the psychoacoustics of beating

On another thread Mike Battaglia said: "The properties of what causes beating is a line of research that has been explored extensively in the psychoacoustics literature. For pure tones, beatlessness corresponds to the tones being far apart, not their being aligned to some just ratio. Beating is caused by critical band interactions in the cochlea, not f0 estimation as taking place in the brain."

I'm very interested in the phenomena of beating. I posted some tunings with equal & proportional beating effects here some time back. People were very kind & helpful in the subsequent discussion even though my own comments included several mistakes that I didn't always recognize until well after they were pointed out. I got the impression that you all are very knowledgeable & willing to be patient when explaining things to people like me who are less musically or mathematically accomplished than you are.

If you all would provide some links & educational comments regarding psychoacoustic studies of beating, & perhaps also regarding ethnomusics where beating plays an important role, I'd appreciate it.

Restricting ourselves to harmonic instrumental timbres, I'm especially interested in what is required for a non-tonic note to be perceived as beating against a tonic partial such as 5/3 that has one or more odd factors in both numerator & denominator. I don't hear such beating in a dyad where such a non-tonic & the tonic are the only notes played, but the additional non-tonic notes required to make some triad or tetrad occasionally suffice to trigger my perception of beating against that odd-factor-numerator-denominator (OFND) tonic partial.

I'm developing some guesses about how this works but they're not very educated guesses. For instance, I'm not even sure how much physical acoustic energy OFND partials have compared with harmonic overtone partials. Wringing such guesses out of my own ignorant brain could be wasted cranial twisting & squeezing effort if the answers are already extant in the literature.

I might better grasp the critical band concepts discussed by Sethares if you all had a go at explaining them. But, since I'm confining my questions to harmonic timbres, your explanation can be simpler than Sethares' own, which addresses the full timbral range.

Please help if you can! Thanks.
==
MLV aka Manu Phonic

On Fri, Aug 12, 2011 at 7:34 AM, manuphonic <manuphonic@...> wrote:
>
> On another thread Mike Battaglia said: "The properties of what causes beating is a line of research that has been explored extensively in the psychoacoustics literature. For pure tones, beatlessness corresponds to the tones being far apart, not their being aligned to some just ratio. Beating is caused by critical band interactions in the cochlea, not f0 estimation as taking place in the brain."
>
> I'm very interested in the phenomena of beating. I posted some tunings with equal & proportional beating effects here some time back. People were very kind & helpful in the subsequent discussion even though my own comments included several mistakes that I didn't always recognize until well after they were pointed out. I got the impression that you all are very knowledgeable & willing to be patient when explaining things to people like me who are less musically or mathematically accomplished than you are.
>
> If you all would provide some links & educational comments regarding psychoacoustic studies of beating, & perhaps also regarding ethnomusics where beating plays an important role, I'd appreciate it.

You can check out Plomp and Llevelt's work for the basics, and I
believe Terhardt has some good stuff as well. Here's a good site that
seems to explain the basics rather well, from a psychoacoustic
standpoint: http://www.avatar.com.au/courses/PPofM/consonance/percept.html

There are other, more abstract ways in which beating can play a role
in the perception of sounds, such as the work I did with "periodicity
buzz," but you're probably better off putting that off until you get
the basics first. Periodicity buzz is more or less an enhanced version
of the equal beating stuff you were working on.

> Restricting ourselves to harmonic instrumental timbres, I'm especially interested in what is required for a non-tonic note to be perceived as beating against a tonic partial such as 5/3 that has one or more odd factors in both numerator & denominator. I don't hear such beating in a dyad where such a non-tonic & the tonic are the only notes played, but the additional non-tonic notes required to make some triad or tetrad occasionally suffice to trigger my perception of beating against that odd-factor-numerator-denominator (OFND) tonic partial.

If the timbres are sine waves, the only thing that will matter is how
close the two notes are to one another. If they're harmonic, you have
to treat each harmonic of each timbre as its own "note" and figure out
how close each of those are to one another as well. It all depends on
the proximity of the two tones to one another. You can play two notes
1230 cents apart with sine waves and you won't notice ay beating at
all if there are sine waves, for instance.

-Mike

Hi Mike, thanks. The avatar.com link you gave hints at something interesting:

> Now consider the same beating effect between two tones in terms of
> their separation in fractions of a critical bandwidth. As we have seen,
> at 500 Hertz the critical band is somewhere around 100 Hertz wide.
> Let us define 100 Hertz as "unity" and consider fractions of that band.

From that it would seem that the critical bandwidth varies with the tonic frequency. Otherwise experts would simply say the critical band is around 100 Hz wide without specifying a tonic at 500 Hz first. But how does the critical bandwidth vary with frequency? I'll guess logarithmically, like most things in the realm of pitch perception, but I wish I had more than a guess.

