Re: Digest Number 135

>>5-limit JI is a big place, present in some form in many meantone tunings.
>
>Strictly speaking, only 1/3 or 1/4 of the notes in a meantone tuning can
>coincide with 5-limit JI.

By huge margin the two most popular meantones in history :)

>>The stone only kills one bird -- it's not my fault that shorest-route
>>metrics have nothing to do with dyadic consonance and dissonance!
>
>Your fault?

Well, you said "could it hurt me", so I was just saying "no, but I can't
help it".

>Anyway, it seems that Tenney and I both tried to kill the two birds with
one >stone and came to similar strategies. The differences are probably
because I >assumed octave-equivalence and Tenney didn't.

I don't have any references on Tenney, other than Divisions of a
Tetrachord. Also, do you have a new algorithm that reflects your new
rectangular, octave-specific way?

>>The shortest-route metrics are equivalent to change-making problems. What
>>is the least number of coins you can use to make 33 cents change? Forget a
>>quarter, nickel, and three pennies, Erlich mints a 33-cent piece on the
>>spot!
>
>Eh?

A shortest-route triangular-lattice metric says "at the octave-equivalent
7-limit, 35/24 is made of a 7/6 and a 5/4". So the customer (listener) is
burdened with two coins in his pocket (ear). The rectangular-lattice
version of this would be 7/4 plus 2/3 plus 5/4, so the customer has 3
coins. I prefer the triangular version because I consider 7/6 a primary
consonant interval at the 7-limit.

Your log of odd limit, while more accurately measuring dyadic dissonance,
only measures shortest routes when the odd-limit is infinity (coins of any
denomination are accepted). Thus 35/24 is represented by one coin. I
don't know what you've done to it since you got hot on octave-specifics,
but your orginal algorithm assumed a triangular lattice (in the above
example, the /3 was in the coin).

Carl