Complexity and inaudible primeness (was: 64:75:96)

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> > > Complexity(a:b) = a*b
>
> Of course, there are many different definitions of complexity:
>
> harmonic distance, indigestibility, harmonic complexity, harmonic
> entropy, harmonicity, gradus suavitatis
>
> (http://www.ixpres.com/interval/dict/complex.htm)

As you know, except for harmonic entropy (which is not a mere complexity
measure) all these and more are compared in my spreadsheet
http://dkeenan.com/Music/HarmonicComplexity.zip

I would prefer to say that these are all different definitions of "harmonic
complexity". Didn't both Vogel and Wilson call their different functions
"harmonic complexity"? Monz only lists Wilson's.

But there is apparently only one definition of complexity of interest to
non-musical mathematicians, as evidenced by its appearance in Eric
Weistein's (ill-fated) MathWorld (I think?). It appeared in connection with
the Stern-Brocot tree. That definition of (purely mathematical) complexity
(a*b) was due to Pierre Lamothe.

It may also be due to Denny Genovese. In my spreadsheet I have "Genovese
DF" calculated as GOF(a)*GOF(b), GOF = Greatest Odd Factor = Remove factors
of 2. But then I have plotted it (or rather it's square root) on a chart of
octave-specific measures. I seem to remember Carl Lumma told me that
"Genovese DF" was simply a*b. Can anyone clarify this?

But as far as I know, only Lamothe proved its interesting mathematical
properties in relation to the Stern-Brocot tree (which, I seem to remember,
also happen to translate to musical properties for small integer ratios).

> I think Dave K. is getting at the idea that there are three factors
> which determine the consonance/dissonance of an interval,
>
> complexity
> tolerance
> span

I wasn't, but it's always good to keep in mind. Especially what Monz fails
to mention in his dictionary, that harmonic complexity measures are only
intended to approximate dissonance in the case of _simple_ ratios narrower
than say two octaves. A ratio with very high complexity can in fact be very
consonant due to being within tolerance of a simple ratio or by being wide.

And what about the (admittedly minor) effect of "rootedness".

> and I think I've convinced Dave K. that, in this context, Tenney's
> complexity, or a monotonic function of it such as a*b, is the best
> measure of complexity.

Yes. But I'll rephrase it. You've convinced me that Lamothe complexity, or
a monotonic function of it such as Tenney's harmonic distance, is the best
measure of harmonic complexity for dyads.

> And I was led there by Graham Breed. So perhaps we
should
> call a*b the Tenney/Breed/Erlich/Keenan complexity or something,
just to
> avoid confusing with Wilson's harmonic complexity or other
functions.

I propose that henceforth "complexity" without qualification be understood
as Lamothe complexity (a*b) since this is the only one that can
legitimately drop the adjective "harmonic", and is also the most reliable
for ranking the disonance of narrow-to-medium-span simple-ratio dyads.

I too am convinced that there is no audible effect or affect due to the
introduction of ratios with a new prime factor. I hear that ratios of 9 are
as distinct from ratios of 3 as they are from ratios of 5 and 7. One
possible reason for this confusion is that it is extremely difficult to
construct a scale that allows accurate ratios of 3 without also allowing
ratios of 9.

I believe the "prime" distinction is another example of a supposed musical
distinction which is in fact, purely mathematical. I did not come to that
conclusion easily. You may remember I tried various prime-weighting schemes
to preserve it in the face of what I was hearing. In the end, the mere size
of the numbers was enough to explain everything.

Here is another possible reason for the confusion:

It is a mathematical fact that, of two numbers of similar size, the number
that has the smaller prime factors (and hence has _more_ factors) will form
_more_ simple ratios with other numbers. This is because it has more
factors that may cancel. But this does not affect the fact that the
consonance of any such simple ratio is dependent only on the _size_ of the
numbers when the ratio is expressed in lowest terms (plus the other
non-prime related factors mentioned above).

Primeness is inaudible, and if the exponents (number of times each prime
may occur as a factor) are not limited, then it is also immeasurable. If
anyone claims that despite being inaudible, limited-exponent primeness has
a subliminal effect on some listeners, then it is their responsibility do
demonstrate this by suitable experiments.

Regards,

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>Didn't both Vogel and Wilson call their different functions
>"harmonic complexity"?

I was not aware of Vogel's definition. What is it?

>But there is apparently only one definition of complexity of interest to
>non-musical mathematicians, as evidenced by its appearance in Eric
>Weistein's (ill-fated) MathWorld (I think?). It appeared in connection with
>the Stern-Brocot tree. That definition of (purely mathematical) complexity
>(a*b) was due to Pierre Lamothe.

