Why is equal temperament based on n-th root of 2?

Why is Equal temperament based on defining the smallest ratio as the n-th root of 2, 2^(1/n). Or how did someone arrive at conclusion, that he/she/they must use 2? Is it because A5 is A4*2?

• Musical pitch is not additive, but multiplicative if you check the frequencies, and that's why dividing an interval equally involves computing roots and not just division. Commented Apr 20, 2023 at 12:30

It's because the frequency ratio of the octave is 2:1. So to divide the octave into n equal parts, you take the nth root of 2.

So, essentially, yes: A5 = A4 * 2. And, more generally, XY = X(Y-1) * 2, where X is some pitch name, and Y is the octave designation.

The standard 2^(1/n) is based on the octave (a factor of 2), but 2 is just a choice. It's possible to define equal temperaments using other intervals, for example the Bohlen-Pierce scale which divides the twelfth into 13 equal parts: 3^(1/13).

If you compare music using the Bohlen-Pierce scale and the standard octave-based scale (and equal temperaments based on other than 12), you'll understand why the vast majority of music uses 2^(1/12).

There are other octave-based equal temperaments that provide better approximations of just intonation, but they are not as practical or well-established as the 12-note scale.

• "you'll understand why the vast majority of music uses 2^(1/12)": you can also understand this if you construct a scale based on the equal division of the fifth into seven semitones. If you do that, the octave is 3.35 cents sharp. It might also be worth mentioning that the 12-note scale was developed before equal temperament. Commented Apr 20, 2023 at 9:08
• @phoog no, you wouldn't notice much if the octave is 3.35 cents sharp, that's too small of a discrepancy. With Bohlen-Pierce however, the closest interval to an octave is a whopping 30 cents flat. Basically, octaves should just be considered to not exist at all in BP... which is of course a rather tricky exercise for composers and listeners, since everyone is so used to having the octave as almost an identity relation. Commented Apr 20, 2023 at 16:17
• On the other hand, if you spend a lot of time listening to good music in other divisions of the octave, you'll start to wonder why the vast majority of music uses 2^(1/12) after all. There are definite concrete advantages (accurate 5ths, reasonable major 3rds, not too many notes, ability to modulate key), but it's also a question of what your ears are used to, and other scales offer other possibilities. Commented Apr 21, 2023 at 3:49
• @leftaroundabout you would notice it because of acoustic interference. If you play A4 and A5 together you will hear beating at a rate of about 1.7 per second or 102 per minute. Commented Apr 21, 2023 at 9:46
• @phoog no, except the most minimalistic, distraction-free contexts. The point where fine tuning actually starts to matter lies somewhere between 5 ct and 20 ct. The 12-edo fifths and the stretched octaves on a piano are perfect for all practical intends. The 15-edo sharp thirds are borderline (but still usually accepted), but 30 ct is in a completely different category. Commented Apr 21, 2023 at 12:10

How did someone arrive at conclusion, that he must use 2? Is it because A5 is A4*2?

Yes.

More precisely, it's because "equal temperament" is shorthand for "equal divisions of the octave" in addition to the fact that the octave is a doubling of the frequency.

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

The reason for the exponentiation is of course the logarithmic relationship between pitch and frequency. If you picked some other base than 2, you would be dividing some other interval into equal steps.

For example, the acoustically pure major third is a ratio of 5:4 or 1.25:1. You can divide that into four equal half steps with a ratio of 1.25^(1/4).

(If you do that then 12 of those half steps have a ratio of 1.25^(12/4) = 1.25^3 = 1.953125. If you tune A=440 Hz in this system, the 12th half step above will be 859.375 Hz instead of 880 Hz.)

It is exactly because of that. Note that intervals are additive, while frequency ratios are multiplicative. If in an additive system you wanted to divide an octave in 12 equal spaces you’d divide the distance by 12. In a multiplicative system you’d rather take the 12-th root of an octave, which is the number x so that x^12 (which is 12 steps up) is an octave, which is 2.

If you want you can think of intervals as logarithms of the frequency ratios. Then 1 octave = 2, so the interval in octaves is the base 2 logarithm of the frequency ratio. Multiply this value by 12 and you’ve got semitones. In the other direction: To get from semitones to frequency ratios we need to divide them first by 12 and then take that power of 2. So 1 semitone is 1/12 octaves is 2^(1/12) frequency ratio is the 12th root of 2.

A different interpretation: Suppose you’ve got 12 different (just) semitones s1, ..., s12. Then we hope that these 12 semitones for an octave, so s1·s2·...·s12 = 2. Then the 12th root of 2 is the so called geometric mean of these 12 semitones, so a single interval that on average behaves the same way as s1,...,s12 multiplicatively in the sense that if you replace all s1,...,s12 by that interval the product of all does not change.

