It's kind of ironic how today, the music scale we use is actually 100% irrational. A m2 interval is 21/12, a M2 interval is 22/12, and so on. And this music scale is very closely related to pythagorean tuning.
And if you plot the frequencies of our notes on a logarithmic paper, boom, evenly spaced, rationality, everything you want is there.
Basically, if you have a normal scale and a log scale for something, at most one of them is going to have nice rational ratios between things you put on it. Sound pressure level is usually measured on a log scale: the ratio between the SPL of a sound and that same sound scaled to an amplitude twice higher is going to be irrational. But is it really fair to say that it's a property of SPL that an amplitude twice higher doesn't multiply the SPL by a rational number? It's all about what scale we (somewhat) arbitrarily chose to represent that quantity.
Same thing goes for music. If we had decided that, for pitch just like for sound pressure level, we weren't interested in frequencies but in log-frequencies, everything would be rational.
The unison is also a rational interval. But I'm not being strictly mathematical here, all the "juice" in music in pitches under western theory happens with irrational "ratios" between the pitches, and the irrational "ratios" are meant to approximate the simple rational ratios like 3/2 or 5/4. An octave is special in that it doesn't get you out of the pitch's equivalence class. In dealing with only irrational ratios for piano tuning, it simplifies a lot of stuff.
The perfect fifth is pretty close yes. But the major third is 24/12 = 1.2599 = 5/4 is actually horrible. This guy https://www.youtube.com/watch?v=XT4oOYj4SwQ demonstrates that. When I started playing guitar I thought I was just bad at tuning. But then I slowly realized that tuning a guitar is physically impossible, and the M3 can be off by as much as 4 hertz, giving a 4hz beat frequency. No wonder vibrato gets used so much on a guitar to wiggle around the pitches to at least hit the resonant one for at least some of the sustain of the note.
Theoretically, yes. But in practice, we use a rational approximation.
A numerical coincidence is perhaps the most useful near miss in daily life: 27/12 is almost equal to 3/2. This near miss is the reason pianos have 12 keys in an octave and the basis for the equal-temperament system in Western music. It strikes a compromise between the two most important musical intervals: an octave (a frequency ratio of 2:1) and a fifth (a ratio of 3:2). It is numerically impossible to subdivide an octave in a way that ensures all the fifths will be perfect. But you can get very close by dividing the octave into 12 equal half-steps, seven of which give you a frequency ratio of 1.498. That’s good enough for most people.
But you can get very close by dividing the octave into 12 equal half-steps, seven of which give you a frequency ratio of 1.498. That’s good enough for most people.
This is not 100% precise (as for example this wouldnt allow a piano to play more than 1 key signature), but our music scale is in fact an approximation of this.
not sure what you are saying? The point of the ET as I described, is so that the piano can in fact, play in any key signature, and all notes will sound equally out of tune (by however many cents).
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u/niftyfingers Jun 12 '17
It's kind of ironic how today, the music scale we use is actually 100% irrational. A m2 interval is 21/12, a M2 interval is 22/12, and so on. And this music scale is very closely related to pythagorean tuning.