Dave, in mathematics (namely, rational approximation theory) the complexity
of a ratio p/q is usually defined as q.

>But as far as I know, only Lamothe proved its interesting mathematical
>properties in relation to the Stern-Brocot tree (which, I seem to remember,
>also happen to translate to musical properties for small integer ratios).

Never followed the translation all the way through. If you remember better,
could you clarify?

>I too am convinced that there is no audible effect or affect due to the
>introduction of ratios with a new prime factor. I hear that ratios of 9 are
>as distinct from ratios of 3 as they are from ratios of 5 and 7. One
>possible reason for this confusion is that it is extremely difficult to
>construct a scale that allows accurate ratios of 3 without also allowing
>ratios of 9.

>Here is another possible reason for the confusion:

>It is a mathematical fact that, of two numbers of similar size, the number
>that has the smaller prime factors (and hence has _more_ factors) will form
>_more_ simple ratios with other numbers. This is because it has more
>factors that may cancel. But this does not affect the fact that the
>consonance of any such simple ratio is dependent only on the _size_ of the
>numbers when the ratio is expressed in lowest terms (plus the other
>non-prime related factors mentioned above).

Thanks, Dave. It seems very difficult to convert most people on this point
-- I hope this argument will help clear away the fog a little bit.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Dave Keenan wrote,
>
> >
> >I too am convinced that there is no audible effect or affect due
to the
> >introduction of ratios with a new prime factor. I hear that ratios
of 9 are
> >as distinct from ratios of 3 as they are from ratios of 5 and 7.
One
> >possible reason for this confusion is that it is extremely
difficult to
> >construct a scale that allows accurate ratios of 3 without also
allowing
> >ratios of 9.
>
> >Here is another possible reason for the confusion:
>
> >It is a mathematical fact that, of two numbers of similar size,
the number
> >that has the smaller prime factors (and hence has _more_ factors)
will form
> >_more_ simple ratios with other numbers. This is because it has
more
> >factors that may cancel. But this does not affect the fact that the
> >consonance of any such simple ratio is dependent only on the
_size_ of the
> >numbers when the ratio is expressed in lowest terms (plus the other
> >non-prime related factors mentioned above).
>
> Thanks, Dave. It seems very difficult to convert most people on
this point
> -- I hope this argument will help clear away the fog a little bit.

Paul and David,

A little bit of the fog bank still remains for me. If "there is no
audible effect or affect due to the introduction of ratios with a new
prime factor", does mentioning "prime" or "odd" limits have any
meaning for defining scales other that the mathematical proportions
behind the tuning? If higher primes can't be perceived, what purpose
do these labels serve for us other than a mathematical one?

Pardon if this has been addressed. Just seeking some clarification.
Sound the fog horn here guide me back from the error of my Prime Ways!

Thanks,

Jacky Ligon

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Dave Keenan wrote,
>
> >Didn't both Vogel and Wilson call their different functions
> >"harmonic complexity"?
>
> I was not aware of Vogel's definition. What is it?

Both of these harmonic complexities are available in Scala. I
understand they are both based on summing all occurrences of prime
factors on both sides of the ratio. According to my spreadsheet (which
I don't trust too much), Wilson's leaves out factors of 2 entirely.
Vogel's (more sensibly in my opinion) weights factors of 2 at half the
significance of all the other primes. Manuel, please confirm or deny.

> Dave, in mathematics (namely, rational approximation theory) the
complexity
> of a ratio p/q is usually defined as q.

Bummer. I guess we call it Lamothe complexity then.

> >But as far as I know, only Lamothe proved its interesting
mathematical
> >properties in relation to the Stern-Brocot tree (which, I seem to
remember,
> >also happen to translate to musical properties for small integer
ratios).
>
> Never followed the translation all the way through. If you remember
better,
> could you clarify?

No. Sorry. You could always email Pierre.

Regards,
-- Dave Keenan

Dave wrote
>Both of these harmonic complexities are available in Scala. I
>understand they are both based on summing all occurrences of prime
>factors on both sides of the ratio. According to my spreadsheet (which
>I don't trust too much), Wilson's leaves out factors of 2 entirely.
>Vogel's (more sensibly in my opinion) weights factors of 2 at half the
>significance of all the other primes. Manuel, please confirm or deny.

Correct, in Vogel's measure factors of 2 are weighted with 1, and in
Wilson's weighted with 0 (ignored). Both weigh other primes with the
prime factor itself.
The Stereo Dekany is amazing.

Manuel