Quarter-Comma Meantone (Part 5)

I’ve been talking about the quarter-comma meantone tuning system, which dominated western music from about 1550 to 1690. So far I’ve been drawing it using a circle of fifths, as above. This is a diagram where as we go clockwise each note is a fifth higher than the previous one.

But why is this system called ‘meantone’? Briefly, it’s because the size of a whole step between notes in this tuning system—called a ‘tone’—is the geometric mean of the two kinds of whole step in Pythagorean tuning!

Let’s figure this out. We could figure out the size of a tone rather directly. But it will pay extra dividends to do a bit more. Let’s first figure out the frequency ratio of two neighboring notes separated by a half step, called a ‘semitone’, and then use that to work out the size of a tone.

We can figure it out using pictures. Let’s start by getting rid of the blue arrows for major thirds, which just clutter things up now:

Next, let’s take this diagram and reorder the notes so they go up the chromatic scale: C, C♯, D, E♭, E, etc. This turns our circle of fifths into a ‘star of fifths’:

Now let’s figure out the frequency ratios of neighboring notes! First, lets go up from C to C♯.

Follow the path from C to C♯ in the picture above. As you do, you’ll go up 7 quarter-comma fifths, but you’ll also hit the lesser diesis. Multiplying all these numbers you get

This is a lot bigger than 1, so you’ve gone up to some C♯ that lies octaves above our original C. You should divide by an appropriate power of 2 to get the C♯ right next to our original C. Namely, you should divide by 16. This gives

So this is how much you multiply the frequency of C to get the C♯ right above it! This number is called the quarter-comma diatonic semitone.

Next, let’s go up from C♯ to D.

With your finger or eye, follow the path from C♯ to D:

As you do, you go up 7 quarter-comma fifths—and this time, you don’t hit the lesser diesis! So this time the numbers along the path multiply to

Again we need to divide by 16, getting

So this is how much we multiply the frequency of C♯ to get the D right next to it. This number is called the quarter-comma chromatic semitone.

If you go through all the notes of the scale you’ll see they’re all spaced by diatonic or chromatic semitones… with one exception, namely the tiny space between F♯ and G♭, which is just the lesser diesis.

Indeed, we get this pattern of semitones:

To see it better, we can remove the scaffolding of fifths:

Note the pleasant alternating pattern of diatonic and chromatic semitones, except for two diatonic semitones right next to C, and two chromatic ones directly opposite it, sandwiching the lesser diesis.

By the way, since the diatonic semitone includes the lesser diesis while the chromatic semitone does not, we get this relation:

Okay, but what does all this have to do with the term ‘meantone’? Well, in music a ‘tone’ is two semitones. Since the semitones in quarter-comma meantone generally alternate between diatonic and chromatic, a tone is usually equal to

quarter-comma chromatic semitone ×
quarter-comma diatonic semitone
=

Fans of the golden ratio will notice that this number is 1/2 less than that!

But the name ‘meantone’ arises because this number is the geometric mean of the two kinds of tones in just intonation, namely 9/8 and 10/9:

While the tones in just intonation hop back and forth between the larger 9/8 and the smaller 10/9, in quarter-comma meantone they generally stay at the mean of those two… which you might even call the golden mean!

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.

• Part 3. What ‘quarter-comma’ means in the phrase ‘quarter-comma meantone’: most of the fifths are lowered by a quarter of the syntonic comma.

• Part 4. Omitting the diminished fifth or augmented fourth from quarter-comma meantone. The relation between the Pythagorean comma, lesser diesis and syntonic comma.

• Part 5. The sizes of the two kinds of semitone in quarter-comma meantone: the chromatic semitone and diatonic semitone. The size of the tone, and what the ‘meantone’ means in the phrase ‘quarter-comma meantone’.

• Part 6. What happens to quarter-comma meantone when you change it from a 13-tone scale to a more useful 12-tone scale by removing the diminished fifth.

• Part 7. Why it’s better to start the quarter-comma meantone scale on D rather than C.

For more on equal temperament, read this series:

• Equal temperament.