Okay so sounds can be broken down into individual tones called sine waves. The math that lets us do this doesn’t care about how tonal or noisy the sound is. It takes arbitrary input. However, human brains and ears (as well as those of many other creatures) seem to optimize for tonality of some type.
The simplified explanation is that we like when the frequencies of the tones that make up a sound are in whole number ratios (the harmonic series). However, there’s a tolerance for frequencies which are close to those ratios but not perfect. And when harmonics don’t fall perfectly within the harmonic series, we can instead prefer intervals between notes which are slightly “out of tune” compared to what the harmonic series would dictate. For instruments like strings and woodwinds where the vibration of the air happens along a more or less straight line, the harmonics tend to be close enough to the harmonic series for this not to matter a ton. But for instruments with different resonant features (bells are a common example), the effects of this are more pronounced.
There is also some math which makes tuning instruments solely to the harmonic series impractical. This combined with the tolerance for consonance I mentioned before has led to a rich sea of different traditions which play around with tuning in different ways. The western tradition alone has a long history with how a twelve note chromatic scale ought to be tuned. It turns out that equally diving the octave into twelve notes just so happens to be a good approximation of a lot of harmonic series intervals, but some intervals are less perfect than others. It’s all a series of compromises.
I’ve been a clarinet player since I was about 9. But when I was like 16, I had a physics class on the science of acoustics. We used slinkies to create standing waves either fixed at both ends or moving at one end to understand the concept of a standing wave, and then diagrams to demonstrate different harmonics. The connections were so amazing to learn. Seeing the diagrams of a closed vs open pipe, and of a string, and the standing waves that can form in it, was so enlightening. I finally knew why my instrument overblew at an octave and a fifth but other instruments overblow at an octave.
It was years later thanks I’m pretty sure to a Reddit comment, that I additionally learnt that a cone also experiences every harmonic, just like an open pipe. Though I unfortunately never learnt the intuition behind how to explain that detail.
Of course, this all goes somewhat out the window when you realise that it’s possible on many instruments to bend the pitch smoothly and play out of tune—or correct your tuning. No idea how the physics behind that works with the model of standing waves with fixed nodes and antinodes.
Okay so sounds can be broken down into individual tones called sine waves. The math that lets us do this doesn’t care about how tonal or noisy the sound is. It takes arbitrary input. However, human brains and ears (as well as those of many other creatures) seem to optimize for tonality of some type.
The simplified explanation is that we like when the frequencies of the tones that make up a sound are in whole number ratios (the harmonic series). However, there’s a tolerance for frequencies which are close to those ratios but not perfect. And when harmonics don’t fall perfectly within the harmonic series, we can instead prefer intervals between notes which are slightly “out of tune” compared to what the harmonic series would dictate. For instruments like strings and woodwinds where the vibration of the air happens along a more or less straight line, the harmonics tend to be close enough to the harmonic series for this not to matter a ton. But for instruments with different resonant features (bells are a common example), the effects of this are more pronounced.
There is also some math which makes tuning instruments solely to the harmonic series impractical. This combined with the tolerance for consonance I mentioned before has led to a rich sea of different traditions which play around with tuning in different ways. The western tradition alone has a long history with how a twelve note chromatic scale ought to be tuned. It turns out that equally diving the octave into twelve notes just so happens to be a good approximation of a lot of harmonic series intervals, but some intervals are less perfect than others. It’s all a series of compromises.
I’ve been a clarinet player since I was about 9. But when I was like 16, I had a physics class on the science of acoustics. We used slinkies to create standing waves either fixed at both ends or moving at one end to understand the concept of a standing wave, and then diagrams to demonstrate different harmonics. The connections were so amazing to learn. Seeing the diagrams of a closed vs open pipe, and of a string, and the standing waves that can form in it, was so enlightening. I finally knew why my instrument overblew at an octave and a fifth but other instruments overblow at an octave.
It was years later thanks I’m pretty sure to a Reddit comment, that I additionally learnt that a cone also experiences every harmonic, just like an open pipe. Though I unfortunately never learnt the intuition behind how to explain that detail.
Of course, this all goes somewhat out the window when you realise that it’s possible on many instruments to bend the pitch smoothly and play out of tune—or correct your tuning. No idea how the physics behind that works with the model of standing waves with fixed nodes and antinodes.