224edo: Difference between revisions

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224edo is a very strong [[13-limit]] system. 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.

As an equal temperament, 224et [[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.

As an equal temperament, 224et [[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. [[217edo]], only a bit smaller, has a worse 13-limit, but it achieves a much higher [[consistency limit]], almost [[31-odd-limit|31-odd]].

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.

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; that said, the use of both types of fifth enables creation of a closed circle of 24 notes per octave, generated as 16 patent fifths plus 8 bb fifths (as in [[quadrant]] temperament)<ref>[http://www.youtube.com/@Xen-p6p @Xen-p6p] (2026), YouTube post on [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024).</ref>, although a different distribution than the quarter-octave distribution specified by quadrant might be desired for a well-tempered 24 note tuning system.

=== Prime harmonics ===

Line 14:

=== Subsets and supersets ===

Since 224 factors into {{nowrap| 25 × 7 }}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.

Since 224 factors into primes as {{nowrap| 25 × 7 }}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.

== Intervals ==

Line 22:

== Notation ==

=== Sagittal ===

224edo can be written in Sagittal using almost the entire Athenian extension ~~(except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]])~~, by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian~~<ref name=":0">[[Ragismic microtemperaments#Brahmagupta]]</ref>~~. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>https://sagittal.org/sagittal.pdf p. 11</ref>

224edo can be written in Sagittal using ''almost'' the entire Athenian extension, by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>[https://sagittal.org/sagittal.pdf Sagittal – A Microtonal Notation System] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>

{| class="wikitable"

|+ Sagittal notation

! ~~224edosteps~~

! colspan="2" |Steps

! 0

! 1

Line 50:

