The next time you play a musical instrument, spend a few minutes trying to understand its physics. After recently buying a trombone, I did just this and was surprised to find myself exploring various rabbit holes to understand just how the instrument works.

Since lots of today’s music is produced and consumed electronically, it’s easy to forget that humans used to make melodies with all sorts of oddly shaped things called instruments. These come in all sorts of shapes and sizes: woodwind, brass, percussion. Trying to understand why these instruments were constructed the way they were, reveals much about the fundamental physics behind sound.

Here’s a rough sketch of how we process sound. Sound is fundamentally oscillations in air pressure; these oscillations cause hair-like cells in our inner ear (called cilia) to move back and forth. These cilia are nerve cells, and like all nerve cells, they send information to the brain via electrochemical impulses. In the case of the ear, it’s the cochlear nerve that eventually carries these signals to the brain. Our brain then magically interprets these signals as sound.

Certain sounds are much more pleasant than others. An organ sonata might send shivers down your spine, while screeching tires might make you plug your ears. But why is it that some sounds sound better than others? If you’re musical, you might know the horrible tritone. But why does it sound so horrible? Why does a fifth sound uplifting, but a minor third sounds ominous? The answers to these questions boil down to the regularity of the vibration created in the inner ear, by the sound waves in the air. I won’t go into this in detail here, but here’s some further reading: https://music.stackexchange.com/questions/30091/why-do-some-arrangements-of-notes-make-a-good-melody-and-some-dont.

(As an aside, if you want to hear some music which is composed by using frequencies which are divorced from the chromatic scale found in Western music, having a listen to this: https://shingles.bandcamp.com/album/generosity-of-the-suns)

Now, let’s go back to instruments, in particular, ones with strings. These are easier to understand than wind instruments (though we can carry across many principles from stringed instruments to wind instruments). Consider a guitar string. This has the simple constraint that it’s fixed at two ends. If we pluck the string exactly from its center, we will create what is called a standing wave: each point along the string will oscillate back and forth with the same amplitude (in this case, the wavelength will be twice the length of the string). This is called the fundamental or the first harmonic. If we pluck the guitar string precisely one quarter along its length, we will create a different standing wave in the string which has a wavelength equal to the length of the string - this is the second harmonic. Here’s a gif which illustrates the first three fundamentals: https://www.physicsclassroom.com/Class/sound/u11l5b1.gif. Because the string is constrained by being fixed at either end, there are finite ways in which the string can oscillate (this is called ‘simple harmonic motion’). The series of possible waves that can be created is called the harmonic series. If we pluck the string at some arbitrary position, we will create some combination of the frequencies in the harmonic series.

Here’s a deeper explanation of how standing waves in air differ from those on a string: https://opencurriculum.org/5537/standing-waves-and-wind-instruments/. I won’t go into too much detail but it suffices to say that the standing-waves-on-a-string exactly describes the oscillations in pressure that occur in standing waves in a tube. Here’s a nice visualization of this: http://energywavetheory.com/wp-content/uploads/2014/11/standing-wave-2.gif

Just as the length of the guitar string constrains the frequencies of the waves that can form on it, the standing waves in a wind instrument are constrained by the length of the instrument. In other words, the length of the tube of a wind instrument determines the wavelengths of the harmonic series. This is probably consistent with your intuition: larger instruments (like the tuba) produce lower pitches because the standing waves they support have longer wavelengths. Smaller instruments (like the piccolo) support standing waves with a smaller wavelength, and hence produce higher pitches.

Wind instruments have more interesting geometry: the bell. A simple cylinder is only capable of playing an odd harmonic series (where each possible standing wave is an odd multiple of the fundamental - the longest wave that fits in the tube). Having a conical bell actually enables a greater harmonic series than a simple cylindrical tube (the non trivial details of which are explored here: http://hyperphysics.phy-astr.gsu.edu/hbase/Music/brassa.html). The bell also creates a wider surface area at the end of the instrument which means sound can radiate more effectively from the instrument.

While this isn’t a dissertation, I hope I’ve sketched how looking at something through a scientific lens can be very insightful. I’m reminded of an anecdote in Richard Feynman’s The Pleasure of Finding Things Out: Feynman has a debate with an artist friend of his about the beauty of a flower. Feynman’s friend claims that Feynman cannot fully appreciate the beauty of a flower because he cannot appreciate the aesthetics an artist has developed. Feynman disagrees by explaining how his understanding of the evolution of life informs his understanding of why the flower is beautifully coloured: the flower is beautifully vibrant so it can attract bees, which will take pollen from this flower and pollinate another, and so on.

So the next time you hear someone playing an instrument, ask yourself, how does that make me hear this?

Resources: