Aural Nonlinearities restrict Analysis

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*.

This effect results from the non-linear slope (in log-log decibel coordinates)
of the *perceived loudness* of a sinusoidal tone existing at the same frequency
(and secondary "frequency dependent" effects measured for a sinusoidal tone
existing at different frequencies), relative to the applied sound pressure level
(SPL) in Decibels (dB).

This effect is approximately equal to a value of ( (SPL)b / (SPL)a ) ^ (2/3)
at SPL levels greater than 40 Phons (in SPL), around 40 dB above the
lowest threshold of human hearing. This represents an approximate 1.5:1
"compression ratio" over an 60 dB "input dynamic range" (up to 100 Phons),
resulting in approximately 20 db of non-linear "loudness reduction" occuring.

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).
_______________________________________

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?"

___________________________________

Recalling the experience in:

/tuning/topicId_23887.html#23898
[/tuning/topicId_23887.html#23898]

PE: Again, Sethares, working with me, was unable to ascertain exactly how
to use his formula. And his formula fails pretty badly for chords of
three or more notes, where it fails to reproduce our observations
that otonal chords are more consonant than utonal chords.

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.
_______________________________

In "harmonic_entropy" message #28:

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 . . . and what about negative dB
values? They're present in any waveform, their extent depending on your
arbitrary choice of a zero-dB point, but they contribute perversely to the
normalization above (the dB numbers could easily sum to zero, or to a
negative number) . . . and you don't really mean "energy" above, since
energy goes as the square of the amplitude, while dBs measure log-amplitude --
correct?

Sethares:
>All the comments above, though, only make sense within
>a limited range... certainly using dB to approximate amplitude
>is good to first approximation

PE: You say "amplitude" but of course dB is logarithmic in amplitude . . . the
two are in no way interchangeable, as Ed Borasky's comments show . . . do
you really mean "loudness"?

_______________________

In "harmonic_entropy" message #29

Sethares:
surely we must be using our numbers to mean loudness,
which is normally measured in dB, with respect to
some arbitrary point at the threshold of hearing...
youre right that we cant be adding negative "loudnesses"
but I do agree that there is a scaling issue here...

Im not sure what the best "zero-point" is.
Maybe zero should be equated to the noise floor?
Now I understand why you refrenced to the smallest
partial, because in a synthetic example the noise floor
is essentially infinite?

So I guess the proper normalization would involve
equal loudnesses among the two sounds?
__________________________________

In harmonic_entropy message #530, J Gill's asked some questions regarding the
conditions under which the "Harmonic Entropy" model is applicable for "complex"
or "harmonic dyads".

> > PE: ..."(b)the multiplicity of partials will represent several
> > independent
> > sources of information for the same ratio-interpretations"

> JG: Would not (b) directly above imply that - for the equivalence
> of "ratio-interpretations" to exist from "several independent sources
> of information" (those sources being the individual complex tones
> from which the dyad is constructed) - the spectral amplitudes of each
> of the overtones of the individual fundamental frequencies of such
> individual complex tones must be equivalent. That is -the (steady
> state, as well as the transient) frequency spectrums of the (complex)
> tones number 1 and number 2 from which the dyad is constructed must
> be identical (or nearly identical) in order for your assumption (b)
> directly above to be valid?

JG: Additionally, due to the nonlinear amplitude transfer function of
perceived loudness levels [shown as approximately equal to a value of
( (SPL)b/(SPL)a )^(2/3) at SPL levels greater than40Phons (in SPL),
which is equal to 0.1 SONE (in LOUDNESS UNITS, being 20 dB above the
threshold of human hearing)] [from "Music, Physics, and Engineering",
Olson, 1967, page 252], is it not also true that, in order that "the
spectral amplitudes of each of the overtones of the individual
fundamental frequencies of such individual complex tones" remain
identical (or nearly identical) relative to the amplitudes of those
fundamental frequencies - that the two identical (steady-state as
well as transient) frequency spectrums of the (complex) tones number
1 and number 2 from which the dyad is constructed ALSO be of equal
(or near equal) scalar amplitude throughout the time period during
which each of the individual complex tones are sounded(steady-state
as well as transient)?
NOTE: The phrases "steady-state" as well as "transient" are utilized
above in an inclusive manner to describe all phases of the process of
time-varying amplitude envelopes multiplying each of the values of
the individual complex tones (existing for some given time duration)
from which the dyad is constructed.
______________________________

In harmonic_entropy mesage 531, Paul Erlich (responding to J Gill's questions
regarding the conditions under which the "Harmonic Entropy" model is applicable
for "harmonic dyads") wrote:

>JG: That is - the (steady
> state, as well as the transient) frequency spectrums of the (complex)
> tones number 1 and number 2 from which the dyad is constructed must
> be identical (or nearly identical) in order for your assumption (b)
> directly above to be valid?

PE: For the assumption to be valid, both tones would have to have a fairly
complete complement of overtones, present with decent amplitude, in the
frequency range in question.
________________________

J Gill: For reasons other than amplitude nonlinearities, as well as for reason
of amplitude nonlinearities, it appears (to me) that the application of
"Harmonic Entropy" to a ("complex" or "harmonic") dyad (where both tones of the
dyad contain harmonics of the individual fundamental frequencies of the
individual notes' pitch) is premised (in the ideal sense of the model) upon the
two "complex" notes to be considered having:

(1) Identical measured *loudness* levels for each of the fundamental frequency
components; and

(2) Identical relative *loudnesses* for each member of the set of the
"harmonics" existing at integer multiples of the individual fundamental
frequency components.

The sinusoidal case of applying the "Harmonic Entropy" concept removes
consideration '(2)' directly above, but appears to, as well, be subject to
consideration '(1)' directly above.
_______________

In "tuning" message #23898, Paul appears to affirm the considerations '(1)' and
'(2)' listed directly above:

/tuning/topicId_23887.html#23898
[/tuning/topicId_23887.html#23898]

>JG: It appear that all such systems attempting to map our
(nonlinear) "aural minds", and particularly the more descriptive
>approaches (which attempt to account for timbre over and above the
>sinusoid) are inherently subject to such nonlinearities in the case
>of *differing* amplitudes between two (or more) simultaneously
>sounded tones (whether sinusoidal or complex) to be.

Paul responds:

>(a) I woudln't call that a "nonlinearity".

JG: Would you call it ("aural loudness" perception) a *linear system*?

Paul's response (continued):

(b) While Sethares' roughness is very sensitive to amplitude EVEN
HOLDING THE AMPLITUDES OF THE TWO TONES EQUAL TO ONE
ANOTHER, harmonic entropy is not.

JG: It seems that to your statement should be added : "... as long as the
perceived 'loudness levels' of each of the fundamental frequencies (as well as
the perceived 'loudness levels' of each and every one of the partials occuring
at all of the various harmonic multiples of each of the fundamental frequencies)
remains *matched* in value at all times considered in analysis.

JG: It also seems that, due to the "frequency-selectivity" of the family of
"Fletcher-Munsen" curves further complicating the picture (as I indicated
above), arestriction of frequencies considered of around 300 - 3000 Hz is
imperative, in order to avoid further errors in approximating an "equal
loudness" of the values of all of the individual frequency components of each of
the complex (fundamental plus harmonics) tones sounded simultaneously.

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

J Gill

I replied to this at harmonic_entropy@yahoogroups.com . . .