A Thousand Words: II
When I got up this morning and re-read last night's post (Well, the post of earlier this morning), I thought I was onto something, but I didn't go into it deeply enough. So, I found an article that is excellent for introducing non-mathmaticians to the concepts of fractal geometry: It is called Fractals: Useful Beauty, and I have put a link to it in the Articles section of my sidebar.
Many musicians object to the over-rationalization of music through math, and I can understand this, because I know exactly where that sentiment usually comes from. Intelligence as an inate aspect of an individual is not one overriding talent, but an amalgam of various talents that one possesses to a greater or lesser degree (I'm betting President Bush got lousy verbal scores). Many musicians are like me: Abjectly hopeless at thinking in terms of numbers. I'm a "shapes guy." Back in highschool I took some sort of standardized test, and it was quite enlightening: I scored in the top 1% in the categories of abstract reasoning, mechanical reasoning, and spatial-relationship perception; but I only got to the sixty-seventh percentile in verbal, and received a dismal forty-second percentile score in numerical ability. This means the average "Joe Sixpack" guy can add, subtract, multiply, and divide with more facility than I can, and most reasonably intelligent college grads can write better than I can (And, I'm sure you've noticed that I can't spell my way out of a wet paper sack).
Because of this, when the fractal geometry article speaks in terms of irrational numbers and mathmatical formulas, my eyes glaze over and I retain nothing of value. On the other hand, when the article explains things in terms of self-similarity, I understand immediately.
Here are some examples of self-similarity from that article.
A Triangle with self-similar triangles within it:
A snowflake made up of "Stars of David"(Actually, the Star of Christ, but you really don't want to get me started on theology):
Finally, here is a fern leaf that combines self-similarity with a growth series:
When we percieve beauty in a human female (Or handsomeness in a male), such as the redhead in the photo I posted below, we are having a positive aesthetic reaction to the nearness-to-perfection expressed by their proportions and self-similarity. Since human beings have a bi-fold symmetry, that is one of the things we look for as a positive attribute: Symmetry.
Music is as close to a living organism as you can get in art: It takes a person to create it, and execute it: Not just once - as in a painting or a sculpture - but every time it is heard (Recordings are, of course, simply sonic records of these living performers "doing their thing").
Back to the fugal process, it is obvious that we are talking in terms of self-similarity with at least the subject and answer, but also many times with the countersubjects and counter-answers. Back when I wrote the Art of Fugue style string quartet fugue, I was totally absorbed with fractal geometry and self-similarity. As a result, even though their are no countersubjects and counter-answers technically defined in that piece, the "free" voices are actually either made up of diminuted and inverted fragments of the fugue subject itself, or they share self-similar elements among themselves. I believe that is an overriding unifying factor that makes that fugue a success (If you want to see the score and/or hear that fugue, it is on my .Mac FileShare Page with the filenames of QUART_4Vox_Fugue.pdf/.mid in both PDF and MIDI formats. There's even an orchestrated version of it in there now).
I didn't plan that fugue out using mathematical formulas (Except for the placement of the pitch climax and overall archetecture), but I rather worked with the shapes within the subject and answer intuitively and applied their variants within the laws of musical motion. If you look at the music of Bach, it is positively filled-to-overflow with self-similar elements. And I don't mean just the fugues: Many of his preludes and chorale harmonizations make use of the technique of harmonic canon, which I've addressed before. Of course, the sonata process is also made up of returning themes, so that too is a musical process that is deeply involved with self-similarity. Even simple melodic sequencing is a self-similar fractal process!
On a sorta/kinda related note: I think Ferrari could design a sexy toilet seat if commissioned to do so. Those designers are definately aware of self-similarity.
Many musicians object to the over-rationalization of music through math, and I can understand this, because I know exactly where that sentiment usually comes from. Intelligence as an inate aspect of an individual is not one overriding talent, but an amalgam of various talents that one possesses to a greater or lesser degree (I'm betting President Bush got lousy verbal scores). Many musicians are like me: Abjectly hopeless at thinking in terms of numbers. I'm a "shapes guy." Back in highschool I took some sort of standardized test, and it was quite enlightening: I scored in the top 1% in the categories of abstract reasoning, mechanical reasoning, and spatial-relationship perception; but I only got to the sixty-seventh percentile in verbal, and received a dismal forty-second percentile score in numerical ability. This means the average "Joe Sixpack" guy can add, subtract, multiply, and divide with more facility than I can, and most reasonably intelligent college grads can write better than I can (And, I'm sure you've noticed that I can't spell my way out of a wet paper sack).
Because of this, when the fractal geometry article speaks in terms of irrational numbers and mathmatical formulas, my eyes glaze over and I retain nothing of value. On the other hand, when the article explains things in terms of self-similarity, I understand immediately.
Here are some examples of self-similarity from that article.
A Triangle with self-similar triangles within it:
A snowflake made up of "Stars of David"(Actually, the Star of Christ, but you really don't want to get me started on theology):
Finally, here is a fern leaf that combines self-similarity with a growth series:
When we percieve beauty in a human female (Or handsomeness in a male), such as the redhead in the photo I posted below, we are having a positive aesthetic reaction to the nearness-to-perfection expressed by their proportions and self-similarity. Since human beings have a bi-fold symmetry, that is one of the things we look for as a positive attribute: Symmetry.
Music is as close to a living organism as you can get in art: It takes a person to create it, and execute it: Not just once - as in a painting or a sculpture - but every time it is heard (Recordings are, of course, simply sonic records of these living performers "doing their thing").
Back to the fugal process, it is obvious that we are talking in terms of self-similarity with at least the subject and answer, but also many times with the countersubjects and counter-answers. Back when I wrote the Art of Fugue style string quartet fugue, I was totally absorbed with fractal geometry and self-similarity. As a result, even though their are no countersubjects and counter-answers technically defined in that piece, the "free" voices are actually either made up of diminuted and inverted fragments of the fugue subject itself, or they share self-similar elements among themselves. I believe that is an overriding unifying factor that makes that fugue a success (If you want to see the score and/or hear that fugue, it is on my .Mac FileShare Page with the filenames of QUART_4Vox_Fugue.pdf/.mid in both PDF and MIDI formats. There's even an orchestrated version of it in there now).
I didn't plan that fugue out using mathematical formulas (Except for the placement of the pitch climax and overall archetecture), but I rather worked with the shapes within the subject and answer intuitively and applied their variants within the laws of musical motion. If you look at the music of Bach, it is positively filled-to-overflow with self-similar elements. And I don't mean just the fugues: Many of his preludes and chorale harmonizations make use of the technique of harmonic canon, which I've addressed before. Of course, the sonata process is also made up of returning themes, so that too is a musical process that is deeply involved with self-similarity. Even simple melodic sequencing is a self-similar fractal process!
On a sorta/kinda related note: I think Ferrari could design a sexy toilet seat if commissioned to do so. Those designers are definately aware of self-similarity.
posted by Hucbald at 12:27 PM
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