Diminished (temperament): Difference between revisions

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'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] ~~with a 1/4-[[octave]] [[period]] [[generator|generated]] by a [[~]][[3/2]] perfect fifth or more generally by anything~~ that ~~melodically qualifies as a fifth. As a 5-limit temperament it~~ [[tempering out|tempers out]] the diminished comma, [[648/625]], and [[36/35]] and [[50/49]] in the [[7-limit]]. ~~A far more accurate interpretation is as~~ a ~~2.3.19 temperament that makes~~ [[~]][[~~19/16~~]] ~~= 1\4 = 300{{cent}}, though its~~ [[~~MOS~~]] ~~structure of~~ [[~~4L 4s~~]] ~~is very flexible, so one could use~~ 3~~\4 minus 8~~/~~7 as a ~670{{cent}}~~ fifth ~~for a 2.7.19 subgroup version of diminished, for example~~. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned. [[12edo]] is the simplest nontrivial tuning. Other possible tunings include [[16edo]] and [[28edo]], both of which having the interesting feature of being good in the 2.7.19 subgroup, so that the fifth is approximately [[28/19]]. [[28edo]] is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[strongly consistent circle]] of them.

'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] that [[tempering out|tempers out]] the diminished comma, [[648/625]], in the 5-limit, and [[36/35]] and [[50/49]] in the [[7-limit]]. It has a 1/4-[[octave]] [[period]] and is [[generator|generated]] by a [[~]][[3/2]] perfect fifth. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned.

See [[Dimipent family #Diminished]] for technical data ~~on the 5-limit temperament~~.

It can be extended to the 2.3.5.7.19-[[subgroup]] where the 1/4-octave period stands in for both ~6/5 and ~19/16 since this ~19/16 is more accurate, though its mos structure of 4L 4s is very flexible, so one could use 3\4 minus ~8/7 as a ~670{{cent}} fifth for a 2.7.19 subgroup version of diminished, for example.

[[12edo]] is an obvious tuning. Other possible tunings include [[16edo]] and [[28edo]], both of which having the interesting feature of being good in the 2.7.19 subgroup, so that the fifth is approximately [[28/19]]. [[28edo]] is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[strongly consistent circle]] of them.

See [[Dimipent family #Diminished]] for technical data.

== Interval chain ==

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-'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] with a 1/4-[[octave]] [[period]] [[generator|generated]] by a [[~]][[3/2]] perfect fifth or more generally by anything that melodically qualifies as a fifth. As a 5-limit temperament it [[tempering out|tempers out]] the diminished comma, [[648/625]], and [[36/35]] and [[50/49]] in the [[7-limit]]. A far more accurate interpretation is as a 2.3.19 temperament that makes [[~]][[19/16]] = 1\4 = 300{{cent}}, though its [[MOS]] structure of [[4L 4s]] is very flexible, so one could use 3\4 minus 8/7 as a ~670{{cent}} fifth for a 2.7.19 subgroup version of diminished, for example. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned. [[12edo]] is the simplest nontrivial tuning. Other possible tunings include [[16edo]] and [[28edo]], both of which having the interesting feature of being good in the 2.7.19 subgroup, so that the fifth is approximately [[28/19]]. [[28edo]] is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[strongly consistent circle]] of them.
+'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] that [[tempering out|tempers out]] the diminished comma, [[648/625]], in the 5-limit, and [[36/35]] and [[50/49]] in the [[7-limit]]. It has a 1/4-[[octave]] [[period]] and is [[generator|generated]] by a [[~]][[3/2]] perfect fifth. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned.
-See [[Dimipent family #Diminished]] for technical data on the 5-limit temperament.
+It can be extended to the 2.3.5.7.19-[[subgroup]] where the 1/4-octave period stands in for both ~6/5 and ~19/16 since this ~19/16 is more accurate, though its mos structure of 4L 4s is very flexible, so one could use 3\4 minus ~8/7 as a ~670{{cent}} fifth for a 2.7.19 subgroup version of diminished, for example.
+[[12edo]] is an obvious tuning. Other possible tunings include [[16edo]] and [[28edo]], both of which having the interesting feature of being good in the 2.7.19 subgroup, so that the fifth is approximately [[28/19]]. [[28edo]] is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[strongly consistent circle]] of them.
+See [[Dimipent family #Diminished]] for technical data.
 == Interval chain ==