Reply to JG and PE

Hi guys, guess Im coming in the middle of this, so forgive if I misunderstand
something:

J Gill: It appears that various (amplitude dependent) nonlinear effects [as detailed
in "Music, Physics, and Engineering", Second Edition, Olson, 1967
(Figures 7.8 - Loudness versus loudness levels, 7.9 - Loudness versus Intensity
Level, and 7.10 - Contour lines of equal loudness for normal ears, after Fletcher
and Munson, pages 252-253)], may present (completely unavoidable) complications
for the accurate determination of the "musical spectrum" *as perceptually experienced*.

Yes, its absolutely true that whats important is the perceived loudness,
rather than the absolute magnitude... but the situation is even
more complicated than you suggest. At one point I had "prewarped"
the loudness of the partials to account for the Fletcher Munson curves,
and found that this didnt make much difference in the results...
whats really important is not the actual amplitudes of partials,
but the amplitudes of the beating... to take a ridiculously extereme
case, suppose one tone has a partial at 40,000 Hz (inaudible)
and another at 40,100 Hz, also inaudible. However, if the two are
sounded simultaneously, then there is a beating at 100 Hz,
which is audible (this is kind of how a theremin works, btw).
So, the Fletcher-Munson curves (which would say to put the
perceived volume of those partials in both sounds to zero)
doesnt really help...

_______________________________________

JG: Paul, evidently referring to applying a (dyadic or tetradic) "discordance" algorithm
which appears to have been generated by Bill Sethares (and referred to in a thread
entitled "Sethares algorithm) in "harmonic_entropy" message #27:

PE: "... if I use a fixed waveform and simply adjust the volume up
or down (by adding a constant to all the dBs), I get very different curves.
Is that the way it's supposed to work, Bill?"

Yes. Again, to take an extreme case, if the volumes were zero,
then the dissonance should be zero (the most consonant sound is silence!),
whereas the dissonance increases for louder sounds - which
strikes me as plausible, though arguable.
Unfortunately, there are few concrete psychoacoustic studies
that pin down the effects of amplitude. In my algorithm from JASA,
for instance, I just used a multiplicative term that
seemed to work OK for midrange values, and that also
has the right limits as volume goes to zero.
This, along with masking effects, are probably the two
most significant simplifications.

___________________________________

PE: The problem Sethares and I were finding was that, depending on the
decibel level you assumed (yes, by "amplitude" Sethares really
means "decibel level"), you can get a wildly different dissonance
curve.

I do use "decibel-level" as an approximation to "perceived loudness"
(and as Ive been chided for before, have often called it "amplitude").
This could clearly be refined by using a better measure of sones
(perceived loudness).

_______________________________

Sethares:
>I think the normal thing to do when comparing different waveforms is
>to equalize the energy in the signal (add up all the dB numbers, then
>divide by the total to normalize to "1", for instance).
>Then youve got a "fair" comparison between the two sounds.

PE: hmm . . . but the results will be very different depending on whether you
normalize to "1" or to some other constant...

There is a scaling issue, and what Ive usually done is, as stated
above, to call the loudest partial as "1" and express all the others
as fractions - so a partial half as loud would be "0.5", and a quarter
as loud would be "0.25". When combining partials with different amplitudes,
the beating is attenuated - the model really only captures this
qualitatively.

_______________________________

JG: Wishing that this "aural analysis" business was simpler, but accepting the fact that it is (by its inherently complicated nature) *not*,

Amen to that. Though if it were simpler, things might
sound pretty strange!

Bill Sethares

--- In tuning@y..., SETHARES@E... wrote:

> to take a ridiculously extereme
> case, suppose one tone has a partial at 40,000 Hz (inaudible)
> and another at 40,100 Hz, also inaudible. However, if the two are
> sounded simultaneously, then there is a beating at 100 Hz,
> which is audible (this is kind of how a theremin works, btw).

