Understanding Harmony As Geometry

This is the first video in the Music As Geometry series teaching how to visualize harmony as geometry.

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[Chalisque]

To add another mathematical perspective, this time coming from number theory rather than geometry, there is something called 'modular arithmetic'.

The usual example is a clock, since everybody knows how to tell the time. There are 12 hours on the clock face, so if we start at 11 o'clock, and go forward 2 hours, we get to 1 o'clock. You can divide the 12 hours exactly into 6 two-hour periods, or 4 3-hour periods, or 3 4-hour periods.

You'll notice that this is _exactly_ the same as what you're describing above, as, in mathematical terms, we're talking about 'arithmetic mod 12'. That is, two numbers 'mod 12' are considered the same if their difference is a whole multiple of 12.

For example 13 and 1 are considered the same number 'mod 12'. Now, we say that two numbers a and b are 'coprime' if their greatest common divisor gcd(a, b) is 1, that is, the largest whole number that exactly divides both a and b is 1. Now gcd(1, 12) = 1, gcd(2, 12) = 2, gcd(3, 12) = 3, gcd(4, 12) = 4, gcd(5, 12) = 1, gcd(6, 12) = 6, gcd(7, 12) = 1, and you can work out the rest. Notice that the _only_ numbers between 1 and 12 that are coprime to 12 are 5 and 7 and, note also that since -7 = 5-12, we see that 7 = -5 'mod 12'. In musical terms, an inversion of a perfect 4th is a perfect 5th.

When converting from traditional musical notation to the mathematical perspective, we need to understand that, when doing 'modular arithmetic', it makes sense to start counting at 0.

unison = 0 (middle C and middle C are 0 semitones apart)
minor 2nd = 1 (C and C# are are 1 semitone apart)
major 2nd = 2 (C and D are 2 semitones apart)
and so on.

Now if we start at 0 and go in steps of 1, we get the chromatic scale. If we start at 0 and go in steps of 2 we get _a_ whole tone scale (of which there are 2, each comprising 6 notes -- because 12/6 = 2). If we start at 0 and go in steps of 3 we get _a_ diminished 7th chord (of which there are 3). If we start at 0 and go in steps of 4, we get an augmented triad, of which there are 4. (In group theory, another related area of maths, these different augmented triads are 'cosets' of one another, but I'm not going to try to explain group theory here.)

If we start at 0 and go in steps of 5, we get all 12 semitones before we get back to where we started: this is the circle of perfect 4ths, the inversion of the circle of perfect 5ths. And if we start at 0 and go in steps of 7, we get the circle of perfect 5ths. (Note that since '-5 = 7 mod 12', these two circles are the exact opposite of each other -- go backwards around the circle of 5ths and you get the circle of 4ths -- because, working mod 12, stepping down by 5 is equivalent to stepping up by 7.)

There are hints of arithmetic 'mod 7' happening when we discuss intervals in a major/minor scale. (I'll use 'scale tone' to mean e.g. a tone if, in the scale of C major, For example, a unison is 0, a second is 1, a third is 2, a fourth is 3, a fifth is 4, a sixth is 5 and a seventh is 6 (this off-by-one is because traditional language for intervals starts counting at 1, being from a time before modern mathematics showed the utility of the number 0 and of counting from it instead of 1). So then an octave is 7, and 7=0 mod 7; and a ninth is 8 scale-tones up from the root, and since 8=1 mod 7, we see that the ninth is the same pitch class as the second.

Much of the above is better demonstrated by example rather than trying to describe in theory -- once you have learned by example, then it is easier to explain the theory, and how things generalise to other moduli than 12 and 7.

[julianman]

I love this concept, but I think you left a lot out, so I'm looking forward to seeing where you go with this series!

[Chalisque]

Maths and music, and indeed physics and computer science, go together in so many beautiful ways. I imagine there are nice ways to teach mathematics to musicians starting from what they already know from their musical backgrounds, and likewise teaching music to mathematicians.