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 1–13 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	36/25
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

* 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.

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.