I’m having trouble understanding these ideas and want to know if my understandings are correct. I’m starting to learn piano again and am super interested in the math/physics of music theory. My understanding based off reddit and viewing some YouTube videos is that equal temp. means that all the 12 notes on a piano are equally distanced from each other frequency wise by a factor of the 12th root of 2 to each other. This is so when a note repeats, its double the frequency the next time up. So theoretically if note A where defined as 550hz, then note B’s frequency would be 550hz multiplied by 12th root of 2 which equals 582.7047019. Okay, easy math to understand this idea. This tuning is again, intended to make the next A note double the frequency to make an octave. But why? Because by the laws of nature and harmonics, pairing a chosen frequency with its harmonic overtones just sounds good. But by making every half step a 12th root of 2 separate from each other, you make every note out of tune equally. This is because a 12th root of 2 is an irrational number.
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Now, here comes just intonation to fuck me all up because I just don’t fucking get it…………. I think. So basically, just int. is going off the principle of strictly following harmonics. To make sure all the harmonies are aligned in place to sound musically perfect, you pick a frequency and have every note tailor made to follow this frequency’s overtones. But by doing this, you jeopardize every other note on a traditional 12 key keyboard unless it’s an overtone of said starting frequency. In other words, just intonation makes an instrument that is only made to sound good for a certain key that corresponds to the chosen frequency. So the reason we have gravitated towards equal temperament is because its more practical for music even if most of it isn’t perfectly harmonic. We try to follow the harmonic rules as close as possible. And so far, it has worked out given all the amazing music we have and will continue to make.
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Your understanding is basically correct. I’d add that “12th root of 2 is an irrational number” is not the fundamental reason for the incompatibility between equal and just temperaments.
The fundamental problem is that the fifths series ((3/2)^m the base frequency in just intonation) and octaves series ((2/1)^n the base frequency) never meet; by that, I mean that you can never build an n-th octave out of pure fifths. The frequencies of consecutive fifths in just intonation goes 3/2, 9/4, 27/16, 81/32, 243/64, … and it will never be a power of 2 (a n-th octave).
So even if we divided the octave with equal temperament using something else than 12 notes, we’d never get pure 3/2 fifths.
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