On harmonic entropy

I would suggest a system to avoid the use of the Farey series in the measure of tonalness. I think the proposed new approach is more accurate and, what is very important, much more intuitive.

Let's begin with the Stern-Brocot tree instead of Farey series. Farey series is an ordered set of all fractions in lowest terms (at any given higher limit for denominator value). This series has a crucial property: given any set of three consecutive terms, the middle one equals the mediant between the third and the first (the mediant between a/b and c/d is (a+c)/(b+d)). Stern-Brocot tree takes the concept of mediant not as a consequence but as a constructing law; the generation of the set of fractions in lowest terms is then a consequence. Taken to infinity, both approaches are equivalent, but in limited terms are not. In Farey Series you introduce the fractions with denominator 1 first, then those with denominator 2, then those with denominator 3, and so on. In Stern-Brocot tree fractions with the same denominator are generated at different levels.

The main advantage of Stern-Brocot tree is that it introduces a rank of simplicity of fractions on objective terms, prior to any calculation of the "room" surrounding any of these. As for music, you find the unison first, then the octave, then the fifth, then the fourth, then the major third, and so on:

Level 0: 0/1, 1/0

Level 1: 1/1

Level 2: 1/2, 2/1

Level 3: 1/3, 2/3, 3/2, 3/1

Level 4: 1/4, 1/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4/1

etc.

Calculations are simpler. When you are aware that 3/2 has been generated by 1/1 and 2/1, you find no difficulty in determine the nearest neighbours at next level, 4/3 and 5/3. And then you can interpole indefinitely next "nearest neighbours" at the higher levels: 7/5 and 8/5, 10/7 and 11/7, 13/9 and 14/9, and so on, without any need to develop the whole tree.

Let's now define the "room of an interval at a level n" by means of the following mathematical expression:

R(i,n)=[i(f)-i(p)](n-2)

where

i is any interval (mathematically, any fraction in lowest terms);

n is any level such that is superior to the level at which i has been generated;

R(i,n) is the room value of interval i at level n;

i(f) is the immediate following interval of i at level n;

i(p) is the immediate precedent interval of i at level n.

For any interval i, the set of room values at different levels is quickly convergent. For example, for 3/2 we obtain:

level 4: [(5/3)-(4/3)]*(4-2)=.667

level 5: [(8/5)-(7/5)]*(5-2)=.600

level 6: [(11/7-(10/7)]*(6-2)=.571

level 7: [(14/9)-(13/9)]*(7-2)=.556

level 8: [(17/11)-(16/11)]*(8-2)=.545

level 9: [(20/13)-(19/13)]*(9-2)=.538

level 10: [(23/15)-(22/15)]*(10-2)=.533

level 11: [(26/17)-(25/17)]*(11-2)=.529

level 12: [(29/19)-(28/19)]*(12-2)=.526

...

Similarly, for 4/3 we get .450, .357, .318, .297, .283, .274, .267, .262, ...

The room values obtained at a reasonably high level can be taken as an index of probability or determinism, and for this reason, of the tonalness of the interval.

If we take absolute values of intervals (not logarithmic in cents) and room values (at a reasonable level of accuracy) to a coordinate graphic, we'll get a curve very akin to those of Paul Erlich, but abolutely symmetric.

A graphic obtained at only a level of 12 gives a very valable "at first glance" description of the tonalness of most usual intervals.

hi albert! thanks for chiming in!

> I would suggest a system to avoid the use of the Farey series in
>the measure of tonalness. I think the proposed new approach is more
>accurate and, what is very important, much more intuitive.

i use the "tenney" series rather than the farey series for most
harmonic entropy calculations.

> Let's begin with the Stern-Brocot tree instead of Farey series.

can you generalize the stern-brocot tree to triads?

