22edo: Difference between revisions

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=== Septimal vs syntonic comma ===

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80, and therefore is not a system of [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, 31edo~~, …~~ do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]] and [[53edo]].

Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80, and therefore is not a system of [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]] and [[53edo]].

The diatonic scale it produces is instead derived from [[superpyth]] temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22et. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.

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=== 5-limit commas ===

Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the [[diatonic tritone]] [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.

=== 7-limit commas ===

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.

=== 11-limit commas ===

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| Bilorugu

| Kalisma

|-

| 13

| [[65/64]]

| {{monzo| -6 0 1 0 0 1 }}

| 26.84

| Thoyo

| Wilsorma

|-

| 13

| [[78/77]]

| {{monzo| 1 1 0 -1 -1 1 }}

| 22.34

| Tholuru

| Negustma

|-

| 13

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* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.

* [http://lumma.org/tuning/erlich/erlich-decatonic.pdf Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']

*~~[[Category:Listen]] [[Category:Twentuning]] [[Category:Alpharabian]] [[Category:Quartismic]]~~ [http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music]

* [http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music]

* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]

== References ==

@@ Line 313: / Line 313: @@
 === Septimal vs syntonic comma ===
-Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80, and therefore is not a system of [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, 31edo, … do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]] and [[53edo]].
+Possibly the most striking characteristic of 22edo to those not used to it is that it does '''not''' temper out the syntonic comma of 81/80, and therefore is not a system of [[meantone]] temperament. This means that 22 distinguishes a number of Pythagorean and 5-limit intervals that 12edo, 19edo, and 31edo do not distinguish, such as the two whole tones 9/8 and 10/9. Indeed, these distinctions are exaggerated in comparison to 5-limit JI and many more accurate temperaments such as [[34edo]], [[41edo]] and [[53edo]].
 The diatonic scale it produces is instead derived from [[superpyth]] temperament, which despite having the same melodic structure as meantone's diatonic scale (LLsLLLs or, [[5L 2s]]), has thirds approximating 9/7 and 7/6, rather than 5/4 and 6/5. This means that the septimal comma of 64/63 vanishes, rather than the syntonic comma of 81/80, which is one of the core features of 22et. Superpyth is melodically interesting for having a quasi-equal pentatonic scale (as the large whole tone and subminor third are rather close in size) and a more uneven heptatonic scale, as compared with 12et and other meantone systems: step patterns 4 4 5 4 5 and 4 4 1 4 4 4 1, respectively.
@@ Line 321: / Line 321: @@
 === 5-limit commas ===
-Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the [[diatonic tritone]] [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.
+Other 5-limit commas 22edo tempers out include the diaschisma, [[2048/2025]] and the magic comma or small diesis, [[3125/3072]]. In a diaschismic system, such as 12et or 22et, the diatonic tritone [[45/32]], which is a major third above a major whole tone representing [[9/8]], is equated to its inverted form, [[64/45]]. That the magic comma is tempered out means that 22et is a magic system, where five major thirds make up a perfect fifth.
 === 7-limit commas ===
-In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.
+In the 7-limit 22edo tempers out certain commas also tempered out by 12et; this relates 12et to 22 in a way different from the way in which meantone systems are akin to it. Both [[50/49]], (jubilee comma), and 64/63, (septimal comma), are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritones of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and an otonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the [[septimal kleisma]], so that the septimal kleisma augmented triad is a chord of 22et, as it also is of any meantone tuning. A septimal comma not tempered out by 12et which 22et does temper out is 1728/1715, the [[1728/1715|orwell comma]]; and the [[orwell tetrad]] is also a chord of 22et.
 === 11-limit commas ===
@@ Line 585: / Line 585: @@
 | Bilorugu
 | Kalisma
+|-
+| 13
+| [[65/64]]
+| {{monzo| -6 0 1 0 0 1 }}
+| 26.84
+| Thoyo
+| Wilsorma
+|-
+| 13
+| [[78/77]]
+| {{monzo| 1 1 0 -1 -1 1 }}
+| 22.34
+| Tholuru
+| Negustma
 |-
 | 13
@@ Line 1,428: / Line 1,442: @@
 * [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosakaidiphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for the Twenty-two Equal Divisions of the Octave]''. 2011.
 * [http://lumma.org/tuning/erlich/erlich-decatonic.pdf Erlich, Paul, ''Tuning, Tonality, and Twenty-Two Tone Temperament'']
-*[[Category:Listen]] [[Category:Twentuning]] [[Category:Alpharabian]] [[Category:Quartismic]] [http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music]
+* [http://porcupinemusic.weebly.com/ "Porcupine Music" - Website Focused on the Development of 22 EDO music]
+* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
 == References ==