TUNING digest 590

> Sure. Maybe you can tell us how your program works, what is the
> expression you are optimising. Is it linear or nonlinear?
> What are the 12 note weights you mentioned?
>
> Manuel Op de Coul coul@ezh.nl

The objective function I minimize the sum of squared errors of
actual interval values minus desired interval values.

errsum = 0 ;
for n = 0 to 11 {
v1 = note_cents[n] ;
w1 = note_weight[n] ;
for i = 0 to 6 {
int n2 = n+i+1 ;
if (n2 < 12) {
v2 = note_cents[n2] ;
w2 = note_weight[n2] ;
} else {
v2 = note_cents[n2-12]+1200. ;
w2 = note_weight[n2-12] ;
}
err = interval_weight[i]*w1*w2*((v2-v1) - desired_interval[i]) ;
errsum = errsum + err*err ;
}
}


The "parameters" to be fit for this function are in the note_cents array
which contains the cents values for the 12 notes in the scale (i.e. 0,100,
200,...,1100) for equal spaced half steps.

The optimization routine doesn't really know if the function is linear or
nonlinear, and, frankly, I don't either (haven't had time to think it
through entirely.) The optimization will converge, and seems to produce
reasonable values.

The weights are entirely arbitrary. For example use a weight of 1 for all
12 notes and 7 intervals and you'd get the equivalent of an unweighted fit.

On the other hand you could give C,D,E,F,G,A and B weights of 100 and leave
C#,D#,F#,G#,A# with weights of 1 and you'd get an optimized set of note values
for the C tonic scale.

Similarly you can weight the intervals so the fifths are nearly perfect, but
the minor 3rds get pretty bad, or whatever.

I gotta go do a christmas party (best excuse I've had all year to cut a
mail message short).

More on this later. I'll attempt to get the code and sample data out to
anyone who requests.

Jeff



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