< Mike's lectures on regular temperament theory

EPISODE 2: The Fundamental Laws of Tempering: First Law

If you missed the first episode, please read it here first!

If you have seen it, then hopefully you're a masters of vectors and covectors now. Many of you knew vectors anyway, and now you know what these mysterious things called covectors are. Now let's move on to bigger and grander things.

First, before getting into exterior algebra itself, I'll introduce a simple mathematical theorem which is basically like the First Law of Tempering. For this, all you need to know is the dimensionality of the original, pre-tempered JI space you're working in. For example, the 3-limit is 2D, the 5-limit is 3D, the 7-limit is 4D, the 11-limit is 5D.

Before I state the law, however, I should probably get some terms straight:

The dimension of a tuning, whether JI or a temperament, is the number of generators needed to hit every note in it. So 12-EDO is a 1D temperament, meantone is 2D, etc. This is sometimes called the rank of the tuning.
A temperament is considered to be a musical entity eliminating one or more commas. All of the various tunings of a temperament are classified as tunings of the same fundamental temperament. For instance 1/4-comma meantone, 1/3-comma meantone, TOP meantone, etc. are all different tunings of the overarching meantone temperament.^{[note 1]}
The word comma has two meanings, the first of which is more relevant to what I'm talking about here:
- In the context of temperaments, a comma is an interval you're deciding to temper out. For instance, the comma being tempered out in meantone temperament is 81/80, and the comma being tempered out in dicot temperament is 25/24.
  - You'll note that there's nothing in this definition stopping you from tempering out something like 5/4 and calling that the "comma" for your temperament, much like there's nothing stopping you from purchasing an 18-wheeler and driving it off of a cliff. The latter may yield more perceptually interesting results than the former.
- In the context of JI, the term "comma" also denotes a small musical interval in JI, one which you may not want to temper out at all. Archetypical examples of commas include 81/80, the syntonic comma, and 64/63, Archytas' comma. Sometimes small intervals are also called other things, such as kleisma, schisma, diesis, etc. I'm not concerning myself with all of these ad hoc names for size ranges, but you should know it so you don't get confused.

Now then, PREPARE FOR LAW

2.1: The First Law of Tempering

Given an initial JI space of dimension d, if you temper out n independent commas, there is exactly one unique temperament eliminating only those commas, and it will be of dimension (d − n).

2.2: Aftermath (pun intended)

Hm. What does that mean? Formulated more concisely,

Dimension of original JI space − Number of commas tempered out = Dimension of resulting temperament

Or, using fancy pseudo-algebraic terms,

if d = Dimension of JI space

and if n = Number of commas tempered out,

then Dimension of resulting temperament = d − n.

If this still looks like the equivalent of [math]\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)[/math] to you, then maybe some examples will help make it clearer.

So, the first thing this law makes reference to is the dimension of the initial JI space, which is being stored in a variable called d. If you're not used to thinking about JI in this way, this may seem a strange concept, at first, but it's really simple—the "dimension" is just the number of primes. If you're working in the 5-limit, for instance, your primes are 2/1, 3/1, and 5/1, so 5-limit JI is a 3D space. Therefore, as far as the formulae above are concerned, for 5-limit JI, d = 3.
The second thing we're supposed to know is the number of commas we're tempering out. This is simple enough. So say we're in 5-limit JI, and we want to temper out 81/80. That's one comma, so n = 1.
Finally, using this information, we can figure out what the dimension of the resulting temperament is. It's going to be equal to d − n = 2.

Therefore, after going through the simple process above, we can see that the resulting temperament is going to be a 2D temperament, meaning it's going to have two generators. Anyone who's played with meantone above knows that this is right.

You'll note also that the law states that "there is exactly one unique temperament eliminating only those commas." What does that mean? Well, it just means that there's only one temperament eliminating 81/80 and nothing else, which you know as meantone temperament. In other words, there's no other temperament that eliminates 81/80 and nothing else; there are no bizarro-meantones or anything like that. There's just one, and it's meantone.

