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Topic: Music and Math  (Read 1665 times)

Kapellmeister27

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Music and Math
on: February 18, 2005, 10:02:35 PM
i am doing a paper in a math class where we have to take general topics (such as music or piano playing) and i was wondering if anyone could point me to some good essays on the subject.  possible subjects could include:

ratios within harmonic or rhythmic analysis

physical feats (e.g. if an average person can play scales in 16th notes at 60 b.p.m within one year, and 100 in two, 120 in three, 132 in four etc, there is a pattern here, what would the graph look like?)

interesting sequences in written music (golden ration/fibonacci, i know there is something about bach using 14s in  his goldberg var.  (14=2+1+3+8(b+a+c+h))

or any other topic you might know about

Offline pianonut

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Re: Music and Math
Reply #1 on: February 18, 2005, 10:50:03 PM
i have always been interested in this subject, myself.  the correlation between music and math and composers and mathmatic formulas.  mozart used to throw dice occasionally to determine a modulation (from what i've read).  i wish that i had written down what i read about bach.  from what i remember, the letter numbers of his name (B=2, A=1, C=3, H=8 total 14).  and, he used this to advantage in some prelude or fugue and then again in the 13th variation (which would actually be 14 in order if you count the aria) of the goldberg variations.  the 13th variation has a length of 14 notes at the starting phrase and keeps repeating (i think) this length of phrase.  you can read about this in a recent Clavier magazine.  spring 2005.

mike may has an article entitled "did mozart use the golden mean?"  under www.americanscientist.org/template    this article explains the golden section as a "precise way of dividing a line, music or anything else" and it shows up a lot in mathematics.  it goes back at least as far as 300 bc, when euclid described it in his major work, the elements.  the pythagoreans knew about it in 500 bc.  the oldest examples of this principle appear in nature itself (proportions).

to describe the golden section, "imagine a line that is one unit long.  then divide the line in two unequal segments, such that the shorter one equals x, the longer one equals (1-x) and the ratio of the shorter segment to the longer one equals the ratio of the longer segment to the overall line; that is x/(1-x) = (1-x)/1  that equality leads to a quadratic equation that can be used to solve for x.  anyway, you can read more about this if you like the idea. 

i believe 'putz' was a student of mozart's and also a mathematician.  he taught mozart some things and wrote "in mozart's time, the sonata form was conceived in two parts - exposition and dev./recapitulation. these are unequal lengths - as mentioned above - with the exposition being shorter, and the dev/recap longer.  if you count measures you could come up with some idea of what mozart actually did.  i haven't gone that far.
do you know why benches fall apart?  it is because they have lids with little tiny hinges so you can store music inside them.  hint:  buy a bench that does not hinge.  buy it for sturdiness.

Offline pianonut

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Re: Music and Math
Reply #2 on: February 19, 2005, 12:41:25 AM
i found my scribbled notes.  the C major fugue of Bach's also has a 14 note long subject.  the article i read was from Clavier (january 2005) and entitled "The Goldberg Mystery" by Walter Schenkman.

Here's a math word problem:

Clementi charged one guinea (equivalent to 10 guilden) for each lesson.  On the other hand, Mozart charged six ducats (27 guilden) for a dozen lessons.  (Quite obvious who made more - but still good math problem)

from mozart's letter to his father on 23 january 1782; "i have three pupils now, which brings me in eighteen ducats a month; for i no longer charge for twelve lessons, but monthly.  i learnt to my cost that my pupils often dropped out for weeks at a time; so now, whether they learn or not, each of them must pay me six ducats.  i shall get several more on these terms, but i really need only one more, because four pupils are quite enough. with four, i should have 24 ducats."

another interesting math concept in music is music in asymentric groupings such as 5, 7, or 11 and requiring time signatures of 5/4, 7/8, or 11/4  or 2+2+3/8.  also, there is polyrhythm, where we hear 3/8 meter in the upper staff and 4/4 in the bass.

frequencies are another thing to look up.  they are interesting to study.
do you know why benches fall apart?  it is because they have lids with little tiny hinges so you can store music inside them.  hint:  buy a bench that does not hinge.  buy it for sturdiness.

