Quarter-Comma Meantone (Part 3)

Last time I explained the tuning system shown above. But I didn’t say why it’s called ‘quarter-comma meantone’.

Today I’ll finally tell you what a ‘quarter comma’ is. Not only will this shed new light on this particular tuning system, it’ll prepare you for understanding other meantone temperaments, like ‘1/3-comma’, ‘1/6-comma’ and so on.

As for the word ‘meantone’, that requires a whole other explanation. But one thing at a time!

We’ve already seen that there’s a conflict built deep into the heart of music: a conflict between wanting lots of just perfect fifths, which are pairs of tones with frequency ratios of 3/2, and wanting lots of just major thirds, which are pairs of tones with frequency ratios of 5/4.

You can’t have lots of both, though you can come close. The problem is that the fourth power of 3/2 is a bit more than 5:

Their ratio is called the syntonic comma, and sometimes I’ll call it σ:

As a result, going up 4 just perfect fifths is a bit more than going up a just major third and 2 octaves. The former increases the frequency by

while the latter increases it by

The ratio of these is the syntonic comma.

In quarter-comma meantone, we make our perfect fifths a bit smaller to get our major thirds to be just. That is, we replace 3/2 by a slightly smaller number whose fourth power is exactly 5. Unsurprisingly, this number is none other than

But here’s another way to think about it: we take 3/2 and divide it by the fourth root of the syntonic comma! That way, when we raise the result to the fourth power, the syntonic comma cancels out the problem and we get exactly 5. So, I’m saying

If this isn’t obvious from what I’ve already said, do the math—it’ll be good for you.

Musicians will say we’ve lowered our fifth by a quarter comma, since they implicitly take logarithms: when I say we’re dividing by the fourth root of the syntonic comma, they say we’re subtracting a quarter of a comma. I won’t be talking that way, but I can still draw a picture of the quarter-comma meantone tuning system that shows these ‘quarter commas’:

Notice that we’ve got 12 of these quarter commas, one between every pair of notes except the devilish F♯ and G♭. That’s a total of 3 commas. We can imagine systems where these 3 commas are distributed in other ways, and we’ll be seeing a bunch of them soon, when we get to well-tempered tuning systems. What quarter-comma meantone does is spread out these commas as evenly as possible.

But what about those notes F♯ and G♭? As I’ve said before, we usually leave out one of those, to get a scale with 12 more or less equally spaced notes. Next time I’ll show you what happens then.

For more on Pythagorean tuning, read this series:

• Pythagorean tuning.

For more on just intonation, read this series:

• Just intonation.

For more on quarter-comma meantone tuning, read these:

• Part 1. Review of Pythagorean tuning. How to obtain quarter-comma meantone by expanding the Pythagorean major thirds to just major thirds.

• Part 2. How Pythagorean tuning, quarter-comma meantone and equal temperament all fit into a single 1-parameter family of tuning systems.

For more on equal temperament, read this series:

• Equal temperament.