Keenan Pepper's explanation of vals

[math] \def\hs{\hspace{-3px}} \def\vsp{{}\mkern-5.5mu}{} \def\llangle{\left\langle\vsp\left\langle} \def\lllangle{\left\langle\vsp\left\langle\vsp\left\langle} \def\llllangle{\left\langle\vsp\left\langle\vsp\left\langle\vsp\left\langle} \def\llbrack{\left[\left[} \def\lllbrack{\left[\left[\left[} \def\llllbrack{\left[\left[\left[\left[} \def\llvert{\left\vert\left\vert} \def\lllvert{\left\vert\left\vert\left\vert} \def\llllvert{\left\vert\left\vert\left\vert\left\vert} \def\rrangle{\right\rangle\vsp\right\rangle} \def\rrrangle{\right\rangle\vsp\right\rangle\vsp\right\rangle} \def\rrrrangle{\right\rangle\vsp\right\rangle\vsp\right\rangle\vsp\right\rangle} \def\rrbrack{\right]\right]} \def\rrrbrack{\right]\right]\right]} \def\rrrrbrack{\right]\right]\right]\right]} \def\rrvert{\right\vert\right\vert} \def\rrrvert{\right\vert\right\vert\right\vert} \def\rrrrvert{\right\vert\right\vert\right\vert\right\vert} [/math][math] \def\val#1{\left\langle\begin{matrix}#1\end{matrix}\right]} \def\tval#1{\left\langle\begin{matrix}#1\end{matrix}\right\vert} \def\bival#1{\llangle\begin{matrix}#1\end{matrix}\rrbrack} \def\bitval#1{\llangle\begin{matrix}#1\end{matrix}\rrvert} \def\trival#1{\lllangle\begin{matrix}#1\end{matrix}\rrrbrack} \def\tritval#1{\lllangle\begin{matrix}#1\end{matrix}\rrrvert} \def\quadval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrbrack} \def\quadtval#1{\llllangle\begin{matrix}#1\end{matrix}\rrrrvert} \def\monzo#1{\left[\begin{matrix}#1\end{matrix}\right\rangle} \def\tmonzo#1{\left\vert\begin{matrix}#1\end{matrix}\right\rangle} \def\bimonzo#1{\llbrack\begin{matrix}#1\end{matrix}\rrangle} \def\bitmonzo#1{\llvert\begin{matrix}#1\end{matrix}\rrangle} \def\trimonzo#1{\lllbrack\begin{matrix}#1\end{matrix}\rrrangle} \def\tritmonzo#1{\lllvert\begin{matrix}#1\end{matrix}\rrrangle} \def\quadmonzo#1{\llllbrack\begin{matrix}#1\end{matrix}\rrrrangle} \def\quadtmonzo#1{\llllvert\begin{matrix}#1\end{matrix}\rrrrangle} \def\rbra#1{\left\{\begin{matrix}#1\end{matrix}\right]} \def\rket#1{\left[\begin{matrix}#1\end{matrix}\right\}} \def\vmp#1#2{\left\langle\begin{matrix}#1\end{matrix}\,\vert\,\begin{matrix}#2\end{matrix}\right\rangle\vsp} \def\wmp#1#2{\llangle\begin{matrix}#1\end{matrix}\,\vert\vert\,\begin{matrix}#2\end{matrix}\rrangle} [/math] A val is a function that assigns a whole number to a JI interval in a consistent way.

Here's an example of what "consistent" means: If v is a val and x and y are just intervals, then v(x + y) = v(x) + v(y), where x + y means the composition of the intervals x and y (you add their cent values, which means you multiply their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is "homomorphism".

Because of the consistency property, if you want to write down a specific val, you don't need to give its value on every possible JI interval you might want to apply it to. Instead, you only need to give the value for primes, because every other interval can be decomposed into primes. For example, if I tell you that v(2/1) = 12, v(3/1) = 19, and v(5/1) = 28, then you can figure out the value of v applied to any other interval of 5-limit JI. For example, 25/24 = 5^2 / (2^3 * 3), so v(25/24) = 2 * v(5) − 3 * v(2) − v(3) 2 * 28 − 3 * 12 − 19 = 1, so 25/24 must be mapped to 1 by this val. The standard way to write this val in symbols is

[math]\val{12 & 19 & 28}[/math]

The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.

Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is [math]\val{12 & 19 & 28}[/math].

Now, consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though it is not the best direct approximation of 625/512 available in 12edo. In other words, the result of tempering with a val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for composite intervals.

