Piano Tuning
This is certainly a fun way to introduce twelve-tone equal temperament to someone who knows enough math to understand logarithms but is new to music theory. It works well, because the only elementary music theory claims that we need to accept without explanation are that there are twelve different note names that repeat in the same order and that each note of a given name is twice the frequency of the previous note of the same name.<p>Okay, not quite. You actually also need to know that the frequency ratios of all pairs of keys n keys apart should be the same, for all n. This article kind of sneaks that one in silently. But once you’ve accepted that, it leaves you with only one option: tuning the keys to frequencies spaced evenly along the logarithmic axis.<p>Of course, there are so many more details on every side of this topic. Why 12 divisions per octave (this certainly isn’t the case for all musical traditions)? Why do we want all pairs of notes n steps apart to have the same ratio (this certainly isn’t the case for some instruments)?<p>And, most importantly to me, since it’s the question I can find the least solid information on: why do we take for granted that one note with twice the frequency of another note sounds so similar that we call both notes by the same name? I have a pretty good understanding of other details of music theory, but I’ve never gotten a straight answer on why (and indeed to what extent) octave equivalence exists.<p>I’ve always speculated that it has something to do with our auditory systems evolving to interpret sounds from roughly simple harmonic oscillators, because those exist in important roles in nature (like the vibrating chords and air columns of human and animal vocalization).<p>But that’s still not a great explanation to me. Yeah, there’s some important noise-makers in nature that have roughly the overtone series. But why would it be important to hear the first overtone as equivalent (in some strong but not absolute sense) to the fundamental? Could it have something to with hearing vocalizations from a distance such that the fundamental would be more attenuated than overtones? That’s really grasping at straws.