot) |
| Higher complexity | Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot) |
| Straddle-3 | A-Team (alter-tricot) • Machine (alter-monocot) |
| No-3 | Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph) |
| No-octaves | Sensamagic (monogem) |
| Exotemperament | Dicot • Mavila • Father |
| Higher-rank | |
| Rank-3 | Hemifamity • Marvel • Parapyth |