This may seem obvious until you think about the implications for tempering out more than one comma. For instance, if you're in the 5-limit, and you eliminate both 81/80 and 128/125, the first law says you're going to get a temperament with a resulting dimension of 3 − 2 = 1. A 1D (sometimes called "rank-1") temperament is an equal temperament, so you know that the result's going to be one of those. However, the first law also states that there's only one temperament eliminating both 81/80 and 128/125 and nothing else. And, since you know that this temperament must be a 1-dimensional equal temperament, this means that there is only one equal temperament eliminating both 81/80 and 128/125. (oh my god, what is it?)^{[note 2]}

2.3: More examples

Let's go over some more examples to make some more sense of this all.

Say you're starting in 5-limit JI, which is a 3D space. Now let's say you temper out 81/80. You started with a 3D space and tempered out one comma. The first law tells us that there is exactly one temperament eliminating 81/80 and nothing else, and that it's of dimension 3 − 1 = 2. You know this temperament as meantone.
Say you're in the 5-limit again, and this time you're tempering out 81/80 and 128/125. There is only one temperament eliminating these commas and nothing else, and in this case it's of dimension 3 − 2 = 1. It just so happens that if you work it out, the rank-1 temperament in question is 12-EDO, specifically [math]\val{12 & 19 & 28}[/math] and its multiples, all of which "contain" [math]\val{12 & 19 & 28}[/math].
Say you're in the 7-limit and you eliminate 81/80 and nothing else. There is only one temperament eliminating only 81/80 in the 7-limit, and it's of dimension 4 − 1 = 3. Note that if you start with the 7-limit and temper out just 81/80, you end up with a different temperament than if you start with 5-limit JI and do the same thing—you now have a 3D temperament, as opposed to a 2D temperament! If you work it out, you get something that's effectively meantone but with an extra 7/1 generator that's totally outside the circle of fifths.
Say in the 7-limit, you temper out 126/125 and 225/224. There is only one temperament eliminating these commas and nothing else, and it's rank 4 − 2 = 2. It is septimal meantone, where augmented sixths (e.g. C–A♯) are 7/4. Now you've tempered out an additional comma to relate it that 7/4 back to the circle of fifths, and brought it down to 2D. (If it's not obvious to you that tempering out 126/125 makes 7/4 an augmented sixth, don't worry about that for now.)
Say in the 7-limit, you temper out 81/80 and 36/35. The only temperament eliminating both of these and nothing else will be of rank 4 − 2 = 2, and it's called dominant temperament. (Although it may not be obvious just yet, it makes 7/4 equal to a normal minor seventh.)

Now, what does it all mean?

2.4: What it all means

OK, hopefully you get it. It's a simple idea, but it leads to some very strong implications right away.

The first important implication is that 1D temperaments are no different from 2D temperaments, even though 2D temperaments tend to have fancy names involving animals and 1D temperaments don't. That they're no different means that if you start in the 5-limit or in any 3D system (like 3.5.7-limit JI, for instance), and temper out 2 commas, you get a single unique 1D temperament, just like you get a single unique 2D temperament if you temper out only 1 comma from 5-limit JI.
The second important realization is that if you care about temperaments being 2D, e.g. things producing MOS's, then: it requires 1 comma to be tempered out from the 5-limit to get to 2D; it requires 2 commas to be tempered from the 7-limit to get to 2D; it requires 3 commas to be tempered out from the 11-limit to get to 2D, etc.
The third important realization is that if you start in something like the 7-limit (which is 4D) and temper out one comma, then you get to a 4 − 1 = 3D temperament. And then, if you temper out ANOTHER comma on top of that, you get to a 4 − 2 = 2D temperament. And then, if you temper out ANOTHER comma, you get to a 4 − 3 = 1D temperament.
The fourth important realization is there's no conceptual difference at all between something like the dimensionality of a 4D JI lattice and the dimensionality of a 4D temperament—whichever you start with, if you temper out an additional comma, you lose a dimension and go down to 3D. A 4D temperament and some 4D JI system are both just different lattices that require four generators, and are hence both "4D."

