Simplicity in ratio

Hi.

I know that consonance is directly related to simplicity of ratio.
It's possible to express mathematically what is "simple"?

For example I know that 16/15 is more complex than 3/2.
Why? Is semplicity based upon the sum of the two numbers?
If so 3/1 is more consonant than 3/2 which is true to me.
But 6/1 is less consonant...Uhm...

Or maybe sum is a wrong way. I can count how many waves of the root I
need to begin a new cycle. But in this case 7/4 and 5/4 are equally
consonant and my ear says that this is'nt true.

What does it mean simple?

Lorenzo

--- In tuning@yahoogroups.com, "gekovivo" <gekovivo@y...> wrote:
> Hi.
>
> I know that consonance is directly related to simplicity of ratio.
> It's possible to express mathematically what is "simple"?

You may be interested in looking over the harmonic entropy list:
/harmonic_entropy/

> For example I know that 16/15 is more complex than 3/2.
> Why? Is semplicity based upon the sum of the two numbers?
> If so 3/1 is more consonant than 3/2 which is true to me.
> But 6/1 is less consonant...Uhm...
>
> Or maybe sum is a wrong way. I can count how many waves of the root
I
> need to begin a new cycle. But in this case 7/4 and 5/4 are equally
> consonant and my ear says that this is'nt true.
>
> What does it mean simple?
>
> Lorenzo

For various reasons, I agree with Benedetti who, way back in the
Renaissance, proposed the *product* for ranking -- which would make
6/1 and 3/2 equally consonant. More recently, James Tenney proposed
that the log of the product be used to get a more quantitative
comparison, and for various reasons, I follow him on that. But read
on.

It's actually very difficult to compare the consonance of such
different-sized intervals -- one is so much wider than the other --
which means that besides COMPLEXITY, SPAN is another important factor
in assessing consonance.

The numbers in the ratio cease to be a useful guide when the numbers
get too large, or when the ratio isn't rational to begin with. For
example, if you detune the perfect fifth so that its ratio is
3001:2000, it will still be more consonant than 16:15; and if you
detune it (as in 12-equal) so that it's 2^(7/12), there is no
numerator and denominator for you to work with at all, yet you'll
probably still find it more consonant than 16:15.

So one has a certain amount of TOLERANCE for mistunings of the
simplest ratios, which makes more complex ratios meaningless in this
context -- so now we have COMPLEXITY, SPAN, and TOLERANCE.

Now timbre can inflence consonance judgments, especially when the
partials are inharmonic, so we have COMPLEXITY, SPAN, TOLERANCE, and
TIMBRE to consider.

Context is a huge factor too -- the same interval can sound dissonant
or consonant depending on the musical context, for example in
conventional common-practice Western music the augmented second is
dissonant despite being acoustically identical to the consonant minor
third -- so really we have to consider COMPLEXITY, SPAN, TOLERANCE,
TIMBRE, and CONTEXT.

And probably much more: REGISTER, LOUDNESS, CULTURE, TRAINING . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

/tuning/topicId_51040.html#51057

> --- In tuning@yahoogroups.com, "gekovivo" <gekovivo@y...> wrote:
> > Hi.
> >
> > I know that consonance is directly related to simplicity of ratio.
> > It's possible to express mathematically what is "simple"?
>
> You may be interested in looking over the harmonic entropy list:
> /harmonic_entropy/
>
> > For example I know that 16/15 is more complex than 3/2.
> > Why? Is semplicity based upon the sum of the two numbers?
> > If so 3/1 is more consonant than 3/2 which is true to me.
> > But 6/1 is less consonant...Uhm...
> >
> > Or maybe sum is a wrong way. I can count how many waves of the
root
> I
> > need to begin a new cycle. But in this case 7/4 and 5/4 are
equally
> > consonant and my ear says that this is'nt true.
> >
> > What does it mean simple?
> >
> > Lorenzo
>
> For various reasons, I agree with Benedetti who, way back in the
> Renaissance, proposed the *product* for ranking -- which would make
> 6/1 and 3/2 equally consonant. More recently, James Tenney proposed
> that the log of the product be used to get a more quantitative
> comparison, and for various reasons, I follow him on that. But read
> on.
>
> It's actually very difficult to compare the consonance of such
> different-sized intervals -- one is so much wider than the other --
> which means that besides COMPLEXITY, SPAN is another important
factor
> in assessing consonance.
>
> The numbers in the ratio cease to be a useful guide when the
numbers
> get too large, or when the ratio isn't rational to begin with. For
> example, if you detune the perfect fifth so that its ratio is
> 3001:2000, it will still be more consonant than 16:15; and if you
> detune it (as in 12-equal) so that it's 2^(7/12), there is no
> numerator and denominator for you to work with at all, yet you'll
> probably still find it more consonant than 16:15.
>
> So one has a certain amount of TOLERANCE for mistunings of the
> simplest ratios, which makes more complex ratios meaningless in
this
> context -- so now we have COMPLEXITY, SPAN, and TOLERANCE.
>
> Now timbre can inflence consonance judgments, especially when the
> partials are inharmonic, so we have COMPLEXITY, SPAN, TOLERANCE,
and
> TIMBRE to consider.
>
> Context is a huge factor too -- the same interval can sound
dissonant
> or consonant depending on the musical context, for example in
> conventional common-practice Western music the augmented second is
> dissonant despite being acoustically identical to the consonant
minor
> third -- so really we have to consider COMPLEXITY, SPAN, TOLERANCE,
> TIMBRE, and CONTEXT.
>
> And probably much more: REGISTER, LOUDNESS, CULTURE, TRAINING . . .

***This is a really interesting post. Paul, this one should go in
your "book" for sure...

J. Pehrson