Magic: Difference between revisions

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{{~~interwiki~~

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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 [[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.)

'''Magic''' is a [[regular temperament|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 interval|prime]] [[7/1|7]], which makes two generators equivalent to [[14/9]] by tempering out [[225/224]]. [[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]], [[63edo]] and [[104edo]].

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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).

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

* 5-limit intervals are simpler than ~~other~~ 7-limit intervals.

* 5-limit intervals are generally simpler than 7-limit intervals.

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 [[Rothenberg propriety|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).

* 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 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 [[telepathy]] temperament). In 22edo they are identical.

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

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

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-{{interwiki
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 | de = Magische Temperaturen
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 {{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 [[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.)
+'''Magic''' is a [[regular temperament|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 interval|prime]] [[7/1|7]], which makes two generators equivalent to [[14/9]] by tempering out [[225/224]]. [[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]], [[63edo]] and [[104edo]].
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 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).
+* It is only slightly more complex than [[meantone]] (both work well with a 19-note gamut).
-* 5-limit intervals are simpler than other 7-limit intervals.
+* 5-limit intervals are generally simpler than 7-limit intervals.
 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 [[Rothenberg propriety|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).
+* 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 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 [[telepathy]] temperament). In 22edo they are identical.
+* [[3L 13s]]: LssssLssssLsssss, where L is a neutral second, which can be taken to represent [[12/11]]~[[13/12]] (in magic temperament) or [[11/10]] (in the related [[telepathy]] temperament). In 22edo they are identical.
 For technical information, see [[Magic family #Magic]]. For a discussion on alternative 11- and 13-limit extensions, see [[Magic extensions]].