Magic: Difference between revisions

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'''Magic''' is a linear temperament in which the ~380 cent generator represents [[5/4]], and five of those make a [[3/1]]. This implies that the [[magic comma]] [[3125/3072]] is tempered out, making it a member of the [[magic family]]. This article also assumes the default mapping for the prime 7, which tempers out [[225/224]] and makes two generators equivalent to [[14/9]]. [[7/4]] can be reached by 12 generators in this mapping. (There is an alternative mapping for 7 known as [[~~Magic family #Muggles|~~muggles]], which may be better melodically for small ~~MOSes~~ due to the smaller generator making the small step a bit larger, but there is little reason to use it unless you~~'re~~ using [[19edo]], in which case it's identical to magic anyway.)

'''Magic''' is a [[linear temperament]] in which the ~380 cent [[generator]] represents [[5/4]], and five of those make a [[3/1]]. This implies that the [[magic comma]] [[3125/3072]] is [[tempering out|tempered out]], making it a member of the [[magic family]]. This article also assumes the default mapping for the prime 7, which tempers out [[225/224]] and makes two generators equivalent to [[14/9]]. [[7/4]] can be reached by 12 generators in this mapping. (There is an alternative mapping for 7 known as [[muggles]], which may be better melodically for small [[mos scale]]s due to the smaller generator making the small step a bit larger, but there is little reason to use it unless you are using [[19edo]], in which case it is identical to magic anyway.)

~~EDOs~~ that contain good magic scales include [[19edo]], [[22edo]], [[41edo]], [[60edo]] and [[104edo]].

Edos that contain good magic scales include [[19edo]], [[22edo]], [[41edo]], [[60edo]] and [[104edo]].

Magic has certain properties that commend it as a step up in complexity from traditional harmony:

* It is the simplest mapping capable of tuning every [[9-odd-limit]] interval better than in [[12edo]].

* It is only slightly more complex than meantone (both work well with a 19 note gamut).

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It is not a panacea because:

* It has no proper mos scales with between 3 and 16 notes over a single period per octave.

* It has no proper ~~MOS~~ scales with between 3 and 16 notes over a single period per octave.

* It is more complex than meantone (higher complexity and badness).

* The 3/2 approximation is 5 times as complex as the 5/4 approximation (the generator) so modulation by fifths is more constrained than you may be used to.

Because the generator is so close to 1\3 of an octave, and the interval left over is accordingly so small, all small magic ~~MOSes~~ consist of three large intervals alternating with three groups of this small interval. Specifically, there are the following ~~MOSes~~, where s always represents the characteristic small interval, which simultaneously represents [[128/125]], [[36/35]], [[28/27]], and [[25/24]].

Because the generator is so close to 1/3 of an octave, and the interval left over is accordingly so small, all small magic mos scales consist of three large intervals alternating with three groups of this small interval. Specifically, there are the following scales, where s always represents the characteristic small interval, which simultaneously represents [[128/125]], [[36/35]], [[28/27]], and [[25/24]].

* [[3L 4s]]: LsLsLss, where L represents 6/5

* [[3L 7s]]: LssLssLsss, where L represents 7/6

* [[3L 10s]]: LsssLsssLssss, where L represents 9/8

* [[3L 13s]]: LssssLssssLsssss, where L is a neutral second, which can be taken to represent 12/11 (in magic temperament) or 11/10 (in the related [[~~Magic family #Telepathy|~~telepathy]] temperament). In 22edo they are identical.

* [[3L 13s]]: LssssLssssLsssss, where L is a neutral second, which can be taken to represent 12/11 (in magic temperament) or 11/10 (in the related [[telepathy]] temperament). In 22edo they are identical.

For technical information see [[Magic family]].

For technical information, see [[Magic family #Magic]]. For a discussion on alternative 11- and 13-limit extensions, see [[Magic extensions]].

== Interval chain ==

{| class="wikitable center-all"

|-

! Cents~~{{asterisk}}~~

! Cents*

| 0

| 380.352

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| (12/11)

|}

<nowiki />* In 7-limit [[POTE tuning]]

<nowiki/>* In 7-limit [[POTE tuning]]

The generator chain val for 13-limit magic is {{val| 0 5 1 12 -8 18 }}, so that five generators give an approximate 3, twelve 14, minus eight 11/64, and eighteen 52.

