Bird's eye view of temperaments by accuracy: Difference between revisions
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Cata is a very efficient 5-limit and 2.3.5.13-subgroup temperament with a generator of a very slightly sharpened [[6/5]], two of which make [[13/9]] and thus three of which make [[26/15]] which is made into half of [[3/1]] so that its octave complement of [[15/13]] is half of [[4/3]]. It is amazing for its combination of accuracy and simplicity, because making six [[~]][[6/5]] generators equal to a fourth or fifth (up to octave-reduction) is the simplest equivalence possible without incurring a lot of damage. Its 7-note scale of [[4L 3s]] is usable, and its interpretation is accurately {[[25/24]], [[6/5]], [[5/4]], [[36/25]][[~]][[13/9]], [[3/2]], [[26/15]], [[2/1]]} so that it is at the simplest structural level well-supplied with plausible harmony, as this structure will persist and be duplicated in every superset/derived MOS scale, such as the likely more useful 15-note one, whose tuning range is at broadest in the [[15edo]] to [[19edo]] range, corresponding to the small step being at least half the size of the large step so that it has [[Rothenberg propriety]] (for those that care about this property). | Cata is a very efficient 5-limit and 2.3.5.13-subgroup temperament with a generator of a very slightly sharpened [[6/5]], two of which make [[13/9]] and thus three of which make [[26/15]] which is made into half of [[3/1]] so that its octave complement of [[15/13]] is half of [[4/3]]. It is amazing for its combination of accuracy and simplicity, because making six [[~]][[6/5]] generators equal to a fourth or fifth (up to octave-reduction) is the simplest equivalence possible without incurring a lot of damage. Its 7-note scale of [[4L 3s]] is usable, and its interpretation is accurately {[[25/24]], [[6/5]], [[5/4]], [[36/25]][[~]][[13/9]], [[3/2]], [[26/15]], [[2/1]]} so that it is at the simplest structural level well-supplied with plausible harmony, as this structure will persist and be duplicated in every superset/derived MOS scale, such as the likely more useful 15-note one, whose tuning range is at broadest in the [[15edo]] to [[19edo]] range, corresponding to the small step being at least half the size of the large step so that it has [[Rothenberg propriety]] (for those that care about this property). | ||
Cata admits an elegant extension to prime 7 called [[Catakleismic]], at the cost of some accuracy, a higher complexity and a smaller valid tuning range. | |||
This extension can be observed based on an [[S-expression]]-based comma list of: {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]](, [[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}, which is notable as making use of the record prime gap between 23 and 29 for an opportune no-11's 13-limit tempering opportunity [[~]][[28/27]][[~]][[27/26]][[~]][[26/25]][[~]][[25/24]], which as shown, implies tempering many notable commas, the most accurate of which is the [[ragisma]] (S25/S27), corresponding here to having an interval [[~]][[14/13]][[~]][[27/25]][[~]][[13/12]], and (arguably) the most interesting of which is making use of the exceptional numerical coincidence that [[676/675|S13/S15 = S26]]. The tuning range for catakleismic is approximately [[53edo]] to [[72edo]] - which are both reasonable tunings for it, with 53edo more accurate on the full subgroup and 72edo more accurate in the [[7-limit]]. | |||
=== [[Sensipent]] === | === [[Sensipent]] === | ||