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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]].  
'''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]].  


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


{{Primes in edo|224|prec=3}}
{{Primes in edo|224|prec=3}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -355 224 }}
| [{{val| 224 355 }}]
| +0.053
| 0.0534
| 1.00
|-
| 2.3.5
| 32805/32768, {{monzo| -5 -32 24 }}
| [{{val| 224 355 520 }}]
| +0.122
| 0.1059
| 1.98
|-
| 2.3.5.7
| 4375/4374, 16875/16807, 32805/32768
| [{{val| 224 355 520 629 }}]
| +0.018
| 0.2009
| 3.75
|-
| 2.3.5.7.11
| 540/539, 1375/1372, 4000/3993, 32805/32768
| [{{val| 224 355 520 629 775 }}]
| -0.012
| 0.1899
| 3.54
|-
| 2.3.5.7.11.13
| 540/539, 625/624, 729/728, 1375/1372, 4096/4095
| [{{val| 224 355 520 629 775 829 }}]
| -0.035
| 0.1805
| 3.37
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 43\224
| 230.36
| 8/7
| [[Gamera]]
|-
| 1
| 59\224
| 316.07
| 6/5
| [[Counterkleismic]]
|-
| 1
| 65\224
| 348.21
| 11/9
| [[Eris]]
|-
| 1
| 71\224
| 380.36
| 56/45
| [[Quanharuk]]
|-
| 1
| 93\224
| 498.21
| 4/3
| [[Helmholtz]] / [[pontiac]] / [[ponta]]
|-
| 1
| 103\224
| 551.79
| 11/8
| [[Emkay]]
|-
| 2
| 31\224
| 166.07
| 11/10
| [[Pogo]]
|-
| 2
| 33\224
| 176.79
| 448/405, 195/176
| [[Quatracot]]
|-
| 2
| 39\224
| 208.93
| 44/39
| [[Abigail]]
|-
| 2
| 43\224
| 230.36
| 8/7
| [[Hemigamera]]
|-
| 4
| 15\224
| 80.36
| 22/21
| [[Quasithird]]
|-
| 4
| 37\224<br>(19\224)
| 198.21<br>(101.79)
| 28/25<br>&nbsp;
| [[Quadrant]]
|-
| 7
| 97\224<br>(1\224)
| 519.64<br>(5.36)
| 27/20<br>&nbsp;
| [[Brahmagupta]]
|-
| 7
| 93\224<br>(3\224)
| 498.21<br>(16.07)
| 4/3<br>&nbsp;
| [[Septant]]
|-
| 8
| 3\224
| 16.07
| 100/99
| [[Octoid]]
|-
| 8
| 93\224<br>(9\224)
| 498.21<br>(48.21)
| 4/3<br>(36/35)
| [[Octant]]
|-
| 28
| 3\224
| 16.07
| 126/125
| [[Oquatonic]]
|}


== Music ==
== Music ==

Revision as of 00:44, 18 July 2021

224EDO is the equal division of the octave into 224 parts of 5.3571 cents 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 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 octoid temperament 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 zeta integral edo.

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

Script error: No such module "primes_in_edo".

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-355 224⟩ [⟨224 355]] +0.053 0.0534 1.00
2.3.5 32805/32768, [-5 -32 24⟩ [⟨224 355 520]] +0.122 0.1059 1.98
2.3.5.7 4375/4374, 16875/16807, 32805/32768 [⟨224 355 520 629]] +0.018 0.2009 3.75
2.3.5.7.11 540/539, 1375/1372, 4000/3993, 32805/32768 [⟨224 355 520 629 775]] -0.012 0.1899 3.54
2.3.5.7.11.13 540/539, 625/624, 729/728, 1375/1372, 4096/4095 [⟨224 355 520 629 775 829]] -0.035 0.1805 3.37

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 43\224 230.36 8/7 Gamera
1 59\224 316.07 6/5 Counterkleismic
1 65\224 348.21 11/9 Eris
1 71\224 380.36 56/45 Quanharuk
1 93\224 498.21 4/3 Helmholtz / pontiac / ponta
1 103\224 551.79 11/8 Emkay
2 31\224 166.07 11/10 Pogo
2 33\224 176.79 448/405, 195/176 Quatracot
2 39\224 208.93 44/39 Abigail
2 43\224 230.36 8/7 Hemigamera
4 15\224 80.36 22/21 Quasithird
4 37\224
(19\224) 		198.21
(101.79) 		28/25
  		Quadrant
7 97\224
(1\224) 		519.64
(5.36) 		27/20
  		Brahmagupta
7 93\224
(3\224) 		498.21
(16.07) 		4/3
  		Septant
8 3\224 16.07 100/99 Octoid
8 93\224
(9\224) 		498.21
(48.21) 		4/3
(36/35) 		Octant
28 3\224 16.07 126/125 Oquatonic

Music

Dreyfus play by Gene Ward Smith

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