Is it mathematically possible to create an equal temperament which matches just interval ratios?

Question

I've coded a Python app which lets us have any number of semitones in an octave in order to experiment with microtonal music. I would like to ask if there's a possibility to create such an equal temperament which will match just interval ratios?

https://en.wikipedia.org/wiki/Equal_temperament#/media/File:Equal_Temper_w_limits.svg

As we can see on the figure above, there are 9 different equal temperaments, and 72-tet is closer to matching with just interval ratios than 12-tet. Is it mathematically possible to divide an octave the way it will perfectly match just interval ratios? For instance, 100-tet, 150-tet, etc.? Many thanks in advance!

Answer 1

By definition this is not possible.

Just intonation ratios are rational numbers, N/M where N, M are integers.

Equal temperament is based on defining the smallest ratio as the n-th root of 2, 2^(1/n).
For 12TET n = 12.

What you are basically asking is if an irrational number can be made to exactly match a ratio of integers. This will never be possible.

Since you are dealing with computers and code you probably know that 2^1/12 can not be expressed in binary with finite precision. This poses an even more interesting question in that realm. The real question is can you generate an equal tempered tuning in s/w that will match just to within a certain error tolerance? And to that the answer may be yes, but a purist would argue that the approximation is not truly equal tempered! The pragmatist would realize that no matter how hard we try we cannot ensure that instruments are tuned so that f(n+1/2)/f(n) = 2^(1/12) so the point is moot. And finally, at some point the human ear cannot tell the difference since there are physical limits to the resolution of our ear+brain system.

If you are willing to track down the limits of human resolution and account to finite precision in computer arithmetic then you might be able to generate an approximate 'TET' algorithm that provides you with Just frequency ratios that are both within the limit of human pitch discrimination and equal to within some error tolerance. That is the best you can hope for.

Answer 2

The other answers approach this from dividing the octave and showing that equal divisions must be irrational. Another way of looking at this is to consider whether we can compose an octave by successive multiplications with a rational number. The result is of course the same: we can't.

Start with the Fundamental Theorem of Arithmetic:

every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

Along with that, we need the definition of irreducible fraction:

Every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers, and b > 0.

Two numbers are "coprime" when they have no prime factor in common. Thus, a rational number can be expressed as the set of prime factors (with exponents) that defines its unique irreducible expression. For example, 81:64 may be expressed as 3⁴ * 2⁻⁶. When you multiply ratios, you add the exponents of their prime factors. So the product of 3:2 and 5:4 (2⁻¹ * 3¹ and 2⁻² * 5¹) is 2⁻³ * 3¹ * 5¹, or 15:8.

You're looking for a ratio R that equally divides an octave into N parts, which means that R^N equals 2. Can we identify such ratios?

The classic example is that of the perfect fifth, a ratio of 3:2. Other intervals may be found by raising that ratio to a certain power, adjusting the octave by multiplying or dividing by a power of 2. For example, the major second can be 9:8, which is the square of (3:2)²/2. The major third can be 81:64, which is (3:2)⁴/4. To generate all the pitches in the circle of fifths, keep multiplying. When you get back to C (which some authors will call B♯), you end up with a pitch slightly higher than seven octaves above than the one you started with. The ratio of those two frequencies is 3¹²:2¹². You can't arrive precisely at the same pitch class because the prime factorization includes 3 with a non-zero exponent.

By generalizing, we can show that the same is true for every ratio R that is not itself a power of 2. (If R is a power of 2, then you have defined one-tone equal temperament, a system in which there is only one pitch class and in which the base interval is the octave or a multiple thereof, which is not interesting. This is the same as dividing the octave using the first root of 2, which is of course 2.)

Consider ratio R with at least one prime factor P unequal to 2. As with the example of the perfect fifth, every time you multiply a frequency by R, the magnitude of P's exponent in the resulting frequency is greater than it was in the original frequency. The goal is to achieve a result where P's exponent is zero, but each multiplication takes us farther from that result. It is therefore impossible.