The last point is worth mentioning again. The number of "generators" a temperament needs is the same basic concept whether the temperament is an actual temperament, or whether it's JI. It can be very useful, in fact, to think of something like JI as itself being a special type of "temperament" which tempers out 0 commas (and, consequently, has 0.0 cents of error), and to think of temperaments as "deriving" from it by tempering out additional commas. Carl has sometimes called JI the "identity temperament" before for this reason.^{[note 3]}

More to come...

Footnotes

↑ Note that others have at time used the term "temperament" differently—historically, some authors have used the term "temperament" in the sense that 1/4-comma meantone is a different "temperament" than 1/3-comma meantone. This use differs from the one presented above, which I think is a more useful definition for the purpose of classifying and organizing the infinite set of new tuning systems we have at our disposal, whatever you want to call them.
↑ You guessed it - this temperament happens to be the familiar 12 tone equal temperament, using the obvious mapping where you consider the 700 cent interval to be 3/2, and the 400 cent interval to be 5/4. In val notation, this is the val ⟨12 19 28]. Multiples of this val also count, for instance ⟨24 38 56], which is 24-EDO—they're considered to be "contorsions" of ⟨12 19 28], with extra unrelated notes that don't relate back to JI in any way at all. Much like all of the different tunings of a temperament are classified as the same temperament, all of the different "contorsions" of a temperament are also considered to have the same underlying temperament powering them. This is only a brief intro to this concept, which we'll touch more on later.
↑ Much like the definition of "temperament," this is one of those things that sometimes gets people up in arms, possibly due to artists feeling that certain choices of definition can interfere with their thought processes. Regardless of what semantic choices you make to explain this concept, or what definition of "temperament" or whatever words you prefer to use, you should understand the underlying idea, because it's important: the same mathematical objects in this tuning that represent temperaments also represent JI, with JI tempering out "zero" commas and containing 0.0 cents of error. Tempered spaces are just like JI spaces and work in exactly the same way, with the temperament itself being a "map" from JI into the tempered space. This isn't just a random philosophical idea, but is quite true mathematically: the wedgie for 5-limit JI, for instance, is [math]\tritval{1}[/math], and the mapping matrix for 5-limit JI is the identity matrix: $$ \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} $$ both of which you'll learn about later.

[1] Note that others have at time used the term "temperament" differently—historically, some authors have used the term "temperament" in the sense that 1/4-comma meantone is a different "temperament" than 1/3-comma meantone. This use differs from the one presented above, which I think is a more useful definition for the purpose of classifying and organizing the infinite set of new tuning systems we have at our disposal, whatever you want to call them.

[2] You guessed it - this temperament happens to be the familiar 12 tone equal temperament, using the obvious mapping where you consider the 700 cent interval to be 3/2, and the 400 cent interval to be 5/4. In val notation, this is the val ⟨12 19 28]. Multiples of this val also count, for instance ⟨24 38 56], which is 24-EDO—they're considered to be "contorsions" of ⟨12 19 28], with extra unrelated notes that don't relate back to JI in any way at all. Much like all of the different tunings of a temperament are classified as the same temperament, all of the different "contorsions" of a temperament are also considered to have the same underlying temperament powering them. This is only a brief intro to this concept, which we'll touch more on later.

[3] Much like the definition of "temperament," this is one of those things that sometimes gets people up in arms, possibly due to artists feeling that certain choices of definition can interfere with their thought processes. Regardless of what semantic choices you make to explain this concept, or what definition of "temperament" or whatever words you prefer to use, you should understand the underlying idea, because it's important: the same mathematical objects in this tuning that represent temperaments also represent JI, with JI tempering out "zero" commas and containing 0.0 cents of error. Tempered spaces are just like JI spaces and work in exactly the same way, with the temperament itself being a "map" from JI into the tempered space. This isn't just a random philosophical idea, but is quite true mathematically: the wedgie for 5-limit JI, for instance, is [math]\tritval{1}[/math], and the mapping matrix for 5-limit JI is the identity matrix: $$ \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} $$ both of which you'll learn about later.

[note 1]

[note 2]

[note 3]

Mike's lecture on the first fundamental law of tempering

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