Sorry for the long response. Those who are interested can read on. In =
a
nutshell, I've wish shakuhachi players would start working more on the
beautiful ways to transition between two tones and less on playing a =
static
tone "in tune." Our music theories are simply to pathetic to articulate
these types of subtleties - and that is the bulk of our shakuhachi
repertoire. If you cannot make elegant transitions between two tones, =
how
can you ever play a phrase of more then two notes? In fact, I know some
folks who are so obsessed with the proper flatness of a meri tone that =
they
have yet to even deal with what it means to "connect tones" and "shape
phrases." OK - Here comes the rant about pitch and our perception of it.
-------
I am not a math expert, but over the years I have given the subject of =
audio
perception vs. music theory some serious attention, and for 5 years =
taught a
course at the College of William and Mary on Music and the Mind. When
talking about the difference between our theories of music or pitch
relations (i.e. measuring pitch relations in absolute terms like cents) =
and
practice (how we actually make the music we're trying to describe) one =
must
consider the neurological limits of human hearing, the fact that the =
brain
tends to construct extremely elaborate beliefs and theories that grow =
out of
an abstract understanding of the language of mathematics and secondarily =
out
of "fuzzy" perceptual experience. The brain has evolved so as to =
quickly
process incomplete sensory information, sensory information that is too
complex for the nervous system to accurately "map". Then the language =
part
of the brain "rounds off" the relations to "learned scales" (these =
scales
exist in auditory memory for most of us, but in language memory for
theorists.) So we can appreciate the complex math of the various
temperament systems, theories that have been of great utility to =
academics'
careers, but at the same time we must bow to the practicing musicians =
who
really produce this amazing complexity. Since practicing musicians tend =
to
spend more time playing and less time theorizing, they tend get =
intimidated
by academics, spend more time than they should trying to explain what =
they
do in theoretical language. They even adopt the brain-creations of =
theorists
when explaining what they do. This helps performers attain status in =
the
academic community. Noneltheless, their amazing neurobiological =
performance
practices still remain mysteries for cognitive scientists to unravel. =
There
remains no consensus in the Music Cognition world regarding how
pitch-relations are transformed within the brain. When considering the
perception and musical reproduction of pitch intervals, one must =
consider
two important truths. =20
(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. =20
(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."
Dan Gutwein
Professor of Music Theroy and Composition(retired)
College of William and Mary
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