Magic - Revision history

Overthink: /* Scales */ 13-limit interpretations

2026-05-27T04:11:17Z

Scales: 13-limit interpretations

← Older revision		Revision as of 04:11, 27 May 2026
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	{{See also\| Magic Tetrachords }}		{{See also\| Magic Tetrachords }}

	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]].		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]], as well as [[33/32]] and [[27/26]] in tridecimal magic.
	* [[3L 4s]]: LsLsLss, where L represents 6/5;		* [[3L 4s]]: LsLsLss, where L represents 6/5;
	* [[3L 7s]]: LssLssLsss, where L represents 7/6;		* [[3L 7s]]: LssLssLsss, where L represents 7/6;

Overthink: /* Interval chain */ more interpretations

2026-05-27T04:09:52Z

Interval chain: more interpretations

← Older revision		Revision as of 04:09, 27 May 2026
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	\| 3		\| 3
	\| 1141.4		\| 1141.4
	\| 27/14		\| 27/14, 35/18, 48/25
	\|-		\|-
	\| 4		\| 4
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	\| 8		\| 8
	\| 643.7		\| 643.7
	\| (13/9, '''16/11''')		\| 35/24, (13/9, '''16/11''')
	\|-		\|-
	\| 9		\| 9
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	\| 13		\| 13
	\| 145.9		\| 145.9
	\| (12/11, 13/12)		\| 35/32, (12/11, 13/12)
	\|}		\|}
	<nowiki/>* In 7-limit CWE tuning		<nowiki/>* In 7-limit CWE tuning

Hkm at 04:12, 26 May 2026

2026-05-26T04:12:45Z

← Older revision		Revision as of 04:12, 26 May 2026
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	Magic has certain properties that commend it as a step up in complexity from traditional harmony:		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 the simplest mapping capable of tuning every [[9-odd-limit]] interval better than in [[12edo]].
			* It is only slightly more complex than [[septimal meantone]] (both work well with a 19-note gamut).
	* 5-limit intervals are generally simpler than 7-limit intervals.		* 5-limit intervals are generally simpler than 7-limit intervals.

Hkm at 04:09, 26 May 2026

2026-05-26T04:09:34Z

← Older revision		Revision as of 04:09, 26 May 2026
Line 28:		Line 28:
	Magic has certain properties that commend it as a step up in complexity from traditional harmony:		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 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).
	* 5-limit intervals are generally simpler than 7-limit intervals.		* 5-limit intervals are generally simpler than 7-limit intervals.

FloraC: /* Tunings */ rephrasing. + missing 2/11-comma mark

2026-04-23T05:13:35Z

Tunings: rephrasing. + missing 2/11-comma mark

← Older revision		Revision as of 05:13, 23 April 2026
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	! Comments		! Comments
	\|-		\|-
	\|[[16edo\|5\16]]		\| [[16edo\|5\16]]
	\|		\|
	\|375.000		\| 375.000
	\|~~Major thirds slightly flatter than this fall under~~ [[submerged]]		\| Flatter tunings may be analysed as [[submerged]]
	\|-		\|-
	\|		\|
	\|[[25/24]]		\| [[25/24]]
	\|376.443		\| 376.443
	\|1/3-comma		\| 1/3-comma
	\|-		\|-
	\|		\|
	\|[[125/72]]		\| [[125/72]]
	\|377.853		\| 377.853
	\|2/7-comma		\| 2/7-comma
	\|-		\|-
	\|		\|
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	\|		\|
	\|-		\|-
	\|		\|
	\|[[45/32]]		\| [[45/32]]
	\|380.929		\| 380.929
	\|		\| 2/11-comma
	\|-		\|-

	\| [[63edo\|20\63]]		\| [[63edo\|20\63]]
	\|		\|
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	\|-		\|-
	\|		\|
	\|[[75/64]]		\| [[75/64]]
	\|382.083		\| 382.083
	\|1/7-comma		\| 1/7-comma
	\|-		\|-
	\|		\|
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	\|		\|
	\|-		\|-
	\|[[25edo\|8\25]]		\| [[25edo\|8\25]]
	\|		\|
	\|384.000		\| 384.000
	\|		\| Sharper tunings may be analysed as [[anthoine]]
	\|-		\|-
	\|		\|
	\| [[5/4]]		\| [[5/4]]
	\| 386.314		\| 386.314
	\| Untempered~~, major thirds slightly sharper than 5/4 fall under 25&28 or [[würschmidt]]~~		\| Untempered
	\|}		\|}
	<nowiki/> * Besides the octave		<nowiki/> * Besides the octave

FloraC: /* Scales */ the n2 is actually 11/10~14/13 in telepathy, which is hopeless.

2026-04-18T08:31:58Z

Scales: the n2 is actually 11/10~14/13 in telepathy, which is hopeless.

← Older revision		Revision as of 08:31, 18 April 2026
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	* [[3L 7s]]: LssLssLsss, where L represents 7/6;		* [[3L 7s]]: LssLssLsss, where L represents 7/6;
	* [[3L 10s]]: LsssLsssLssss, where L represents 9/8;		* [[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]]~[[13/12]] (in tridecimal magic~~) or [[11/10]] (in the related [[telepathy]] temperament). In 22edo they are identical~~.		* [[3L 13s]]: LssssLssssLsssss, where L represents [[12/11]]~[[13/12]] in tridecimal magic.