If my guess is correct then the dyadic sine-wave timbre dissonance discussed at the avatar.com link is maximal at about 84 cents & the width of the critical band is about 316 cents. Plausible, I suppose. Be that as it may....

Harmonic timbres, not sine waves, are where I want to go with this. Given a tonic note in some harmonic timbre I want to understand what's required for a non-tonic note to be perceived as beating against a tonic partial that's a just interval but not a simple overtone or undertone, a ratio like 6/5 or 15/11 with odd factors in both numerator & denominator (OFND). My exploration suggests that a triad or tetrad must be sounded to make such beating perceptible.

You stated:

> If the timbres are sine waves, the only thing that will matter is how
> close the two notes are to one another. If they're harmonic, you have
> to treat each harmonic of each timbre as its own "note" and figure out
> how close each of those are to one another as well.

This would suggest that OFND beating only seems to be against a tonic partial but is really against a partial of some non-tonic note in the chord that's required to make the beating perceptible. For instance, if beating against the tonic's 5/3 is audible in a chord with a non-tonic near 4/3, maybe what it's really beating against is the 5/4 overtone of that non-tonic.

That idea poses this challenge. I know how fast a non-tonic note would beat against 5/3 of the tonic if that were a tonic partial with a significant acoustic energy peak. Let's call that beating rate R. Now play another non-tonic note near 4/3 of the tonic, the timbre of all notes played being harmonic. When I run the numbers, I cannot satisfy myself whether the first non-tonic note should beat against 5/4 of the second non-tonic note at R or 2R.

In my explorations I'm hearing R, but that could be true even at 2R if the chord includes another beat at rate R that reinforces alternating waves of the 2R beat.

Any insights?
==
MLV aka Manu Phonic

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Aug 12, 2011 at 7:34 AM, manuphonic <manuphonic@...> wrote:
> >
> > On another thread Mike Battaglia said: "The properties of what causes beating is a line of research that has been explored extensively in the psychoacoustics literature. For pure tones, beatlessness corresponds to the tones being far apart, not their being aligned to some just ratio. Beating is caused by critical band interactions in the cochlea, not f0 estimation as taking place in the brain."
> >
> > I'm very interested in the phenomena of beating. I posted some tunings with equal & proportional beating effects here some time back. People were very kind & helpful in the subsequent discussion even though my own comments included several mistakes that I didn't always recognize until well after they were pointed out. I got the impression that you all are very knowledgeable & willing to be patient when explaining things to people like me who are less musically or mathematically accomplished than you are.
> >
> > If you all would provide some links & educational comments regarding psychoacoustic studies of beating, & perhaps also regarding ethnomusics where beating plays an important role, I'd appreciate it.
>
> You can check out Plomp and Llevelt's work for the basics, and I
> believe Terhardt has some good stuff as well. Here's a good site that
> seems to explain the basics rather well, from a psychoacoustic
> standpoint: http://www.avatar.com.au/courses/PPofM/consonance/percept.html
>
> There are other, more abstract ways in which beating can play a role
> in the perception of sounds, such as the work I did with "periodicity
> buzz," but you're probably better off putting that off until you get
> the basics first. Periodicity buzz is more or less an enhanced version
> of the equal beating stuff you were working on.
>
> > Restricting ourselves to harmonic instrumental timbres, I'm especially interested in what is required for a non-tonic note to be perceived as beating against a tonic partial such as 5/3 that has one or more odd factors in both numerator & denominator. I don't hear such beating in a dyad where such a non-tonic & the tonic are the only notes played, but the additional non-tonic notes required to make some triad or tetrad occasionally suffice to trigger my perception of beating against that odd-factor-numerator-denominator (OFND) tonic partial.
>
> If the timbres are sine waves, the only thing that will matter is how
> close the two notes are to one another. If they're harmonic, you have
> to treat each harmonic of each timbre as its own "note" and figure out
> how close each of those are to one another as well. It all depends on
> the proximity of the two tones to one another. You can play two notes
> 1230 cents apart with sine waves and you won't notice ay beating at
> all if there are sine waves, for instance.
>
> -Mike
>

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
>
>
> From that it would seem that the critical bandwidth varies with the tonic frequency. Otherwise experts would simply say the critical band is around 100 Hz wide without specifying a tonic at 500 Hz first. But how does the critical bandwidth vary with frequency? I'll guess logarithmically, like most things in the realm of pitch perception, but I wish I had more than a guess.
>

I like this presentation of the critical band:

http://www.sfu.ca/sonic-studio/handbook/Critical_Band.html

because it shows in an easily understood graphical way how our common musical intervals correspond with it and the limit of frequency discrimination. As a jazz writer, I also find it interesting because the level of discrimination of half-tones is only in a fairly specific part of the lower frequency range, which happens to correspond just about exactly with some modernistic chord voicing styles that put half-steps in the same range.

