August 10, 2009

The Sound of Music - Music of the Spheres Part 2

Hello all! Finally, another post is ready (or rather I’m working on it, but by the time you read this, it’ll be ready . . . yeah, ok, you got it right?).

Moving on to musical notes and their sounds . . .

Pythagoras (yes, the infamous Pythagoras that devised the Pythagorean Theorem used in math) loved music too. At least, I suppose he did, for he investigated it quite a bit. And I mean, who would investigate something so profoundly unless he/she liked it? Well, actually, I just read (on Wiki) that he thought Greek music wasn’t really nice and wanted to make it sound better, using his super-mathematical-scientific-mystical-etc-power.

Anyways, he figured out that the pitch of a musical note depends on the length of the string being plucked (think about the lyre). So let’s say the musician plucks his string, and it produces a LA note. Well, if the string’s cut exactly at midpoint and the musician plucks that cord, then it’s also going to produce a LA note, but one octave higher (meaning 2 times higher). Pretty spiffy, huh?

Why does it work that way? Because the first string (the longer one), vibrates at a certain frequency which is ½ slower than the frequency of the shorter string, or, in other words, the second cord vibrates 2 times faster! This relationship is expressed mathematically as a frequency ratio of 1:2.

Of course, there are other ratios which Pythagoras thought were of the utmost importance, namely:

(1) the perfect fifth: frequency ratio of 2:3
(2) the perfect fourth: frequency ratio of 3:4

And these are the basis of musical harmony.

So, the key to this whole message, is that Music = Math.

Awesome, isn’t it? Basically, I postulate that anyone who likes music likes math! Yep, that’s right. Aren’t you amazed? You’re all innate mathematicians! Isn’t that one of the best things you could have ever found out about yourself? :)

With that, I’ll let you go!

--Alessa

7 comments:

  1. I enjoy the beauty of math, but hate reading or solving math problems. Music is meticulously constructed math, without the requirement of the recipient to think or understand it. Music is a translation of math--a medium universal enough for every human to enjoy. I have one question for you: If music is math, then is Michael Jackson the King of Math?

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  2. No. Because he wasn't the king of music, just of one type. I'd say he has potential to be considered pre-calculus. Or maybe geometry :)

    But I totally agree with you, gotfrank, music is a different approach to math that can be solved in a different way (through hearing, or vibrations), but it doesn't make it any less mathematical for it :)

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  3. Salut Cousine !

    It's me - you know your cousin Marc the violin player. Your mom told me about your blog, so here am I. Cool blog, and it's funny I ran on your post on math & music because I've also been fascinated by this connection for many years.

    I dove into more details on this a few years ago when I was preparing a course on harmony. I'll send you some materials for fun, just in case you wouldn't have figured out yet what's in it (but I doubt...) - like why are there 7 notes in the diatonic scale or 12 notes in the chromatic scale, the pythagorean scale and the pythagorean comma, the zarlino scale, the need for temperament, etc ...

    But I confess I was never able to read more than 10 pages of the "Gödel Escher Bach" book I was offered. Did you read it ?

    Any way there's no doubt strong affinities between scientists/mathematicians and musicians.
    I think it has to do mostly with the taste for abstraction. Music is one of the most abstract form of art because the emotions and pleasures derived from it do not rely on any visual stimulus or any specific associations with objects of the real world.

    I've met plenty of physicists/mathematicians/engineers/doctors with whom I played much music. When I was a student, I got an internship at the CERN in Geneva because (I learned afterwards) the Physicist who hired me specifically looked for students that were string players to organize quartet sessions at his home. When he sent me the letter to tell me I was hired, he specifically asked me to bring my violin along, and on my first day of work, he spoke for like 20 min. about the job, an then asked me : "are you free tonight to play quartet ?"

    So, when do you write us a fugue ?

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  4. Hello Cousin!

    I'm glad you liked this entry on math and music. It's definitely something that has picked my interest and now I wish I'd taken that particular class when in college but I was too busy (maybe then I would have been able to write you a fugue^^).

    Oh, please send me the material you have on harmony. I still have no idea why there are 7 notes in the diatonic scale and I didn't even know there was a pythagorian scale!!! See, compared to you, I'm still a real novice in this topic, but I'd LOVE to learn more :)

    I haven't even heard of the Godel Escher Bach book. Is it devoted to math and music as well?