! 21

|-

| ~~Revo~~

! rowspan="2" |Symbol

| rowspan="2" | {{sagittal||//|}}

! Evo

| rowspan="2" | {{sagittal||(}}

| rowspan="2" | <big>{{sagittal||//|}}</big>

| rowspan="2" | {{sagittal|)|(}}

| rowspan="2" | <big>{{sagittal||(}}</big>

| rowspan="2" | {{sagittal|~|(}}

| rowspan="2" | <big>{{sagittal|)|(}}</big>

| rowspan="2" | {{sagittal|/|}}

| rowspan="2" | <big>{{sagittal|~|(}}</big>

| rowspan="2" | {{sagittal||)}}

| rowspan="2" | <big>{{sagittal|/|}}</big>

| rowspan="2" | {{sagittal||\}}

| rowspan="2" | <big>{{sagittal||)}}</big>

| rowspan="2" | {{sagittal|(|(}}

| rowspan="2" | <big>{{sagittal||\}}</big>

| rowspan="2" | {{sagittal|//|}}

| rowspan="2" | <big>{{sagittal|(|(}}</big>

| rowspan="2" | {{sagittal|/|)}}

| rowspan="2" | <big>{{sagittal|//|}}</big>

| rowspan="2" | {{sagittal|/|\}}

| rowspan="2" | <big>{{sagittal|/|)}}</big>

| {{sagittal|(|)}}

| rowspan="2" | <big>{{sagittal|/|\}}</big>

| {{sagittal|(|\}}

| {{sagittal|#}}{{sagittal|\!/}}

| {{sagittal|)||(}}

| {{sagittal|#}}{{sagittal|\!)}}

| {{sagittal|~||(}}

| {{sagittal|#}}{{sagittal|\\!}}

| {{sagittal|/||}}

| {{sagittal|#}}{{sagittal|(!(}}

| {{sagittal|||)}}

| {{sagittal|#}}{{sagittal|!/}}

| {{sagittal|||\}}

| {{sagittal|#}}{{sagittal|!)}}

| {{sagittal|(||(}}

| {{sagittal|#}}{{sagittal|\!}}

| {{sagittal|~~//|~~|}}

| {{sagittal|#}}{{sagittal|~!(}}

| {{sagittal|/||)}}

| {{sagittal|#}}{{sagittal|)!(}}

| {{sagittal|~~/||\~~}}

| {{sagittal|#}}{{sagittal|!(}}

| {{sagittal|#}}

|-

~~| Evo~~

! Revo

| {{sagittal|#}}~~{{sagittal|\!~~/}}

| <big>{{sagittal|(|)}}</big>

| {{sagittal|~~#}}{{sagittal~~|\!)}}

| <big>{{sagittal|(|\}}</big>

| {{sagittal|~~#}}{{sagittal~~|~~\\!~~}}

| <big>{{sagittal|)||(}}</big>

| {{sagittal|~~#}}{{sagittal~~|(!(}}

| <big>{{sagittal|~||(}}</big>

| {{sagittal|#}}~~{{sagittal|!~~/}}

| <big>{{sagittal|/||}}</big>

| {{sagittal|~~#}}{{sagittal~~|!)}}

| <big>{{sagittal|||)}}</big>

| {{sagittal|~~#}}{{sagittal~~|\!}}

| <big>{{sagittal|||\}}</big>

| {{sagittal|~~#}}{{sagittal~~|~!(}}

| <big>{{sagittal|(||(}}</big>

| {{sagittal|~~#}}{{sagittal~~|~~)!(~~}}

| <big>{{sagittal|//||}}</big>

| {{sagittal|~~#}}{{sagittal~~|!(}}

| <big>{{sagittal|/||)}}</big>

| {{sagittal|#}}

| <big>{{sagittal|/||\}}</big>

|}

Because it uses the entire Athenian system (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), it allows no accidental enharmonic respellings

=== Ups-and-downs notation ===

Line 380:

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| [[Barium]]

|}

<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==

; [[Mercury Amalgam]]

* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning

; [[Gene Ward Smith]]

* ''Dreyfus'' (archived 2010) – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning

~~; [[Mercury Amalgam]]~~

== References ==

* [https://www.youtube.com/watch?v=~~iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning~~

[[Category:Listen]]

[[Category:Canopic]]

[[Category:Indra]]

[[Category:~~Listen]]~~

[[Category:Shibi]]

~~[[Category:Mirkwai~~]]

[[Category:Octoid]]

