Complexity measures

I also wanted to reply earlier but had other things taking my attention.
Paul E. pointed out rightly that my triangular prime lattice algorithm
is indeed different from Paul H.'s definition.
Like Dave said, prime factoring metrics don't correlate with
consonance. I consider the triangular logarithmic harmonic complexity
similar to Barhloo's harmonicity/indigestability value, but then with
prime "shortcuts", i.e. not counting some prime by way of a higher
prime masking one lower prime if they are at opposite sides of the
division sign.
Whether the extra effort is worth the trouble when not many values
are different from Paul E.'s log of highest odd factor I don't know.
I don't want to endorse it as a useful metric because I haven't got
enough interest in it at the moment to make a study.
But to compare: when we take 11/10, 11/9, 11/8, 11/7, 11/6 then Paul's
values are all the same, while mine gives a higher value for only
11/9, which might be considered strange from an odd-limit standpoint.
On the other hand, for 9/8, 9/7, 9/5 my values are all different and
Paul's are the same which might be considered strange from a
prime-limit standpoint.
When we modify Bahlouw's inverse harmonicity by setting the weights
for 2 to zero and for the other primes to their log, then my
triangular logarithmic complexity is always between this value and
Paul's log of largest odd factor. And Paul Hahn's version between mine
and Paul Erlich's.
BTW, I've put Paul E.'s metric in Scala for the next version.

Manuel Op de Coul coul@ezh.nl

"Paul H. Erlich" <PErlich@Acadian-Asset.com> wrote:

>Which is the only one worth considering? Hahn's algorithm for any given odd
>limit? Or only for infinite odd limit (which yields complexity of an
>interval = the log of the odd limit of the interval)?

Only the unlimited odd one = log of odd limit = yours.

>Lattices are useful for visualizing the harmonic relationships between
>notes, for composition algorithms utilizing those relationships, etc. etc.

Absolutely.

>Marion was espousing this viewpoint on the list some time ago. I think it's
>not true at all. Frequency resolution is very register-dependent. I (and
>Partch) agree that _in a given register_ what you are saying is correct. But
>I don't think there's a simple mathematical formula that works across
>registers.
>
>There is no single period (like 50ms) that works for all registers. That
>would imply that every octave you go up, the "limit" for consonant intervals
>doubles. Now it is recognized that 5-limit intervals are dissonant in the
>bass register, and on a good day I might call a high 9:7 consonant. But far
>from doubling every octave, that looks like a limit that doubles (at best)
>over 4-5 octaves, and certainly does not increase further after that.

Right! Of course. Silly me. I'm sure glad you're on this list Paul.

-- Dave Keenan
http://dkeenan.com

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