	=== Scala files ===		=== Scala files ===

FloraC: Move scale analysis to the proper section

2026-04-18T08:28:47Z

Move scale analysis to the proper section

← Older revision		Revision as of 08:28, 18 April 2026
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	* 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.		* 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]]~[[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]].		For technical information, see [[Magic family #Magic]]. For a discussion on alternative 11- and 13-limit extensions, see [[Magic extensions]].
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	{{See also\| Magic Tetrachords }}		{{See also\| Magic Tetrachords }}

			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]]~[[13/12]] (in tridecimal magic) or [[11/10]] (in the related [[telepathy]] temperament). In 22edo they are identical.

			=== Scala files ===
	; Mos 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]]. The boundary of propriety is 41edo.

	; Transversal scales		; Transversal scales

FloraC: Misc. improvements in the intro

2026-04-02T09:41:33Z

Misc. improvements in the intro

FloraC: Cleanup on infobox

2026-02-09T09:23:07Z

Cleanup on infobox

← Older revision		Revision as of 09:23, 9 February 2026
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	{{interwiki		{{interwiki
			\| en = Magic
	\| de = Magische Temperaturen		\| de = Magische Temperaturen
	~~\| en = Magic~~
	\| es =		\| es =
	\| ja = マジック		\| ja = マジック
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	\| Title = Magic		\| Title = Magic
	\| Subgroups = 2.3.5, 2.3.5.7		\| Subgroups = 2.3.5, 2.3.5.7
	\| Comma basis = [[3125/3072]] (~~2.3.~~5); <br>[[225/224]], [[245/243]] (~~2.3.5.~~7)		\| Comma basis = [[3125/3072]] (5-limit); <br>[[225/224]], [[245/243]] (7-limit)
	\| ~~Generator~~ = ~~5/4~~		\| Edo join 1 = 19 \| Edo join 2 = 22
	\| Mapping = 1; 5 1 12		\| Mapping = 1; 5 1 12
			\| Generators = 5/4
			\| Generators tuning = 380.5
			\| Optimization method = CWE
	\| Pergen = (P8, P12/5)		\| Pergen = (P8, P12/5)
	\| Color name = Laquinyoti		\| Color name = Laquinyoti
	~~\| Edo join 1 = 19 \| Edo join 2 = 22~~
	~~\| Optimization method = CTE~~
	~~\| Generator tuning = 380.7~~
	\| MOS scales = [[3L 4s]], [[3L 7s]], …, [[3L 16s]], [[19L 3s]]		\| MOS scales = [[3L 4s]], [[3L 7s]], …, [[3L 16s]], [[19L 3s]]
	\| Odd limit 1 = 5 \| Mistuning 1 = 5.9 \| Complexity 1 = 7		\| Odd limit 1 = 5 \| Mistuning 1 = 5.9 \| Complexity 1 = 7

FloraC: /* Chords and harmony */ chords of magic isn't the main article of this section (and therefore not a subpage)

2026-02-04T12:17:28Z

Chords and harmony: chords of magic isn't the main article of this section (and therefore not a subpage)

← Older revision		Revision as of 12:17, 4 February 2026
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	== Chords and harmony ==		== Chords and harmony ==
	{{~~Main~~\| Chords of magic }}		{{See also\| Chords of magic \| Functional harmony in rank-2 temperaments }}
	~~{{See also~~\| Functional harmony in rank-2 temperaments }}

	The fundamental otonal consonance of magic, voiced in a roughly {{w\|tertian harmony\|tertian}} manner, is 4:5:6:7:9, available in a 13-tone mos. To start with, consider the the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2). In terms of generator steps, they are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of [[meantone]]. Two approaches to functional harmony thus arise.		The fundamental otonal consonance of magic, voiced in a roughly {{w\|tertian harmony\|tertian}} manner, is 4:5:6:7:9, available in a 13-tone mos. To start with, consider the the just major triad (1–5/4–3/2) and just minor triad (1–6/5–3/2). In terms of generator steps, they are 0–1–5 and 0–4–5; this is similar, but also in clear contrast to the 0–4–1 and 0–(−3)–1 of [[meantone]]. Two approaches to functional harmony thus arise.

← Older revision		Revision as of 09:41, 2 April 2026
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	{{~~interwiki~~		{{Interwiki
	\| en = Magic		\| en = Magic
	\| de = Magische Temperaturen		\| de = Magische Temperaturen
Line 22:		Line 22:
	{{Wikipedia\| Magic temperament }}		{{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]].		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:		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 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 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.		* 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]].		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 4s]]: LsLsLss, where L represents 6/5
	* [[3L 7s]]: LssLssLsss, where L represents 7/6		* [[3L 7s]]: LssLssLsss, where L represents 7/6
	* [[3L 10s]]: LsssLsssLssss, where L represents 9/8		* [[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]].		For technical information, see [[Magic family #Magic]]. For a discussion on alternative 11- and 13-limit extensions, see [[Magic extensions]].