Lucy tuning
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[math]\displaystyle{ \def\val#1{\left\langle\begin{matrix}#1\end{matrix}\right]} }[/math] Lucy tuning is the meantone tuning advocated by Charles Lucy, with a fifth of precisely 600 + 300/π = 695.493 cents. This is close to the 88edo fifth of 695.455 cents, and shares its general characteristics such as being a good tuning for mothra and euterpe temperaments, providing a mothra generator, an approximate 8/7, of 200+100/π cents, which extends 5-limit meantone Lucy tuning to a version with a mothra (1/3 meantone fifth) generator. As a tuning for 5-limit meantone, it has the softer quality characteristic of the flatter fifth meantones between 50edo and 19edo. It has a major third of 1200/π cents, or 1/π of an octave, 381.972 cents, 4.342 cents flat of 5/4, but 2.827 cents sharp of 1/3-comma meantone's major third.
A reasonable mapping for 11-limit extended Lucy tuning would be
- [math]\displaystyle{ \val{1200 & 1800 + \frac{300}{π} & 2400 + \frac{1200}{π} & 3400 - \frac{100}{π} & 4400 - \frac{800}{π}} }[/math]
This tempers out the mothra commas of 81/80, 1029/1024, 99/98 and 385/384. While Charles Lucy himself does not seem to consider the possibility of extending Lucy tuning, it should be noted that the mothra mapping above gives a 7/4 of 1000-100/π, a mere 0.659 cents flat of a just 7/4. Another way to extend Lucy tuning is meanpop:
- [math]\displaystyle{ \val{1200 & 1800 + \frac{300}{π} & 2400 + \frac{1200}{π} & 2400 + \frac{3000}{π} & 5400 - \frac{3900}{π}} }[/math]
Since this does not involve splitting the generator into thirds, it is closer to Lucy's 5-limit perspective.