Quarter-Comma Meantone (Part 1)

I’ve spent the last few weeks drawing pictures of tuning systems, since I realized this is a good way to show off their mathematical beauty. Now I’ll start deploying them!

I’ve already written about the first two hugely important tuning systems in Western music:

• Pythagorean tuning.

• Just intonation.

It’s time to introduce the third: ‘quarter-comma meantone’.

But first, remember the story so far!

Pythagorean tuning may go back to Mesopotamia, but it was widely discussed by Greek mathematicians—perhaps including Pythagoras, whose life is mainly the stuff of legends written down centuries later, but more certainly Eratosthenes, and definitely Ptolemy. It was widely used in western Europe in the middle ages, especially before 1300.

The principle behind Pythagorean tuning is to start with some pitch and go up and down from there by ‘Pythagorean fifths’—repeatedly multiplying and dividing the frequency by 3/2—until you get two pitches that are almost 7 octaves apart. Here I’ll do it starting with C:

But there are some problems. The highest tone is a bit less than 7 octaves above the lowest tone! Their frequency ratio is called the Pythagorean comma. And we get a total of 13 tones, not 12.

To deal with these problems, we can simply omit one of these two tones and use only the other in our scale. There are two ways to do this, which are mirror images of each other:

Each breaks the symmetry of the scale. And each gives one fifth that’s noticeably smaller than the rest. It’s called a ‘wolf fifth’—because it’s so out of tune it howls like a wolf!

What can we do? One solution is simply to avoid playing this fifth. You’ve probably heard the old joke. A patient tells his doctor: “It hurts when I lift my arm like this.” The doctor replies: “So don’t lift your arm like that!”

This worked pretty well for medieval music, where the fifth and octaves were the dominant forms of harmony, and people didn’t change keys much, so they could avoid the wolf fifth. But in the late 1300s, major thirds became very important in English music, and soon they spread throughout Europe. A major third sounds perfectly in tune—or technically, ‘just’—when it has a frequency ratio of

But the major thirds in Pythagorean tuning are bigger than this!

Let’s see why. This will eventually lead us to the solution called ‘quarter-comma meantone’ tuning.

To go up a major third in Pythagorean tuning, we take any tone and go up 4 fifths, getting a tone whose frequency is

times as high. Then we go down 2 octaves to get a tone whose frequency is

times that of our original tone. This is called a Pythagorean major third. It’s close to the just major third, 5/4 = 1.25. But it’s a bit too high!

Let’s see what what these Pythagorean major thirds look like, and where they sit in the scale. To do this, let’s take our original ‘star of fifths’:

and reorder the notes so they form a ‘circle of fifths’:

Here we see two wolf fifths, each containing one of the notes separated by a Pythagorean comma (namely G♭ and F♯). As we’ve seen, if we omit either one of these notes we’re left with a single wolf fifth. But this breaks the left-right symmetry of the above picture, so let’s leave them both in for now.

Now let’s draw all the Pythagorean thirds in blue:

A pretty, symmetrical picture. But not every note has a blue arrow pointing out of it! The reason is that not every note has some other note in the scale a Pythagorean third higher than it. We could delve into this more….

But instead, let’s figure out what to do about these annoyingly large Pythagorean thirds!

Historically, the first really popular solution was to use ‘just intonation’, a system based on simple fractions built from the numbers 2, 3 (as in Pythagorean tuning) but also 5. It was discussed by Ptolemy as far back as 150 AD. But it became widely used from roughly 1300 to at least 1550—starting in England, and then spreading throughout Europe, along with the use of major thirds.

Just intonation makes a few important thirds in the scale be just, but not as many as possible. Around 1523 another solution was invented, with more just thirds: ‘quarter-comma meantone’. It became popular around 1550, and it dominated Europe until about 1690. Let’s see what this system is, and why it didn’t catch on sooner.

The idea is to tweak Pythagorean tuning so that all the Pythagorean thirds I just showed you become just thirds! To do this, we’ll simply take the Pythagorean system:

and shrink all the blue arrows so they have a frequency ratio of 5/4.

Unfortunately this will force us to shrink the black arrows, too, In other words, to make our major thirds just, we need to shrink our fifths. It turns out that we need fifths with a frequency ratio of

This is only a tiny bit less than a ‘Pythagorean’ fifth, namely 1.5. It is not a nasty wolf fifth: it sounds pretty good. In fact it’s quite wonderful that the fourth root of 5 is so close to 3/2. So, using some fifths like this may count as an acceptable sacrifice if we want just major thirds.

Here’s what we get:

This tuning system is called quarter-comma meantone.

You’ll note that by shrinking the blue and black arrows—that is, the thirds and fifths—we’ve now made the note F♯ lower than G♭, rather than higher, as it was in Pythagorean tuning. Their frequency ratio is now

which is yet another of those annoying little glitches: this one is called the lesser diesis.

So that’s quarter-comma meantone tuning in a nutshell. But there’s a lot more to say about it. For example, I haven’t explained all the numbers in that last picture. Where do and the lesser diesis 128/125 come from??? I haven’t even explained why this system called ‘quarter-comma meantone’. These issues are related. I’ll explain them both next time, but I’ll give you a hint now. I told you that the Pythagorean major third

is a bit bigger than the just major third:

But how much bigger? Their ratio is

This number, yet another of those annoying glitches in harmony theory, is called the syntonic comma. And this, not the Pythagorean comma, is the comma that gives ‘quarter-comma meantone’ its name! By taking the syntonic comma and dividing it into four equal parts—or more precisely, taking its fourth root—we are led to quarter-comma meantone. I’ll show you the details next time.

Quarter-comma meantone is dramatically different from the earlier tuning systems I’ve discussed, since it uses an irrational number: the fourth root of 5. I think this is why it took so long for quarter-comma meantone to be discovered. After all, irrational numbers were anathema in the old Pythagorean tradition relating harmony to mathematics.

It seems that quarter-comma meantone was discovered in a burst of more sophisticated mathematical music theory in Renaissance Italy—along with other meantone systems, but I’ll explain what that means later. References to tuning systems that could possibly refer to meantone were published as early as the 1496 text Practicae musica by Franchinus Gaffurius. Pietro Aron unmistakably discussed quarter-comma meantone in his 1523 book Toscanello in musica. However, the first mathematically precise descriptions appeared in the late 16th century treatises by the great Gioseffo Zarlino (Le istitutioni harmoniche, 1558) and Francisco de Salinas (De musica libri septem, 1577). Those two also talked about ‘third-comma’ and ‘two-sevenths-comma’ meantone 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.