Response to Daniel Wolf

>(1) Does your central pitch processor depends on harmonic partials? Real
>instrumental sounds deviate significantly from a ideal series.

These deviations are small compared with the resolution of the central pitch
processor (~1%). Actually, the central pitch processor works best when the
relationships are slightly stretched, but this is a small effect.

>Does your
>processor make tolerance decisions at both the individual instrument
>(pitch) and the instrumental ensemble (chord) levels?

Yes, absolutely. A chord consisting of several "inharmonic series" could be
analysed in this way. For a chord consisting of harmonic or near-harmonic
series, the partials add no conflicting information as to the root should
be. However, the partials can aid in making this judgment, especially if the
chord tone fundamentals are rather low in pitch.

>It was for precisely
>this complication that I made my initial assumptions of a sine wave
>orchestra.

Do you agree now that this assumption is incompatible with your treatment of
subharmonic chords?

>(2) Are amplitude and phase differences among the partials ignored?

Yes, but phase differences do not affect the central pitch processor.
Amplitude differences may, but as long as the fundamentals are similar in
amplitude and have similar overtone structures, the ratio-interpretation
should be correct.

>(3) Would it be terribly difficult to couple your root-finding function
>with some harmonic distance function in order to address chord
>progressions?

I don't see why it would be difficult. However, the character of chord
progressions seems to me far more subjective and culturally influenced, or
at least far more difficult to model, than the perception of virtual pitch.
Although certainly general statements can be made, such as compounds of
simple harmonics are more important for chord progressions, while higher
harmonics are more important for chord construction.

>(4) The limits on amplitude and durational aspects limit the algorithm to a
>part of the worlds music; would you define this more precisely? In
>general, I find this to be the most difficult part of your work (and of
>other research along these lines) in that you have reified aspects
>particular to a small musical repertoire as psychoacoustical facts.

My theory is only intended to predict the perception of chords under
circumstances of long duration and equal amplitude -- the ideal
circumstances for perceiving complex ratios. In other circumstances, the
parameter that controls the precision of the central pitch processor's input
signals can be adjusted to reflect less precision, and the results will show
a preference for simpler relationships.


Some comments:

>>(1) Naturally, if a triad is tuned 5:6:7, the fundmental frequency of the
>>series on 1 is reinforced;

>what happens, when it is tuned /7:/6:/5, or in a temperament where the
>approximation of 7 is poor?

Then one has an essentially rootless sound, since no simple interpretation
in terms of harmonics gets much certainty. For example, in 22-tet, the a
pitch resolution of 1% leads to 0 6 11 being heard as 7:6:5 with a relative
certainty of 38.5, while 0 5 11 has no such certain interpretation; the most
likely interperetation is as 24:20:17 with a relative certainty of 8.70.
(This is using the version of the algorithm that treats all inversions and
extensions as equivalent).

> If I recall correctly, Boomsliter and Creel
>demonstrated that in harmonic contexts both trained and untrained subjects
>had a preference for the real interval of 225/128 over 7/4, thus showing
>that a lower position within a harmonic series was not necessarily
>preferable to some other, more complex relationship.

They were probably just targeting a slighlty stretched 7/4. Terhardt
addresses some possible explanations for this preference for stretching.

Might I suggest that their results were also far from unbiased.

>(2) My musical experience suggests, however, that diminished, augmented,
>and other intervallically symmetrical pitch complexes function in tonal
>music as ambiguous - neither major nor minor - structures. For this reason,
>I introduced the _neutral_ characterization for pitch spacings that are
>neither harmonic nor subharmonic.

This may often be useful.

>(3) 54:64:81 is a chord that I am able to identify immediately in
>relationship to a previously known tonic of 2^n. I recognize it in both
>terms of harmonic distance from 2^n, from the size of the melodic intervals
>which I have learned to distinguish (thus at least part of my recognition
>is cultural), and from beating.

Yes, but most people have not learned to distinguish these things.

>Moreover, my recognition of this chord cues
>me in on the entire field of relationships including this chord and the
>chord proceeding it. It is this field that seems to determine musically
>dynamic characteristics: chord _progression_; tonicity; _resolution_;
>surprise, etc..

Don't let me dissuade you from trying to formulate a wide-ranging theory of
JI compostition. Sounds good to me.

>(4) The melodic difference between 9/8 and 10/9 is too real to be dismissed
>as something "handled on paper". Even if the listener doesnt get the
>melodic difference immediately (something heard all the time in good string
>playing or unaccompanied vocal music), eventually the comma difference will
>either manifest itself in a _bad_ fifth, which is immediately _heard_ as
>such, or a _return_ to a tonic that has missed its mark by a comma (the
>ear is much faster at picking this up than the eye anyways).

Not necessarily. There are other possibilities:

(1) An apparently held tone was actually adjusted by a comma during a non-p5
chord change.
(2) Tempering.

>(5) Which interpretation does this support? Your own analysis of the minor
>triad prefers a subharmonic description. You can not have it both ways. For
>this reason, I find it sufficent to base my spacing description upon
>fundamental frequencies and a single calculated fundamental or guiding
>tone.

Please re-read what I have said. The tonicity algorithm form only a part of
the considerations of consonance and tuning. The other part is roughness,
which I feel has been dealt with adequately in the research.

>(6) The harmonic and subharmonic series are two transfinite series with
>different memberships; the second may be constructed from the first, but a
>one to one mapping of the second onto the first is not possible.

Sure it's possible. n->1/n.

>This is
>however, coincidental to my point, that we process subharmonic structures
>less well than harmonic due to conceptual and computational difficulties.
>Example: it is much easier to compare three objects with one object than it
>is to compare one object with thirds of an object.

This analogy is metaphysical and of limited value compared with an
understnding of the virtual pitch phenomenon. Who's to say that the "object"
is frequency and not period or string length.

>(7) Prof. Hennix has worked specifically with musical issues; most closely
>with the late John Myhill and with La Monte Young. (Her closest purely
>mathematical collaborations were with Myhill and with A.S. Yesinin-Volpin.)
>She may represent a different school of mathematical foundations from your
>own, but her work is anything but drivel; indeed intuitionist proof demands
>are the most rigorous.

I am sympathetic with an intuitionist philosophy of mathematics. However,
the one subject to which I have devoted more study that microtonality is the
foundations of quantum mechanics. The mathematics of quantum mechanics is
eminently classical, though its metaphysics and epistemology are most
certainly not. So assuming that mathmatics goes funny at the very small due
to quantum effects would be viciously circular. Besides, if quantum effects
were associated with musical tones, it would be with those of high pitch;
low-pitch tones are low-energy-per-quantum waves and therefore behave
classically.


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