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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 = 13
| Odd limit 1 = 5 | Mistuning 1 = 5.9 | Complexity 1 = 7
| Odd limit 2 = 9 | Mistuning 2 = 5.9 | Complexity 2 = 41
| Odd limit 2 = 9 | Mistuning 2 = 5.9 | Complexity 2 = 13
}}
}}
{{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 [[septimal 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]].
* [[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.


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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| 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
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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.  
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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]], as well as [[33/32]] and [[27/26]] in tridecimal magic.
* [[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 represents [[12/11]]~[[13/12]] in tridecimal magic.
=== 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
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! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
| [[16edo|5\16]]
|
| 375.000
| Flatter tunings may be analysed as [[submerged]]
|-
|
| [[25/24]]
| 376.443
| 1/3-comma
|-
|
| [[125/72]]
| 377.853
| 2/7-comma
|-
|-
|  
|  
| 5/3
| [[5/3]]
| 378.910
| 378.910
|  
| 1/4-comma
|-
|-
| '''6\19'''
| '''[[19edo|6\19]]'''
|  
|  
| '''378.947'''
| '''378.947'''
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|-
|-
|  
|  
| 9/5
| [[9/5]]
| 379.733
| 379.733
|  
| 2/9-comma
|-
|-
| 19\60
| [[60edo|19\60]]
|  
|  
| 380.000
| 380.000
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|-
|-
|  
|  
| 15/14
| [[15/14]]
| 380.093
| 380.093
|  
|  
|-
|-
| 32\101
| [[101edo|32\101]]
|  
|  
| 380.198
| 380.198
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|-
|-
|  
|  
| 7/5
| [[7/5]]
| 380.228
| 380.228
|  
|  
|-
|-
|  
|  
| 21/20
| [[21/20]]
| 380.279
| 380.279
|  
|  
|-
|-
|  
|  
| 3/2
| [[3/2]]
| 380.391
| 380.391
| 5-, 7- and 9-odd-limit minimax
| 5-, 7- and 9-odd-limit minimax, 1/5-comma
|-
|-
| 13\41
| [[41edo|13\41]]
|  
|  
| 380.488
| 380.488
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|-
|-
|  
|  
| 21/16
| [[21/16]]
| 380.634
| 380.634
|  
|  
|-
|-
|  
|  
| 7/4
| [[7/4]]
| 380.735
| 380.735
|  
|  
|-
|-
| 33\104
| [[104edo|33\104]]
|  
|  
| 380.769
| 380.769
|  
|  
|-
|-
| 20\63
|
| [[45/32]]
| 380.929
| 2/11-comma
|-
| [[63edo|20\63]]
|  
|  
| 380.952
| 380.952
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|-
|-
|  
|  
| 7/6
| [[7/6]]
| 380.982
| 380.982
|  
|  
|-
|-
|  
|  
| 15/8
| [[15/8]]
| 381.378
| 381.378
|  
| 1/6-comma
|-
|-
| '''7\22'''
| '''[[22edo|7\22]]'''
|  
|  
| '''381.818'''
| '''381.818'''
| '''Upper bound of 9- and 15-odd-limit diamond monotone'''
| '''Upper bound of 9- and 15-odd-limit diamond monotone'''
|-
|
| [[75/64]]
| 382.083
| 1/7-comma
|-
|-
|  
|  
| 9/7
| [[9/7]]
| 382.458
| 382.458
|  
|  
|-
| [[25edo|8\25]]
|
| 384.000
| Sharper tunings may be analysed as [[anthoine]]
|-
|-
|  
|  
| 5/4
| [[5/4]]
| 386.314
| 386.314
|  
| Untempered
|}
|}
<nowiki/> * Besides the octave
<nowiki/> * Besides the octave
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* [[Lumatone mapping for magic]]
* [[Lumatone mapping for magic]]
* [[5edt]], an equal tuning in which a stack of five ~5/4's is exactly 3/1
* [[5edt]], an equal tuning in which a stack of five ~5/4's is exactly 3/1
* [[Marvel–sensamagic equivalence continuum]] – equivalence continuum of septimal magic


== External links ==
== External links ==

Latest revision as of 04:11, 27 May 2026

Magic
Subgroups 2.3.5, 2.3.5.7
Comma basis 3125/3072 (5-limit);
225/224, 245/243 (7-limit)
Reduced mapping ⟨1; 5 1 12]
ET join 19 & 22
Generators (CWE) ~5/4 = 380.5 ¢
MOS scales 3L 4s, 3L 7s, …, 3L 16s, 19L 3s
Ploidacot alpha-pentacot
Pergen (P8, P12/5)
Color name Laquinyoti
Minimax error 5-odd-limit: 5.9 ¢;
9-odd-limit: 5.9 ¢
Target scale size 5-odd-limit: 7 notes;
9-odd-limit: 13 notes
English Wikipedia has an article on:

Magic is a 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 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 scales 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.

