[Squeakland] music and math

Thu Nov 29 08:24:40 PST 2007

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 at comcast.net wrote:
> >Date: Tue, 27 Nov 2007 21:06:01 -0500
> >From: "David T. Lewis" 
> <<http://mailcenter.comcast.net/wmc/v/wm/474DD3ED0001F7050000103B2200750744CFCE0A0404070303?cmd=ComposeTo&adr=lewis%40mail%2Emsen%2Ecom&sid=c0>lewis at mail.msen.com>
> >Subject: Re: However ...Re: [Squeakland] Panel discussion: Can the
> >AmericanMind be Opened?
> >To: subbukk 
> <<http://mailcenter.comcast.net/wmc/v/wm/474DD3ED0001F7050000103B2200750744CFCE0A0404070303?cmd=ComposeTo&adr=subbukk%40gmail%2Ecom&sid=c0>subbukk at gmail.com>
> >Message-ID: 
> <<http://mailcenter.comcast.net/wmc/v/wm/474DD3ED0001F7050000103B2200750744CFCE0A0404070303?cmd=ComposeTo&adr=20071128020601%2EGA75166%40shell%2Emsen%2Ecom&sid=c0>20071128020601.GA75166 at 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
><mailto:mmille10 at comcast.net>mmille10 at comcast.net
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