RE: More on spacing: response to PAULE

I am going to respond to this as I read it, so this won't be a perfectly
formulated reply. I will post the derivation of my tonalness algorithm
seperately, as it is mathematically painful and will be difficult to notate
in ASCII.

A general remark: The central pitch processor is the mechanism by which we
perceive a set of harmonic partials as a single note -- the virtual pitch --
with an associated timbre. Just about any instrument in the symphony
orchestra serves as a perfect example. Whether this process is inborn or
acquired, some claim prenatally, is a matter of debate. Its existence is
not.

>how is
>it that you are able to decide that sine waves are "distinguished and
>digestable" when described in harmonic context but suddenly not so when
>renotated as subharmonic?

A large or infinite harmonic series of sine waves will be interpreted by the
central pitch processor as a single complex tone, having a clear
fundamental. All combination tones of the sine waves will fall into this
same harmonic series, reinforcing the sensation. A large subharmonic series
will confuse the central pitch processor, as each pair of tones implies a
different fundamental. Often the perfect fifth will win out due to its
importance in defining the virtual pitch. The combination tones of the
subharmonic series will not conform to the subharmonic series; in the case
of an large subharmonic series, they will fill up nearly all of pitch space,
confusing the central pitch processor further and creating lots of
roughness.

>(2) Or do you define as subharmonic any complex without 2^n as the lowest
>member?

No. A subharmonic complex is one whose relative periods can be expressed
with smaller integers than the relative frequencies. Clearly there is no
distinction for two-note complexes.

>A spacing theory would compare the notations to determine which
>characterization is more likely (if neither is, then the neutral
>characterization is chosen).

Yes, I agree -- if roughness between harmonic partials comes into play. For
sine waves, the subharmonic representaion is irrelevant.

>Your algorithm - and I do wish to examine it
>in detail - strikes me as a root direction theory (and more in the
>Hindemith than in the Rameau direction).

Yes, I believe it partially is that. I have found that there are already
successful algorithms for computing the roughness (Plomp & Levelt) aspect of
consonance (K&K, for example). However, I did not come across any algorithms
that applied the root-finding function of the central pitch processor to
chords in a mathematically consistent way, and tonalness -- or clearness of
root perception -- is the other component of consonance. So I put such an
algorithm together, and it's a total idealization, but is intended to be a
good approximation as long as some partials fall within the optimal
frequency range for the central pitch processor. Given more psychoacoustical
data, such as how the resolution of the central pitch processor varies for a
given set of frequencies and amplitudes, the algorithm can be easily made
more flexible.

>(3) You cite several psychoacoutistical certainties in support of your
>algorithm, I am especially curious to learn of experimental results
>supporting (a) "in a purely psychoacoustical sense, a chord played will
>evoke certain interpretations in terms of the harmonic series"; this may be
>what your algorithm tells me, and is a conjecture that I would like to
>support, but I would like to find experimental evidence in this direction
>(and I have encountered evidence to the contrary: see Boomsliter and Creel
>on the 7th partial);

My musical experience certainly differs from this; for example, how do you
explain the fact that a 5:6:7 diminished triad evokes the missing root of
the dominant seventh chord, if not for the importance of the 7th partial?
Terhardt has done much work in this area as well.

>(b) "the ears central pitch processor simply doesnt
>go that far"; I am completely confused by this: what do you mean by
>"central pitch processor", and how far, exactly does it go? The harmonic
>progression I presented, and its Pythagorean interpretation are simply too
>obiquitous to be dismissed in this way.

You yourself stated that the fundamental of a harmonic relationship can't be
too low in pitch for the relationship to be perceived; even with that
simplistic a model, how can you expect 54:64:81 to be perceived as such, and
with what root?

>(4) The sequential procedure would be something like the following: given
>the complex 500 Hz,
>600 Hz 750 Hz, the first ratio interpreted would be the 3/2 fifth of
>750/500, the initial interpretation would be 3/2 in a harmonic series over
>fundamental 250Hz; however, not finding an intermediate tone between 2 and
>3, successive fifths in the series are scanned to find a similar complex,
>examining each n(2,3) until an intermediate tone is located that matches
>the complex. When, for example 10,12,15 over a fundamental of 50 Hz is
>selected, then it is compared with its subharmonic notation /6,/5,/4. If
>this characterization is more simple, then it is chosen.

Once again, there is no psychacoustic mechanism by which subharmonic
relationships of sine waves can be heard!

>(Need I add the
>fact that the difference tones of this complex are all within the spectrum
>of 50 Hz?

And this supports which interpretation, pray tell?

