224edo: Difference between revisions

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224edo is a very strong [[13-limit]] system, tempering out [[32805/32768]] in the [[5-limit]]; [[4375/4374]], 16875/16807 and 65625/65536 in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned [[essentially tempered chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic triad]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. It is the twelfth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]].

224edo tempers the [[syntonic comma]] to 1/56th of the octave and as a corollary supports the [[barium]] temperament.

224edo tempers the [[syntonic comma]] to 1/56th of the octave and as a corollary supports the [[barium]] temperament. In addition, since 224 = 4 x 56, flattening the fifth by one step of 224edo produces a tuning for the [[quarter-comma meantone]]. Such generator however reduces to [[112edo]], being 65\112.

=== Prime harmonics ===

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 edo is a very strong [[13-limit]] system, tempering out [[32805/32768]] in the [[5-limit]]; [[4375/4374]], 16875/16807 and 65625/65536 in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned [[essentially tempered chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic triad]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. It is the twelfth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]].
-edo tempers the [[syntonic comma]] to 1/56th of the octave and as a corollary supports the [[barium]] temperament.
+edo tempers the [[syntonic comma]] to 1/56th of the octave and as a corollary supports the [[barium]] temperament. In addition, since 224 = 4 x 56, flattening the fifth by one step of 224edo produces a tuning for the [[quarter-comma meantone]]. Such generator however reduces to [[112edo]], being 65\112.
 === Prime harmonics ===