Magic

Magic

Subgroups	2.3.5, 2.3.5.7
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
Comma basis	3125/3072 (5-limit); 225/224, 245/243 (7-limit)
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

Magic (19 & 22) is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5. It is a 3-cluster temperament, as indicated by the edo join (22 - 19 = 3).

Interval chain

In the following table, odd harmonics 1–15 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, 31/16
4	321.8	6/5, 29/24
5	702.3	3/2
6	1082.7	15/8, 28/15
7	263.2	7/6
8	643.7	36/25
9	1024.1	9/5, 29/16
10	204.6	9/8
11	585.0	7/5
12	965.5	7/4
13	145.9	35/32

* In 7-limit CWE tuning

Extensions

Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 = 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.

This can be seen from the following S-expression for the Magic comma:

3125/3072
= (25/24)^2/(16/15)
= (25/24)*(25/24)/(32/31*31/30)
= (25/24)S25*S26*S27*S28*S29*S30/(32/31)
= (S25*S26*S27*S28*S29*S30)^2*S31
= (S15*S28*S29*S30)^2*S31

structurally inducing the above 2.3.5.7.29.31 extension.

Scales

This page or section is a work in progress. It may lack sufficient justification, content, or organization, and is subject to future overhaul.

MOS scales

Because three generators just barely fall short of an octave, the MOS scales generated by Magic are quite commatic. The Magic Diesis (a small step simultaneously representing 25/24, 28/27, 36/35, and 128/125) is the s step of all MOS scales up to the 22-form

7-form

An example of the Magic Cadence (III - I) in 22edo, written in SATB format with native fifths notation.

The 7-note Magic scale has the pattern 3L 4s, sometimes called Mosh, with a large step of 6/5. The disparity in size between the two types of steps grants a quality to stepwise melodies that some find to sound awkward or lurching; while some composers may prefer or intend such a sound, those who do not are cautioned to avoid long stepwise runs for fear of creating an aimless and chromatic-sounding melodic core.

The 4:5:6 triad can be found on the tonic in two of the modes: LsLsLss and LsLssLs; these modes place the triad on degrees 1, 3, and 4. Generalizing from this, we can see that the other main type of triad that occurs on these degrees is 1/1 - 5/4 - 9/7, with an additional 1/1 - 16/15 - 9/7 triad in the ssLsLsL mode.

The LsLsLss mode has the clearest utility, with a clear cadence from the III chord to the I chord; we may consider this motion to be Magic temperament's analog to the Perfect Cadence from Meantone[7]. This cadence creates contrary motion, with the 6 and 5 of the III chord resolving respectively to the 8 and 4 of the I chord, and the 3 sustained between both chords creates a strong sense of continuity.

The LsLssLs mode contains an inverse version of this cadence, which can be seen as a Magic analog to Meantone's Plagal Cadence.

Other scale forms

Due to the juxtaposition of the Magic Diesis against significantly larger step sizes, it is often desirable to use a non-MOS structure in Magic, such as Blackdye or the 5-odd Diamond.

List of tunings

EDO patent vals

The following patent vals support 2.3.5 Magic. Vals that are contorted in 2.3.5 are not included.

Edo	Extension to 7	Generator	Fifth
3		400.000	800.000
25		384.000	720.000
22	19 & 22	381.818	709.091
107		381.308	706.542
85	19 & 22	381.176	705.882
63	19 & 22	380.952	704.762
104		380.769	703.846
41	19 & 22	380.488	702.439
60	19 & 22	380.000	700.000
79		379.747	698.734
19	19 & 22	378.947	694.737
35		377.143	685.714
16		375.000	675.000

Target-tunings

The following are some examples of useful target tunings available in Magic temperament.

Just interval	Comma fraction	Generator	Fifth
5/4	0-comma	386.314	731.569
15/8	1/6-comma	381.378	706.891
3/2	1/5-comma	380.391	701.955
9/5	2/9-comma	379.733	698.665
5/3	1/4-comma	378.910	1694.552
25/24	1/3-comma	376.443	682.213