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Correlation between music and mathematics.
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| Stef |
Basically i have a final paper coming up and this was the topic i picked out of the ones available. I remember reading stuff that BT wrote about the connection but still have come up short trying to find those articles.
If anyone has any articles they'd like to share or have any input on the topic it would be quite cool.
So what do you guys, how strong is the connection between music (production and just listening) and mathematics? |
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| kadomony |
I'd say it's pretty much everything.
Sound is generated by waves which can be replicated through mathematical expressions (ie digitally)
Each scale also has set defined mathematical intervals between the notes as well.
Back in the day I knew this guy who used his knowledge geometry to create his own scales. |
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| MrJiveBoJingles |
| quote: | Originally posted by MrJiveBoJingles
[This is mainly for those of you who (like me) are interested in the mathematical qualities of music, but it's good information for anyone.]
Producers of electronic music deal quite a bit with frequencies; cutting them out, shifting them over time, boosting them, changing the proportions between them, and so on. "Sound waves" are our way of conceptualizing the motion of air molecules that occurs when some mechanical or electrical event (which we refer to as "striking a note") happens, and frequency is a measure how fast these waves of molecules move up and down. Frequency is specified in "Hertz (Hz)," which means "cycles per second."
Low frequency wave (top) and high frequency wave (bottom):
In terms of human hearing, lower and higher frequencies mean lower and higher "pitches," and musicians and musical theorists in the past made organized systems of pitches to designate what frequencies their instruments should play; they based these systems of pitches both on what was pleasing to the human ear and on what was mathematically "beautiful," and we still use these systems today.
What a tuning system specifies is how the physical frequencies designated by musical "notes" relate mathematically to one another. In Western music, the "reference frequency" used to tune most instruments is called "A 440" or "A4." That is, "A4" on a keyboard or other instrument is defined as the key which, when pressed, will cause the instrument to release a sound wave with a frequency of 440 Hz. When a note (call it "X") is an octave above or below another note (call it "Y"), that means that note X has a frequency either twice that or half that of note Y, respectively. A half-step is the distance between one note and the next (ratio between the two frequencies) in a given Western tuning system.
All the stuff mentioned in the last paragraph is universal in traditional Western music and in all non-experimental modern pop music, which includes most electronic music. Where things get complicated is in how specific tuning systems derive the frequencies of individual notes from octave to octave. The most popular tuning system in modern Western music is called equal temperament tuning (more specifically, twelve-tone equal temperament tuning). It gradually became the most popular because it contains no "wolf intervals" (very harsh-sounding frequency ratios which make it unfeasible to use an acoustic instrument to play in multiple keys without retuning it). Here is how equal temperament tuning works:
Equal Temperament Tuning
Twelve-tone equal temperament tuning, or 12-TET for short, specifies the frequency difference from one note to the next by using a constant ratio. This ratio is derived in the following manner:
Remember from above that two notes being an octave apart means that the pitch of one is twice that of another. Consider two notes, A4 and A5, which are an octave apart. We can represent their frequencies in the following way:
Frequency of A4 = f(A4)
Frequency of A5 = f(A5)
Since A5 is twice the frequency of A4, the ratio of f(A5) to f(A4) is 2 to 1 (2:1).
Now remember that there are twelve tones in our desired tuning system, and we want the tones to increase or decrease using a constant ratio from frequency to frequency. We will designate this ratio "R." If our system is going to have twelve tones, then it must be the case that R is a number that when multipled by itself twelve times will equal two: otherwise we could not use it to specify the twelve intervals that go to make up an octave in a Western tuning system. Another way of saying "R multiplied by itself twelve times" is just R^12 ("R to the twelfth power").
So this means that:
Ratio of f(A5) to f(A4) = 2:1 = R^12.
How do we remove the exponent from R^12? Easy: we take the twelfth-root of both terms in the following equation:
2/1 = R^12
...so...
Twelfth Root of 2 = R
The twelfth root of 2 is approximately 1.05946309. So this is the number we will use to specify the ratio of the frequency of one note to the frequency of the note below it. Consider A4; its frequency is 440 Hz. To derive Bb4, the next highest note in the tuning system, we would multiply 440 by 1.05946309. This gives us 466 Hz, which will be the frequency of Bb4 in our system. To get the next highest note, B, we would multiply 466 by 1.05946309 (or 440 by 1.05946309 times 1.05946309, which amounts to the same thing). This will give us 494 Hz, the frequency of B.
By now you can see the pattern: to get a note that is X half-steps up from the "root" note, we must multiply the frequency of the root note by R^X. If you have any trouble understanding this process, the following chart should help clear things up:
How would you derive the frequencies of notes below A4? Easy: instead of multiplying in order to go up, you would divide to go down, using the exact same pattern -- X semi-tones down from A4 = 440 / (R^X). |
http://www.tranceaddict.com/forums/...threadid=361540 |
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| DigiNut |
Kind of a weak topic IMO.
Musical tones correspond to frequencies, and musical timbres correspond to various harmonics. In some cases, musical concepts like octaves are related by mathematical formulas.
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Yeah... so? Waveforms and signals are concepts from physics, not mathematics, and even the notion that music relates to physics is a bit of a raised-eyebrow hypothesis.
Sure, you can apply certain mathematical concepts to music, same as you can apply mathematical concepts to biology or to football. Math is just a syntax for creating abstract models of the physical world, and music is one of many different things you might want to model.
I suppose you could compare them on the basis that "music" as we think of it is also an abstract model, that notes and scales and chords are just abstract representations of an underlying reality that's actually much more complex, but we work with so many abstractions of so many things that it would be hard to make a convincing case for that particular parallel being somehow unique. Or you could focus on modern music production and how it's all based on signal processing which is highly mathematical in nature, but I think that's a weak hypothesis (you could say the same thing about animation).
Personally, I think most of the "music is math" fluff is just rhetoric and stoner-talk from people who haven't had a lot of experience in one or the other. I have, and I've never noticed any interesting analogies that I would classify as more than coincidental. But you say you've already chosen the topic so... good luck. I think you'll need it. |
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| Subtle |
| Isnt Aphex Twin one of those who uses mathematics for music ? |
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| MrJiveBoJingles |
| My impression is that Aphex Twin is more of a hands-on, "take my synth apart and get out my soldering iron" kind of guy than a "write equations on my computer" kind of guy. Although the two aren't always mutually exclusive, of course. |
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| gr8ape |
| quote: | Originally posted by DigiNut
Kind of a weak topic IMO.
Musical tones correspond to frequencies, and musical timbres correspond to various harmonics. In some cases, musical concepts like octaves are related by mathematical formulas.
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Yeah... so? Waveforms and signals are concepts from physics, not mathematics, and even the notion that music relates to physics is a bit of a raised-eyebrow hypothesis.
Sure, you can apply certain mathematical concepts to music, same as you can apply mathematical concepts to biology or to football. Math is just a syntax for creating abstract models of the physical world, and music is one of many different things you might want to model.
I suppose you could compare them on the basis that "music" as we think of it is also an abstract model, that notes and scales and chords are just abstract representations of an underlying reality that's actually much more complex, but we work with so many abstractions of so many things that it would be hard to make a convincing case for that particular parallel being somehow unique. Or you could focus on modern music production and how it's all based on signal processing which is highly mathematical in nature, but I think that's a weak hypothesis (you could say the same thing about animation).
Personally, I think most of the "music is math" fluff is just rhetoric and stoner-talk from people who haven't had a lot of experience in one or the other. I have, and I've never noticed any interesting analogies that I would classify as more than coincidental. But you say you've already chosen the topic so... good luck. I think you'll need it. |
Well said im in math in college, took some signal processing classes, and played piano for almost 10 years and I can tell you, although you can EXPLAIN some musical stuff with math, music isnt math lol |
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| Kismet7 |
I think the coorelation of music to trees is more important. /what a hippy would say.
And I would agree. |
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| DJ RANN |
I'm split about this.
As an engineer, you really realise how much of music IS math and vice versa, and even though part of me says music is random, and creative thought without constraints, so is math, once you get beyond the standard laymen and high school definition. The chaos that math can have and the fact it governs everything and that nothing is truly original or random, makes it seem so close to music.....
............but as Elvis (the other one) says:
"writing about music is like dancing about architecture" |
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| Felix Hoo |
| If you consider composed experimental music as music, then you can look into Barry Truax who created Riverrun using real time implementation of granular synthesis.He uses uniform random distributions to control the grain parameters, producing a stochastic music based on probabilities. |
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| flutlicht junky |
| Music is basically number in time therefore yes music 'is' maths :tongue3 |
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| palm |
| aphex twin is one of those who live in a bulletproof tank |
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