~~[[Category:Quartismic]]~~

~~[[Category:Shibi]]~~

@@ Line 5: / Line 5: @@
 edo is a very strong [[13-limit]] system. 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.
-As an equal temperament, 224et [[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.
+As an equal temperament, 224et [[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. [[217edo]], only a bit smaller, has a worse 13-limit, but it achieves a much higher [[consistency limit]], almost [[31-odd-limit|31-odd]].
-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.
+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; that said, the use of both types of fifth enables creation of a closed circle of 24 notes per octave, generated as 16 patent fifths plus 8 bb fifths (as in [[quadrant]] temperament)<ref>[http://www.youtube.com/@Xen-p6p @Xen-p6p] (2026), YouTube post on [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024).</ref>, although a different distribution than the quarter-octave distribution specified by quadrant might be desired for a well-tempered 24 note tuning system.
 === Prime harmonics ===
@@ Line 14: / Line 14: @@
 === Subsets and supersets ===
-Since 224 factors into {{nowrap| 2<sup>5</sup> × 7 }}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
+Since 224 factors into primes as {{nowrap| 2<sup>5</sup> × 7 }}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
 == Intervals ==
@@ Line 22: / Line 22: @@
 == Notation ==
 === Sagittal ===
-edo can be written in Sagittal using almost the entire Athenian extension (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian<ref name=":0">[[Ragismic microtemperaments#Brahmagupta]]</ref>. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>https://sagittal.org/sagittal.pdf p. 11</ref>
+edo can be written in Sagittal using ''almost'' the entire Athenian extension, by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>[https://sagittal.org/sagittal.pdf Sagittal – A Microtonal Notation System] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>
 {| class="wikitable"
 |+ Sagittal notation
-! 224edosteps
+! colspan="2" |Steps
 ! 0
 ! 1
@@ Line 50: / Line 50: @@
 ! 21
 |-
-| Revo
+! rowspan="2" |Symbol
-| rowspan="2" | {{sagittal||//|}}
+! Evo
-| rowspan="2" | {{sagittal||(}}
+| rowspan="2" | <big>{{sagittal||//|}}</big>
-| rowspan="2" | {{sagittal|)|(}}
+| rowspan="2" | <big>{{sagittal||(}}</big>
-| rowspan="2" | {{sagittal|~|(}}
+| rowspan="2" | <big>{{sagittal|)|(}}</big>
-| rowspan="2" | {{sagittal|/|}}
+| rowspan="2" | <big>{{sagittal|~|(}}</big>
-| rowspan="2" | {{sagittal||)}}
+| rowspan="2" | <big>{{sagittal|/|}}</big>
-| rowspan="2" | {{sagittal||\}}
+| rowspan="2" | <big>{{sagittal||)}}</big>
-| rowspan="2" | {{sagittal|(|(}}
+| rowspan="2" | <big>{{sagittal||\}}</big>
-| rowspan="2" | {{sagittal|//|}}
+| rowspan="2" | <big>{{sagittal|(|(}}</big>
-| rowspan="2" | {{sagittal|/|)}}
+| rowspan="2" | <big>{{sagittal|//|}}</big>
-| rowspan="2" | {{sagittal|/|\}}
+| rowspan="2" | <big>{{sagittal|/|)}}</big>
-| {{sagittal|(|)}}
+| rowspan="2" | <big>{{sagittal|/|\}}</big>
-| {{sagittal|(|\}}
+| <small>{{sagittal|#}}{{sagittal|\!/}}</small>
-| {{sagittal|)||(}}
+| <small>{{sagittal|#}}{{sagittal|\!)}}</small>
-| {{sagittal|~||(}}
+| <small>{{sagittal|#}}{{sagittal|\\!}}</small>
-| {{sagittal|/||}}
+| <small>{{sagittal|#}}{{sagittal|(!(}}</small>
-| {{sagittal|||)}}
+| <small>{{sagittal|#}}{{sagittal|!/}}</small>
-| {{sagittal|||\}}
+| <small>{{sagittal|#}}{{sagittal|!)}}</small>
-| {{sagittal|(||(}}
+| <small>{{sagittal|#}}{{sagittal|\!}}</small>
-| {{sagittal|//||}}
+| <small>{{sagittal|#}}{{sagittal|~!(}}</small>
-| {{sagittal|/||)}}
+| <small>{{sagittal|#}}{{sagittal|)!(}}</small>
-| {{sagittal|/||\}}
+| <small>{{sagittal|#}}{{sagittal|!(}}</small>
+| <small>{{sagittal|#}}</small>
 |-
-| Evo
+! Revo
-| {{sagittal|#}}{{sagittal|\!/}}
+| <big>{{sagittal|(|)}}</big>
-| {{sagittal|#}}{{sagittal|\!)}}
+| <big>{{sagittal|(|\}}</big>
-| {{sagittal|#}}{{sagittal|\\!}}
+| <big>{{sagittal|)||(}}</big>
-| {{sagittal|#}}{{sagittal|(!(}}
+| <big>{{sagittal|~||(}}</big>
-| {{sagittal|#}}{{sagittal|!/}}
+| <big>{{sagittal|/||}}</big>
-| {{sagittal|#}}{{sagittal|!)}}
+| <big>{{sagittal|||)}}</big>
-| {{sagittal|#}}{{sagittal|\!}}
+| <big>{{sagittal|||\}}</big>
-| {{sagittal|#}}{{sagittal|~!(}}
+| <big>{{sagittal|(||(}}</big>
-| {{sagittal|#}}{{sagittal|)!(}}
+| <big>{{sagittal|//||}}</big>
-| {{sagittal|#}}{{sagittal|!(}}
+| <big>{{sagittal|/||)}}</big>
-| {{sagittal|#}}
+| <big>{{sagittal|/||\}}</big>
 |}
+Because it uses the entire Athenian system (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), it allows no accidental enharmonic respellings
 === Ups-and-downs notation ===
@@ Line 380: / Line 382: @@
 | [[Barium]]
 |}
-<nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Music ==
+; [[Mercury Amalgam]]
+* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning
 ; [[Gene Ward Smith]]
 * ''Dreyfus'' (archived 2010) – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning
-; [[Mercury Amalgam]]
+== References ==
-* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning
+[[Category:Listen]]
+[[Category:Canopic]]
 [[Category:Indra]]
-[[Category:Listen]]
+[[Category:Shibi]]
-[[Category:Mirkwai]]
 [[Category:Octoid]]
-[[Category:Quartismic]]
-[[Category:Shibi]]