Yahoo Tuning Groups Ultimate Backup tuning Nonlinear Effects on Linear Analysis

/tuning/topicId_23887.html#23898
[/tuning/topicId_23887.html#23898]Graham Breed wrote:

> Sethares mentioned both papers, but used his own formula that's
> qualitatively the similar but simpler to calculate. I could dig it
> out
> for you, so could he :)

Paul Erlich responds:

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.

> GB: I worked this stuff out a while back. I couldn't reproduce K&K's
> quantitative results. I got good curves from Sethares' formula,
> but this
> wasn't with real-world timbres.

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.

> GB: All these measures will break down if you throw too many partials
> at them.
> One of the important parts of Sethares' method is to simplify the
> spectrum to only the most important partials.

PE: Yes -- he justifies this in terms of masking.

> GB: This is a subjective
> process, and so may be a problem with these automatic adaptive
> tuners
> people have in mind.

PE: Factors like vibrato, room acoustics, equipment fidelity, and of
course volume will often be more important variables than timbre for
the perception of consonance and dissonance.

> GB: Partch didn't have a formula that I'm aware of, other than the odd-
> limit.
> He drew what he thought made sense. I thought it was a bit sneaky
> of
> Paul to veer toward a discussion of harmonic entropy ;)

PE: I see the smiley, but I should clarify: I predicted back around '97
(see the On Harmonic Entropy webpage) that an octave-equivalent
formulation of harmonic entropy would resemble the One-Footed Bride.
It did so a lot better than I even had hoped!

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/tuning/topicId_23937.html#23937
[/tuning/topicId_23937.html#23937]

Bill Sethares wrote:

<Partch's one footed bride and dissonance curves:

I havent actually done it, but I'll bet you could get very close
to the one footed bride using the sensory dissonance methods.
While some people dont like the fact that the method
is dependent on the details of the spectrum of the sound, one
advantage is that by using different spectra you can shape
the curve in many ways. In this case, you pretty clearly want to
use a harmonic spectrum, but you are left with a number of degrees
of freedom as regards the amplitudes of the partials, and the
pitch of the fundamental.>>

___________________________________________________

/tuning/topicId_26941.html#26941
[/tuning/topicId_26941.html#26941]

Margot Schulter wrote:

<Setharean tuning/timbre systems alike leads me to favor a subtle and
complex approach to questions of consonance/dissonance. Such concepts
as sensory consonance, Paul Erlich's harmonic entropy, and stylistic
concord/discord may all address important dimensions of the question.

Here I would like especially to address two basic points. The first is
that the "sensory consonance" of Sethares -- a factor or dimension I
might term smoothness/roughness -- can interact with Erlich's harmonic
entropy or what I'll call simplicity/complexity in assorted
permutations.>>

___________________________________________________

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

> > PE: ...<< 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 than 20 PHONS (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: It appears that the 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*.

Returning again to Paul Erlich's comments (excerpted from quoted statements
above):

Returning again to Paul Erlich's statements regarding the "Harmonic Entropy"
concept:

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 amplitudes 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.

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.

As a result, not only will results be "amplitude dependent" (requiring
evaluation at numerous loudness levels in order to "complete the picture"), but
any distinct amplitude level "mixes" between the (sinusoidal or complex) tones
to be analyzed must also be evaluated (individually, as well).

Such requirements, then, appear to require the necessary addition of human ears
"in the test loop" for representative and accurate results to be determined in
the course of testing any "conceptual model" (whether it involve sin/cos signals
or complex signals with harmonic energies).

Humbled every day, J Gill :)

Nonlinear Effects on Linear Analysis

🔗J Gill <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>