[Squeakland] music and math

Fri Nov 30 22:23:26 PST 2007

Hi Alan.

I think you're right about the character of music theory I had. I took two courses in it, both freshman level about 14 years ago. The first talked more about the history of music, and a little about structure. We learned about the different kinds of chords. I remember that much, though we didn't get into the mathematics, just characteristics. I guess there was some "subtle math" going on anyway, despite what was being taught, because we talked some about the patterns, and about how you can change a chord from one type to another just by making some adjustments in half steps and whole steps.

The second course got more into "keyboard" harmony, as you put it. We learned about keys (ie. C major, etc.), identifying notes and chords by sound and making associations, learning "keyboarding" on a piano (doing scales and chords). The most interesting part to me was learning about the choral scheme. For the first time we discussed "music with constraints". We did a tad of composition, writing our own chords, using these constraints, and playing them. Most of what we did with this was analyze choral sheet music. Our professor wanted us to pick out certain characteristics. The only one I remember is key changes. It was interesting because I felt like it taught us about keys at a deeper level, and had us thinking "outside the box". It was the first time I had seen that the author didn't have to pay attention to the key markers at the beginning of each staff in order for the piece to hold together. I had seen this before in the orchestra classes I took in jr. high and high school, b
ut I didn't understand what was going on. The writer could deviate into a new key by just manipulating some of the notes. Analyzing it felt like working with set theory. Once I hit a point where "Note X is not in the set of Key A", then I'd see if "Notes X, Y, Z" were in the set of another key. Sometimes I'd see that the new key the writer used was really a union between two key templates. I'm sure this confused some students, because they wouldn't see it as a change if most of the notes in the new key were the same as the old one. I believe it was in this context that the professor made the "mysterious" association between "good at math" and "good at music theory". No, I don't think he understood math and what it's about.

What you said about music is very interesting. The idea that even the tuning scheme can be adjusted is news to me. I'm pretty sure we used the "equal tuning" scheme for everything when I had orchestra.

Music theory showed me a little of the "mind" of the composer, but the way it was presented did not convey much of the math, except what us students could glean from the presentation.

---Mark
mmille10 at comcast.net

-------------- Original message -------------- 
From: Alan Kay <alan.kay at vpri.org> 
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 solid ify 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" < lewis at mail.msen.com>
>Subject: Re: However ...Re: [Squeakland] Panel discussion: Can the
>AmericanMind be Opened?
>To: subbukk < subbukk at gmail.com>
>Message-ID: < 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:
>& gt; 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 restaur ants 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 at comcast.net
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