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== Scales ==

; MOS scales

* [[Magic7]] ~~– Improper~~ [[3L 4s]]

* [[Magic7]] – improper [[3L 4s]]

* [[Magic10]] ~~– Improper~~ [[3L 7s]]

* [[Magic10]] – improper [[3L 7s]]

* [[Magic13]] ~~– Improper~~ [[3L 10s]]

* [[Magic13]] – improper [[3L 10s]]

* [[Magic16]] ~~– Improper~~ [[3L 13s]]. The boundary of propriety is 19edo.

* [[Magic16]] – improper [[3L 13s]]. The boundary of propriety is 19edo.

* [[Magic19]] ~~– Proper~~ [[3L 16s]]. The boundary of propriety is 22edo.

* [[Magic19]] – proper [[3L 16s]]. The boundary of propriety is 22edo.

* [[Magic22]] ~~–~~ [[19L 3s]] scale. The boundary of propriety is 41edo.

* [[Magic22]] – [[19L 3s]] scale. The boundary of propriety is 41edo.

; Transversal scales

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== Music ==

; [[Cameron Bobro]]

* [https://web.archive.org/web/20201127015243/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Magical_Daydream_CBobro.mp3 ''Magical Daydream''] ~~–~~ ''A brief demonstration of the near-Just musical temperament which flattens the pure major third of 5:4 by a few cents, such that 5 major thirds does not exceed 3:1 (a pure fifth + 1 octave), but meets it precisely. In a purely tuned system, the thirds would exceed 3:1 by what is known as the small diesis, (a ratio 3125/3072, about thirty cents). This temperament, then, brings (almost) pure thirds and pure fifths together.''

* [https://web.archive.org/web/20201127015243/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Magical_Daydream_CBobro.mp3 ''Magical Daydream''] – ''A brief demonstration of the near-Just musical temperament which flattens the pure major third of 5:4 by a few cents, such that 5 major thirds does not exceed 3:1 (a pure fifth + 1 octave), but meets it precisely. In a purely tuned system, the thirds would exceed 3:1 by what is known as the small diesis, (a ratio 3125/3072, about thirty cents). This temperament, then, brings (almost) pure thirds and pure fifths together.''

* [https://web.archive.org/web/20201127014810/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 ''Evening Horizon''] ~~–~~ ''The earliest implementation (by happy accident, it seems) of this temperament was, to my knowledge, by Paul von Janko over a century ago. More recently, an online tuning community has elaborated many precise variations, calling the temperament "magic". This piece is a demonstration of the array of pitches created by using 22 generators (the slightly tempered 5:4) within the octave, an approach which creates a "moment of symmetry", with all pitches separated by the same two intervals. This has many curious repercussions, creating some musical possibilities and restricting others.''

* [https://web.archive.org/web/20201127014810/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 ''Evening Horizon''] – ''The earliest implementation (by happy accident, it seems) of this temperament was, to my knowledge, by Paul von Janko over a century ago. More recently, an online tuning community has elaborated many precise variations, calling the temperament "magic". This piece is a demonstration of the array of pitches created by using 22 generators (the slightly tempered 5:4) within the octave, an approach which creates a "moment of symmetry", with all pitches separated by the same two intervals. This has many curious repercussions, creating some musical possibilities and restricting others.''

; [[Graham Breed]]

* [http://x31eq.com/music/dingsheng.mp3 ''Golden Age''] ~~– Disco~~ involving magic comma pumps.

* [http://x31eq.com/music/dingsheng.mp3 ''Golden Age''] – disco involving magic comma pumps.

* [http://x31eq.com/music/dingshi.mp3 ''Extravagant Food''] ~~– A~~ single magic comma pump in under 60 seconds in 60edo.

* [http://x31eq.com/music/dingshi.mp3 ''Extravagant Food''] – a single magic comma pump in under 60 seconds in 60edo.

