224edo: Difference between revisions

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'''~~224EDO~~''' is the [[EDO|equal division of the octave]] into 224 parts of 5.3571 [[cent]]s each. It 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 [[Quartisma|117440512/117406179]] 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 chords. It defines the [[optimal patent val]] for [[Ragismic_microtemperaments #Octoid|octoid temperament]] in the 7-, 11- and 13-limit, and for [[Mirkwai_family|mirkwai]], the 7-limit planar temperament tempering out 16875/16807. It also provides an excellent tuning for [[Mirkwai_family #Indra|indra]] and [[Mirkwai_family #Shibi|shibi]] temperaments. It is the twelfth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]].

The '''224 equal divisions of the octave''' ('''224edo'''), or the '''224(-tone) equal temperament''' ('''224tet''', '''224et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 224 parts of 5.3571 [[cent]]s each.

~~224~~ = 32 * 7, and ~~has divisors 2~~, 4, 8, ~~16, 32~~, 7, 14, ~~28, 56~~, and ~~112~~.

== 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 chords. 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]].

224 = 32 × 7, and has divisors 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.

=== Prime harmonics ===

== Regular temperament properties ==

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-'''224EDO''' is the [[EDO|equal division of the octave]] into 224 parts of 5.3571 [[cent]]s each. It 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 [[Quartisma|117440512/117406179]] 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 chords. It defines the [[optimal patent val]] for [[Ragismic_microtemperaments #Octoid|octoid temperament]] in the 7-, 11- and 13-limit, and for [[Mirkwai_family|mirkwai]], the 7-limit planar temperament tempering out 16875/16807. It also provides an excellent tuning for [[Mirkwai_family #Indra|indra]] and [[Mirkwai_family #Shibi|shibi]] temperaments. It is the twelfth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]].
+The '''224 equal divisions of the octave''' ('''224edo'''), or the '''224(-tone) equal temperament''' ('''224tet''', '''224et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 224 parts of 5.3571 [[cent]]s each.
-= 32 * 7, and has divisors 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.
+== 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 chords. 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]].
-{{Primes in edo|224|prec=3}}
+= 32 × 7, and has divisors 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.
+=== Prime harmonics ===
+{{Primes in edo|224}}
 == Regular temperament properties ==