You gain so much more appreciation for music when you actually *hear* this kind of stuff going on.

]]>Depends what you mean by equal :-) Musically, yes… physically, no.

If they were physically the same, you’d have what is called “equal temperament”, where the ratio of frequencies between notes a semitone apart is 1+2^(1/12):1 Do that 12 times, and you end up with a ratio of 2:1 for twelve semitones, which is equal to one octave (i.e. the gap between two notes of the same name). However, music played on a piano tuned that way will have a “bland” feel. The problem is that if you take an interval of one fifth (7 semitones), it sounds best with a ratio of 3:2 rather than 1+2^(7/12):1 (although the difference is small). Now, you can try and tune a piano so that all notes seven semitones apart are tuned with a ratio of 3:2… but then you’ll find that the octaves don’t work. (Take a low C on the piano, go up a fifth twelve times, and you’ll end up on a C again, seven octaves higher). If you do the maths, you’ll find that (3/2)^12 = 129.74 is very close to 2^7=128.

So, what piano tuners have to do is pick a set of ratios for each note in the scale that preserve octaves (always), and try and keep as many fifths “correct” (or nearly so) as possible – with a bias towards those fifths that occur more often in written music (e.g. C-G rather than G#-D#). A given set of ratios is called a “temperament”… which one you end up with will depend mostly on your piano tuner’s preferences. (Also, different temperaments go in and out of fashion.)

(If you were to only ever play music in one key, you can use “just temperament”, where all the ratios are integers and powers of 2, 3 or 5. That’s also the temperament into which unaccompanied singers will gravitate)

Hope this interests you!

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