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<span style="display: block; text-align: right;">Other languages: [[:de:Magische_Temperaturen-x-7-Limit-magisch|Deutsch]]</span>
{{Interwiki
| en = Magic
| de = Magische Temperaturen
| es =
| ja = マジック
}}
{{Infobox regtemp
| Title = Magic
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[3125/3072]] (5-limit); <br>[[225/224]], [[245/243]] (7-limit)
| Edo join 1 = 19 | Edo join 2 = 22
| Mapping = 1; 5 1 12
| Generators = 5/4
| Generators tuning = 380.5
| Optimization method = CWE
| Pergen = (P8, P12/5)
| Color name = Laquinyoti
| MOS scales = [[3L 4s]], [[3L 7s]], …, [[3L 16s]], [[19L 3s]]
| Odd limit 1 = 5 | Mistuning 1 = 5.9 | Complexity 1 = 7
| Odd limit 2 = 9 | Mistuning 2 = 5.9 | Complexity 2 = 13
}}
{{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|magic comma]] 3125/3072 is tempered out, making it a member of the [[Magic_family|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]], but there's basically no reason to use it unless you're using [[19edo|19edo]], in which case it's 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|19edo]], [[22edo|22edo]], [[41edo|41edo]], [[60edo|60edo]] and [[104edo|104edo]].
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:
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.


<ul><li>Every non-trivial 7-limit interval is better tuned than in [[12edo|12edo]].</li><li>It is the simplest mapping with the above property.</li><li>It is only slightly more complex than meantone (both work well with a 19 note gamut).</li><li>5-limit intervals are simpler than other 7-limit intervals.</li></ul>
It is not a panacea because:
* 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]]).
* 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.


It fails to be a panacea because:
For technical information, see [[Magic family #Magic]]. For a discussion on alternative 11- and 13-limit extensions, see [[Magic extensions]].


<ul><li>It has no proper MOS scales of between 3 and 16 notes.</li><li>It is more complex than meantone</li><li>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.</li></ul>
== Interval chain ==
In the following table, odd harmonics 1–13 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! #
! 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)
|}
<nowiki/>* In 7-limit CWE 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.
 
== Chords and harmony ==
{{See also| Chords of magic | 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.
 
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.


Because the generator is so close to 1\3 of an octave, and the interval left over (which represents both 128/125 and 25/24) 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 of 128/125~25/24.
=== Scala files ===
; Mos scales
* [[Magic7]] – improper 3L 4s
* [[Magic10]] – improper 3L 7s
* [[Magic13]] – improper 3L 10s
* [[Magic16]] – improper 3L 13s. The boundary of propriety is 19edo.
* [[Magic19]] – proper [[3L 16s]]. The boundary of propriety is 22edo.
* [[Magic22]] – [[19L 3s]]. The boundary of propriety is 41edo.


<ul><li>[[3L_4s|3L 4s]]: LsLsLss where L = 6/5</li><li>[[3L_7s|3L 7s]]: LssLssLsss where L = 7/6</li><li>[[3L_10s|3L 10s]]: LsssLsssLssss where L = 9/8</li><li>[[3L_13s|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#Magic-Telepathy|telepathy]] temperament). In 22edo they are identical.</li></ul>
; Transversal scales
* [[Magic19trans37]]
* [[Magic19trans37ex]]
* [[Magic22trans37]]
* [[Magic22trans37ex]]


==Interval chain==
; Others
* [[Paulsmagic]]


{| class="wikitable"
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
|-
| | 0.
! Constrained
| | 380.352
! Constrained & skewed
| | 760.704
! Destretched
| | 1141.056
| | 321.408
| | 701.76
| | 1082.112
| | 262.464
| | 642.816
| | 1023.168
| | 203.52
| | 583.872
| | 964.224
| | 144.576
|-
|-
| | 1/1
! Tenney
| | 5/4
| CTE: ~5/4 = 380.4994{{c}}
| | 14/9
| CWE: ~5/4 = 380.2194{{c}}
| | 48/25~125/64
| POTE: ~5/4 = 380.0585{{c}}
| | 6/5
| | 3/2
| | 15/8
| | 7/6
| | (16/11)
| | 9/5
| | 9/8
| | 7/5
| | 7/4
| | (12/11)
|}
|}


The generator chain val for 13-limit magic is &lt;0 5 1 12 -8 18|, so that five generators give an approximate 3, twelve 14, minus eight 11/64, and eighteen 52.
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~5/4 = 380.6512{{c}}
| CWE: ~5/4 = 380.4576{{c}}
| POTE: ~5/4 = 380.3520{{c}}
|}


=Spectrum of Magic Tunings by Eigenmonzos=
=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~5/4 = 380.391{{c}}
| 3/2
| ~5/4 = 379.968{{c}}
| {{Monzo| 0 3 -1 }}
|-
| 7-odd-limit
| ~5/4 = 380.391{{c}}
| 3/2
| ~5/4 = 380.506{{c}}
| {{Monzo| 0 1 -7 15 }}
|-
| 9-odd-limit
| ~5/4 = 380.391{{c}}
| 3/2
| ~5/4 = 380.384{{c}}
| {{Monzo| 0 36 -23 32 }}
|}
 
=== Tuning spectrum ===
This tuning spectrum reflects 7-limit magic; for the tuning spectrum of tridecimal magic, see [[Magic extensions#Magic|the page on magic extensions]].


