224edo: Difference between revisions

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== Theory ==

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 chords]] 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 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 chords]] 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]] and is the second-smallest EDO after [[87edo|87]] to approximate all of the first 16 harmonics of the harmonic series with [[minimal consistent EDOs|no greater than 25%]] relative error.

224edo tempers the [[syntonic comma]] to 1/56th of the octave (4 steps) and as a corollary supports the [[barium]] temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for [[meantone]] and is very close (0.15 cents) to the [[quarter-comma meantone]] fifth. The generator however reduces to [[112edo]], being 65\112.

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 {{Infobox ET}}
-{{EDO intro|224}}
+{{ED intro}}
 == Theory ==
-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 chords]] 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 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 chords]] 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]] and is the second-smallest EDO after [[87edo|87]] to approximate all of the first 16 harmonics of the harmonic series with [[minimal consistent EDOs|no greater than 25%]] relative error.
 edo tempers the [[syntonic comma]] to 1/56th of the octave (4 steps) and as a corollary supports the [[barium]] temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for [[meantone]] and is very close (0.15 cents) to the [[quarter-comma meantone]] fifth. The generator however reduces to [[112edo]], being 65\112.