Vals supporting temperaments

A specific equal temperament (as opposed to an EDO) is represented by a single val. There are infinitely many vals, and therefore technically speaking infinitely many equal temperaments, corresponding to each EDO, but usually only a handful are anywhere near good enough to make sense musically. For example, [math]\val{12 & 18 & 27}[/math] is a different 5-limit temperament with 12 equal steps to the octave, which maps 3/2 to 600 cents, and 5/4 to 300 cents, so it's musically useless. But in some cases there are different equal temperaments with the same number of steps that are about equally good. For example, 17edo could be used as two different equal temperaments in the 5-limit: one corresponds to the patent val [math]\val{17 & 27 & 39}[/math], in which 5/4 is very flat at 353 cents, and one corresponds to the [math]\val{17 & 27 & 40}[/math] val (sometimes called "17c"), in which 5/4 is very sharp at 424 cents. Although these could be realized musically as exactly the same notes (17 equal divisions of 2/1), they are nevertheless different temperaments, because a 4:5:6 chord, for example, would be represented differently.

A val v is said to temper out a comma c whenever v(c) = 0. For example, [math]\val{17 & 27 & 40}[/math] tempers out 81/80 (exercise for the reader), but the patent val for 17edo does not. Thus, if you played a meantone comma pump (which depends on 81/80 being tempered out to return to the same pitch) in 17edo using the [math]\val{17 & 27 & 40}[/math] val, it would work, whereas the version using the patent val would not return to the same pitch. We summarize this by saying "[math]\val{17 & 27 & 40}[/math] supports meantone temperament" or "[math]\val{17 & 27 & 40}[/math] is a meantone val".

Temperaments other than equal temperaments (that is, rank 2 and higher) can be constructed out of vals. This operation can be denoted "v₁ ^ v₂" or "v1 & v2". One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals.

For example, consider the statement "5-limit meantone is 12 & 19". Here's a list of the simplest commas tempered out of those two 5-limit equal temperaments:

• 12: 81/80, 128/125, 648/625, 2048/2025, 6561/6400...
• 19: 81/80, 3125/3072, 6561/6400, 15625/15552...

In 12-tone equal temperament, all of the commas in the first list are tempered out (mapped to 0). In 19, all of the commas in the second list are tempered out. In the temperament you get from combining them, "12 & 19", only the commas common to both lists are tempered out. These are 81/80, 6561/6400 = (81/80)², (81/80)³, (81/80)⁴... In other words, it works out that 81/80 is the only basic, fundamental comma that vanishes in both 12 and 19—all the other commas are powers of 81/80, so they automatically temper out if 81/80 is tempered out. So we say that the 5-limit temperament "12 & 19" is the same thing as "the 81/80 temperament" or "meantone".

In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, type "12 19" into the equal temperaments field, and type "5" into the limit field. It tells you that the resulting temperament is called "meantone", it has 81/80 as its only "unison vector" (aka comma), and other information you might find useful.

Practical consequences

Any piece of music can be performed without "comma issues" in any temperament in which all the appropriate commas vanish. It's okay if more vanish unnecessarily, but never less, or else issues arise. Some examples:

• A piece written in 12edo (assuming the patent val because it's the only one that makes musical sense), that actually takes advantage of more than one comma vanishing, cannot be played in any significantly different temperament without arbitrarily modifying the music. The octave stretch can be changed, for example, but it has to be a 12-note circulating temperament, with 12 roughly equal steps, or else some of the puns / comma pumps will not work.
• On the other hand, a piece written in meantone temperament (for example a piece written in conventional notation with everything "spelled" properly and no enharmonic puns) can not only be played in 12edo, but also in 19edo, 31edo, 43edo, or any equal temperament in which 81/80 is tempered out. It could even be performed in 17edo using the 17c val–it will sound quite different, but everything will still work out in a logical way. The same goes for 7edo–here, major and minor triads both become neutral triads, but that applies in a uniform way to all chord progressions and everything still "works".
• However, a piece written in meantone temperament cannot be performed in a non-meantone temperament like 22edo (using the patent val), or just intonation, without serious comma issues arising. It's possible to do it, but you'd basically be writing a new version of the piece from scratch rather than mechanically and faithfully translating it. You could play it in 22edo using a much worse val ([math]\val{22 & 35 & 52}[/math] rather than [math]\val{22 & 35 & 51}[/math]), but the chords would be far from the most accurate approximations that 22edo offers and would sound unnecessarily out-of-tune.
• A piece written in strict JI can be played in any regular temperament whatsoever. The more accurate, the better, of course.