Good link, nicely informative, thanks. The diagrams there, & Appendix E one click away, show clearly that my guess was wrong. The critical bandwidth for distinguishing simultaneous tones varies with center frequency but not logarithmically, in a complex near-linear curve that's evidently a function of the basilar membrane geometry in the inner ear.

At the lower end of the typical human hearing range critical bandwidth is 0.67 of the center frequency at 50 Hz; that fraction swiftly drops through 0.29 as the center freq rises through 350 Hz; then the drop rate slows as tbe fraction nears 0.2 at 700 Hz; from 1370 Hz till 2500 Hz the critical bandwidth remains near 0.15 of center freq; then the fraction slowly rises again as the center freq rises above 2900 Hz; the highest center freq shown is 13500 Hz, where the critical bandwidth is 0.26 of the center frequency. Or so says Appendix E:

http://www.sfu.ca/sonic-studio/handbook/Appendix_E.htmL

As the author points out, stitch the critical bandwidths together with minimal overlap & you get only 24 maximally distinctive center frequencies in the whole audible range.

Cheers!
==
MLV aka Manu Phonic

--- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@> wrote:
> >
> >
> > From that it would seem that the critical bandwidth varies with the tonic frequency. Otherwise experts would simply say the critical band is around 100 Hz wide without specifying a tonic at 500 Hz first. But how does the critical bandwidth vary with frequency? I'll guess logarithmically, like most things in the realm of pitch perception, but I wish I had more than a guess.
> >
>
> I like this presentation of the critical band:
>
> http://www.sfu.ca/sonic-studio/handbook/Critical_Band.html
>
> because it shows in an easily understood graphical way how our common musical intervals correspond with it and the limit of frequency discrimination. As a jazz writer, I also find it interesting because the level of discrimination of half-tones is only in a fairly specific part of the lower frequency range, which happens to correspond just about exactly with some modernistic chord voicing styles that put half-steps in the same range.
>

> As the author points out, stitch the critical bandwidths together
> with minimal overlap & you get only 24 maximally distinctive
> center frequencies in the whole audible range.

The first of those center frequencies is given as 50 Hz but, for it, the critical bandwidth is undefined. The other 23 center freqs, as multiples of the 50 Hz first freq, are at:

3, 5, 7, 9, 57/5, 14, 84/5, 20, 117/5, 137/5, 32, 37, 43, 50, 58, 68, 80, 96, 116, 140, 170, 210 &, lastly, 270.

Now I'm wondering what musics could be made & how they'd sound using only those 24 maximally distinctive center freqs as notes. Timbrally these critical bandwidth findings are based on pure sine waves. In such timbres there'd be virtually no dyadic roughness, but other timbres would likely present themselves to the improviser or composer.

Cheers!
==
MLV aka Manu Phonic

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
>
> Good link, nicely informative, thanks. The diagrams there, & Appendix E one click away, show clearly that my guess was wrong. The critical bandwidth for distinguishing simultaneous tones varies with center frequency but not logarithmically, in a complex near-linear curve that's evidently a function of the basilar membrane geometry in the inner ear.
>
> At the lower end of the typical human hearing range critical bandwidth is 0.67 of the center frequency at 50 Hz; that fraction swiftly drops through 0.29 as the center freq rises through 350 Hz; then the drop rate slows as tbe fraction nears 0.2 at 700 Hz; from 1370 Hz till 2500 Hz the critical bandwidth remains near 0.15 of center freq; then the fraction slowly rises again as the center freq rises above 2900 Hz; the highest center freq shown is 13500 Hz, where the critical bandwidth is 0.26 of the center frequency. Or so says Appendix E:
>
> http://www.sfu.ca/sonic-studio/handbook/Appendix_E.htmL
>
> As the author points out, stitch the critical bandwidths together with minimal overlap & you get only 24 maximally distinctive center frequencies in the whole audible range.
>
> Cheers!
> ==
> MLV aka Manu Phonic
>
>
> --- In tuning@yahoogroups.com, "bigAndrewM" <bigandrewm@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@> wrote:
> > >
> > >
> > > From that it would seem that the critical bandwidth varies with the tonic frequency. Otherwise experts would simply say the critical band is around 100 Hz wide without specifying a tonic at 500 Hz first. But how does the critical bandwidth vary with frequency? I'll guess logarithmically, like most things in the realm of pitch perception, but I wish I had more than a guess.
> > >
> >
> > I like this presentation of the critical band:
> >
> > http://www.sfu.ca/sonic-studio/handbook/Critical_Band.html
> >
> > because it shows in an easily understood graphical way how our common musical intervals correspond with it and the limit of frequency discrimination. As a jazz writer, I also find it interesting because the level of discrimination of half-tones is only in a fairly specific part of the lower frequency range, which happens to correspond just about exactly with some modernistic chord voicing styles that put half-steps in the same range.
> >
>