Magic

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 temperament

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

• 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.

Transversal scales

• Magic19trans37
• Magic19trans37ex
• Magic22trans37
• Magic22trans37ex

Others

• Paulsmagic

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

• Golden Age – disco involving magic comma pumps.
• Extravagant Food – a single magic comma pump in under 60 seconds in 60edo.
• Gene's Jitterbug – 9-odd-limit harmony, may not require magic.

Jake Freivald

• Little Magical Object (2013) – play | SoundCloud – Magic[19] in 41edo tuning

Andrew Milne

• Magic Traveller (2008) – Magic[10] with 379.8-cent generator

Chris Vaisvil (site)

• Chromatic piece in magic 16 (2012) – Magic[16] in 145edo tuning (→ magic16)
• A Piece in Paulsmagic (2012) – in paulsmagic
• The Magic of Belief (2013) – Magic[19] in 41edo tuning

Xenllium

• Magical life (2023) – Magic[19] in pure-fifth tuning

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

External links

• Magic Temperament – Graham Breed's documents