Okay, I see the problem, but I'm still pretty disturbed.
I never learned this [ ⟩ angle bracket notation when I was studying linear algebra in college. We always just wrote things with plain brackets [ ]. Why not just use those for generator-count vectors?
I never learned it in math either, but I learned the Dirac notation (or bra-ket notation) in physics where we needed to distinguish vectors in 3-dimensional physical space (or 4D-space-time) from those in more abstract mathematical spaces of quantum mechanics. And so when Gene introduced them from Exterior algebra, I actually thought this was a good notation. Now, I'm not so sure.
When you use square brackets in linear algebra, you are supposed to use the correct 2D arrangement of the numbers. If you don't list the numbers vertically, then it is not a vector but a (single-row) matrix. You can however use the superscript "T" for transpose. So
⎡a⎤ = [a b⟩ could be written [a b]T
But sure. You can use whatever brackets you want, so long as you spell out what they mean in your particular context.
Hm. Those look to me like just parenthetical notes re: the harmonics and how many generators it took to reach them. Besides, in a generator-count vector, doesn't the period come first, then the generator(s)?
Even if George was somehow using G in this way, would you recommend it? I think it would be really cumbersome to put P G₀ G₁ ... after each term in a generator-count vector as a means only to distinguish it from a monzo.
OK. Bad example. Forget George's "G"s.
Yes the period comes first. But the period is also a generator. It's only special because it's chosen so you can make up the (possibly approximate) interval-of-equivalence by using it alone.
No I wasn't suggesting putting anything after each individual number. As per my example in quotes, I suggest putting the word "generators", or the abbreviation "gens", or even just a big "G", after the whole vector.
You'd do better to call them P G1
... so P can also be considered as G0
I didn't realize until you said that that you're suggesting monzos/vals are bad. That surprises me because in my ~15 years of experience, microtonal music is impossible without them. They're all I've ever known. So what you're proposing is an alternative to them?
Good grief no. As I've said before, it's the terminology
, the actual words "monzo" and "val" that are completely unnecessary and obscure the relationship to the math that at least some
people (like yourself) are likely to have already learned.
Or are you just proposing we stop calling them monzos and vals...
.../mappings and instead call them column vectors and matrices?
Not that either. But at least that's closer to what I'm proposing. In the sense that it's in the right direction, but it goes too far toward pure math instead of stopping with the math as applied to our specific domain. I don't understand how you could have missed what I've actually been calling them in this thread. "prime exponent vectors", "generator count vectors" and "mappings" (or "mapping matrices").
Saying "column vector" in linear algebra is redundant. All vectors are columns. If it's a row, its a matrix, or a "linear form". But because we've gotten it all mixed up with exterior algebra terminology and notation, thanks to Gene, I felt I had to add "column" to be clear here.
And an equal temperament's (rank-1) matrix would always just have a single row. In that case perhaps you're not changing anything about my understanding about any of this works, just renaming everything.
I wouldn't be re
naming everything. I'd be un-renaming them. Those were the terms we were using before Gene turned up and renamed them in an obscure manner. And now we even have people who should know better, like Mike, calling something a monzo just because it's a vector. We should ask Joe Monzo what a monzo
And the word "val" wasn't some pre-existing math term that Gene applied. It's just a word Gene made up — to replace the already perfectly serviceable word "mapping", which can be shortened to "map" if you really need it to be one syllable and 3 letters. And "patent val"? "Patent" here just means "obvious". The "patent val" was previously called the "standard mapping".
I'll admit that there's still something deeply mysterious and unsettling to me about how those six numbers arranged that way represent "meantone", first of all because meantone is just one of those things that I've read about hundreds of times and had explained to me dozens of times but I just can't ever really seem to understand it well enough to retain it or explain it to anyone.
That's because there are way too many numbers for a first pass at what meantone is. If you assume octave equivalence and an octave period, you only really need two numbers in the mapping [1, 4]. That says that 1 generator is octave-equivalent to prime 3, or 3/2, and 4 generators are octave-equivalent to prime 5, or 5/4. That tells you that the generator (which can always be made less than an octave) must be somewhere between a pure fifth and whatever octave-extension of 5 you get close to when you stack 4 pure fifths. 3^4/2^4 = 81/16 = 5.065. So the generator is between 3/2 = 1.5 = 702.0¢ and ⁴√5 = 1.495 = 696.6¢.
OK. Not quite true. You have to also look at the ratio between
primes 5 and 3. The mapping of 5/3 is 4-1 = 3 generators. If you stack 3 fifths you get something near 10/3. So we could have a generator that was as small as ³√(10/3) = 1.494 = 694.8¢.
Maybe what weirds me out the most is that with a rank-1 temperament I can figure out how big a step is by dividing 1200¢ by the first term, the one for prime 2. But with a rank-2 or more, how are you supposed to know how big each step is? I dunno. I'm probably a hopeless case on this one. I just don't think I've liked or understood a piece of music using a linear temperament well enough to get a grip on it. This probably isn't the forum to try for the upteenth time to explain it to me. Hopefully other people shouldn't have this sort of reaction to this material.
To get the step sizes. You have to choose a specific size of generator (say a middling one like 697¢) and decide how many notes you're going to have. This is not a given like it is with an EDO. Then you have to multiply the size of your generator by every integer from 0 up to one less than your number of notes, reduce all values to the first octave, and sort them into ascending order. Then you can add 1200¢ to the end of the list and subtract each one from the next to see what the step sizes are.
If you want to know what numbers of notes give you a distributionally-even scale, you can ask Wolfram Alpha to give you the convergents of the generator divided by the period (both in cents),
https://www.wolframalpha.com/input/?i=c ... 697%2F1200
and look at the denominators.