* [http://x31eq.com/music/jitter.ogg ''Gene's Jitterbug''] ~~–~~ 9-odd-limit harmony, may not require magic.

* [http://x31eq.com/music/jitter.ogg ''Gene's Jitterbug''] – 9-odd-limit harmony, may not require magic.

; [[Jake Freivald]]

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; [[Andrew Milne]]

* [https://soundcloud.com/andrew-j-milne/magic-traveller ''Magic Traveller''] (2008) ~~–~~ Magic[10] with 379.8-cent generator

* [https://soundcloud.com/andrew-j-milne/magic-traveller ''Magic Traveller''] (2008) – Magic[10] with 379.8-cent generator

; [[Chris Vaisvil]] ([http://www.chrisvaisvil.com/ site])

* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 ''Chromatic piece in magic 16''] (2012) ~~–~~ Magic[16] in 145edo tuning (~~→~~ [[magic16]])

* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 ''Chromatic piece in magic 16''] (2012) – Magic[16] in 145edo tuning (→ [[magic16]])

* [http://micro.soonlabel.com/22-ET/daily20120128-pauls-magic.mp3 ''A Piece in Paulsmagic''] (2012) ~~– In~~ [[paulsmagic]]

* [http://micro.soonlabel.com/22-ET/daily20120128-pauls-magic.mp3 ''A Piece in Paulsmagic''] (2012) – in [[paulsmagic]]

* [http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 ''The Magic of Belief''] (2013) ~~–~~ Magic[19] in 41edo tuning

* [http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 ''The Magic of Belief''] (2013) – Magic[19] in 41edo tuning

; [[User:Xenllium|Xenllium]]

* [https://youtube.com/watch?v=5iBXKRWE-_U ''Magical life''] (2023) ~~– In~~ 19~~-tone~~ pure-fifth ~~magic scale~~

* [https://youtube.com/watch?v=5iBXKRWE-_U ''Magical life''] (2023) – Magic[19] in pure-fifth tuning

