# [ELI5] What is equal temperament in music?

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[ELI5] What is equal temperament in music?

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So I’m not trained in music theory but I’ve recently watched a video that explains this really well, linked here: https://youtu.be/QEjANevZVfw

It means the actual frequency of the notes on our musical scale are equally spaced out (logarithmically). Each note is 2 to the 12th root times more than the note before on the system we usually use (a 12-tone equal temperament).

Equal Temperament means that the ratio of the frequencies of a certain interval is always the same, no matter where on the scale that interval lies.

As an example, in equal temperament the fifth between C-G and that between F#-C# both have a frequency ratio of ca. 1.4983 (the 12th root of 2 to the 7th power, to be exact). A song played in C major sounds exactly the same as one in F#-major, just higher (or lower). That might sound obvious (and today it’s how the vast majority of instruments are tuned), but historically, different temperaments were used where some fifths were exactly 1.5 while others were farther off, which led to some scales sounding more harmonic while others sounded more dissonant.

Most wind instruments have one chamber where sound is created. By pressing buttons, you open up passages along the chamber, effectively shortening it or lengthening it. Trombones quite literally do lengthen and shorten their chamber with a slide. Other brass instruments with keys may have a few different chambers, but there are still only a few. Nowhere near as many chambers as notes that a player can produce. Even most stringed instruments have 3 or 4 (6 for guitars) strings that are responsible for every note.

Pianos, xylophones, marimbas, other tuned bell-type instruments have a unique string or bar for every single note. The main effect is that you can tune these instruments to any scale, on any octave. But the instruments above are designed to play in a certain ranges. The further away from that range you get, the larger the error starts to build up. Pianos, xylophones… Have equal temperament tuning. Every not is the exact same ratio to the frequency of the one before it, all the way up the scale, and every octave jump is perfectly even.

Slightly less eli5. The harmonics used to tune wind instruments rely on a vary neat and tidy ratio like 1:2, 2:3, 3:4, 4:5… They can be written with finite digits. Unfortunately, no matter which one you pick, some things will be off. Let’s say you want to walk down the keys of a piano and strike every 4th key (skipping 3 keys between each press). There are 12 keys in an octave, so after the striking 3 keys, we should hit a perfect octave (3*4=12). We’re also playing along with a saxophone which uses harmonics to tune (I don’t actually know which harmonics, but the example will work well enough).

The saxophone plays a C, and the piano plays a C. They start at the same frequency because instruments are often tuned to middle C. Whatever frequency that is, 4 half-steps up (that’s the interval I mentioned, every key on the piano is called a half-step, and we’re playing every 4th key), the saxophone plays a E which has a harmonic ratio of 4:5. The next note is G# so one more 4:5 ratio. We can just multiply the two ratios together like this: (4*4):(5*5) = 16:25. One more jump takes us back to C again, but one octave higher and the total ratio is 4³:5³ or 64:125 and look at that, the second number is really close to double the first. 64*2=128 were only 3 off (≈2.34%). But a true octave is supposed to be exactly double. So the saxophone is starting to sound a little bit flat compared to the piano, it’s lagging behind.

And this is true no matter which neat and tidy ratio you use. You’ll get further and further from the true octave the further you go out. The only way to get a true octave is to use the ratio 1:2, but how do we evenly subdivide that ratio into 12 keys, the 12 half-steps of the octave? That number is 1:¹²√2 but the twelfth root of two cannot be expressed as a ratio with finite digits like 4:5 can. So we lose a bit of the harmonic interactions within an octave, but they’re usually close enough that you can’t tell the difference. This way, any key on the piano can be played, and it will sound good enough to the vast majority of people. Whereas harmonic tuning will only play well for a small range.