A theremin works by combining the two tones in a non-linear way, and
what you hear is not beating but a difference tone. The two phenomena
are quite distinct perceptions, and it is a very common error to
confuse or identify them, possibly because they often occur together
at the same frequency. In the case of 40,000 Hz and 40,100 Hz tones,
this beating would _not_ be audible, but if non-linearities occur
because of high loudness or other distortion, a 100 Hz _difference
tone_ may be heard.
>
> Sethares:
> >I think the normal thing to do when comparing different waveforms
is
> >to equalize the energy in the signal (add up all the dB numbers,
then
> >divide by the total to normalize to "1", for instance).
> >Then youve got a "fair" comparison between the two sounds.
>
> PE: hmm . . . but the results will be very different depending on
whether you
> normalize to "1" or to some other constant...
>
> There is a scaling issue, and what Ive usually done is, as stated
> above, to call the loudest partial as "1" and express all the others
> as fractions - so a partial half as loud would be "0.5", and a
quarter
> as loud would be "0.25". When combining partials with different
amplitudes,
> the beating is attenuated - the model really only captures this
> qualitatively.

Yes, only qualitatively, and a bit arbitratily, you'd admit?

--- In tuning@y..., SETHARES@E... wrote:

> JG: Wishing that this "aural analysis" business was simpler, but accepting the fact that it is (by its inherently complicated nature) *not*,
>
> Amen to that. Though if it were simpler, things might
> sound pretty strange!
>
> Bill Sethares

J Gill:

Sethares, in the course of discussing mathematical algorithms which endeavor to predict the perceived aural "dissonance" of combinations of various simultaneously sounded musical tones (which may contain harmonics of the fundamental frequency components present, known as "partials") points out that (from his experience):

"... whats really important is not the actual amplitudes of partials,
but the amplitudes of the beating... to take a ridiculously extereme
case, suppose one tone has a partial at 40,000 Hz (inaudible)
and another at 40,100 Hz, also inaudible. However, if the two are
sounded simultaneously, then there is a beating at 100 Hz,
which is audible (this is kind of how a theremin works, btw).
So, the Fletcher-Munson curves (which would say to put the
perceived volume of those partials in both sounds to zero)
doesnt really help...".

Erlich, in the course of responding to Sethares' post quoted above, points out the necessary qualitative distinction between what is referred to as "beating", and what is known as "difference tones":

"A theremin works by combining the two tones in a non-linear way, and
what you hear is not beating but a difference tone. The two phenomena
are quite distinct perceptions, and it is a very common error to
confuse or identify them, possibly because they often occur together
at the same frequency. In the case of 40,000 Hz and 40,100 Hz tones,
this beating would _not_ be audible, but if non-linearities occur
because of high loudness or other distortion, a 100 Hz _difference
tone_ may be heard.".

JG: Reviewing some research data previously communicated:

The resultant *perceived partials* generated by such non-linearity (at 500 Hz) are the 2nd harmonic (at 40 Phons), the 3rd harmonic (at 60 Phons), the 4th harmonic (at 70 Phons), and the 5th harmonic (at 80 Phons). At around 250 Hz, these "compression thresholds" begin about 10-20 dB lower levels (in Phons) than are listed above. At 1000 Hz (and higher in frequency), these "compression thresholds" begin about 10 dB higher levels (in Phons) than are listed above ("Music, Physics, and Engineering", Second Edition, Olson, 1967, after Wegel and Lane, page 256).

It appears that (if the reported data above is correct) because the "aural mind" (for lack of a more inclusive term) creates "phantom frequency components" (which do not appear in a spectral analysis of the physical sound pressure level, alone) at relatively *low* levels of SPL [such as at 40 Phons, which is 40 dB (a factor of 100) greater than the "lowest threshold of hearing", but is 60 dB (a factor of 1000) *less* than the aural "threshold of pain", at around 100 Phon],
the *distinctions* between the generic term "beat frequency" (where no non-linearities are implied), and the (spectrally perceived, as opposed to physically measurable) existence of "sum and difference frequencies" (which are the resultant spectral "products" of passing a signal through a non-linear medium) are (at best) significantly "blurred" in discussion, due to their "intangibility".

Respectfully Humbled by Nature, J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> the *distinctions* between the generic term "beat frequency" (where
>no non-linearities are implied), and the (spectrally perceived, as
>opposed to physically measurable) existence of "sum and difference
>frequencies" (which are the resultant spectral "products" of passing
>a signal through a non-linear medium) are (at best)
>significantly "blurred" in discussion, due to their "intangibility".

I don't see any intangibility to the distinctions. What exactly do
you mean?