>That certain tuning
>systems, temperaments among them, may map onto harmonic series is probably
>true, but it requires a definition of intervallic tolerance that seems to
>me to be musically (i.e. culturally, and thus limited to particular times
>and places) defined, and probably outside of the domain of psychoacoustics
>proper:

I disagree. The definitions of intervallic tolerance are nothing more and
nothing less than a charcterization of the psychoacoustic phenomena which
distinguish just intervals -- namely, consonance. This is subdivided into
roughness, whose tolerance has to do with critical bands and beating; and
tonalness, whose tolerance has to do with the precision with which frequency
information is transmitted to the central pitch processor. You are correct
that both aspects are not independent of amplitude and duration, but that
just means that an algorithm such as mine would ideally have to take these
things into account. In principle, it is still the right way of looking at
things.

>Your statement _the ear doesnt continue its harmonic series math from
>one chord to the next_ is unsupported, and probably unsupportable.

I did not mean for it to be taken in such a strong sense. Clearly there are
strong elements of commonality between one chord and the next when their
relationship is a simple one; the dominant-to-tonic progression is a case in
point. And clearly the central pitch processor can extract some information
on virtual pitch from non-simultaneous tones -- the virtual pitch mechanism
is not dependent on phenomena associated with exact simultaneity, such as
difference tones. But while the difference between a 9:8 and a 10:9 is easy
to perceive harmonically, when these intervals represent the melodic
intervals between the roots of non-adjacent chords in a progression, the
difference comes down to how the progressions are handled on paper.

>(7) I cannot resist a few more remarks on the subharmonic. First, although
>for musical purposes we never need get past fairly small numbers, while
>every harmonic complex may be renotated as subharmonic and vice versa, due
>to one of those nice paradoxes in the fundamentals of mathematics - and
>beyond my ability to explain - the number of members in the harmonic series
>is greater than that in the subharmonic series.

What does this sentence mean? And I have a good knowledge of mathematics,
fundamental and otherwise -- what on earth are you talking about?

>Second, the _descent to the
>infinitesimal_ represents to me a series of increasingly complex - indeed,
>counterintuitive - calculations, hence my speculation about increased
>mental activity. The ascription of quantum effects to the infinitesimal
>comes from the intuitionist C. Hennix, resident mathematician at the
>Institute for Psychoanalysis in Paris;

Is this in a context relevant to music, or any other real-world
applications? I'll bet my hat it's pseudo-scientific drivel.

>Third, musical examples of the
>subharmonic are readily available in the repertoire, cf Martin Vogels book
>on the _Tristan_ chord, or this little example from the Rondo K.494, m
>164-165, where the left hand sustains for two measures the chord c4 d4 f4
>ab4, which moves (does not resolve) to c4 f4 a4, i.e subharmonic seventh
>chord from c moving to f major 6-4. Mozart uses this chord, here and
>elsewhere, in a voice leading distinctive from his use of diminished and
>dominant seventh chords. By my spacing characterization, the fourth c-f
>would be too important to overlook, and the traditional (in American
>practice) _half diminished_ description seems not very useful. If this
>were in just intonation, I would choose a subharmonic characterization /8
>/7 /6 /5 over the harmonic 105 120 140 175. I am curious to learn how
>Pauls algorithm will interpret this chord (16 18 21 26? or 24 27 32 38?).

I strongly agree that subharmonic characterizations can be useful in musical
practice and to the future development of music. But the essence of this
characterization lies in the partials, which acheive minimum roughness when
the fundamentals are arranged in _either_ a harmonic or a subharmonic
series. My algorithm deals not with roughness but with tonalness, and
ascribes no harmonic interpretation of c4 d4 f4 ab4 more than a negligible
amount of certainty. For a coarse enough frequency resolution, 9:10:12:14
would win out, but still by a negligible amount. Both the 3:4 interpretation
of c4 f4 and the 5:8 interpretation of c4 ab4 will be more powerful;
therefore the chord will not be heard as a single harmonic complex.

Let me reiterate my view of the equal-tempered minor triad with harmonic
partials. I would tend to characterize it as a subharmonic complex, since
(a) the level of roughness is similar to that of a major triad, and thus a
near-local-minimum, due to the individual intervals being close to 3:2, 5:4,
and 6:5 and (b) several possible harmonic representations accodring to my
algorithm (10:12:15, 16:19:24) lead to essentially the same interpretation:
that of a perfect fifth with an added non-harmonic note. Since the exact
numbers in (a) are clearly more important than the exact numbers in (b), the
former should be included in an overarching representation, thus:
1/6:1/5:1/4. If a chord has favorable roughness characteristics but only
tolerable tonalness characteristics, it just might be a subharmonic complex.


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..................................continued from last week.

