I am not a physics or a math expert, but over the years I have given the
subjects of audio cognition vs. music theory some serious attention.
When talking about the difference between theory and practice (and that
goes for all theories of music in all cultures) one must consider the
neurological limits of human hearing and the fact that minds (with
respect to all sensory phenomena) tend to construct extremely elaborate
beliefs and theories based upon incomplete sensory information or
sensory information that is too complex for the nervous system to
accurately take in. Quite often, music theories describe pitch relations
that exceed measurable neurobiological thresholds, theories that have
been of great utility to academics but have been far less useful to
practicing musicians who develop their own practical strategies for
dealing with complexity. Since practicing musicians tend to spend more
time playing and less time theorizing, they tend to adopt the
explanations of theorists regarding what they do. This can give them
status in the academic community but their amazing neurobiological
strategies remain mysteries for cognitive scientists to unravel 20-30
years later. When considering the perception and musical reproduction
of pitch intervals, one must consider two important truths.
(1) The accuracy of perception changes in relation to register because
the basilar membrane provides us with less effective spatial maps of
pitch fundamentals as pitches move toward the lower and upper registers.
I am deliberately avoiding the notion that the mechanics of hearing has
anything to do with comparing pitches, because that entire process takes
place in the mind and is completely subject to top-down neuro-cognitive
revisionism. We won't even deal with the fact that the basilar membrane
less effectively maps pitch fundamentals when it is confronted with a
number of complex spectra at once, spectra emanating from different
sources (i.e. polyphonic music, or music played in a noisy environment.)
Consider this problem. Let us divide the octave (simply the difference
between any pitch and its 1st harmonic - i.e. double the frequency) into
1200 equal parts or cents. That gives us the usual 100 cents between 12
equally tempered half steps. For our example we will use A 220Hz and A
440Hz - the lower octave of a 2.3 to 2.4 shakuhachi. The entire octave
is only 220 cycles wide, therefore each half step (100 cents) is a
difference of 5.45 cycles (1200/220). A quarter tone would be 2.72
cycles and half of that would be 1.36 cycles. In order to "perceive"
the difference between tuning systems or between close pitches belonging
to different systems we must perceive fundamentals that differ by these
small amounts. How is this done? By hearing "beat frequencies" - the
amplitude swells produced by two acoustic waves canceling each other
out. Beat frequencies permit hearing the differences between
simultaneous pitches that are really quite close in frequency. If the
beat frequencies are slower than 20 cps, (as in the case of the
differences describe above and the difference we are talking about) then
the basilar membrane cannot even represent them, and at that point were
into the new world of cognitive studies. This leads to paragraph 2 where
monophonic and polyphonic pitch discrimination merge.
(2) If a performer, theorist, or your average Joe is asked to compare
pitches when listening to a single monophonic source (for example solo
shakuhachi playing honkyoku), Joe's brain (not his inner ear) must
construct a memory of pitch no.1 while at the same time the wave-lengths
of the evolving wave-train continue to enter his inner-ear. Even though
there is a location on the auditory cortex (A1 or Heschel's gyrus) that
physically maps the basilar membrane, Joe's mental construction is not
(I repeat) is not located there. Remember, the incoming data is located
there. Joe's mental map of pitch no. 1 is not an audio tape or the
brain's version of a digital image-map in any sense. The vast majority
of neuroscientists and cognitive scientists agree that these memories
are nothing more than neurological "switches" that "turn on" learned
concepts - received theories! In other words, I hear note no. 1.
(Sound keeps coming in.) A1 continues to register gradual changes in
the wave-length of the spectrum entering the ear. At some point
(determined by my unique biology) a threshold is exceeded, my brain
flips the switch, (a top-down concept does not flip the switch - biology
flips it) and my belief system concludes that the pitch difference
actually IS the Pythagorean comma, tsu vs. tsu-no-meri, or some other
thing I've learned.
I'll never forget the time my young niece was on a drive with us out in
the country and we passed one lonely cow out in the distance - she said
"Aunt Chris - look at the doggy."
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