Tonalness, Farey series, entropy

[Paul Erlich:]
>The conceptual advantage of the Farey series (which leads to a denominator
>rule) is that is seems reasonable to believe that the brain has a
>"template" of sorts of the harmonic series up to a certain limit. Then
>all intervals within that template are possible interpretations, and
>others aren't. If this is the way it works, it doesn't matter if you are
>holding the lower note, upper note, arithmetic mean, or geometric mean
>constant.

Doesn't the Plomp/Levelt result for two sine waves eliminate the
possibility of any such "template" (and maybe even the whole idea of
tonalness and entropy based on ratios). As I understand it, if such a
template exists it should work just as well for sine waves as for harmonic
timbres. Don't tonalness and your harmonic entropy merely assume that there
are no inharmonic partials, but don't require harmonic ones. I understand
that with low amplitude sine tones very little happens at the octave or
fifth (let alone anthing more complex), and what does happen can be
attributed to combination tones generated by non-linearities.
-- Dave Keenan
http://dkeenan.com

>Doesn't the Plomp/Levelt result for two sine waves eliminate the
>possibility of any such "template" (and maybe even the whole idea of
>tonalness and entropy based on ratios).

It would except there have been other experiments (Joos) which found
that ratios like 5:4 are preferred over mistuings even for low-amplitude
sine waves. The effects are also much more pronounced when there are
more than two tones (just as in the illusory pitch-shift effects we've
been discussing), but there were only two in Plomp/Levelt.

>and what does happen can be
>attributed to combination tones generated by non-linearities.

Absolutely not! Remember the Goldstein experiment we were discussing
recently, where the virtual pitch was measured for off-harmonic
sine-wave complexes? The virtual pitch is _not_ a combination tone!

Let me quote the abstract of Terhardt's May 1974 paper "Pitch,
consonance, and harmony":

*****
Comparison of recent psychoacoustic data on consonance with those on
roughness reveals that "psychoacoustic consonance" merely corresponds to
the absence of roughness and is only slightly and indirectly correlated
with musical intervals. Thus, psychoacoustic consonance cannot be
considered as the basis of the sense of musical intervals. The basisi of
that sense seems to be provided by the _concept of virtual pitch_. This
concept is introduced with a model. The concept accounts for many
psychoacoustic and musical phenomena as, e.g., the ambiguity of pitch of
complex tones, the "residue," the pitch of inharmonic signals, the
dominance of certain harmonics, pitch shifts, the sense for musical
intervals, octave periodicity, octave enlargement, "stretching" of
musical scales, and the "tonal meaning" of chords in music.
*****

So before putting all one's eggs in the Plomp/Levelt/Sethares basket,
one has to at least consider all the phenomena listed above, how well
they can be explained by roughness, how well they can be explained by
virtual pitch, and their musical importance. My own feeling is that
while not all the phenomena that Terhardt lists support his conclusions,
enough do so that virtual pitch cannot be ignored from any informed
discussion of consonance. If nothing else, comparing otonal and utonal
chords of, say, the 11-limit should be evidence enough that roughness
isn't everything.

Consider this clincher from the second page of the same Terhardt
article:

*****
[a long list of examples, this last one is really good:]. . . single
isolated complex tones are the elements of music and never are regarded
to be dissonant. However, a single complex tone can sound very rough,
e.g., when it is produced by a pulse-like sound sourse of low repetition
rate.

Hence it must be concluded:

(1) The experimental results on consonance and roughness appear
to be significant and consistent, and thus provide a solid babsis of _a
certain kind of consonance_, i. e., of _psychoacoustic consonance_. The
psychoacoustic consonance of any sound is related to the absence of
roughness. To this extent, Helmholtz's (1863) consonance theory is
strongly supported.