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 septimal meantone (both work well with a 19-note gamut).
  • 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 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.

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

Interval chain

In the following table, odd harmonics 1–13 and their inverses are in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 380.5 5/4
2 760.9 14/9
3 1141.4 27/14, 35/18, 48/25
4 321.8 6/5
5 702.3 3/2
6 1082.7 15/8, 28/15
7 263.2 7/6
8 643.7 35/24, (13/9, 16/11)
9 1024.1 9/5
10 204.6 9/8
11 585.0 7/5
12 965.5 7/4
13 145.9 35/32, (12/11, 13/12)

* In 7-limit CWE tuning

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

Chords and harmony

See also: Chords of magic and Functional harmony in rank-2 temperaments

The fundamental otonal consonance of magic, voiced in a roughly 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.

First, we can use the triads above as the basis of harmony, but swapping the roles of 3 and 5 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is 5/4 over tonic; a "subdominant" chord is 5/4 under tonic. This leads to an approach closely adherent to mos scales. The 7-tone mos contains a tonic and a "dominant" chord. The 10-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions. The suspended chord of meantone is made of two generators stacked, and doing the same in magic, we also have the augmented triad (1–5/4–14/9) as the magic analog of the suspended chord of meantone.

Second, we can use the same triads as the basis of harmony, and keeping the role of the chain of fifths as the spine on which the functions are defined. This means dominant is still 3/2 over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. This is essentially working in JI, but using the commas tempered out in some way to lock into the identity of the temperament.

Scales

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, as well as 33/32 and 27/26 in tridecimal magic.

  • 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 represents 12/11~13/12 in tridecimal magic.

Scala files

Mos scales
Transversal scales
Others

Tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 380.4994 ¢ CWE: ~5/4 = 380.2194 ¢ POTE: ~5/4 = 380.0585 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 380.6512 ¢ CWE: ~5/4 = 380.4576 ¢ POTE: ~5/4 = 380.3520 ¢

Target tunings

Target tunings
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
5-odd-limit ~5/4 = 380.391 ¢ 3/2 ~5/4 = 379.968 ¢ [0 3 -1⟩
7-odd-limit ~5/4 = 380.391 ¢ 3/2 ~5/4 = 380.506 ¢ [0 1 -7 15⟩
9-odd-limit ~5/4 = 380.391 ¢ 3/2 ~5/4 = 380.384 ¢ [0 36 -23 32⟩

Tuning spectrum

This tuning spectrum reflects 7-limit magic; for the tuning spectrum of tridecimal magic, see the page on magic extensions.

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
5\16 375.000 Flatter tunings may be analysed as submerged
25/24 376.443 1/3-comma
125/72 377.853 2/7-comma
5/3 378.910 1/4-comma
6\19 378.947 Lower bound of 9- and 15-odd-limit diamond monotone
9/5 379.733 2/9-comma
19\60 380.000
15/14 380.093
32\101 380.198 101cd val
7/5 380.228
21/20 380.279
3/2 380.391 5-, 7- and 9-odd-limit minimax, 1/5-comma
13\41 380.488
21/16 380.634
7/4 380.735
33\104 380.769
45/32 380.929 2/11-comma
20\63 380.952
7/6 380.982
15/8 381.378 1/6-comma
7\22 381.818 Upper bound of 9- and 15-odd-limit diamond monotone
75/64 382.083 1/7-comma
9/7 382.458
8\25 384.000 Sharper tunings may be analysed as anthoine
5/4 386.314 Untempered

* Besides the octave

Music

Cameron Bobro
  • 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.
  • 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
Jake Freivald
  • Little Magical Object (2013) – play | SoundCloud – Magic[19] in 41edo tuning
Andrew Milne
Chris Vaisvil (site)
Xenllium

See also

External links

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