Quarter-Comma Meantone (Part 2)

Azimuth 2023-12-18

Last time I introduced ‘quarter-comma meantone’, the tuning system shown above. Today let’s see if we can understand all the numbers in this picture.

You see 13 circles with letters in them standing for tones in a scale, connected by arrows labeled by numbers. The numbers are the frequency ratios between these tones:

• The blue arrows are ‘just major thirds’, with a frequency ratio of 5/4. It’s the quest for a 12-tone scale with a lot of just major thirds that led people to quarter-comma meantone.

• The red arrows are ‘quarter-comma fifths’, with a frequency ratio of

$\displaystyle{ \sqrt[4]{5} \approx 1.49534878122\dots}$

This is remarkably close to a ‘just perfect fifth’, with a frequency ratio of 3/2. We’d love to have a 12-tone scale with lots of just major thirds and just perfect fifths, but alas it’s impossible!

• The purple arrow from F♯, to G♭ is the ‘lesser diesis’, with a frequency ratio of 128/125. Ideally this ratio would equal 1, so we’d have a scale where the augmented fourth (F♯) is the same as the diminished fifth (G♭). But we can’t have this without something else going wrong.

Let’s analyze what’s going on with these three numbers by treating them as variables:

Here T stands for ‘third’ and it should be a number close to 5/4. F stands for ‘fifth’ and it should be close to 3/2. x should be close to 1.

If you look at the picture above you’ll see one blue arrow takes you as far as four red arrows. You might naively think this implies F = T⁴ but that can’t be true: for quarter-comma meantone we have $F = \sqrt[4]{5}$ and T = 5/4. So in fact

$T = F^4/4$

The point is that the notes in the picture represent not specific pitches but ‘pitch classes’: pitches modulo powers of 2. So, for example, you start at C and go up four fifths you get to E, but this E has frequency 4 times that of the E you get by going up a third from C.

Similarly, to go all the way around the circle you go along 12 red arrows and one purple arrow. You might naively think this means xF¹² = 1. But in fact you’ve gone up 7 octaves: the frequency has gone up by a factor of 2⁷. So in fact

$x F^{12} = 2^7$

These are all the equations we need to get ourselves a scale:

$F^4 = 4T, \qquad x F^{12} = 2^7$

We have 3 unknowns and just 2 equations, so we get a 1-parameter family of scales this way.

Delightfully, this idea gives three of the most popular tuning systems in the last millennium of western music! They are the three most obvious choices: we can either take F = 3/2, or T = 5/4, or x = 1.

Pythagorean tuning

If we love perfect fifths and want these to be just, we must take F = 3/2. Then our equations force

$\displaystyle{ T = \frac{81}{64}, \qquad F = \frac{3}{2}, \qquad x = \frac{3^{12}}{2^{19}} }$

This gives Pythagorean tuning! In this system x is actually less than 1. Its reciprocal is called the Pythagorean comma. I may sometimes call this p:

p = Pythagorean comma = 3¹²/2¹⁹ = 531441/524288 ≈ 1.0136

Since x is less than 1, the diminished fifth (G♭) is actually below the augmented fourth (F♯). This makes something funny happen if we write the notes so that the frequencies keep going up as we go clockwise. Ponder this:

Quarter-comma meantone

If we love major thirds and want these to be just, we must take T = 5/4. Then our equations force

$\displaystyle{ T = \frac{5}{4}, \qquad F = \sqrt[5]{4}, \qquad x = \frac{2^7}{5^3} }$

This gives quarter-comma meantone! Now x is greater than 1, and it’s called the lesser diesis. I’ll sometimes call it δ:

δ = lesser diesis = 2⁷/5³ = 128/125 = 1.024

Here’s what quarter-comma meantone looks like:

Equal temperament

If we want our augmented fourth to be the same as the diminished fifth, we must take x = 1. Then our equations force

$\displaystyle{ T = \sqrt[3]{2}, \qquad F = 2^{7/12}, \qquad x = 1 }$

This gives 12-tone equal temperament:

This scale has more symmetry than the other two, but all the frequency ratios are irrational except for octaves.

What’s next?

Pythagorean tuning ruled western music, or at least western music theory, from at least 1000 to 1300 AD. Quarter-comma meantone was dominant from about 1550 to 1690. Equal temperament ruled from about 1790 to now.

You’ll notice some gaps in that chronology! Just intonation flourished from about 1300 to 1550, and this system was based on a different idea: trying to get as many frequency ratios as possible to be simple fractions. I’ve discussed it in detail starting here. A rich and interesting variety of ‘well-tempered’ systems competed from about 1690 to 1790; these tweak the idea of quarter-comma meantone in various ways, and I’ll talk about them later.

I hope you understand the math underlying quarter-comma meantone a bit better now. But I still haven’t said why it’s called ‘quarter-comma meantone’! For that we’ll need to dig deeper into ‘commas’. We’ll need to understand those well to appreciate the well-tempered systems.

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.