When I mention to people that I love both the sciences and the humanities, one of the most common responses is that it makes sense, since mathematics and music are intertwined. They absolutely are—like any other sound, music doesn’t make sense without distinct frequencies, which we hear and tease apart in our own inner ears.
Whether you’re looking at anatomical details or performance theory, the link between music and math is a popular topic, which made me wonder about the history of using math and physics to explain music. Were musical pitches fixed even before we had the technology to measure sound frequencies? How did those frequencies become associated with the musical notation (A-G) we use today?
For mathematical music theory, we have to go back to Pythagoras, who was very into the music-math connection. It’s impossible to know if everything that’s attributed to him is true, but tradition says Pythagoras noticed in the sixth century BC that the length and tension of a string was related to the pitch of the note produced when it was plucked.
Pythagoras’s observation goes for all instruments that rely on strings and tubes. Without going into great detail, producing a note requires a standing sound wave, with points that stay fixed, called nodes, and points of maximum displacement, called antinodes. The length of a string or tube is related to the largest wavelength that will create a standing wave in that string or tube. That wavelength gives the lowest resonant frequency or first harmonic, and smaller wavelengths that also fit the tube or string are higher harmonics. The set of harmonics is unique to the length of tube or string, and determines the note we hear when it’s sounded.
Here’s a quick illustration standing waves in strings and tubes! (Note that strings actually vibrate in transverse waves as I’ve illustrated, but all the sound waves are actually longitudinal. It’s wrong to illustrate them as transverse waves like I am here, but it’s really hard to show longitudinal standing waves in still images. You should really check out this cool animation (not mine) to get a more accurate idea!)
In all these cases, you can see that the way to get a smaller base wavelength would be to shorten the pipe or string (or change the tension of a string or the diameter of a pipe).
It’s a big step from understanding that shorter pipes equal higher pitches to noting specific pitches in music. The letters we use commonly in Western musical notation show up first in the sixth century AD, when the philosopher Boethius used the first fourteen letters of the alphabet to represent two full diatonic (seven-note) scales. Later this was altered to using the first seven letters, repeated with different notation for each octave.
Here we come to an interesting question: The key or starting note that produces a major scale using only natural notes with no accidentals (sharps or flats) is C, not A. Many of us are used to thinking in terms of major scales, and C is often treated as a base or central note (like middle C, which is about as central as it gets). Why weren’t the notes named to reflect that, maybe making A the root of the accidental-free major scale?
Unlike in modern music, nobody used major scales (also called Ionian mode) when the letters were assigned. As far as I can tell, the best guess is that A was simply the lowest note that was comfortable for most male singers, and the rest of the letters were assigned according to the popular modes of the time. In any case, the letters were in use for about a thousand years before we had any scientific way to record what they represented in terms of frequency.
That isn’t to say that people couldn’t tune music reliably. The human ear is pretty impressive and can tell the difference between pitches just a few Hz (cycles per second) apart. For reference, at the very low end of human hearing a whole octave spans about 20 Hz, and at the high end it spans about 10 kHz. Especially for those higher notes, we can match pitches pretty darn well! (I’ll talk more about specific frequencies in another post, but it has to do with logarithms and how they help us measure octaves and other intervals.)
However, the pitches represented by musical notation did vary from place to place, partly driven by the limits of instruments. Organ pipes, for example, would wear down over time and have to be trimmed and hammered back into shape, resulting in the pipes getting shorter and the whole instrument going up in pitch (see above). This was just a fact of life, and composers and performers adapted with a lot of transposing. Frequency and pitch weren’t studied extensively until Galileo and Newton, and actually standardizing pitches wasn’t really an option until the invention of the tuning fork in 1711, which made it possible to quantify the difference between pitches.
Once we had a way of measuring sound wave frequencies and recording exact pitches numerically, surely variations in pitch became a thing of the past, right? Well, not exactly. In fact, the 19th century saw a particularly widespread epidemic of what was called “pitch inflation.”
Pitch inflation was more or less exactly what it sounds like—the same note names continued to be used, but the actual frequencies got higher and higher. This trend was encouraged by the prominence of orchestras and instrumental ensembles over singers, as well as the growth of concert halls and audience expectations. Basically, audiences wanted to hear vibrant, brilliant music, and a great way to outshine the competition was to create instruments that played higher pitches than everyone else’s.
This trend went on until people started to pay attention to the poor singers. Key changes are bad enough sometimes in normal music, so imagine the effect when you’re talking about opera, which is famous for straining the limits of human vocal performance. Vocal cords can’t be tuned up like strings, so eventually something had to be done to save people’s voices.
There were several attempts to set standard concert pitches, mostly based on the frequency of the A above middle C, also called A4. For the sake of brevity, I’ll spare you the details, but you should read the whole saga (or here’s another source with a timeline)! It involves a few false starts and confusions over whether standard frequencies were set at specific temperatures (if you were ever in band, you know instruments change pitch in extreme heat and cold).
One of the interesting suggestions that didn’t stand the test of time for general use was scientific pitch, which is used occasionally in scientific contexts. Instead of starting with a value for A4, it focused on middle C, setting it to 256 Hz. This is mathematically pleasing because all Cs have frequencies that are powers of 2 (that’s where the logarithms come in—stay tuned for more on that!). In fact, some people think this pitch standard, which results in A4 at about 432 Hz, has cosmic resonances and significance.
If you’re interested, check out Omega432, which is pretty weird but saner than some (he does say 440 Hz music has “a [sic] element of negative spacetime that lends itself to the undertone series,” but also denounces any claims that it was introduced by Joseph Goebbels, so he’s got that going for him).
After about a century, a technical standard of A4 = 440 Hz was agreed on by the International Organization for Standardization, though it had already been widely used for decades by that point. Today, the standard concert pitch remains at A440 (the notation for A = 440 Hz), so we can all rest easy knowing that pitches are finally standardized.
…except that the Boston Symphony Orchestra uses 441 Hz, the New York Philharmonic uses 442 Hz, and many European symphony orchestras use 443 Hz. Old habits die hard.
(Yes, this does mean perfect pitch is usually relative. Still impressive!)
For more on the history of pitch, here’s another source in addition to the ones I linked above. As usual, Wikipedia also has a lot of information! Also, check out this general timeline of the understanding of sound waves and frequencies.
For more of the science of music, here’s a resource on the physics of musical instruments. Enjoy!