    I totally agree with you, that music is really subjective. Yet somehow we mostly all react the same way to music. Surely it must mean something, right?

    Did you get to work at CERN while they were building the particle accelerator that would create mini-black holes? It's amazing that you went there! I'm really impressed! And did you get to play the bugs bunny song for the Physicist too? :)

    Looking forward to learning more about music, math and harmony from you!

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  5. Hi Alessa,

    You get that in your mailbox very soon - I'm curious to hear your comments :-)

    But here's a foretase: the pentatonic scale (C-D-E-G-A) - only 5 notes

    It is remarkable to observe that many cultures in the world have this scale as basis of their music, for example African and Asian music use it all the time. Rock music as well: most guitar players when they need to do a solo on a 12-bar blues will just use the pentatonic scale.
    (ask your guitarist friends: they must know about the pentatonic scale).

    So how come they all have "decided" on the same scale ? Probably through empirical practice and tradition, but how exactly ?

    Here's the most convincing explanation (nobody knows the truth for sure) I've heard.

    If you need to tune an instrument that produces several fixed notes (e.g. the strings of a lyra, a violin, a guitar, the woodsticks of a xylophon, etc ...), your ear will favor intervals that sound nicely. The octave is very pleasing (very consonant), but it's not very intersting to tune the strings of your instrument in octaves !

    The next most pleasing (consonant) interval to the ear is the fifth. The violin for ex. is tuned in fifth. Bagpipes have humming notes tuned in fifth.

    So if you tune the strings of your instrument, or the woodsticks of your xylophon, or whatever ... by producing each time a fifth with respect to the previous sound, you generate new notes.

    These notes are quite far apart: you leap from one to the next. But you can bring them down by transposing them down in octaves, so that they all are contained in one octave, and produce a nice ascending movement in small steps.

    Example: if you have 5 strings and say you start from F, you get C, then G, D, A.

    If you transpose these notes down to have smooth climbing steps, you get: F-G-A-C-D

    The pentatonic scale !
    Eurêka !

    So you see: the simple fact of tuning an instrument with fixed sounds makes you create notes that form a subdivision of the octave. That's what a scale is: a particular system to chop the octave into pieces.

    If you had 7 strings instead of 5, and tune them in 5th, you'd get: F-C-G-D-A-E-B, or re-arraging them in ascending order and starting from C:
    C-D-E-F-G-A-B

    The diatonic scale ! Eurêka once more !

    This illustrates the close relation that exists between the definition of the notes and the tuning of instruments. And it's not a mistery than that very different cultures found empirically the same solutions.

    This is so fundamental, and yet they never teach this in music schools. I could never figure out why. I think too many musicians have been trained just to reproduce what comes from tradition, and as teachers, they keep on doing the same.

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  6. Oh yes, about "Godel Escher Bach"

    Check this link on wikipedia: http://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

    I was offered this book for my birthday when I lived in NY, by a physics student.

    Looks really interesting, but I dropped out after a few pages

    But maybe you'd love this :-)

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  7. Marc,

    First of all, I'd like to thank you for explaining the pentatonic scales to me. I do wish they'd teach that in music school as well though, to be quite frank, I probably wouldn't have known about it even if they had as I rarely paid attention in class :)

    It is very interesting to know that so many cultures share the same predilection for this particular scale. I wonder why that is. If the way those notes harmonize somehow resonates in a special way with us.

    I'm not sure whether you've read any of my previous posts, but I have mentioned how some people have done research on our own vibrational pattern, whether it's at the EM field or DNA levels. Perhaps (and this is just a hypothesis amongst thousands) the music created with these scales somehow resonates with our own vibrational pattern, which is why it sounds pleasing to us? Whereas if the music was off somehow, then its frequency would go against ours, which would be why we wouldn't like it.

    What do you think?

    You know, I checked out the Wiki info on GEB that you gave me and I believe I've seen that cover before :) I'm kind of like you though, in that even though I'm really interested in the topic, I'm not sure whether I'll be able to read it. But who knows, maybe one day...?

    PS: Can't wait to read the whole documents you've sent me :)
    PS2: Thanks again for the lesson! I LOVE it^^

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