Sounds as though the music teacher didn't understand math and what it is
all about.
(By "music theory" he probably meant the kinds of
("keyboard") harmony that are covered in the first year or two
of college.)
The kinds of thinking here are a lot like what is done in geometry. For
example, one way to think of chords is as shapes whose "side
lengths" are measured in semi-tone intervals (the smallest interval
between two adjacent notes on a piano keyboard). These
"distances" have been made uniform since the late 18th or early
19th century (not without musical penalty, see below).
So a major chord in closed position has the shape 4-3 and a minor chord
has the shape 3-4. So a chord of any kind can be built by starting on any
note and counting intervals. This scheme normalizes chords in the same
way triangles are normalized by their shapes. Scales are normalized in
the same way, and thus this also normalizes keys.
In harmonic theory, we are interested in how melodies can be harmonized
by adding chords, how a sequence of chords (called a "chord
pattern") "works" musically, how movements to other keys
and returns can be made, etc. The first order and second order theories
of these are very different. There is a famous 18th century piece by
Purcell called "The Contest Between Melodie and Harmonie", and
this sums up what Baroque music was all about. The golden age of Jazz
(roughly, the 20th century until the late 50s or so) followed a very
similar pathway in how melody and harmony were thought about (and not
entirely by separate invention).
The first order theory is very much about how tensions are introduced and
relaxed, how the notion of a "key center" can be used to
provide stability (and length) for excursions, how bass lines can be used
to solidify movements, etc. The second order theory was used very
strongly by Bach then less so until roughly Wagner, and then in highly
developed show and pop music (by stage bands, etc.) to try to intertwine
melodic devices (like voice leading) with larger harmonic schemes that
would force "emergent harmonizations" that are not easily
described by the first order theory. What is called "chord
substitution" (alternate harmonizations, sometimes of breathtaking
beauty) in Jazz heavily rely on such mechanisms.
"Mathematics" is a plural because it is about many different
ways of "thinking very carefully" with invented representations
and inference rules. So this kind of thinking about music is
mathematics (i.e. rather than "like math", it is math).
And, within music, there are lots of ways of making generalizations that
help with styles.
For example, my pipe organ and harpsichord have the older tuning schemes
used in the 17th century. Why would anyone revert? Here's the problem (as
first discovered to their horror by the Pythagoreans). Octaves are
multiples of 2. The harmonic 5th is the third harmonic, which is a
multiple of 3. So if you try to make a scale by running out the 5ths (of
5ths etc.) they will never come back to the original note (2 and 3 are
relatively prime). One way of running out the "circle of 5ths"
creates a discrepency of about 1/75th of an octave. The equal tuning
system mentioned above divides out this glitch evenly by making every 5th
a little bit flat (and this results in rather wide 3rds). This works
(sort of) OK on a piano because it doesn't have a lot of harmonics and
most people are not very sensitive to in-tune-ness. Here, every
chord is equally out of tune!
On an organ or harpsichord (which are very rich in harmonics), the result
of equal tuning is that major chords don't hold still, and minor chords
are jangly. The older tuning schemes made some chords much more perfect
and sacrificed others. This results in a harmonic theory that is partly
about "sunlight" and "storms" depending on what keys
you are playing in and how the harmonic progressions are devised. Because
the harmonics are different on organs, harpsichords, clavichords,
fortepianos, etc., it is not unusual for each to have a somewhat
different unequal tuning to deal with the strengths and weaknesses
provided. Some of the greatest music in the world was composed using
these different bases for thinking, and much of this music loses much of
its meaning in a modern tuning scheme.
And, there is a math to these other ways of thinking about how things go
together, but it is a somewhat different math. The analogy is to the many
kinds of geometries - all mathematical - that have been devised starting
in the 19th century.
Referencing back to "art" and "technique". Learning
all this doesn't necessarily make you into a composer or a better player,
just as learning painting technique doesn't necessarily produce art. But
if the artistic impulses are working then all this technique is
tremendously helpful. Unfortunately for education, knowing technique is
often all that is asked of a teacher .... oops!
Cheers,
Alan
At 10:01 PM 11/28/2007, mmille10@comcast.net wrote:
>Date: Tue, 27 Nov 2007
21:06:01 -0500
>From: "David T. Lewis"
<
lewis@mail.msen.com>
>Subject: Re: However ...Re: [Squeakland] Panel discussion: Can
the
>AmericanMind be Opened?
>To: subbukk
<
subbukk@gmail.com>
>Message-ID:
<
20071128020601.GA75166@shell.msen.com>
>Content-Type: text/plain; charset=us-ascii
>
>On Mon, Nov 26, 2007 at 11:38:30PM +0530, subbukk wrote:
>> Coming from a culture steeped in oral tradition, I find 'sounds'
better
>> than 'symbols' when doing math 'in the head'. The way I learnt
to handle
>> numbers (thanks to my dad) is to think of them as a phrase.
324+648 would be
>> sounded out like "three hundreds two tens and four and six
hundreds and four
>> tens and eight. three hundreds and six hundreds makes nine
hundreds, two tens
>> and four tens make six tens and four and eight makes one ten and
two, giving
>> me a total of nine hundreds seven tens and two".
Subtraction was done using
>> complements. So 93-25 would be sounded out as "five more to
three tens, six
>> tens more to nine tens and then three more, making a total of
six tens and
>> eight'. The technique works for any radix - 0x3c would be
"three sixteens and
>> twelve'.
>>
>> In Ind ia, many illiterate shopkeepers and waiters in village
restaurants use
>> these techniques to total prices and hand out change. No written
bills.
>>
>> The advantage with sounds is that tones/stress/volume can be
used to decorate
>> numbers. With pencil and paper, changing colors, sizes or
weights would be
>> impractical.
>
>Subbu,
>
>Thanks for sharing this. I think that it is very interesting that
sound
>and oral skills can be a basis for mathematical thinking. My
cultural
>background is less oral, so I did not even think of this as a
possibility.
>It seems that music and mathematics are somehow connected, but I
never thought
>to extend this to verbal types of music.
>
>Dave
I took a couple music theory courses in college years ago. One of my
professors mentioned that he noticed a correlation between those who were
good at math and those who tended to grasp mus ic theory readily. He had
no explanation for this though.
---Mark
mmille10@comcast.net
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