Friday, August 1, 2014

Mathematics and music

So, I've been keeping this blog thing now for 2 years now, I guess. I started it as way to just get rid
This is note to myself, it comes from this website
of thoughts floating in my head, but no posts from that era exist (I don't have them at least). I deleted them, then used this as a way to chronicle my studies of thermodynamics, which transformed into me studying statistical mechanics. The thing is, this is very math light compared to what I want it to be. I have a few posts on statistical mechanics, and my A to Z posts have some pieces here and there on concepts. Well, I like to think I'm always working to improve something about myself and my work, so now this is over 2 years old and Philadelphia Math Counts is about a year old, it's seems to be time to work on more straight math pieces. Maybe someday I really will be an expert.
I think the two themes I've talked about here is statistical mechanics, for the reasons stated above, and music. As I've mentioned before in other posts, this is because in the physical world I will not shut up about music if people get me started. The history of music, all music, fascinates me; the stories of the musicians, where they got their inspirations, the creations of seemingly creative and spontaneous works of beauty. The act of listening to music and it's affect on psychology is so cool. Music can calm the mind and cause it to focus on a task, or it make the blood flow and elicit an excited state that causes people to panic, riot, and kill. Those silly artists can't make that claim, unless the emotional states are "bored" and "less bored".
I've stayed away from math and music for esteem reasons, namely because it's been done so well by other people. Vihart leaps to mind. Pythagoras had some ideas. Even musicians have done it better. Per Norgard is a god in this respect. Just, so, beautiful. And the music is written using some sort of matrix, and it ends up being based on mathematical concepts. Why shouldn't it? Math lends itself so well to music. Instead of dealing with visual vectors in space or abstract mathematical vectors, music exists as a forth dimensional vector. It is a pattern of sound, and as a vector as well as exhibiting patterns some of the basic geometry of Elements can be applied.
Ratios and rates of speed are applied here. With two separate repeating patterns of sound that approaches a limit in time, such as 4'33", then Zeno's fallacy of motion should be a problem here, shouldn't it? Let's use a simple round, like Row, row, row your boat. Every English speaker should be familiar with it. If not, here's a video:

Sung correctly, person two never catches up to person one. Anyone who has taught this to children has seen that person two can catch up to person two eventually. Music is expressed in time, which is a ratio, funny enough. We could either make children sing this song forever, which would be sadistic, or through the miracle of technology we could record one person singing it and loop it twice. Here's the first question I have on the subject, if music is vector that exists in time, can it act as 2-dimensional lines on a flat surface? Can we apply Euclid's "Elements" to music?
The five postulates (from WolframMathWorld) are:
  1. A straight line segment can be drawn joining any two points
  2. Any straight line segment can be drawn indefinitely into a straight line
  3. Given any straight line segment, a circle can be drawn having the segment as a radius and one endpoint as the center.
  4. All right angles are congruent
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. 
No. On first glance at the postulates, my answer is "No, music does not fall under the same postulates." I'm writing this on a creativity high constructed from a 6 mile run and a small coffee, so this piece is really just flowing. Don't expect a paper that people will still be reading and loving 300 years from now.
Some fit though. Music is a vector of time, and given that hypothisis, a point in space would be equal to point in time. 0'00" would be point one. 4'33" the end point. Any reasonable point in time between them could be chosen as a point.  And as mentioned before, the technology exists to extend this line into infinity, the round could be played as long as power lasts on the music player. For the point of argument, a musical vector exists with two different points: a point in time, as mentioned before, and a musical note. This allows us some freedom in that we judge speed as beats per minute, and we can compare vectors. Musical vectors can intersect and diverge at any of these given points. One piece could have an end point of 0'00", another may start at 2'30". They could end at different times. Note wise, they may start as random chaotic stews at the same time, with different speeds even, then converge at space of music with note of G#, stay together, then split.
3 and 4 give me problems because it describes geometry as a state of shapes, and not what they exist as in the world of abstract numbers and formulas. It may be possible that musical vectors exist in this way, since music can "loop", the music in this paper exists as lines and may be able to connect points in a loop thus making a strange musical triangle with angles. If the speed of an object can be given as a vector in space, and the definitions in this post are either accurate or are improved for accuracy, then music can be written as a vector in 2 dimensional space, and the angles of a musical work can be checked.
This post escaped from me. It was going to be something about primes. And there is so much more work I wan to do on this subject! What is the nature of musical vectors? How do they apply to other principles in "Elements"? Their relation to number theory and primes? You can find symmetry in notes and music, and differential equations can be applied to music. It acts as a vector. But I need to stop for the day. The exercise buzz is wearing off, and the coffee, and I'm actually supposed to be doing a job, I guess?
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Couple things. 1st, if I keep saying that I'm working on trying to change some structural things on my blog, it will get done. It's been put off, and it looks cluttered now. Just get rid of some things, emphasize others, add some code I've been wanting to try. Basics stuff I've been playing with in my free time during the week.
2nd, there are ads. Gonna put the money towards the Dog Lovers, Unite cause. Since I've had ads for about 24 hours now, I haven't really decided how much is going where. After a couple weeks or even a month I should know. So if you click ads here, the small portion that comes to me is going towards something good. Or you click here, the dog picture above, or the page tab and share and donate it.

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