RE: Measures of Scale

>}So if...
>
>}4(P5) - M3 - 2(T) = X
>
>}..then what's a good name for "X"? Size of the syntonic comma?
>
>Yes.
>
>}Is this a
>}good definition of the syntonic comma?
>
>There is another possible definition:
>
>3(P5)-M6-T=X
>
>The two definitions are often equivelent, but in the case of tunings not
>consistent within the 5-limit (such as 20tet) they are unequal. In such a
>case I say that the syntonic comma is not well-defined in that tuning.
>
>}So a meantone tuning is one where
>}the syntonic comma is best represented by zero steps?
>
>Yes!
>
>}And if...
>
>}12(P5) - 7(T) = Y
>
>}..then what's a good name for "Y"? The degree of positive or negetiveness?
>}Isn't this the definition of the Pythagorean comma?
>
>Yes, yes.
>
>}When are these measures the same?
>
>Sometimes, sometimes not.
>
>}Does anybody know of any other useful
>}measures for scales?
>
>The diaschisma [3(T) - 4(P5) - 2(M3)] and schisma [M3 - 8(P5) + 5(T)] can be
>defined in terms of the pythagorean comma and syntonic comma, so they are not
>independently useful. Two measures that are useful in the context of my
>decatonic scales are the septimal comma
>
>2(T) - H7 - 2(P5)
>
>and the septimal sixth-tone
>
>2(M3) - 2(H7) + T
>
>where H7 is the harmonic seventh. Just as with the syntonic comma, other
>definions of these measures are possible, but the alternate definions are
>equivalent for tunings that are consistent within the 7-limit. My decatonic
>scales are possible only if the septimal comma and septimal sixth-tone both
>vanish. Such tunings are the septimal equivalent of meantone tunings, but
>have wide instead of narrow "perfect fifths".


SMTPOriginator: tuning@eartha.mills.edu
From: John Chalmers
Subject: Greek triads, Classification of temperaments
PostedDate: 31-12-97 21:38:05
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