{| class="wikitable"
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| [[16edo|5\16]]
|
| 375.000
| Flatter tunings may be analysed as [[submerged]]
|-
|
| [[25/24]]
| 376.443
| 1/3-comma
|-
|-
! | Eigenmonzo
|
! | Major Third
| [[125/72]]
| 377.853
| 2/7-comma
|-
|-
| | 6/5
|  
| | 378.910
| [[5/3]]
| 378.910
| 1/4-comma
|-
|-
| | 10/9
| '''[[19edo|6\19]]'''
| | 379.733
|  
| '''378.947'''
| '''Lower bound of 9- and 15-odd-limit diamond monotone'''
|-
|-
| | 7/5
|  
| | 380.228
| [[9/5]]
| 379.733
| 2/9-comma
|-
|-
| | 4/3
| [[60edo|19\60]]
| | 380.391 (5, 7 and 9 limit minimax)
|  
| 380.000
|
|-
|-
| | 11/9
|  
| | 380.700 (11 limit minimax)
| [[15/14]]
| 380.093
|
|-
|-
| | 8/7
| [[101edo|32\101]]
| | 380.735
|  
| 380.198
| 101cd val
|-
|-
| | 12/11
|  
| | 380.818
| [[7/5]]
| 380.228
|
|-
|-
| | 14/11
|  
| | 380.875
| [[21/20]]
| 380.279
|
|-
|-
| | 7/6
|  
| | 380.982
| [[3/2]]
| 380.391
| 5-, 7- and 9-odd-limit minimax, 1/5-comma
|-
|-
| | 11/8
| [[41edo|13\41]]
| | 381.085
|  
| 380.488
|
|-
|-
| | 11/10
|  
| | 381.666
| [[21/16]]
| 380.634
|  
|-
|-
| | 9/7
|  
| | 382.458
| [[7/4]]
| 380.735
|  
|-
|-
| | 5/4
| [[104edo|33\104]]
| | 386.314
|
| 380.769
|
|-
|
| [[45/32]]
| 380.929
| 2/11-comma
|-
| [[63edo|20\63]]
|
| 380.952
|
|-
|
| [[7/6]]
| 380.982
|
|-
|
| [[15/8]]
| 381.378
| 1/6-comma
|-
| '''[[22edo|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
|
|-
| [[25edo|8\25]]
|
| 384.000
| Sharper tunings may be analysed as [[anthoine]]
|-
|
| [[5/4]]
| 386.314
| Untempered
|}
|}
<nowiki/> * Besides the octave


=[[Chords_of_magic|Chords of magic]]=
== Music ==
 
; [[Cameron Bobro]]
=[[Magic_Tetrachords|Magic Tetrachords]]=
* [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.''
=Music=
''[http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 Chromatic piece in magic 16]''
 
[[magic16|magic16]]
 
''[http://micro.soonlabel.com/22-ET/daily20120128-pauls-magic.mp3 A Piece in Paulsmagic]''
 
[[paulsmagic|paulsmagic]]
 
''[http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 The Magic of Belief]'' Magic[19] in 41et tuning
 
[http://www.chrisvaisvil.com/ Chris Vaisvil]
 
''[https://soundcloud.com/jdfreivald/little-magical-object Little Magical Object]'' [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/little-magical-object.mp3 play] Magic[19] in 41et tuning by [[Jake_Freivald|Jake Freivald]]


''[http://micro.soonlabel.com/gene_ward_smith/Others/Milne/Magic%20Traveller.mp3 Andrew Milne; magic with 379.8 cent generator]''
; [[Graham Breed]]
* [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/jitter.ogg ''Gene's Jitterbug''] – 9-odd-limit harmony, may not require magic.


''[http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Magical_Daydream_CBobro.mp3 Magical Daydream]''
; [[Jake Freivald]]
* ''Little Magical Object'' (2013) &ndash; [https://web.archive.org/web/20201127014655/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/little-magical-object.mp3 play] | [https://soundcloud.com/jdfreivald/little-magical-object SoundCloud] – Magic[19] in 41edo tuning


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


''[http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 Evening Horizon]''
; [[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/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


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


''[http://x31eq.com/music/dingsheng.mp3 Golden Age] disco involving magic comma pumps.''
== See also ==
* [[Devadoot]] – 5/4-equivalent or twelfth-equivalent magic
* [[Kite Guitar]]
* [[Lumatone mapping for magic]]
* [[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


''[http://x31eq.com/music/dingshi.mp3 Extravagant Food] a single magic comma pump in under 60 seconds in 60-equal.''
== External links ==
* [http://x31eq.com/magic/ Magic Temperament] &ndash; Graham Breed's documents


''[http://x31eq.com/music/jitter.ogg Gene's Jitterbug] 9-limit harmony, may not require magic.''
[[Category:Magic| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Magic family]]
[[Category:Marvel temperaments]]
[[Category:Sensamagic clan]]
[[Category:Keemic temperaments]]
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