== See also ==

@@ Line 7: / Line 7: @@
 {{Wikipedia| Magic temperament }}
-'''Magic''' is a linear temperament in which the ~380 cent generator represents [[5/4]], and five of those make a [[3/1]]. This implies that the [[magic comma]] [[3125/3072]] is tempered out, making it a member of the [[magic family]]. This article also assumes the default mapping for the prime 7, which tempers out [[225/224]] and makes two generators equivalent to [[14/9]]. [[7/4]] can be reached by 12 generators in this mapping. (There is an alternative mapping for 7 known as [[Magic family #Muggles|muggles]], which may be better melodically for small MOSes due to the smaller generator making the small step a bit larger, but there is little reason to use it unless you're using [[19edo]], in which case it's identical to magic anyway.)
+'''Magic''' is a [[linear temperament]] in which the ~380 cent [[generator]] represents [[5/4]], and five of those make a [[3/1]]. This implies that the [[magic comma]] [[3125/3072]] is [[tempering out|tempered out]], making it a member of the [[magic family]]. This article also assumes the default mapping for the prime 7, which tempers out [[225/224]] and makes two generators equivalent to [[14/9]]. [[7/4]] can be reached by 12 generators in this mapping. (There is an alternative mapping for 7 known as [[muggles]], which may be better melodically for small [[mos scale]]s due to the smaller generator making the small step a bit larger, but there is little reason to use it unless you are using [[19edo]], in which case it is identical to magic anyway.)
-EDOs that contain good magic scales include [[19edo]], [[22edo]], [[41edo]], [[60edo]] and [[104edo]].
+Edos that contain good magic scales include [[19edo]], [[22edo]], [[41edo]], [[60edo]] and [[104edo]].
 Magic has certain properties that commend it as a step up in complexity from traditional harmony:
 * It is the simplest mapping capable of tuning every [[9-odd-limit]] interval better than in [[12edo]].
 * It is only slightly more complex than meantone (both work well with a 19 note gamut).
@@ Line 18: / Line 17: @@
 It is not a panacea because:
+* It has no proper mos scales with between 3 and 16 notes over a single period per octave.
-* It has no proper MOS scales with between 3 and 16 notes over a single period per octave.
 * It is more complex than meantone (higher complexity and badness).
 * The 3/2 approximation is 5 times as complex as the 5/4 approximation (the generator) so modulation by fifths is more constrained than you may be used to.
-Because the generator is so close to 1\3 of an octave, and the interval left over is accordingly so small, all small magic MOSes consist of three large intervals alternating with three groups of this small interval. Specifically, there are the following MOSes, where s always represents the characteristic small interval, which simultaneously represents [[128/125]], [[36/35]], [[28/27]], and [[25/24]].
+Because the generator is so close to 1/3 of an octave, and the interval left over is accordingly so small, all small magic mos scales consist of three large intervals alternating with three groups of this small interval. Specifically, there are the following scales, where s always represents the characteristic small interval, which simultaneously represents [[128/125]], [[36/35]], [[28/27]], and [[25/24]].
 * [[3L 4s]]: LsLsLss, where L represents 6/5
 * [[3L 7s]]: LssLssLsss, where L represents 7/6
 * [[3L 10s]]: LsssLsssLssss, where L represents 9/8
-* [[3L 13s]]: LssssLssssLsssss, where L is a neutral second, which can be taken to represent 12/11 (in magic temperament) or 11/10 (in the related [[Magic family #Telepathy|telepathy]] temperament). In 22edo they are identical.
+* [[3L 13s]]: LssssLssssLsssss, where L is a neutral second, which can be taken to represent 12/11 (in magic temperament) or 11/10 (in the related [[telepathy]] temperament). In 22edo they are identical.
-For technical information see [[Magic family]].
+For technical information, see [[Magic family #Magic]]. For a discussion on alternative 11- and 13-limit extensions, see [[Magic extensions]].
 == Interval chain ==
 {| class="wikitable center-all"
 |-
-! Cents{{asterisk}}
+! Cents*
 | 0
 | 380.352
@@ Line 67: / Line 65: @@
 | (12/11)
 |}
-<nowiki />* In 7-limit [[POTE tuning]]
+<nowiki/>* In 7-limit [[POTE tuning]]
 The generator chain val for 13-limit magic is {{val| 0 5 1 12 -8 18 }}, so that five generators give an approximate 3, twelve 14, minus eight 11/64, and eighteen 52.
@@ Line 87: / Line 85: @@
 == Scales ==
 ; MOS scales
-* [[Magic7]] &ndash; Improper [[3L 4s]]
+* [[Magic7]] – improper [[3L 4s]]
-* [[Magic10]] &ndash; Improper [[3L 7s]]
+* [[Magic10]] – improper [[3L 7s]]
-* [[Magic13]] &ndash; Improper [[3L 10s]]
+* [[Magic13]] – improper [[3L 10s]]
-* [[Magic16]] &ndash; Improper [[3L 13s]]. The boundary of propriety is 19edo.
+* [[Magic16]] – improper [[3L 13s]]. The boundary of propriety is 19edo.
-* [[Magic19]] &ndash; Proper [[3L 16s]]. The boundary of propriety is 22edo.