Of course, one of the things about equal temperament is that 2^7/12 is so close to 1.5 that perfect fifths are close enough to pure for most purposes. From the standpoint of ratios, this comes about because 3^12 (531,441) is fairly close in value to 2^19 (524,288). You might find decent approximations by looking for numbers that are similarly close in value to some power of two.

In practice, though, I think people who have explored N-tone equal temperament as an approximation of just intonation have chosen N such that some power of the Nth root of 2 is close in value to 1.25 (the ratio of the just major third). If you're interested in some other interval then you can experiment with values of N to find a close approximation to that interval.

I feel compelled to close with this warning, however: if you have too many divisions of the octave, the system is not useful for human musicians. It's only going to be useful for a computer. If you're looking into such a system as an approximation of variable-pitch just intonation, the programmer (or the program) will have to choose which of the several notes to use. In 53-tone equal temperament, a whole step can be 8/53 or 9/53 of an octave in size. In variable-pitch just intonation, a whole step can be 10:9 ratio or a 9:8 ratio. It's basically the same problem. Why not just program your computer to use variable-pitch just intonation?

Answer 3

As I understand the question, this is pure mathematics:

No it is impossible. No matter, how many divisions you have, say n, the step width will always be nth root of two and therefore an irrational number.

The just relations are rational numbers, so there will always be approximations, but the more you choose, i. e. the higher n is, the closer you will be able to approach.

Answer 4

You can't even get an equal temperament system* in which fifths and octaves are perfect. This is simply because if you had a system of equal steps in which an octave was A steps large and a twelfth was B steps large, it would stand to reason that B octaves would be equal to A twelfths, since both would be an interval of AB steps. This can't happen though: a just twelfth is the ratio 3:1 and an octave 2:1. If you stack multiples of an interval, you raise its ratio to a power - but no positive power of 3 equals a power of 2, given that powers of 3 are all odd and powers of 2 are all even. This reasoning basically applies to all pairs of intervals - two rational numbers sharing a power is a very special property.

Otherwise said: a fifth is equal to log(1.5)/log(2) octaves (about .585) and this number cannot be represented as a ratio of integers. However, you can try to approximate it by rational numbers - using, for instance, the convergents to that ratio (which are, in a sense, the best approximations of up to a given maximum denominator), you would get the following sequence of approximations to the ratio:

0/1, 1/1, 1/2, 3/5, 7/12, 24/41, 31/53, 179/306, ...

Where the number 7/12 would be interpreted to mean that a fifth is about 7 tones in 12TET - which is, of course, a familiar fact. These particular denominators will do far better than other fractions with a similarly large denominator in approximating a fifth - for instance, 7/12 is only off by about 3 parts per 2000 - which is far better than the approximation that log(1.5)/log(2) by rounding to the nearest hundredth: 0.58, despite this latter approximation using a denominator of 100 as 58/100. The approximation 31/53 is only off by about 1 part per 20000, which is pretty good for an approximation whose denominator is just 53.

Of course, it's a bit harder to say what happens when you suddenly want ratios other than fifths and octaves and composites of them - if you wanted just octaves, fifths, and thirds, you would search for a denominator (number of steps) such that both log(3)/log(2) (for twelfths) and log(5)/log(2) (for major third + two octaves) were close to fractions with this denominator - and this not as mathematically straightforward as approximating just a pair of intervals (but still impossible to do perfectly).

(*Notwithstanding that you could expand to multiple dimensions with multiple kinds of equal steps - for instance a Tonnetz or isomorphic keyboard represents exactly this where one dimension has steps of perfect fifths and the other major thirds - which also then leads to minor thirds along another direction. Of course, you lose the linear nature of a keyboard this way, since now you're dealing with two ratios - and you still don't have octaves, though you could imagine adding yet a third axis!)

Answer 5

Other answers do a good job of proving why no non-trivial exact solution can exist. For completeness, I'll note that there is a trivial solution, albeit not especially useful musically - one note per octave. All ratios of pitches differ by some power of two, which is always an integer and thus "just" - trivially so, as only one pitch class is allowed.

Answer 6

This question is 2500 years old, and the answer is no. See my old answer here why # and b