The study of the inner workings of music. Includes things like scales, harmony, rhythm, and why it's fundamentally impossible to tune a piano perfectly.

Music is extremely mathematical, much to my own personal delight. Everything fits into a well-defined system, and can be described in simple, non-ambiguous terms. Until you study jazz, that is.

And yet there are always mysteries that linger. One of my favorites is this: how is it that two notes in the interval of an octave manage be different, and yet in some undiscernable, indescribable way, sound the same? Yes, I know that the wavelength of the lower pitch is exactly twice as long as the higher one, but that still doesn't explain the intangible way that they can sound the same and yet different.

Not to mention that once everything seems to be making sense, jazz comes along and breaks every single rule you ever learned. Set your cat down on the keys of your piano and you just played a jazz chord.

Any geek should love music theory. Especially when you realize that music is much like computer programming: it's beautifully logical, and yet we're still much better at it than any computer. :-)
People seem to be interpreting the whimsical tone of my writeup to mean that I don't know what I'm talking about. Rest assured that I mostly do. :-)

Art Tatum: I don't really understand the point of your analogy. Different shades of the same color don't really compare to octaves--color shades are only a result of varying intensity of the same hue, whereas octaves are a result of two pitches whose frequencies share a certain property (they are in the ratio of 1:2^n). If the color spectrum were periodic in such a way that every period had a color that was like a color in the last period, yet somehow different, then your analogy would apply.

srkorn: Yes, I understand the idea of a fundamental frequency and resulting overtones, but that still does nothing to explain the psychoacoustic reasoning behind human perception of the sound of an octave. How would you explain to a deaf person what an octave sounds like? What could you say, besides "They're different, but somehow they sound the same?"
As to the question of why tones at the interval of an octave sound both the same and different: imagine each pitch as being a color. For example, perhaps E is red. Now, the E in each octave is of a slightly different intensity--the lowest E on a piano keyboard is a very dark brick red, while the highest E is a very light (almost pinkish) red.

On the question of jazz chords: the chords that are used in jazz are just logical extensions of traditional tertial harmony. They did not originate with jazz, however. They first appeared in the music of French impressionists. Hope some of this helped.

klash: Actually, I guess I should explain more completely. Your comment about the visible spectrum being periodic is actually what I intended. I imagine that if we could see above and below the visible spectrum, we would see the visual equivalent of "octaves." That still doesn't really explain it, I guess, but it's the best thing I can come up with.

Any good text on chromaticism will answer many of the questions surrounding the quality of sounds created in jazz or by the various intervals. On any stringed instrument (including the piano), when a note is played, the string vibrates not only at the wavelength of the note played, but also wavelengths half as long, a third as long, a fourth as long, etc., decreasing in intensity as the wavelengths decrease in size. For example, if you play an A at 110Hz, the string will also vibrate at 220Hz, 330Hz, 440Hz, 550Hz, and possibly a few more will be discernable. The first six sounds produced are called the senario of a tone. So, when you play the 110Hz A and the 220 Hz A together, one of the similarities comes from the fact that both are producing the exact same tone of 220Hz, and the tones in the senario of the higher A are being duplicated by the lower one as well. As a side note, the other notes produced by the senario of a note are its perfect fifth and major third; for example, A's 5th is the E at 330Hz, and its major 3, C#, is sounded at 550Hz (this is why the major chord has such a stable sound).

Also important in the sound of an octave interval is the idea of difference tones. Most people know that if you play 440Hz and 439Hz together, you can hear the two being out of tune as the phases will generate a tone of 1 Hz. So, when you play 220Hz and 110Hz together, a difference tone of 110Hz is produced, reinforcing the bottom octave's sound.

It gets much more complicated than all this, however; as noted above, read a book on chromaticism for more information. Most of my knowledge on the subject I got out of Chromaticism: Theory and Practice, by Howard Boatwright.

The development of a fugue:

Is the second section of a fugue.

It is comprised of episodes and entries. episodes contain fragments of the subject and sequences. Thus, it is not tonally stable.

Episodes are used to lead smoothly from one entry of the subject to the next. Entries are in a closely related key, thus, tonally stable.

They contain one or more complete subjects...these multiple subjects can be linked. Devises found in an entry are: augmentation, diminution, melodic inversion, stretto and mixed modes.

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