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15 February 2005 @ 11:24 pm
Algebra of music revisited revisited  
Quick correction: in multiplication, timings must be added, not multiplied.

EDIT: No, that doesn't work either. One possibility: for notes A×B=C, Ctiming = Btiming + Atiming × Bduration. Only problem is that would make multiplication non-commutative, and I was hoping that motives would be a commutative ring, if not a field.
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Current Mood: embarrassedembarrassed
Current Music: Megumi Hayashibara - Cruel Angel's Thesis
Derakonderakon on February 16th, 2005 07:48 am (UTC)
Y'know, you can edit entries...;)
Alun Clewealun_clewe on February 16th, 2005 08:44 am (UTC)
No, actually, if you want augmentation and diminution to be able to be effected by multiplying a motive by a note, the timings do have to be multipled--you were right the first time. (Or right the second time, I suppose, since that was your second post on the subject. ;) )

Personally, I still don't like the unit-dependent definition for multiplication, but eh, I've covered that. At least it's no longer dependent on the units of the pitch. And anyway, the new definition of tone is definitely an improvement IMO.
gwalla: halloweengwalla on February 16th, 2005 07:30 pm (UTC)
Actually, neither works, I just realized. In neither case would augmentation/diminution work properly. If it was just adding timings, the time between notes would never change; if it was multiplying timings, the time between notes would change based on how late the note they're multiplied by appears in in its motive. I need to totally rethink this.

As for unit-dependence, the "units" for time are pretty arbitrary. A "beat" can mean a different span of time depending on the tempo, and even a different written note depending on the time signature. Since my work doesn't deal with tempo or meter, "beat" means nothing, really.
Alun Clewealun_clewe on February 17th, 2005 12:41 am (UTC)
...if it was multiplying timings, the time between notes would change based on how late the note they're multiplied by appears in in its motive.

Well, yes, if you were multiplying a motive by another entire motive. But if multiplication of notes multiplied timings, augmentation/diminution could be effected by multiplying a motive by a single note (with its timing equal to its duration). I don't know what you expect multiplying entire motives to accomplish anyway.

As for unit-dependence, the "units" for time are pretty arbitrary.

That's my point, and why I don't think multiplication should be unit-dependent. The units are arbitrary, so the results of a multiplication shouldn't (IMO) depend on what units one is using (or how one defines a beat).

As an example: The same motives can be expressed, for example, in 3/4 or 6/8 timing. (It may make a difference to a musician which time signature is used, but in a very subtle way; they're still essentially the same motives.) Yet if multiplication is defined in a way that depends on how long a "beat" is, multiplying those motives together could have very different results depending on which time signature you use. This seems to me to be very undesirable.

This is why I recommended having multiplication add the durations instead of multiply them--it removes the unit-dependence of the results. Yes, it would mean multiplication would yield different results than under your definition, but I'm not sure exactly what you're going for or why one result should be preferable to another anyway.

(I also recommended pitches be added instead of multiplied, for the same reason, but now you've eliminated pitches anyway in favor of "tones". Unfortunately, this just introduces a new problem, in that now the result is dependent on the choice of reference tone, which also doesn't strike me as terribly useful. In retrospect, I take back what I said about the change from pitches to tones being an improvement: the use of frequencies was fine; the problem was the decision to multiply them together.)
gwallagwalla on February 17th, 2005 01:28 am (UTC)
Well, the whole thing about beats and tones is that the choice of reference units is arbitrary, but must be the same for all notes, motives, etc. you're dealing with. A length of "one beat" is the same for all motives, likewise tone 0 always refers to the same pitch. It's just that what those lengths and pitches actually are is irrelevant to the math—it's a matter of performance, not calculation. The "units" can be ignored entirely; they're just there to give people a better idea of how the numbers relate to actual music (which they seem to be failing to do).

As for what I'm trying to do with multiplication: I'm trying to define multiplication as the operation used in strict canon (including prolation canon), where one factor is the leader, the other is the pattern of followers (which are all the same as the leader, but time-shifted, transposed, and possibly mensurally augmented or diminished), and the product is the canon itself. For example, a canon in two at the fifth would be the leader motive times a motive consisting of two notes, differing from each other in tone by a fifth.
Alun Clewealun_clewe on February 17th, 2005 11:01 pm (UTC)
Yes, I realize that the choice of reference units is arbitrary, but once chosen it does affect the math, the way you've defined multiplication. It's not just a matter of performance; it does affect the calculation. See my 3/4 vs 6/8 time signature example--you're dealing with the same motives, but depending on how you define a "beat" the multiplication yields very different results.

Here's a simpler example: take a single note, and multiply it by itself. Say that note is one beat long. Its product with itself, if you multiply the durations, is also one beat long. Now, choose a definition of "beat" half as long as before. Now the note is two beats long, and multiplying it by itself yields a note four beats long. Same note, same absolute duration, the only thing that's changed is the choice of what duration to consider one "beat". But with one choice of "beat", the note stays the same duration when multiplied by itself, and with a different choice of beat its duration doubles.

A similar example could be given for tones: say you have a note, multiplied by itself, and that note happens to be the D above Middle C. Now, say the D above Middle C is also your reference tone. Multiply the note by itself, with your current definition of multiplication, and you get back a note with the same tone. But if Middle C is your reference tone, you get back an E. Same note, but for one choice of reference tone it stays the same when multiplied by itself, and for another choice it goes up a whole step.

The way you've defined note multiplication, the units can't be ignored; the choice of reference units directly affects the results of the calculation.
wanderingbhikkhwanderingbhikkh on February 16th, 2005 09:23 am (UTC)
Man, I think you're on to something here.