Magic: Difference between revisions

| ja = マジック

| Subgroups = 2.3.5, 2.3.5.7

| Comma basis = [[3125/3072]] (5-limit); <br>[[225/224]], [[245/243]] (7-limit)

| Mapping = 1; 5 1 12

| Generators = 5/4

| Generators tuning = 380.5

| Pergen = (P8, P12/5)

| MOS scales = [[3L 4s]], [[3L 7s]], …, [[3L 16s]], [[19L 3s]]

| Odd limit 2 = 9 | Mistuning 2 = 5.9 | Complexity 2 = 13

'''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]].

* 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.

* 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 ==

{| class="wikitable center-1 right-2"

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.

{{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 ==

* [[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.

 

; Mos scales

 

; Transversal scales

 

; Others

 

=== 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

! Constrained & skewed

! Destretched

|-

| CTE: ~5/4 = 380.4994{{c}}

| CWE: ~5/4 = 380.2194{{c}}

| POTE: ~5/4 = 380.0585{{c}}

 

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

! Constrained & skewed

! Destretched

|-

| CTE: ~5/4 = 380.6512{{c}}

| CWE: ~5/4 = 380.4576{{c}}

| POTE: ~5/4 = 380.3520{{c}}

 

{| 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

| ~5/4 = 380.391{{c}}

| 3/2

| ~5/4 = 379.968{{c}}

| ~5/4 = 380.391{{c}}

| ~5/4 = 380.506{{c}}

|-

| 9-odd-limit

| ~5/4 = 380.391{{c}}

| 3/2

| ~5/4 = 380.384{{c}}

 

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 center-all left-4"

| [[25/24]]

| 1/3-comma

| [[125/72]]

| 2/7-comma

| [[5/3]]

| 378.910

| 1/4-comma

| '''378.947'''

| '''Lower bound of 9- and 15-odd-limit diamond monotone'''

| [[9/5]]

| 2/9-comma

| [[60edo|19\60]]

| [[15/14]]

| 380.093

| 380.198

| [[7/5]]

| [[21/20]]

| [[3/2]]

| 5-, 7- and 9-odd-limit minimax, 1/5-comma

| [[21/16]]

| 380.634

| [[7/4]]

|-

|-

| [[45/32]]

| 2/11-comma

| [[7/6]]

| [[15/8]]

| 381.378

| '''381.818'''

| [[75/64]]

| 1/7-comma

| [[9/7]]

|-

|-

| 386.314

 

== 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/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/dingshi.mp3 ''Extravagant Food''] – a single magic comma pump in under 60 seconds in 60edo.

 

* ''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

 

; [[Andrew Milne]]

* [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])

 

; [[User:Xenllium|Xenllium]]

 

* [[5edt]], an equal tuning in which a stack of five ~5/4's is exactly 3/1

 

* [http://x31eq.com/magic/ Magic Temperament] &ndash; Graham Breed's documents

 

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