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).

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.