Is someone ever going to give a precise definition? One uploaded to

the files area of the entropy group would be nice. If you can

calculate it, you can define it--if nothing else works, say exactly

what you are calculating.

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

> Is someone ever going to give a precise definition? One uploaded to

> the files area of the entropy group would be nice. If you can

> calculate it, you can define it--if nothing else works, say exactly

> what you are calculating.

Well, there are variations. But it's always a function (meant to

reflect one component of dissonance) of a precise specified input

interval or chord. And entropy is always defined as in information

theory:

sum(p*log(p))

where the sum is over all possible states and p is the probability of

each state.

For harmonic entropy, each "state" is a just dyad (or n-ad in future

versions), i.e., a ratio. The universe of possible ratios is

determined by a rule (such as max(p,q)<N or p*q<N, generally any rule

such that p(j)*q(i) - p(i)*q(j) = 1 for any pair of adjacent ratios p

(i)/q(i), p(j)/q(j), and with N tending toward infinity). The

probability of each dyad is determined by determining the

corresponding area under a normal curve (whose s.d. is an input

parameter) standard, centered around the actual input value. The

width of the "slice" corresponding to each dyad is determined

assuming that it occupies the full "range" between the adjacent

mediants.

My Matlab entropy function is of the form

output = entropy(cents,s,N)

where s is a cents value for the input interval, s is the s.d. of the

normal curve, and N is (these days) typically used for the rule

p*q<N. Why, because:

It was found that the local minima P/Q tend to satisfy P*Q<C no

matter what "rule" was chosen. The curve as a whole has an

overall "slope" unless the Tenney rule (p*q<N) is chosen. Then it is

often found that the entropy for the simpler local minima is

proportional to the Tenney Harmonic Distance, log(P*Q). It is also

found that using 1/sqrt(p*q) as a proxy for each dyad's "range", and

simply multiplying this by the height of the bell curve _exactly_ at

p/q, leads to a nearly identical functional appearance, except that

there is less sensitivity to tiny changes in N.

Open questions:

Is there a function, F(x,y), such that F(entropy,s) is invariant to

changes in N? For s=1%, F(entropy,1%) = exp(entropy/2.3) seemed to

work.

Can we explicitly calculate what this function converges to for N-

>infinity? Or at least prove that is does converge, and calculate the

limit to some computational error?

Can we prove that the observations mentioned above (about the local

minima and about proxying for the width being OK) are in some sense

true?

I prepared a full plan for calculating triadic harmonic entropy. See

the harmonic_entropy@yahoogroups.com archives. How can we optimize

the calculation?

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

> For harmonic entropy, each "state" is a just dyad (or n-ad in

future

> versions), i.e., a ratio. The universe of possible ratios is

> determined by a rule (such as max(p,q)<N or p*q<N, generally any

rule

> such that p(j)*q(i) - p(i)*q(j) = 1 for any pair of adjacent ratios

p

> (i)/q(i), p(j)/q(j), and with N tending toward infinity).

The

> probability of each dyad is determined by determining the

> corresponding area under a normal curve (whose s.d. is an input

> parameter) standard, centered around the actual input value.

The "actual input value" is a certain number of cents, so how can a

normal curve be centered around it?

The

> width of the "slice" corresponding to each dyad is determined

> assuming that it occupies the full "range" between the adjacent

> mediants.

A "dyad" is a fraction reduced to lowest terms, e.g 5/4, if I am

following you. The "adjacent mediants" is not clear, but perhaps you

mean the fractions on each side of the Farey sequence in which

the "dyad" first appears? In the case of 5/4, that would be

1/1 < 5/4 < 4/3, so we would integrate a normal function between 1

and 4/3 to get p?

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

> > The

> > probability of each dyad is determined by determining the

> > corresponding area under a normal curve (whose s.d. is an input

> > parameter) standard, centered around the actual input value.

>

> The "actual input value" is a certain number of cents,

Right.

> so how can a

> normal curve be centered around it?

If the number of cents is c, the curve is

y=1/(s*sqrt(2*pi))*exp((x-c)^2/2*s^2)

where x is the position on the interval axis (have you looked at any

of the harmonic entropy curves)?

> > The

> > width of the "slice" corresponding to each dyad is determined

> > assuming that it occupies the full "range" between the adjacent

> > mediants.

>

> A "dyad" is a fraction reduced to lowest terms, e.g 5/4, if I am

> following you.

Yes.

> The "adjacent mediants" is not clear, but perhaps you

> mean the fractions on each side of the Farey sequence in which

> the "dyad" first appears?

No, I mean the mediants between the fraction and its immediate

neighbors in a Farey sequence of order N.

> In the case of 5/4, that would be

> 1/1 < 5/4 < 4/3, so we would integrate a normal function between 1

> and 4/3 to get p?

You would integrate between two complicated ratios, typically both

very close to 5/4, but far closer to 5/4's immediate successor and

immediate predecessor in the Farey sequence of order N (because N is

large). Thus 5/4 would typically end up with a much greater

probability than its neighbors.