• Thank you very much for your answer. Commented Apr 20, 2023 at 8:12

Doubling the frequency of a note puts it up an octave. If A4 is 440Hz, the next A up (A5) will be 880Hz (and A3, the next one down, will be 220Hz). Can we accept that as a given? So that's why '2'.

You'll notice that the difference between A3 and A4 is 220Hz Between A4 and A5 is 440Hz. We're multiplying here, not adding.

So if we want to have 12 equal divisions between A4 and A5, i.e. multiply the frequency by the same figure (call it x) 12 times, then 'x times x times x times x....(12 times)' needs to come out as '2'. Or, in math language, x needs to be the 'twelfth root of 2'.

Long answer: Sound is a wave. Two tones with frequencies in a 2:1 ratio are, so to speak, the pair that's most in sync of all without being identical; the compound wave has the simplest possible profile that two different frequencies can make.

To a gradually but fast lessening extent, something similar can be said of other pairs where the frequency ratio is a fraction of two small numbers, and indeed, the ratio of 3:2 (or 2:3 going down) forms a fifth in Pythagorean tuning. But an octave, the simplest and purest of them all, is the uncontested queen of all musical intervals.

When we move on from Pythagorean to equal temperament, she shall not be dethroned.

This is a complex question of physics, human physiology, math, technology and history.

• First, the physics part.

A lot of sound sources happen to generate a base frequency and its (almost integer or almost half-integer) multiplies.

• Second, the human ears and the brain between them:

Consequently, human ears are quite precise at detecting these multiplies in order to attribute a complex sound to a single sound source. Sometimes, the base frequency is inaudible for one reason or another so we are equally good at detecting simple ratios as 2/3, 3/4 and so on.

This ability, in fact, dates well before we became humans. Most mammals have it as well (I am not aware of wider taxon featuring this, but it is quite possible that some birds have it, too).

Did you see the "almost" word above? Single sounds that are precise at these ratios sound pretty and sounds that deviate from them sound unpleasant. Going deeper into this is called "harmony".

• Third, the math.

One can build a pitch scale by using the "natural" intervals (1:2 now called octave, 2:3, 3:4 and so on) and seed the intervals between them with additional fixed pitches based on whatever logic or aesthetics one may decide. Human ears are hard-wired for the small-integer ratios, the rest is "in between".

• The history

Long ago, the collective music thinking came up with the Pythagorean scale (actually way older than its namesake). It had 8 intervals of unequal size, some of them good enough to put a half-tone between them, others - not really. Good for a lot of purposes, bad when one wants an universal, transposable instrument like a pipe organ or a piano.

After some tweaking, people arrived to the 12 equal temperament scale by fixing only the octave at an easy frequency ratio (1:2) and spacing the 12 intervals inside equally (ratio-wise). Some of the scale points went almost exactly at the Pythagorean points, others went somewhat off, but the overall result is good enough to be the base of the modern Western and far-East (Chinese) music.

Now, the caveats: 12 equal temperament scale is not the only one in use and itself is somewhat tweaked for one use or another. String instruments and vocals do what they call "intonation" and completely different pitch scales (e.g. microtonal ones, middle-East traditional music that uses equal-ish temperament scales with different number of intervals) do exist as well.

• This doesn't address the question of why n-tone ETs use the nth root. Commented Apr 21, 2023 at 13:16
• @Aaron: If one wants to have positive real values x, y, z, such that xyz=2, there are many xyz triples that could work, such as e.g. {1.25, 1.25, 1.28} but only one where x=y=z, and the value that would need to be shared among x, y, and z would be--by definition, the cube root of 2. For an n-tone equal temperament, the note ratio must be the nth root of 2. For an n-tone unequal temperament, an infinite range of sets of values would be possible. Commented Apr 21, 2023 at 16:08
• @Aaron: If one wanted a three-tone scale, it's possible that the set of frequency ratios {1.25,1.25,1.28} might sound better than {cubert(2),cubert(2),cubert(2)}, if one started on the second note of the scale the frequency ratios would be {1.25,1.28,1.25}, violating the advantage of an equal temperament, which is that a scale which starts on any note should be equivalent to a scale which starts on any other note. Commented Apr 21, 2023 at 16:13
• @supercat I don't mean to be obtuse, but I have no idea what you're trying to explain to me, and I get the impression you either didn't understand my comment here (the post doesn't address the original question), or you didn't ready my answer. Please explain what I'm missing. Commented Apr 21, 2023 at 17:21