Primes versus odds measure

Ciao Paul.

>It's a real pleasure talking to you. Thanks for making this list a
>much more interesting place!

I try to dedicate some time to this group since just intonation
theory is just my hobby. Sometimes this is very hard to do... so I
apologize eventually for my really late answers...
Of course I'm the last here.. So please, be patient with me. I'm
really ignorant about a lot of things. :-)

>> This is not true to my ears but I understand that this could be
>> subjective. I think is less subjective that the chord: 1/1, 3/2,
9/4
>> is much less complex than any other chords which includes 7/5.
>
>But that's not a fair comparison.
>Any such chord will have *two* 3:2s in it!!! So of course it will
>show up as much simpler than any chord which includes 7/5!

This means that 9/4 is a good condition for ratios simplicity.
This is very important. I would call this "relationship consonance".
Certainly, as you wrote, the possibilities to develop a simpler
chord are major in presence of 9/4 instead of 7/5.
We were discussing about complexity and I still consider 9/4 less
complex than 7/5, also for this reason (the sound of the single
intervals is another reason).
Odd numbers complexity measure don't respect these observation
(since 9/4 should be more complex than 7/5) and for these reason I
don't think it is a good way to measure ratios complexity.

>But chord
>complexity and interval complexity are two different things. Each
>interval, and their combined effect, must be included in an
>evaluation of chord complexity.

Yes. But chord complexity and interval complexity are interdependent
aspects.
As I've written this defines 9/4 simpler than 7/5 for the
best "relationship consonance".
This is the reason why the chord 1/1, 7/5 and 3/2 is more complex
than 1/1, 9/4 and 3/2: 9/4 is simpler than 7/5.
So I think that odd numbers ratios complexity measure don't work.
Primes measure works better. And one kind of complexity measure
excludes the other one.

>A third note in a chord provides some "juice" that helps swallow
>complex composite intervals. Without such additional notes, the
>factorability is no aid whatsoever, and the interval dry as toast.

I don't understand this.

>> I think a good complexity measure would have
>> to work in these cases too.
>
> It works exactly as it should, since 15/12 and 5/4 are identical.

If 15/12 and 5/4 would be "identical" odd numbers measure would work
in an "identical" way.
But this is not true since you have to reduce the ratio to obtain a
right complexity measure.
Using primes as measure you don't have this inconvenience. This
confirm me that primes measure works better: in a more generalized
way.

>> Complexity has to do with the "digestibility" of prime numbers.
>> To understand you can think to the growing rhythmic complexity
>> between a couple of notes, a triplet of them, a quintlet (not
>> sure this is the right word...), a septlet (?)....
>
>I don't think the rhythmic analogy holds in the pitch sphere when it
>comes to dyadic complexity. Complexity of larger musical units (such
>as chords or even entire infinite tuning systems) has to be
>considered as a separate and larger problem, and there are indeed
>fruitful lines of investigation in these regards on these lists,
>especially the tuning-math and harmonic_entropy lists.

I think there are two components in perceiving complexity of dyads.
The first is based upon the numbers of common harmonics and the
critical bandwith. I think that this aspect is very related to the
subject of perception.

The second thing is the intrinsecal object complexity.
Considering 2:1, 3:1 and 7:1 i listen to pairs of sounds with all
the harmonics in common. Anyway I hear *different* things. I feel
this difference is based on primes families.

Ciao

Lorenzo

--- In tuning@yahoogroups.com, "frizzerius" <lorenzo.frizzera@l...>
wrote:

> >> This is not true to my ears but I understand that this could be
> >> subjective. I think is less subjective that the chord: 1/1, 3/2,
> 9/4
> >> is much less complex than any other chords which includes 7/5.
> >
> >But that's not a fair comparison.
> >Any such chord will have *two* 3:2s in it!!! So of course it will
> >show up as much simpler than any chord which includes 7/5!
>
> This means that 9/4 is a good condition for ratios simplicity.
> This is very important. I would call this "relationship consonance".

This seems like a very abstract concept, very hard to define and
quantify, but OK.

> >But chord
> >complexity and interval complexity are two different things. Each
> >interval, and their combined effect, must be included in an
> >evaluation of chord complexity.
>
> Yes. But chord complexity and interval complexity are
interdependent
> aspects.
> As I've written this defines 9/4 simpler than 7/5 for the
> best "relationship consonance".
> This is the reason why the chord 1/1, 7/5 and 3/2 is more complex
> than 1/1, 9/4 and 3/2: 9/4 is simpler than 7/5.

But look at the *intervals* in the chords!

chord 1/1, 9/4 and 3/2 has a 3:2, another 3:2, and a 9:4.

chord 1/1, 7/5 and 3/2 has a 7:5, a 15:14, and a 3:2.

See that 15:14 interval? That has *large numbers* in it. Thus this
chord is going to be *considerably rougher* than the other chord.

I had no need, or purpose, to call upon any "relationship consonance"
to reach this conclusion. I simply observed the size of the numbers
in the intervals.

> So I think that odd numbers ratios complexity measure don't work.

It works great in the example above, because 15 is such a large odd
number.

> Primes measure works better.

I reach the opposite conclusion.

> >> I think a good complexity measure would have
> >> to work in these cases too.
> >
> > It works exactly as it should, since 15/12 and 5/4 are identical.
>
> If 15/12 and 5/4 would be "identical"

They're both 386 cents, right?

> odd numbers measure would work
> in an "identical" way.

What do you mean?

> I think there are two components in perceiving complexity of dyads.
> The first is based upon the numbers of common harmonics and the
> critical bandwith. I think that this aspect is very related to the
> subject of perception.

Yes, and the measures based on these quantities point to the size of
the numbers of the (reduced ratios), regardless of the primes
involved, determining the relative concordance of simple ratios. You
can read Helmholtz, Kameoka & Kuriyagawa, or Sethares, and you'll see
the same result for this part of the investigation -- size of the
numbers matters, prime factorization doesn't.

> The second thing is the intrinsecal object complexity.
> Considering 2:1, 3:1 and 7:1 i listen to pairs of sounds with all
> the harmonics in common.

I'm not sure exactly what you mean here. To me, the second thing
consists in how well a chord resembles a harmonic series -- and this
applies even when there are no harmonics and no critical band
roughness to speak of! But since we're talking about dyads, there's
no pressing need to follow this angle anyway . . .

> Anyway I hear *different* things. I feel
> this difference is based on primes families.

Prime factorization is very useful in certain contexts. At least we
agree on that.

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

/tuning/topicId_53868.html#53872
>
> Yes, and the measures based on these quantities point to the size
of
> the numbers of the (reduced ratios), regardless of the primes
> involved, determining the relative concordance of simple ratios.
You
> can read Helmholtz, Kameoka & Kuriyagawa, or Sethares, and you'll
see
> the same result for this part of the investigation -- size of the
> numbers matters, prime factorization doesn't.
>

***Just out of curiousity, what does Monz think about this, or is
this just part of that "old argument" that has been going on and off
for years now... (??)

JP

hi Joe,

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_53868.html#53872
> >
> > Yes, and the measures based on these quantities point to
> > the size of the numbers of the (reduced ratios), regardless
> > of the primes involved, determining the relative concordance
> > of simple ratios. You can read Helmholtz, Kameoka & Kuriyagawa,
> > or Sethares, and you'll see the same result for this part of
> > the investigation -- size of the numbers matters, prime
> > factorization doesn't.
> >
>
> ***Just out of curiousity, what does Monz think about this,
> or is this just part of that "old argument" that has been
> going on and off for years now... (??)
>
> JP

honestly, i haven't really bothered my mind with concordance
measures for at least 5 years now. been too busy digging
into tuning history, and lately working on the software and
webpages.

these days on subjects like this i'm inclined to just agree
with Paul ... especially when Gene, Dave Keenan, and Graham
also all agree with him, which happens more often that you
might think.

i'm still "partial" to the primes (hee hee). i'm well aware
that as soon as one gets not too far up into the prime series
(say around 23 or so, and perhaps even as soon as 13) the
distinctions begin to blur. but for at least the lowest
few primes (3, 5, 7, 11) *WITHIN A RATHER LOW EXPONENT-LIMIT*
(perhaps +/-3), there is a great distinction in sound from one
to the next, and i will stubbornly cling to the belief that
that is important.

-monz

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_53868.html#53872
> >
> > Yes, and the measures based on these quantities point to the size
> of
> > the numbers of the (reduced ratios), regardless of the primes
> > involved, determining the relative concordance of simple ratios.
> You
> > can read Helmholtz, Kameoka & Kuriyagawa, or Sethares, and you'll
> see
> > the same result for this part of the investigation -- size of the
> > numbers matters, prime factorization doesn't.
> >
>
> ***Just out of curiousity, what does Monz think about this, or is
> this just part of that "old argument" that has been going on and
off
> for years now... (??)
>
> JP

Usually Monz isn't trying to rank concordance. Instead, he's talking
about "affect" which is, of course, more elusive and harder to set up
psychological experiments to test. His theory that there is such a
thing as "prime-affect", etc., hasn't been made quantitatively
precise, as far as I know, but Monz tends to separate this issue from
the concordance issue, so that there is no necessary conflict or
argument here.

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

/tuning/topicId_53868.html#54155

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > /tuning/topicId_53868.html#53872
> > >
> > > Yes, and the measures based on these quantities point to the
size
> > of
> > > the numbers of the (reduced ratios), regardless of the primes
> > > involved, determining the relative concordance of simple
ratios.
> > You
> > > can read Helmholtz, Kameoka & Kuriyagawa, or Sethares, and
you'll
> > see
> > > the same result for this part of the investigation -- size of
the
> > > numbers matters, prime factorization doesn't.
> > >
> >
> > ***Just out of curiousity, what does Monz think about this, or
is
> > this just part of that "old argument" that has been going on and
> off
> > for years now... (??)
> >
> > JP
>
> Usually Monz isn't trying to rank concordance. Instead, he's
talking
> about "affect" which is, of course, more elusive and harder to set
up
> psychological experiments to test. His theory that there is such a
> thing as "prime-affect", etc., hasn't been made quantitatively
> precise, as far as I know, but Monz tends to separate this issue
from
> the concordance issue, so that there is no necessary conflict or
> argument here.

***Hmmm. Sounds like "affect" is some kind of metaphysical JI
aura... interesting... :)

JP