Equal temperament

An equal temperament is the regular temperament interpretation of an EDO (or more generally, any equal tuning). As such, it contains not only a set of notes but a way to translate JI intervals into numbers of steps in a consistent manner.

Vals

An equal temperament is notated by a val, a sequence of numbers that specifies how many steps each prime is mapped to in the ET. The val consisting of the edo's best direct approximations of each prime is called that edo's patent val. For example, 12edo's patent val in the 5-limit is ⟨12 19 28], which is 12-ET in the 5-limit, and tells you that the octave is mapped to 12 steps, the third harmonic is mapped to 19 steps, and the fifth harmonic is mapped to 28 steps.

Using a val

To use a val, determine the prime factorization of the interval you want to map to the equal temperament.

Example 1

For example, 125/96 is 5^3 / (3^1 * 2^5). So, we add 3 times the entry for 5 (28*3), and subtract 1 times the entry for 3 (19*1) and 5 times the entry for 2 (12*5). (28*3)-(19*1)-(12*5) = 84-19-60 = 5, and so 125/96 is represented by 5 steps in 12-ET.

Example 2

Another example: 128/125 is 2^7 / 5^3. So, we add 7 times the entry for 2 in the val (12*7), and then subtract 5 times the entry for 5 in the val (28*3). (12*7)-(28*3) = (84-84) = 0. This actually means that 128/125 is tempered out in 12edo - in other words, it's represented by the unison.

Example 3

3125/2048 is 5^5 / 2^11. So, we add 5 times the entry for 5 in the val (28*5) and subtract 11 times the entry for 2 (12*11). (28*5)-(11*12) = 140-132 = 8. Note that this is different from what we would get if we were to approximate 3125/2048 directly (which would be 7 steps).

Conclusions

In our third example, the result from using the val is different from the direct approximation. This follows out of the rules of regular temperaments, discussed on the page on regular temperaments, and while it may seem counter-intuitive, it overall leads to a more clear and easier to work with harmonic structure. Vals essentially allow you to forget about JI and work entirely within your ET of choice.

Wart notation

An arbitrary equal temperament may be notated, if you do not wish to use its full val, by notating its deviation from the patent val of the edo it corresponds to.

To do this, we specify the patent val of n-edo as just n, then we can specify each prime we want to map to the second best approximation by appending a letter, called a wart, after the number:

• adding a means you make the mapping of 2 worse
• adding b means you make the mapping of 3 worse
• adding c means you make the mapping of 5 worse
• adding d means you make the mapping of 7 worse
• etc.

So we can refer to [17 27 40] by "17c" (not to be confused with 17 ¢ (cents)), where we mnemonically think "a, b, c; 3rd letter; 3rd prime is 2, 3, 5; there is one 'c' so we make the mapping of prime 5 worse (further from just) once compared to patent".

The general rules:

• Wart letters specify prime approximations being altered from the patent val. The n-th letter of the alphabet refers to the n-th prime: a~2, b~3, c~5, d~7, e~11 etc.
• A letter which appears m times refers to the (m + 1)-th most accurate mapping for that prime.
• So, if a number representing a val is wartless, it is taken to mean the patent val.
• A wart letter may prefix the number, in which case it specifies the corresponding prime as the interval of equivalence to be divided by the following number. For example, b13 refers to the patent val of 13ed3. The octave is assumed, so "a" is typically not written out.

One can also retroactively assume that wart notations are what are being suffixed with "-ET" when we write, for example, "12-ET". As such, 13ed3's patent val, the equal-tempered form of Bohlen-Pierce tuning, may be called "b13-ET", and if the second-best mapping of 5/4 is used in 12edo (that is, [12 19 27]), then you might write the corresponding equal temperament as 12c-ET.