Didacus: Difference between revisions
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{{Infobox | {{Infobox regtemp | ||
| Title = Didacus | | Title = Didacus | ||
| Subgroups = 2.5.7, 2.5.7.11 | | Subgroups = 2.5.7, 2.5.7.11 | ||
| Comma basis = [[3136/3125]] (2.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11) | | Comma basis = [[3136/3125]] (2.5.7); <br>[[176/175]], [[1375/1372]] (2.5.7.11) | ||
| Edo join 1 = 6 | Edo join 2 = 25 | | Edo join 1 = 6 | Edo join 2 = 25 | ||
| | | Mapping = 1; 2 5 9 | ||
| Generators = 28/25 | Generators tuning = 194.4 | Optimization method = CWE | |||
| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]] | | MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]] | ||
| Odd limit 1 = 2.5.7 7 | Mistuning 1 = 1.22 | Complexity 1 = 13 | |||
| Odd limit 1 = 7 | Mistuning 1 = 1.22 | Complexity 1 = 13 | | Odd limit 2 = 2.5.7.11 11 | Mistuning 2 = 4.13 | Complexity 2 = 19 | ||
| Odd limit 2 = 11 | Mistuning 2 = 4.13 | Complexity 2 = 19 | |||
}} | }} | ||
'''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 size and generator steps. 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). | '''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 size and generator steps. 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). | ||
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=== Optimized tunings === | === Optimized tunings === | ||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style="font-size: 105%; white-space: nowrap;" | | |+ style="font-size: 105%; white-space: nowrap;" | Norm-based tunings | ||
|- | |- | ||
! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean | ! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean | ||