Combination Tones and Optimality

Hey guys, I'm a bit new here but i've been on the XA facebook group for a while, so I'm pretty sure some of you know me. My name is Ryan, but you probably know me as that weirdo that often discusses the 91:90 biome temperament with Mike.

So anyways, on to business. Consider a consonant dyad A:B. What note can we add to turn it into a triad with consonance of a similar degree? This note must not increase the repeat period of the original dyad. In addition, we have to consider the difference tones; if the difference tones created from the addition of a third note compete with the original tones A and B, we have a problem. However if the difference tones have frequencies A and B, they strengthen the original dyad. The only note which meets the above conditions is the combination tone A+B. It keeps the original repeat period of the dyad A:B, and its difference tones support the harmony.

(If A:B is irrational, the repeat period can be thought of as an abstract quality relating to the regularity of the dyad. By adding the combination tone A+B the regularity of the resulting triad is maximized, even if there is no finite repeat period.)

Another hypothetical situation: consider the blackwood fifth of 720 cents, or 3/5 of an octave. Following the previous procedure, we can turn this dyad into a triad of ideal consonance by simply adding the summation tone A+B. In this case the combination tone is 2^(3/5)+1, or 1597.16 cents, about 10 cents sharper than a perfectly just major tenth. Coincidentally this value is really close to the blackwood (20 equal) 400 cent third.

Some other triads using this procedure:

Using the 7-equal fifth:
0, 685.71, [1576.58]

Using the 16-equal fifth:
0, 675, [1570.19]

Using the mavila fourth:
0, 522.86, [1481.09]

So, what do you guys think of creating scales using this technique? I have a few ideas in mind. Someone could possibly start with a rank-1 temperament, convert it into a rank-2 temperament using one of the combination tones as a second lattice axis, and then converting it back into another rank-1 temperament.

Advice and criticism would be appreciated.

-Ryan

On Thu, Jun 9, 2011 at 10:16 PM, domeofatonement
<domeofatonement@...> wrote:
>
> Hey guys, I'm a bit new here but i've been on the XA facebook group for a while, so I'm pretty sure some of you know me. My name is Ryan, but you probably know me as that weirdo that often discusses the 91:90 biome temperament with Mike.

Ryan is the man, except he seems to suggest above that the 91/90 biome
temperament is weird, whereas it is actually awesome. Anyways -

> So anyways, on to business. Consider a consonant dyad A:B. What note can we add to turn it into a triad with consonance of a similar degree? This note must not increase the repeat period of the original dyad. In addition, we have to consider the difference tones; if the difference tones created from the addition of a third note compete with the original tones A and B, we have a problem. However if the difference tones have frequencies A and B, they strengthen the original dyad. The only note which meets the above conditions is the combination tone A+B. It keeps the original repeat period of the dyad A:B, and its difference tones support the harmony.

I think that what you're getting at is optimizing for "periodicity
buzz," which has more to do with having notes spaced apart linearly
from one another than actual nonlinear "combination tones." See this
thread for more information, as well as a visual diagram of what's
going on

http://launch.dir.groups.yahoo.com/group/tuning/message/95699?var=1

Note that "combination tones" aren't what's at work here; it has more
to do with amplitude modulation and beating than anything.

> (If A:B is irrational, the repeat period can be thought of as an abstract quality relating to the regularity of the dyad. By adding the combination tone A+B the regularity of the resulting triad is maximized, even if there is no finite repeat period.)

Don't forget B-A:A:B and A:(A+B)/2:B as well.

> So, what do you guys think of creating scales using this technique? I have a few ideas in mind. Someone could possibly start with a rank-1 temperament, convert it into a rank-2 temperament using one of the combination tones as a second lattice axis, and then converting it back into another rank-1 temperament.

How so do you mean "converting it into a rank-2 temperament?"

But either way, you've sort of arrived at an offshoot to the Wilson
equal-beating approach, where you solve for the choice of generator
that puts the tempered 4:5:6 into some kind of proportional beating
relationship with itself. If you don't mind allowing the temperament
to be irregular, then you might find a superior approach to be to find
recurrent relationships - take these chords for instance as a tuning
of Father temperament around 13-equal

17:20:23:26, 17:20:23:26:29, 17:20:23:26:29:35, 17:20:23:26:29:32:35, etc

These should sound like 13-equal ish chords, but resonant and awesome
in a way that 13-equal itself is not. The reason that they are so
awesome is that when you get into rationally intoned chords like the
above, not only do the fundamentals of each chord beat in sync with
one another (as you demanded with your A:B:A+B example), but if the
overall chord is rational, all of the harmonics will also likewise
beat in sync (assuming a harmonic timbre).

There's a lot more that can be squeezed out of this method, like
finding ways to detune or temper these chords in the least
perceptually disruptive way possible, but I haven't worked it out yet.
If you find the idea interesting maybe it's time to resurrect that
line of inquiry.

-Mike

> Ryan is the man, except he seems to suggest above that the 91/90 biome
> temperament is weird, whereas it is actually awesome. Anyways -

It is weird in a good way. Doesn't "xenharmonic" mean "strange harmony," after all? ;)

> I think that what you're getting at is optimizing for "periodicity
> buzz," which has more to do with having notes spaced apart linearly
> from one another than actual nonlinear "combination tones." See this
> thread for more information, as well as a visual diagram of what's
> going on
>
> http://launch.dir.groups.yahoo.com/group/tuning/message/95699?var=1
>
> Note that "combination tones" aren't what's at work here; it has more
> to do with amplitude modulation and beating than anything.

Hmm, I'll have to look more into it.

> Don't forget B-A:A:B and A:(A+B)/2:B as well.

I was mainly concerned about A:B:A+B because it doesn't increase the effective period length. I suppose B-A:A:B also works, although A:(A+B)/2:B will double the effective period length for the dyad A:B.

> How so do you mean "converting it into a rank-2 temperament?"
>
> But either way, you've sort of arrived at an offshoot to the Wilson
> equal-beating approach, where you solve for the choice of generator
> that puts the tempered 4:5:6 into some kind of proportional beating
> relationship with itself. If you don't mind allowing the temperament
> to be irregular, then you might find a superior approach to be to find
> recurrent relationships - take these chords for instance as a tuning
> of Father temperament around 13-equal

I mean making a lattice with two axes: One axis is an interval from a small EDO, and the other axis is the summation tone resulting from that dyad. Then you turn it into a regular temperament, and from there you can turn it into an EDO. Sorry, I probably confused you by saying rank-2 when I meant rank-3.

And dang, I had the suspicion that my "discovery" had already been investigated. That is the consequence of not doing my homework before I posted.

-Ryan

On Fri, Jun 10, 2011 at 12:11 AM, domeofatonement
<domeofatonement@...m> wrote:
>
> I mean making a lattice with two axes: One axis is an interval from a small EDO, and the other axis is the summation tone resulting from that dyad. Then you turn it into a regular temperament, and from there you can turn it into an EDO. Sorry, I probably confused you by saying rank-2 when I meant rank-3.
>
> And dang, I had the suspicion that my "discovery" had already been investigated. That is the consequence of not doing my homework before I posted.

No, it's not nearly as investigated as well as you might think. In
fact, please go right ahead and investigate it more - all I wanted to
do was give you some theoretical background on what might be going on.
In fact, this topic has been the subject of some contentious debate
recently, with Carl proposing instead that the behavior generated is
more in the brain, although he hasn't put out a concrete hypothesis to
test yet (I've offered to generate listening examples to test it when
he does). But either way, there is something magical about frequencies
that are linearly spaced from one another, and I think it could lead
to lots of improvements in how we handle tuning error if someone could
go into it a bit more.

I don't think that anyone's suggested the rank-2 -> rank-3 approach
you laid out above, although again I'm not sure how perceptually
significant "true" combination tones are, as they often tend to get
confused with AM in this case (and even more often get confused with
VF processing). To me, the greatest asset in putting a chord into a
linearly-spaced arrangement is that it evokes "periodicity" buzz,
which if you understood my above explanation isn't related to
periodicity at all. But if you create something that sounds good, then
I don't care if it comes from critical band effects or inner-ear
nonlinearities.

-Mike