source file: mills2.txt
Date: Thu, 19 Sep 1996 11:18:53 -0700

Subject: RE: Consistency generalized

From: PAULE <ACADIAN/ACADIAN/PAULE%Acadian@mcimail.com>

Paul H.,

	    I believe your original definition leads to consistency for level 
N>1.5. For 1<N<1.5, there is no guarantee that the approximating intervals 
will be consistent with one another.

	    I don't think the concept of consistency has much to do with whether a 
particular JI interval is approximated well enough to be used as a harmonic 
consonance. The absolute accuracy of the approximation is the determining 
factor, not the accuracy relative to the size of the smallest steps in the 
tuning. For example, the 12-tET major third sounds like a 5/4 even if 12-tET 
is considered a subset of 72-tET, where the interval is off by an entire 
step. I have derived an algorithm for determining the certainty with which 
an arbitrary interval is perceived as one or another JI intervals, based on 
J. L. Goldstein's 1973 paper, "An optimum processor theory for the central 
formation of the pitch of complex tones." Manual Op de Coul has ingeniously 
integrated this algorithm into Scala, under the Farey command. I leave it to 
him to provide instructions. Using this algorithm, we find that for any 
listener across the entire gamut of frequency resolution the most prominent 
interpretation of the a 7-tET "third" will be 11/9, and the most prominent 
interpretation 7-tET "tenth" will be as 17/7. Listeners at one extreme of 
the gamut will perceive the 7-tET "sixth" as 5/3, but those at the other 
extreme will hear it as 18/11. Thus the typically considered "imperfect 
consonances" in triadic harmony are not well represented in 7-tET.

	    What consistency offers is a supplement to considerations of how good 
the approximations to JI are. Essentially, composing with a consistent 
tuning will be no more difficult than composing in JI, while in inconsistent 
tunings, complications may arise if one attempts to always use the best 
approximations to JI intervals.

 - Paul E.


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