According to Golstein[22], the precision with which freqency information is
transmitted to the brain's central pitch processor is between 0.6% and 1.2%
within a certain optimal frequency range. Musical tones normally have
partials within this range, and since harmonic partials are integer
multiples of thier fundamentals, they will yield the same
ratio-interpretation for their fundamentals as would the fundamentals
transposed into this range. (This is why we did not worry about overtones in
the first place.) We can therefore model the tones with normal probability
distributions in log-frequency space; the standard deviation (let's call it
s) used can be between 0.6% and 1.2%, or greater to reflect various
non-ideal conditions. The use of log-frequency space corresponds most
closely with psychoacoustic research and musical practice.

Now the "width" in 17,

log(a(j))-log(b(j))

represents a section of log-freqeuncy space and thus a portion of the
distribution lies within this range. The exact amount of probability
contained withing this range is thus the certainty with which u=f(1)/f(2) is
interpreted as p(j)/q(j), and is proportional to

1/(s*sqrt(2pi))*integral(exp(-.5((x-log(u))/s)^2),dx)

in log-frequency space. Now the width in (19) vanishes for large N, but it
remains proportional to 1/L(j), so (replacing the integral by a product
since W is small compared to s) we define the certainty with which u is
intepreted as p(j)/q(j) as

C(u,p(j)/q(j))=1/(s*sqrt(2pi)L(j))*exp(-.5((log(p(j)/q(j))-log(u))/s)^2
) (20)

(20) can be extended to chords of more than two notes as follows: A chord of
n notes contains n(n-1)/2 different intervals. But since the whole chord
must be related to a single fundamental, any individual interval might not
be in lowest terms, and the "limit" should now be the greatest of all the
odd factors of the numbers used to represent the chord. Denote this limit by
L(t); we therefore define

NC(L(j,t))=NC(L(j))L(j)/L(t)

We then have

C(chord)=product over all intervals(C(L(j,t))^(2/n))

where the exponent 2/n is the number of independent degrees of freedom in
tuning the chord, n-1, divided by the number of factors in the product,
n(n-1)/2. This is done becuse multiplying probabilities of n-1 independent
judgments is the probablity of the overall judgment; the exponent is thus a
correction for the fact that the number of intervals increases faster than
the number of notes. In other words, take the geometric mean certainty, and
multiply it by itself a number of times appropriate to the true number of
degrees of freesom. This effectively reduces the standard deviation s by a
fatcor 1/(sqrt(n-1)), as one would guess from general statistical
considerations.

As for finding the roots of chords, Parncutt states,[23]

The pitch pattern of a complex tone may be recognized even if parts of
the pattern are
missing, or extra elements are added. Similarly . . .the root of a
chord may be
perceived if notes corresponding to harmonics of the root are missing,
or if notes not
corresponding to harmonics are added. For example, the root of the C
major triad is
weakened, but not changed, if the note E (which corresponds to the
fifth harmonic of
C) is replaced by Eb (which doesn't correspond to any normally audible
harmonic of C);
the root (C) is maintained by the strong root implication of the fifth
C-G.

So a good guess would be that for a four-note chord, whichever of its two,
three, or four-note subgroups has an interpretation of the gretest C-value
is the subgroup that determines the root; the root itself is the note
interpreted as a power of 2, since it is then octave-equivalent to the
fundamental. If none of the notes of the chord are interpreted as a power of
two, the chord seems unstable, as the sensation of the fundamental is quite
secure.


[13] Adapted from van Eck, C. L. van Panthaleon. 1981. _J. S. Bach's
Critique of Pure Music_. Princo, Culemborg, The Netherlands, Appendix II.

[14] Parncutt, Richard. 1989. _Harmony: A Psychoacoustical Approach_.
Springer-Varlag, New York, p. 70.

[15] Goldstein, J. L., 1973. "An optimum processor theory for the central
formation of the pitch of complex tones," _J. Acoust. Soc. Amer._ Vol. 54 p.
1496.

[16] Terhardt, E. 1974. "Pitch, consonance, and harmony." _J. Acoust. Soc.
Amer._ Vol 55 p. 1061.

[17] Wightmann, F. L. 1973. "The pattern-transformation model of pitch." J.
Acoust. Soc. Amer. Vol. 54 p. 407.

[18] If we were to fix f(q) and vary f(p), we would only have an upper bound
on q, giving us an infinite series of fractions to consider. Fixing the
center of the interval leads to Mann's criterion, p+q<=N. Mann, Chester D.
1990. _Analytic Study of Harmonic Intervals_. Tustin, Calif. Either way, we
still end up with (7) below, so that the relative certainties of various
interpretations of a given interval will remain the same. (5) is chosen
because the series produced by it display a greater degree of octave
equivalence.

[19] Three proofs of this are given in Hardy, G. H. and Wright, E.M. 1960.
_An Introduction to the Theory of Numbers_. Oxford University Press, London,
Ch. 3.

[20] See Mann, p. 163 for one possible justification.

[21] Partch, p. 184

[22] Goldstein, p. 1499

[23] Parncutt, p. 70.


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