Random Thoughts from Mike Battaglia

This page will serve as a scratch pad for some random musings that I have about regular temperament theory. Some of these ideas might end up turning into actual pages on general theory, others might not be notable enough for that. If anyone sees any technical inaccuracy or oversimplification here, let me know and I'll change it.

Relating MOS's to Regular Temperaments

This only makes sense if all of the coefficients of the matrix are positive, so matrices in which the row vectors correspond to equal temperament vals are a good start.

Meantone has been called the "5&7" temperament, which has been thought of as though meantone generates MOS of size 5 and 7. It's been claimed that this is a "separate paradigm" from the regular mapping paradigm, but it's not really. First off, let's take a look at the mapping for meantone:

[<1 1 0]

<0 1 4]>

The row vectors of this matrix are vals which represent generator chains. The top one has 2/1 mapping to 1 step, so we call it the "period," the bottom has 3/2 mapping to 1 step, and that's the "generator." (If you want, the top one can be thought of as 1-EDO, and the bottom one can be thought as 1-ED3/2).

OK, but what are do the column vectors of the matrix represent? Let's flip the matrix around and it becomes obvious:

2/1: [1 0>

3/1: [1 1>

5/1: [0 4>

The column vectors represent "monzos," but just monzos in the tempered space. Instead of the columns representing primes, they just represent generators, with the first coefficient being the period and the second being the generator.

OK, now let's take a look at the mapping for 5p&7p:

[<5 8 12]

<7 11 16]>

Many people usually just think of this like an "unreduced mapping," which is basically useless until you put it into some kind of reduced normal form. But have you ever considered what it actually means? In this case, the rows are still generators, telling you what the mappings for 2 and 3 are. But look at the column vectors now:

2/1: [5 7>

3/1: [8 11>

5/1: [12 16>

These are still tempered monzos, telling you that 2/1 is reachable by 5 of one type of generator plus 7 of the other type of generator. So if you just connect this part of your brain to the MOS part on the left frontal hemisphere, you'll realize this is describing a 5a7b scale (ether 5L7s or 7L5s) with 2/1 as the period, assuming you evenly distribute the steps as much as possible.

So we can see that this temperament forms 5a7b scales if the period is an octave. But if the period is 3/1, we can also see it forms 8a11b scales, and if it's 5/1, it forms 12a16b scales.

If xLys is what you want, then 5a7b corresponds to 5L2s, the only thing which yields 5L7s and 7L5s as children. THEREFORE, the following information specifies meantone and only meantone temperament:

2/1 as period: 5L2s

3/1 as period: 8L3s

5/1 as period: 12L4s

This assumes we're referring to the same "L" and "s" steps for all of these MOS's.

In short, it's been stated that there's no way to relate MOS's like "5L2s" to a regular temperament like "meantone," because the two come from different paradigms. This is only half true, however, because when you talk about 5L2s, you're already mapping at least one interval - the 2/1 as the period. By just mapping the 3/1 and 5/1 as well, you end up throwing in the additional information necessary to describe a regular temperament.

In comparison, here's the matrix for 5c&7c, which represents schismatic temperament:

[<5 8 11]

<7 11 17]>

The column vectors are:

2/1: [5 7>

3/1: [8 11>

5/1: [11 17>

So in schismatic, we end up with

2/1 as period: 5L2s

3/1 as period: 8L3s

5/1 as period: 11L6s

The same MOS's are formed for 2/1 and 3/1, but in schismatic, you end up with 11L6s for 5/1 instead of 12L4s.

This can get tricky if some of the coefficients of the matrix are negative - it's easy to figure out that 5 steps out on one type of generator chain, and 2 steps on out another, can yield a 5a2b MOS if you arrange the steps in a distributionally even manner. But what if it's like, 5 steps of one type of generator chain gets you to 2/1, but then -3 steps on the other chain gets you to 2/1? What's 5L-3s mean? It ends up getting into something called "monotonic ascending order," but I'd rather not deal with that here.