+* [[Magic19]] – proper [[3L 16s]]. The boundary of propriety is 22edo.
-* [[Magic22]] &ndash; [[19L 3s]] scale. The boundary of propriety is 41edo.
+* [[Magic22]] – [[19L 3s]] scale. The boundary of propriety is 41edo.
 ; Transversal scales
@@ Line 314: / Line 312: @@
 == Music ==
 ; [[Cameron Bobro]]
-* [https://web.archive.org/web/20201127015243/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Magical_Daydream_CBobro.mp3 ''Magical Daydream''] &ndash; ''A brief demonstration of the near-Just musical temperament which flattens the pure major third of 5:4 by a few cents, such that 5 major thirds does not exceed 3:1 (a pure fifth + 1 octave), but meets it precisely. In a purely tuned system, the thirds would exceed 3:1 by what is known as the small diesis, (a ratio 3125/3072, about thirty cents). This temperament, then, brings (almost) pure thirds and pure fifths together.''
+* [https://web.archive.org/web/20201127015243/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Magical_Daydream_CBobro.mp3 ''Magical Daydream''] – ''A brief demonstration of the near-Just musical temperament which flattens the pure major third of 5:4 by a few cents, such that 5 major thirds does not exceed 3:1 (a pure fifth + 1 octave), but meets it precisely. In a purely tuned system, the thirds would exceed 3:1 by what is known as the small diesis, (a ratio 3125/3072, about thirty cents). This temperament, then, brings (almost) pure thirds and pure fifths together.''
-* [https://web.archive.org/web/20201127014810/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 ''Evening Horizon''] &ndash; ''The earliest implementation (by happy accident, it seems) of this temperament was, to my knowledge, by Paul von Janko over a century ago. More recently, an online tuning community has elaborated many precise variations, calling the temperament "magic". This piece is a demonstration of the array of pitches created by using 22 generators (the slightly tempered 5:4) within the octave, an approach which creates a "moment of symmetry", with all pitches separated by the same two intervals. This has many curious repercussions, creating some musical possibilities and restricting others.''
+* [https://web.archive.org/web/20201127014810/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 ''Evening Horizon''] – ''The earliest implementation (by happy accident, it seems) of this temperament was, to my knowledge, by Paul von Janko over a century ago. More recently, an online tuning community has elaborated many precise variations, calling the temperament "magic". This piece is a demonstration of the array of pitches created by using 22 generators (the slightly tempered 5:4) within the octave, an approach which creates a "moment of symmetry", with all pitches separated by the same two intervals. This has many curious repercussions, creating some musical possibilities and restricting others.''
 ; [[Graham Breed]]
-* [http://x31eq.com/music/dingsheng.mp3 ''Golden Age''] &ndash; Disco involving magic comma pumps.
+* [http://x31eq.com/music/dingsheng.mp3 ''Golden Age''] – disco involving magic comma pumps.
-* [http://x31eq.com/music/dingshi.mp3 ''Extravagant Food''] &ndash; A single magic comma pump in under 60 seconds in 60edo.
+* [http://x31eq.com/music/dingshi.mp3 ''Extravagant Food''] – a single magic comma pump in under 60 seconds in 60edo.
-* [http://x31eq.com/music/jitter.ogg ''Gene's Jitterbug''] &ndash; 9-odd-limit harmony, may not require magic.
+* [http://x31eq.com/music/jitter.ogg ''Gene's Jitterbug''] – 9-odd-limit harmony, may not require magic.
 ; [[Jake Freivald]]
@@ Line 328: / Line 325: @@
 ; [[Andrew Milne]]
-* [https://soundcloud.com/andrew-j-milne/magic-traveller ''Magic Traveller''] (2008) &ndash; Magic[10] with 379.8-cent generator
+* [https://soundcloud.com/andrew-j-milne/magic-traveller ''Magic Traveller''] (2008) – Magic[10] with 379.8-cent generator
 ; [[Chris Vaisvil]] ([http://www.chrisvaisvil.com/ site])
-* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 ''Chromatic piece in magic 16''] (2012) &ndash; Magic[16] in 145edo tuning (&rarr; [[magic16]])
+* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 ''Chromatic piece in magic 16''] (2012) – Magic[16] in 145edo tuning (→ [[magic16]])
-* [http://micro.soonlabel.com/22-ET/daily20120128-pauls-magic.mp3 ''A Piece in Paulsmagic''] (2012) &ndash; In [[paulsmagic]]
+* [http://micro.soonlabel.com/22-ET/daily20120128-pauls-magic.mp3 ''A Piece in Paulsmagic''] (2012) – in [[paulsmagic]]
-* [http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 ''The Magic of Belief''] (2013) &ndash; Magic[19] in 41edo tuning
+* [http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 ''The Magic of Belief''] (2013) – Magic[19] in 41edo tuning
 ; [[User:Xenllium|Xenllium]]
-* [https://youtube.com/watch?v=5iBXKRWE-_U ''Magical life''] (2023) &ndash; In 19-tone pure-fifth magic scale
+* [https://youtube.com/watch?v=5iBXKRWE-_U ''Magical life''] (2023) – Magic[19] in pure-fifth tuning
 == See also ==

Magic: Difference between revisions

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