Re: complexity

At 07:57 15/03/99 -0500, Joseph L Monzo <monz@juno.com> wrote in private
email:

>Thanks for the compliments on the lattice piece, Dave.
>I did it all in one night, but it did take a lot
>of work.
>
>> I'd prefer if you didn't talk of Keenan's "musical
>> complexity" in future. First, "musical" is too broad,
>> and I prefer "harmonic". And second, it's not mine,
>> since others (Manuel Op de Coul at least) have proposed
>> treating the prime weights as parameters.
>
>OK, Manuel called this "harmonicity", right?
>How come we don't want to use his term?
>(or have I mangled my understanding of this?
>I am brain-dead from too much theorizing)

I expect others are confused too, so forgive me for posting your question
to the list.

The terminology is messy. Barlow's Harmonicity (and Manuel's
parameterisation of it) is a "simplicity" measure (not complexity) i.e. the
numbers increase in the same direction as consonance whereas harmonic
complexity measures increase in the direction of dissonance. Take the
reciprocal of (the absolute value of) Barlow's Harmonicity to convert it to
a complexity measure. I've included this in my spreadsheet and called it
Barlow's Indigestibility, since that's what he called it when applied to
whole numbers.

Most related measures increase with dissonance. Barlow's (and Manuel's
generalisation of it) appear to be the only ones that don't. I prefer the
dissonance direction, since otherwise we may end up with an infinity (1/0)
at the unison (as we do with Barlow's harmonicity).

*** LCM is the same as product, for dyads ***

Something else I just realised (duh!). Euler's Gradus Suavatis and Totient
functions (two COMPLEXITY measures) are first defined for whole numbers,
and then defined for any chord as the Gradus Suavatis or Totient
respectively, of the Least Common Multiple (LCM) divided by Greatest Common
Divisor (GCD) of the numbers in the chord (expressed as n:m:k:j:... where
n,m,k,j,.. are whole numbers. When they are applied to dyads where the
ratio n:d is already in lowest terms, the LCM is the same as the product
n*d (and the GCD is 1), and so these measures suffer from the possible
defect pointed out by Paul Erlich where for example, 15/4 has the same
complexity as 12/5, or 3/2 the same as 6/1. However the disagreement with
perceived dissonance may be taken care of by TOLERANCE in the first case
and SPAN in the second. So they may not amount to a defect in the
COMPLEXITY measure.

Note also that the LCM of a chord corresponds to the "guide tone" while the
GCD (Greatest Common Divisor) corresponds to the "virtual fundamental".
These functions are available in Excel and are also useful for obtaining
prime factorisations.

*** More notes = higher limit ? ***

Another thing. I looks to me like the more notes there are in a chord, the
higher one can go in COMPLEXITY before TOLERANCE takes over and blurrs the
distinctions between the ratios. We seem to agree that ratios of 11 are
approximately the maximum for dyads, and ratios of 19 for triads. Anyone
want to disagree? Anyone want to suggest approximate limits for tetrads,
pentads etc?

*** Update of dyadic complexity spreadsheet ***

My spreadsheet now compares the following 11 harmonic complexity measures
for all ratios up to a maximum sum-of-numerator-and-denominator of 21.

(n+d)/2
Prime weighted (n+d)/2 (choose your own weights)
Integer limit
Odd limit
Prime limit
Barlow's indigestibility /3
Wilson's complexity /2
Vogel's complexity /2
Euler's gradus suavatis /2
Euler's totient /4
Genovese DF (sqrt)

The previous version did not contain Vogel's, and Euler's pair were
seriously incomplete.

I'll add the log-max-prime weighted triangular lattice distance if someone
tells me how to do it. Paul H., Paul E. and Manuel, do you all agree yet?

It's grown significantly in size so I've zipped it.

http://dkeenan.com/Music/HarmonicComplexity.zip 135kB

I also zipped my Sethares dissonance spreadsheet but it didn't shrink much.

http://dkeenan.com/Music/SetharesDissonance.zip 636kB

Regards,
-- Dave Keenan
http://dkeenan.com