Season 9 Episode 4

Maria is back on the show to tell us about the maths for breaking down sound and music and colour into fundamental components.

Watch on YouTube

Further Reading

Frequency Spectrum

The images that Maria showed for the spectrum of frequencies over time for various instruments can be found at Rafael A. Irizarry's Music and Statistics Demo page.

I showed off an app on my phone that does this analysis live. That was Speccy Spectrum Analyzer and it turns out that I paid 59p for that. To be clear, that wasn’t a paid product placement, and there are many free alternatives are available on both the Google Play Store (e.g. Sound Spectrum Analyzer) and the Apple App Store (e.g. Audio Spectrum Analyzer).

The Desmos interactive with colours and waves and sounds is available here.

Fourier Series

The maths for breaking down a signal (any continuous and periodic signal!) into sine and cosine waves is called a Fourier Series. We had Max on OOMC a couple of years ago to talk about the maths involved; Fourier Series | OOMC S3ep8 | YouTube.

There’s a textbook introduction to the theory here; Fourier Series | Math is Fun.

Harmonic Series

You can find out more about the harmonic series at Oberton.

When you saw the harmonic series, did you notice anything about the “shape” of the notes on the sheet music? If not, go and have a look at it and come back here!

The sheet music is a kind of plot with time on the \(x\)-axis and frequency on the \(y\)-axis, where notes are plotted at points corresponding to when they’re played and what the frequency is. Here we’re playing a sequence of notes in order, so there’s a linear relationship between time and the position in the sequence.

Actually, it's not quite correct to say that it’s frequency on the \(y\)-axis. It’s more accurate to say that it’s the logarithm of frequency on the \(y\)-axis, because doubling the frequency corresponds to going up an octave, which is a vertical translation of the musical notes by a fixed amount on the sheet music, corresponding to the law of logarithms \(\log(2f)=\log(2)+\log(f)\). Similarly, going up by a fifth (or any other fixed interval) corresponds to multiplication of the frequency by some fixed amount, and on the sheet music that corresponds to a vertical shift.

The harmonic series follows a simple rule; the frequencies are \(nf\) where \(f\) is some fundamental frequency and \(n\) is one of the numbers 1, 2, 3, 4, ... .

This leads to a natural guess; the frequencies of the harmonic series should be displayed at something like \(\log(nf)=\log(n)+\log(f)\), so a logarithmic curve, scaled appropriately, should go through all the points. Behold, Desmos!

Note that (pun intended) most of the notes in the harmonic series have sharps and flats and worse on them. Those notes for which \(n\) is a power of 2 are OK, because those are just octaves above the fundamental frequency. And if you use Pythagorean tuning, then perhaps frequencies of the form \(3\times 2^k f\) fit into your tuning system too. But if you use equal temperament  then none of the overtones except the powers of two will be an integer number of semitones above your fundamental frequency, so that’s a shame.

For a fun look at the use of the harmonic scale in composition, see this video which describes how Ravel can mimic a pipe organ by combining multiple instruments playing in perfect fifths; Why composers must learn the overtone series | YouTube.

Desmos Tone

This was the video that made me aware that Desmos has a tone feature now; Desmos sounds like a church organ... | YouTube. I think that they’re doing two sorts of synthesis here; layering frequencies from the harmonic scale with particular amplitudes that give an organ sound, and also layering that stack with different fundamental frequencies (shifted according to a chromatic scale) to form chords.

And this is the most impressive use of this technology that I’ve seen; Vocal Synth in Desmos | YouTube.

Astronomy and spectroscopy

The Webb Telescope spectrogram image was taken from this page.

Music Recommendations

Maria wants to recommend Jacob Collier e.g. Jacob Collier - The Audience Choir | YouTube.

James wants to recommend Virtual Riot (e.g. I heard you like polyrhythms | YouTube) and Jerobeam Fenderson (e.g. Core - Oscilloscope music | YouTube).

Maria was on OOMC last year to talk about musical intervals (Sounds Like Maths | OOMC S7ep9) and created a YouTube playlist of other music theory recommendations.