How does Euler's ranking of chords, where chord a:b:c:d would be given a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy rankings of chords? Euler's method gives the same value to otonal as to utonal chords, so it must have some differences. How great are they?

>How does Euler's ranking of chords, where chord a:b:c:d would be >given a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy >rankings of chords? Euler's method gives the same value to otonal as >to utonal chords, so it must have some differences. How great are >they?

I guess we won't really know until we have chordal harmonic entropy.

I remember the totient function, which didn't work at all, even for dyads, but the above measure looks different.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>How does Euler's ranking of chords, where chord a:b:c:d would be >given a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy >rankings of chords? Euler's method gives the same value to otonal as >to utonal chords, so it must have some differences. How great are >they?

they could be very great, in many cases. for example, harmonic entropy is a *continuous* function of the input intervals, defined for irrational as well as rational intervals.

i wish this were euler's actual ranking method, but of course he went even further (off the deep end) with his final formulae for GS.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> i wish this were euler's actual ranking method, but of course he went > even further (off the deep end) with his final formulae for GS.

It's been a while since I read Euler; I got this from a paper by van der Pol in the 1946 Music Review, on "Music and the Elementary Theory of Numbers," which perhaps I should report on. I was going through old stuff at my mother's, and I found this, which van der Pol had sent Dick Lehmer and I acquired after he died. I also found copies of the paper I mentioned from the 80s, which I shall resurrect--it seems it was turned down by a couple of dubious characters named Chalmers and Balzano.