Offline ted

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Re: Music and Math
Reply #3 on: February 19, 2005, 02:35:55 AM
There is one aspect of this which I published in a mathematics magazine many years ago because musical journals weren't interested. I think I did post it somewhere on a forum but can't be bothered finding it now. It was original to me at the time but it's so simple I cannot believe nobody else thought of it.

If the piano keyboard is regarded as a cyclic group of order twelve and pitches reduced to within the octave then the number of possible chord types, disregarding inversions and pitch changes and counting a silence is 352. In fact, the outlook yields the nice, intuitively obvious result that a chord type (e.g. major, minor, ninth, diminished ....) is precisely equivalent to a partition of twelve.

Applying the Polya-Burnside theorem to the cyclic group of order twelve you get detailed breakdowns of the chord types for a given number of notes. For example, there are 19 three note chords, 43 four note chords and so on.  To further analyse, say, the chords of four notes, reapply the Polya-Burnside theorem to each partition of four notes. For example, the half-diminished and the augmented seventh both belong to the partition 4,4,2,2 - being 4,2,4,2 and  4,4,2,2 respectively.

At the time, that would be almost forty years ago now, I thought this analysis could have quite promising application to those interested in serial composition. Whether somebody actually used it I have no idea.

"Mistakes are the portals of discovery." - James Joyce

Offline ted

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Re: Music and Math
Reply #4 on: February 19, 2005, 02:54:02 AM
Another path to explore is the subdivision of the octave. In order to produce an equally tempered scale containing at least the ordinary major scale, that is to say the seven closest related frequencies, thereby ensuring that the new system at least embeds the ordinary scale, it is necessary that 2 to the power of (A/B) is very close to 1/2, where B is the number of notes in the octave and A the position of the required note producing an approxomate fifth.

A simple programme will soon find that 12 (piano keyboard),17,19,24,29... will probably give aurally acceptable results to some degree. The big surprise to me was 29, which gives an extremely close fifth, and I didn't hesitate to write code to test the sounds. To cut a long story short, experienced musicians I sent tapes to couldn't tell they were actually listening to music which modulated through 29 keys instead of 12, a fact which really surprised me. 29 has the disadvantage (?) of being prime, but the major scale, for instance, represented by the partition 5,5,2,5,5,5,2 sounds very good. 

Again, it is comparatively easy these days to produce this music now we have computers, but remarkably few people seem to experiment in any depth.
"Mistakes are the portals of discovery." - James Joyce

Offline ted

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Re: Music and Math
Reply #5 on: February 19, 2005, 03:09:38 AM
Of course these are trivial things compared to the big question of how far algorithmically composed music can engender musical, perhaps emotional reactions in humans. This, to me, is the daddy of all mathematics/music associations and I am going to have another try at it soon (my son has gone overseas and left his powerful computer here - mustn't waste it). I bought a compiler but became all involved in algorithmic art - I'll start down the music track anytime now.

My earlier experiments were very encouraging in that, even in the restricted area of fugues, I did write code which composed things that moved and surprised me. What are we to say of the statement that computer music moves us ? Some people, for reasons I cannot understand, seem to have tremendous philosophical difficulty with this notion. Nonetheless, it is a very young field and again, remarkably few musicians appear to be working on it.
"Mistakes are the portals of discovery." - James Joyce

Offline pianonut

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Re: Music and Math
Reply #6 on: February 19, 2005, 03:22:52 AM
interesting!  there is so much to experiment with in sound.  and, it's neat that you tested out your theories on experienced musicians.  that's the idea in mathematics, i think, is to prove that a theorum is real.  when you can show it on paper as well as in music, it's almost like saying music IS a science as well as an art.

i was reading in the most recent national geographic about perfect pitch.  people who speak mandarin chinese (or any tonal language) are more able to hear pitches exactly (without relation to other pitches). 
do you know why benches fall apart?  it is because they have lids with little tiny hinges so you can store music inside them.  hint:  buy a bench that does not hinge.  buy it for sturdiness.
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