Bird's eye view of temperaments by accuracy: Difference between revisions
add generator tunings for echidna, catakleismic and diaschismic; also correct the targets of catakleismic (13 was missing??) |
add generator tunings for sensible and srutal |
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* 26 for {3, 5, 11, 17, 23, 31, 33, 51, 55, 69, 85, 99(, 115)} ([[19L 8s]]) | * 26 for {3, 5, 11, 17, 23, 31, 33, 51, 55, 69, 85, 99(, 115)} ([[19L 8s]]) | ||
* 35 for adding {9, 15, 25, 93(, 155)} ([[19L 8s]] or 19L 27s) | * 35 for adding {9, 15, 25, 93(, 155)} ([[19L 8s]] or 19L 27s) | ||
[[#Generator tunings|Generator tunings]]: 24\65, 41\111 | |||
Sensible is a fairly accurate (though quite complex) subgroup interpretation of sensi (or extension of [[#Sensipent]]) which is especially accurate in the [[CEE tuning]] (443.115{{cent}}) if avoiding composite odds and intervals of 23 so that under such restrictions it classifies as [[#High accuracy]], meaning it's one of the most accurate temperaments in this category, especially if you use error-cancellations of sharp harmonics to your advantage to construct complex harmonic series chords in order to justify the generally-higher errors of composite harmonics. [[46edo]] is a somewhat reasonable but trivial tuning of it, [[65edo]] is somewhat better, and [[111edo]] is even better, but it works especially well in a more optimized tuning, hence its significance as a rank 2 temperament. The 27-note MOS, though not achieving all of its odd harmonics from a single note, is sufficient, because the missing odds {9, 15, 25, 93(, 155)} also do reasonably occur within the span of the 27-note MOS. Therefore, sensible can be seen as a detempering of [[27edo]] to a more accurate rank 2 temperament on a mostly-unrelated subgroup. | Sensible is a fairly accurate (though quite complex) subgroup interpretation of sensi (or extension of [[#Sensipent]]) which is especially accurate in the [[CEE tuning]] (443.115{{cent}}) if avoiding composite odds and intervals of 23 so that under such restrictions it classifies as [[#High accuracy]], meaning it's one of the most accurate temperaments in this category, especially if you use error-cancellations of sharp harmonics to your advantage to construct complex harmonic series chords in order to justify the generally-higher errors of composite harmonics. [[46edo]] is a somewhat reasonable but trivial tuning of it, [[65edo]] is somewhat better, and [[111edo]] is even better, but it works especially well in a more optimized tuning, hence its significance as a rank 2 temperament. The 27-note MOS, though not achieving all of its odd harmonics from a single note, is sufficient, because the missing odds {9, 15, 25, 93(, 155)} also do reasonably occur within the span of the 27-note MOS. Therefore, sensible can be seen as a detempering of [[27edo]] to a more accurate rank 2 temperament on a mostly-unrelated subgroup. | ||
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Bound-violating intervals: [[13/9]] | Bound-violating intervals: [[13/9]] | ||
[[#Generator tunings|Generator tunings]]: (4\46, 1\2), (7\80, 1\2) | |||
Srutal is an at least no-19's [[23-limit]] temperament, being an extension of [[#Srutal archagall]] to the full [[17-limit]] and finding [[23/16]] as an augmented fourth, that is, as a tritone of ([[~]][[9/8]])<sup>3</sup>. Therefore, familiarizing oneself with srutal archagall is recommendable as the structures and tunings are nearly identical (and their tunings merge meaningfully in [[80edo]]), with the main difference being number of notes and breadth of harmonies targetted. Srutal can find more primes than just those in the no-19's 23-limit but they are more complex so more likely to be used opportunistically in a 34-note MOS, so that this temperament can be seen as a detemperament of [[34edo]]. [[80edo]] is a good tuning for it, though [[46edo]] deals well enough with the no-19's 23-limit part (potentially add-31) at the cost of a variety of distinctions. An alternative extension of srutal archagall to just the full 17-limit which is damages the 2.3.5.17 subgroup slightly more is [[#Diaschismic]]. | Srutal is an at least no-19's [[23-limit]] temperament, being an extension of [[#Srutal archagall]] to the full [[17-limit]] and finding [[23/16]] as an augmented fourth, that is, as a tritone of ([[~]][[9/8]])<sup>3</sup>. Therefore, familiarizing oneself with srutal archagall is recommendable as the structures and tunings are nearly identical (and their tunings merge meaningfully in [[80edo]]), with the main difference being number of notes and breadth of harmonies targetted. Srutal can find more primes than just those in the no-19's 23-limit but they are more complex so more likely to be used opportunistically in a 34-note MOS, so that this temperament can be seen as a detemperament of [[34edo]]. [[80edo]] is a good tuning for it, though [[46edo]] deals well enough with the no-19's 23-limit part (potentially add-31) at the cost of a variety of distinctions. An alternative extension of srutal archagall to just the full 17-limit which is damages the 2.3.5.17 subgroup slightly more is [[#Diaschismic]]. | ||