>Farey series is an ordered set of all fractions in lowest terms (at
>any given higher limit for denominator value). This series has a
>crucial property: given any set of three consecutive terms, the
>middle one equals the mediant between the third and the first (the
>mediant between a/b and c/d is (a+c)/(b+d)). Stern-Brocot tree takes
>the concept of mediant not as a consequence but as a constructing
>law; the generation of the set of fractions in lowest terms is then
>a consequence. Taken to infinity, both approaches are equivalent,
>but in limited terms are not. In Farey Series you introduce the
>fractions with denominator 1 first, then those with denominator 2,
>then those with denominator 3, and so on. In Stern-Brocot tree
>fractions with the same denominator are generated at different
>levels.

with what acoustical or psychophysical justification? level N of the
stern-brocot tree has 2^N fractions in it -- why would this have any
relationship with how we hear?

> The main advantage of Stern-Brocot tree is that it introduces a
>rank of simplicity of fractions on objective terms,

the farey series does this, and the "tenney" series is even better at
it.

>prior to any >calculation of the "room" surrounding any of these. As
>for music, you find the unison first, then the octave, then the
>fifth, then the fourth, then the major third, and so on:
>
> Level 0: 0/1, 1/0
>
> Level 1: 1/1
>
> Level 2: 1/2, 2/1
>
> Level 3: 1/3, 2/3, 3/2, 3/1
>
> Level 4: 1/4, 1/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4/1
>
> etc.

the levels become less intuitive farther down.

> Calculations are simpler. When you are aware that 3/2 has been
>generated by 1/1 and 2/1, you find no difficulty in determine the
>nearest neighbours at next level, 4/3 and 5/3. And then you can
>interpole indefinitely next "nearest neighbours" at the higher
>levels: 7/5 and 8/5, 10/7 and 11/7, 13/9 and 14/9, and so on,
>without any need to develop the whole tree.
>
> Let's now define the "room of an interval at a level n" by means of
the following mathematical expression:
>
> R(i,n)=[i(f)-i(p)](n-2)
>
> where
>
> i is any interval (mathematically, any fraction in lowest terms);

why would you restrict it to be a fraction?

> n is any level such that is superior to the level at which i has
been generated;
>
> R(i,n) is the room value of interval i at level n;
>
> i(f) is the immediate following interval of i at level n;
>
> i(p) is the immediate precedent interval of i at level n.
>
> For any interval i, the set of room values at different levels is
quickly convergent. For example, for 3/2 we obtain:
>
> level 4: [(5/3)-(4/3)]*(4-2)=.667
>
> level 5: [(8/5)-(7/5)]*(5-2)=.600
>
> level 6: [(11/7-(10/7)]*(6-2)=.571
>
> level 7: [(14/9)-(13/9)]*(7-2)=.556
>
> level 8: [(17/11)-(16/11)]*(8-2)=.545
>
> level 9: [(20/13)-(19/13)]*(9-2)=.538
>
> level 10: [(23/15)-(22/15)]*(10-2)=.533
>
> level 11: [(26/17)-(25/17)]*(11-2)=.529
>
> level 12: [(29/19)-(28/19)]*(12-2)=.526
>
> ...
>
> Similarly, for 4/3 we
>get .450, .357, .318, .297, .283, .274, .267, .262, ...

you can do this with the other series too, i don't see what advantage
stern-brocot presents. with the "tenney" series, you can also do an
instant comparison -- at any given level, the "room" occupied by a
ratio is proportional to 1/sqrt(n*d). this generalizes nicely to
chords with any number of terms, the "room" in multi-dimensional
space being inversely proportional to the geometric mean of the terms.

> The room values obtained at a reasonably high level can be taken as
>an index of probability or determinism, and for this reason, of the
>tonalness of the interval.

but what about irrational intervals, particularly those that are
close to simple ratios?

> If we take absolute values of intervals (not logarithmic in cents)
and room values (at a reasonable level of accuracy) to a coordinate
graphic, we'll get a curve very akin to those of Paul Erlich, but
abolutely symmetric.

can i see your curve please? what symmetry exactly are you referring
to? my curves are absolutely symmetric around 1/1, which is why i
don't bother showing the left half of the graph.

it sounds like you're only addressing the "complexity" part of the
dissonance measure. any reasonable dissonance measure must have
a "tolerance" part as well . . .

thanks for your thoughts,
paul