If you look at a piano keyboard you'll see groups of 2 black notes alternating with groups of 3. So the pattern repeats after 5 black notes, but if you count you'll see there are also 7 white notes in this repetitive pattern. So: the pattern repeats each 12 notes.

Some people who never play the piano claim it would be easier if had all white keys, or simply white alternating with black. But in fact the pattern makes it easier to keep track of where you are - and it's not arbitrary, it's musically significant.

(2/n)

Starting at any note and going up 12 notes, we reach a note whose frequency is almost exactly double the one we started with. Other spacings correspond to other frequency ratios.

I don't want to overwhelm you with numbers. So I'm only showing you a few of the simplest and most important ratios. These are really worth remembering.

(3/ n)

We give the notes letter names. This goes back at least to Boethius, the guy famous for writing The Consolations of Philosophy before he was tortured and killed at the order of Theodoric the Great. (Yeah, "Great".) Boethius was a counselor to Theodoric, but he really would have done better to stay out of politics - he was quite good at math and music theory.

Boethius may be the reason the lowest note on the piano is called A. We now repeat the names of the white notes as shown in the picture: seven white notes A,B,C,D,E,F,G and then it repeats.

[Whoop, Lisa is making me get up, make breakfast and go to the gym. I'll continue this later.]

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So the scale used to start at A, using only white notes. But due to the irregular spacing of white notes, a scale of all white notes sounds different depending on where you start. Starting at A gives you the "minor scale" or "Aeolian mode", which sounds kinda sad. Now we often start at C, since that gives us the scale most people like best: the "major" scale.

(Good musicians start wherever they want, and get different sounds that way. But "C major" is like the vanilla ice cream of scales - now. It wasn't always this way.)

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From the late 1100s to about 1600 people called describe pitches that lie outside 7-tone system "musica ficta" ("false" or "fictitious") notes. But gradually these notes - the black keys on the piano if you're playing in C major - became more accepted.

To keep things simple for mathematicians, I'll usually denote these with the "flat" symbol, ♭. For example, G♭ is the black note one down from the white note G.

(Musicians really need both flats and sharps, and they'd also call G♭ something else: F♯. I'll actually need both G♭ and F♯ at some points in this talk!)

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Since starting the scale with the letter C takes a little practice, I'll do it a different way that mathematicians may like better. I'll start with 1 and count up. Musicians put little hats on these numbers, and I'll do that.

For example, we'll call the fifth white note up the scale the "fifth" and write it as a 5 with a little hat.

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Now for the math of tuning systems!

The big question is: how do we choose the frequency of each note? This is literally how many times per second the air vibrates, when we play that note.

Since 1850, by far the most common method for tuning keyboards has been "12-tone equal temperament". Here we divide each octave into 12 equal parts.

What do I mean by this, exactly? I mean that each note on the piano produces a sound that vibrates faster than the note directly below it by a factor of the 12th root of 2.

But we can contemplate "N-tone equal temperament" for N = 1, 2, 3, .... - and some people do use these other tuning systems!

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Here's a picture of the most popular modern tuning system: 12-tone equal temperament. As we march around clockwise, each note has a frequency of 2^{1/12} times the note directly before it.

When we go all the way around the circle, we've gone up an octave. That is, we've reached a frequency that's TWICE the one we started with.

But a note that's an octave higher sounds "the same, only higher". So in a funny way we're back where we started.

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But now for a big question: why do we use a scale with 12 notes?

To start answering, notice that we actually use three scales: one with 5 notes (the black keys), one with 7 (the white keys) and one with 12 (all the keys).

As mathematicians we can notice a pattern here.

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What's so good about scales with 5, 7 or 12 notes?

A crucial clue seems to be the "fifth". If you go up to the fifth white note here, its frequency is about 3/2 times the first. This is one of the simplest fractions, and it sounds incredibly simple and pure. So it's important. It's a dominant force in western music.

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We can make a chart to see how close an approximation to the fraction 3/2 we get in a scale with N equally spaced notes.

N = 5 does better than any scale with fewer notes!

N = 7 does better than any scale with fewer notes!

N = 12 does better than any scale with fewer notes! And it does *way* better. To beat it, we have to go all the way up to N = 29 - and even that is only slightly better.

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Here's a chart of how close we can get to a frequency ratio of 3/2 using N-tone equal temperament.

See how great 12-tone equal temperament is?

There are also some neat patterns. See the stripes of even numbers and stripes of odd numbers? That's not a coincidence. For more charts like this, and much more cool stuff along these lines, go here:

https://johncarlosbaez.wordpress.com/2023/10/19/perfect-fifths-in-equal-tempered-scales-part-2/

(13/n)

@johncarlosbaez I don't know much about music but I am guessing these issues arise because we need to divide octaves into individual frequencies on the basis that this is necessary to create, for example, "keys" on instruments like pianos. You can't have an infinite number of keys in an octave on a piano.

But what about producing music through a computer? How the notation for such a thing would work, I have no clue but in theory I imagine every "chord" could be perfect?

@rozeboosje @johncarlosbaez

A lot of computer music doesn't exactly have notation. Instead it has a user interface.

Let's say you have a computer and want to make a very simple chord over A. You can go look for the frequency of the 5th note in the scale or you could just multiply 440 by 3/2. This direct use of fractions is called "just intonation".

The problem with doing this is that the third note in the scale in 12tet is way off from the ideal just intoned fraction. So if you have a perfectly in tune chord on C and then your melody goes to E, that E is going to sound really wrong.

So you could also just make all the notes in the scale just intoned, but if your music goes onto a key change, it can also sound off, unless you retune all your notes to be the just notes relative to the new key. All your As are now a new frequency. This can sound great, but its fiddly.

Changing all the notes in every chord relative to the bottom note in the chord also often sounds wrong and kind of unmoored. Its often better to stick with the scale that belongs to your key because the notes are more stable and you can more easily tell when you're getting to an important note.

There are many composers who write this way but its extra work so most people stick with 12tet.

@johncarlosbaez

I need to look through my notes and can't today and theres a risk I may forget in the chaos I expect later today. I'll make a note, bit if you want to remind me that would be great.

I ran into these problems with my own experimentation and then talked about it with Ellen Fullman who told me about the unmoored thing and somewhere I do know where she learned this.

There's also keyboard tuning software from the late 80s and early 90s in MAX that had an octave of the keyboard set aside to tune the key, but this is another thing I need to search to find.