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.

List of patent vals

The following patent vals support 2.3.5.7 Magic. Contorted vals are not included.

Edo	Generator	Fifth
22	381.818	709.091
85	381.176	705.882
63	380.952	704.762
104	380.769	703.846
41	380.488	702.439
60	380.000	700.000
19	378.947	694.737