(2) _However_, psychoacoustic consonance is distinctly different
from another kind of consonance which plays a basic role in tonal music.
The universal importance of harmonic intervals in music cannot be
explained satisfactory [sic] by the concept of psychoacoustic
consonance.
*****

Two things should be interjected at this point: Since the virtual pitch
phenomenon is clearly a psychoacoustic one, the term "psychoacoustic
consonance" for non-virtual-pitch-related consonance is unfortunate.
Thankfully, this term has been replaced by "sensory consonance" in the
recent literature. Secondly, the emphasis on "musical intervals" in the
abstract above is misplaced. A sense of intervals and associated melodic
functions is found in the vast majority of musical cultures and has its
own logic, informed by the sense of similarity at the octave, fifth, and
fourth, but otherwise having nothing to do with consonance.

Terhardt "derives" a model like my Farey series model (in his case, a
Farey series of order 8, although Goldstein found virtual pitch effects
involving higher harmonics) and also comes up with a probabilistic
formulation for the interpretation of the stimulus:

*****
. . . in many cases there exist _several_ maxima of the
virtual-pitch-cue distribution. Hence it is defined that the probability
with which a particular virtual pitch is ascribed to the stimulus
depends on the height of the corresponding maximum.
*****

Terhardt's model is based on a learning process and by considering known
non-linearities (that is, shifts) in the pitch-perception process, his
model accounts for pitch shifts in the template which lead to a
preference for slightly enlarged intervals, something my harmonic
entropy model ignores but Brian McLaren has emphasized fervently.

Terhardt dismisses combination tones as a basis for musical harmony:

*****
as is revealed by the results of, e. g., Plomp (1965), in the sounds of
music, difference tones only rarely have a chance to be perceptually
relevant. In most cases they remain below the threshold of hearing and
in most of the remaining cases are masked with high probability by the
musical sound itself. Moreover, when in special cases difference tones
become audible (e. g., when two flutes are playing a duo), their effect
in general is annoying instead of "harmonic."
*****

This last point I think is especially important -- there are still
people around (e. g., Heinz Bohlen) who think combination tones are the
basis of harmony but I believe this last example is a death-blow to that
theory.

Terhardt admits that his model is imperfect. He does not consider "pitch
phenomena which are produced by 'quasirandom' signals (e. g.,
'time-separation pitch,' 'repetition pitch,' 'periodicity pitch' of
interrupted noise)." But if anything, these phenomena provide further
evidence of the importance of the virtual pitch concept, and certainly
not of a Helmholtz-type "place" theory of hearing.

Some of Terhardt's conclusions:

*****
The realization of musical sounds seems to be governed by the
two foregoing principles which may be termed _the principle of minimal
roughness_ and _the principle of tonal meanings_. Both principles imply
certain requirements for the fundamental-frequency ratios and spectral
configurations of realized _musical chords_. If the pitch shifts in the
peripheral auditory system would not exist, the requirements of both
principles were well compatible. In particular, both principles would
require the same precise small-integer frequency ratios of the
fundamentals of musical tones. However, since there exist the phenomena
of pitch shifts and, thereby, of interval stretching, both requirements
become conflicting.

Musical _scales_ are established in order to make musical
_intervals_ realizable. With respect to the tuning of musical scales it
follows from the third conclusion that _every_ kind of tuning is a
compromise. If a musical scale is tuned strictly according to the
requirements of the "principle of minimal roughness," one obtains the
_just scale_ [Terhardt not really correct on this, as the well-known
comma problem of the pentatonic and diatonic scales demonstrates].
However, this just scale, which by the "classical" theory of music is
considered as an ideal, disregards the requirements of the "principle of
tonal meanings" (i. e., insofar as interval stretching is involved).
Thus, the new theoretical point of view developed in the present article
reveals that even the just scale is a compromise [I would say, a
compromise beyond that revealed by comma problems].

_The equally tempered scale_, chosen for practical purposes (i.
e., to make it easy to play on keyboard instruments in every key) does
look much less "blasphemic" from the new point of view than it does in
the light of the "classical" theory [again, Terhardt commits the common
error of forgetting about syntonic commas, but his point remains]. On
the basis of the foregoing conclusions, the equally tempered scale can
be considered as just one of several compromises.
*****

More to come . . .

Wow! Thanks Paul E., for all that work in posting such an erudite
explanation. I'm convinced.

-- Dave Keenan
http://dkeenan.com