Magic
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 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 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 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 meantone (both work well with a 19 note gamut).
- 5-limit intervals are simpler than other 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.
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.
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 |
| 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 | (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 | (12/11) |
* 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
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
- 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 scale. The boundary of propriety is 41edo.
- Transversal scales
- Others
Tunings
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 21/13 | 369.747 | ||
| 13/7 | 378.617 | ||
| 5/3 | 378.910 | ||
| 6\19 | 378.947 | Lower bound of 9- to 15-odd-limit diamond monotone | |
| 15/13 | 379.355 | ||
| 13/9 | 379.577 | ||
| 13/10 | 379.660 | ||
| 9/5 | 379.733 | ||
| 13/12 | 379.890 | ||
| 27/20 | 379.968 | 5-odd-limit least squares | |
| 19\60 | 380.000 | ||
| 13/8 | 380.029 | ||
| 15/14 | 380.093 | ||
| 32\101 | 380.198 | 101cde val | |
| 7/5 | 380.228 | ||
| 21/20 | 380.279 | ||
| 13/11 | 380.354 | 13- and 15-odd-limit minimax | |
| [0 56 -31 46 -94 88⟩ | 380.377 | 13-odd-limit least squares | |
| [0 36 -23 32⟩ | 380.384 | 9-odd-limit least squares | |
| [0 58 -29 52 -108 100⟩ | 380.389 | 15-odd-limit least squares | |
| 3/2 | 380.391 | 5-, 7- and 9-odd-limit minimax | |
| 13\41 | 380.488 | ||
| [0 1 -7 15⟩ | 380.506 | 7-odd-limit least squares | |
| 21/16 | 380.634 | ||
| 11/9 | 380.700 | 11-odd-limit minimax | |
| [0 85 -14 52 -68⟩ | 380.714 | 11-odd-limit least squares | |
| 7/4 | 380.735 | ||
| 33\104 | 380.769 | 104ff val | |
| 21/11 | 380.779 | ||
| 11/6 | 380.818 | ||
| 11/7 | 380.875 | ||
| 20\63 | 380.952 | 63f val | |
| 7/6 | 380.982 | ||
| 11/8 | 381.085 | ||
| 15/11 | 381.211 | ||
| 15/8 | 381.378 | ||
| 11/10 | 381.666 | ||
| 7\22 | 381.818 | 22f val, upper bound of 9- to 15-odd-limit diamond monotone | |
| 9/7 | 382.458 | ||
| 5/4 | 386.314 |
Music
- 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.
- 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.
- Little Magical Object (2013) – play | SoundCloud – Magic[19] in 41edo tuning
- Magic Traveller (2008) – Magic[10] with 379.8-cent generator
- 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
- 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
External links
- Magic Temperament – Graham Breed's documents