Magic: Difference between revisions
From Xenharmonic Reference
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= (S15*S28*S29*S30)^2*S31 | = (S15*S28*S29*S30)^2*S31 | ||
making the above 2.3.5.7.29.31 extension a [[canonical extension|structurally induced extension]]. | {{Adv|making the above 2.3.5.7.29.31 extension a [[canonical extension|structurally induced extension]].}} | ||
{{cat|Temperaments}}{{Navbox regtemp}} | {{cat|Temperaments}}{{Navbox regtemp}} | ||
Revision as of 21:31, 5 March 2026
| Magic |
225/224, 245/243 (7-limit)
9-odd-limit: 5.9¢
9-odd-limit: 13 notes
Magic 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.
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:
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
making the above 2.3.5.7.29.31 extension a structurally induced extension.
