Didacus: Difference between revisions
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| Title = Didacus | | Title = Didacus | ||
| Subgroups = 2.5.7 | | Subgroups = 2.5.7 | ||
| Comma basis = | | Comma basis = 3136/3125 (2.5.7) | ||
| Edo join 1 = 6 | Edo join 2 = 25 | | Edo join 1 = 6 | Edo join 2 = 25 | ||
| Mapping = 1; 2 5 9 | | Mapping = 1; 2 5 9 | ||
| Line 9: | Line 9: | ||
| Odd limit 1 = 2.5.7 7 | Mistuning 1 = 1.22 | Complexity 1 = 13 | | Odd limit 1 = 2.5.7 7 | Mistuning 1 = 1.22 | Complexity 1 = 13 | ||
}} | }} | ||
'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out | [[File:Didacus13.png|thumb|Didacus 13-note MOS]] | ||
'''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 a slightly narrowed (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. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant. | |||
[[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]]. | [[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. A more accurate but complex full-7-limit extension is Hemiwurschmidt ({{e|31}} & {{e|37}}) which adds the [[Wurschmidt]] relation (5/4)<sup>8</sup> ~= 3/2. | |||
== Interval chain == | == Interval chain == | ||
In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''. | In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''. | ||
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! rowspan="2" | 2.5.7 intervals !! colspan="3" | Intervals of extensions | ! rowspan="2" | 2.5.7 intervals !! colspan="3" | Intervals of extensions | ||
|- | |- | ||
! Tridecimal | ! Tridecimal Didacus !! [[Luna and hemithirds#Intervals|Hemithirds]] !! Hemiwürschmidt (L11.23) | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 84: | Line 87: | ||
| '''16/13''' | | '''16/13''' | ||
| 128/105 | | 128/105 | ||
| 60/49, 92/75 | | 11/9, 27/22, 60/49, 92/75 | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 165: | Line 168: | ||
<nowiki/>* In [[CWE]] undecimal didacus | <nowiki/>* In [[CWE]] undecimal didacus | ||
== List of patent vals == | |||
See [[Didacus/Patent vals]]. | |||
{{Navbox regtemp}} | {{Navbox regtemp}} | ||
{{cat|temperaments}} | {{cat|temperaments}} | ||
Latest revision as of 01:51, 2 June 2026
| 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 a slightly narrowed (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. In the full 7-limit, Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.
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. A more accurate but complex full-7-limit extension is Hemiwurschmidt (31 & 37) which adds the Wurschmidt relation (5/4)8 ~= 3/2.
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 (L11.23) | |||
| 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.
