At first
thought, most people don't really think about direct ties between mathematics
and music. Sure, there are a finite number of beats per measure, counting is
paramount, and numbers are used to indicate measures especially for instances
of repeats. But who ever considered the idea of representing an entire symphony,
or musical piece in general, by a singular curve on an oscilloscope?[1] Or a
system of musical bars and tables in combination with dice which allows for the
production of thousands of minuets in no time? In this chapter, Melody-Making Machines, the relationship
between math and music is supported by explanations of such aforementioned
systems and devices, among several others as well.
Converting a
symphony to a curve wouldn’t have been possible without the use of the Fourier analysis.
Explained by R. Nave, the analysis is simply, “The process of decomposing a
musical instrument sound or any other periodic function into its constituent
sine or cosine waves”. Curves have the capability of being coded to any desired
precision by numbers, thus allowing us to quantize and translate them into a
chain of numbers. Similarly, Max Stadler in 1779 devised a system whereby any
person, proficient in music or not, could have the capability to compose ten
thousand minuets or trios. The system, inspired by Mozart’s affinity for
mathematical puzzles and musical permutations, is elaborated by Martin Gardner
(The Colossal Book of Mathematics, p.
631):
The “Mozart” system consists of a set of short measures numbered 1
through 176. The two dice are thrown 16 times. With the aid of a chart listing
11 numbers in each of eight columns, the first eight throws determine the first
eight bars of the waltz. A second chart is used for the second eight throws
that complete the 16-bar piece. The charts are constructed so that the waltz
opens with the tonic or keynote, modulates to the dominant, then finds its way
back to the tonic on its final note.
With this system, one can produce nearly 3.7974983e+14 waltzes, each garnished
with an individual Mozartean touch. The very first recordings of these dice
waltzes were made by O’Beirne, a famous mathematician who worked with the
Glasgow firm of Barr and Stroud in Scotland to build the first computer in the
country, the Solidac. This computer was programmed to play pieces in
clarinetlike tones.
Although many intelligent mathematicians have found
ingenious ways to create music with the use of complicated computer programs
and numbers galore, no one has yet found the perfect algorithm to produce even
a simple string of music that flows as well as the tones of human-made melodies.
There is a certain amount of passion and creativity that is present in a
composer’s work that cannot be matched by any computers’ arbitrary tune. Even
so, the fact that a simple system of numbers and tables can be utilized in a
way that produces thousands of combinations of melodies shows that math can
appear in places you’d never expect, and can surprise you in some interesting
ways.
Bibliography
Gardner,
Martin. The Colossal Book of Mathematics. New York: Norton, 2001. Print.
Nave,
R. "Fourier Analysis and Synthesis." N.p. Web. 14 Oct. 2015.
http://hyperphysics.phy-astr.gsu.edu/hbase/audio/fourier.html
[1] An oscilloscope is
a device that measures the change of an electrical voltage / signal
The link between music and mathematics has always fascinated me. Especially in the age of current technology, with music such as dubstep and EDM, math in the form of computer science, and music are overlapping more than ever before. DJ’s are skilled in mixing frequencies and multiple songs using computer programs and by ear alike. I think that is very interesting that a program was developed in order to create millions of different melodies so long ago. It makes mew think that people went through a phase where they were too lazy to write their own music, which I’m sure is not the case.
ReplyDeleteMy main question is how the “Mozart system” was figured out. The numbers of column and bars seem so random. I understand why multiples of four would be used; because four-four time signature is the most common. Would the Mozart System also work for music in different time signatures? Can it create things other than waltzes? I think it was especially interesting how none of the melodies, as Monica puts it, “flow as well as the tones of human-made melodies”. This observation paralleled my thoughts of the industry of robotics. Robots can have human qualities, but they can’t truly express the passion and creativity of the human mind. Sherry Turkle talks about this in the beginning of her book Alone Together
At the beginning of chapter 47, Melody-Making Machines, the author discusses how through numbers you can recreate any poem, painting or symphony ever created. I found this to extremely fascinating. By laying a graph on top of a painting and numbering each individual cell you could recreate the entire painting as precisely if not better than the original painting. The same thing applies to musical symphonies, the whole symphony can be represented by an oscilloscope. “This curve can be coded to any desired precision by numbers, these numbers can be quantized and expressed by a number chain.” (The Colossal Book of Mathematics pg. 628). I was also intrigued by the section talking about all the other machines that have been invented that are able to create their own music. Like the Kaleidacousticon it would shuffle cards and it could compose 214 million waltzes, or the Componium, a pipe organ that played its own compositions. You did a good job explaining the “Mozart” system and how it works. The dice system used to create the bars for the waltz is very interesting, using two dice to determine the first eight bars in the first column with the first eight throws and the second eight throws determine the second eight bars. I never thought you could make music by making charts and rolling dice to pick notes.
ReplyDeleteTo be one hundred percent honest, I had no idea there could be such a distinct connection between music and mathematics. I always knew there was a beat count and a beat measure but I never actually related that to mathematics. Chapter 47 was quite interesting to me personally, but also very confusing at parts. The Fourier analysis about the symphony curve was actually a hard concept for me to grab. I found the concept of mathematicians creating music with intense computer programs quite interesting. One question I have is how exactly the "mozart" system works. This system seemed rather difficult. After reading, I now have a great appreciation for composer's. It seems like there is a lot more to the music than just picking a tune and writing lyrics. Though the chapter itself was quite confusing for me, I also took a great amount of interest in it. Chapter 47 showed how complicated things can be even when they seem so simple. After reading this, it really reminded me of why I am musically challenged, I never realized that music itself has so much mathematics in it. I would be interested to learn more on this topic outside of The Colossal Book of Mathematics.
ReplyDeleteI had no idea about the connection between music and math. Reading this chapter was interesting enough with that in play. Like Tiffany said, I knew about the beats per measure and everything that goes along with it, but I had no idea about the compositions itself being connected to math. What I found interesting was what the Mozart system is and how it was actually devised with the two dice thrown 16 times. I thought it was cool how the first 8 throws determine the first 8 bars of a waltz. And the rest could be produce up to 3.7974 e14 waltzes. I did not know about the Mozart system nor did I know that any system of composure could involve mathematics. It was cool how the same could be applied to poems and pieces of art. I think that even though we have computer programs to create music now, the older compositions are more “timeless” just in the sense that the composers had to use their heads in order to make the pieces we all know of today. Overall I found this chapter to be pretty interesting. I can say I learned another way of looking at the music, art, and poetry of the 1600-1700s.
ReplyDelete