Quarter-Comma Meantone (Part 6)

So far I’ve focused on the quarter-comma meantone system in its mathematically beautiful, symmetrical form above. Today I’ll say more about the scale as actually played: how we trim it down from a 13-note scale to a more practical 12-note scale, and what are the intervals between notes in the resulting scale.

The scale above has:

• a lot of thirds with frequency ratios that are exactly 5/4,

• a lot of fifths that are only slightly flat compared to the ideal 3/2,

• a big problem that we need to deal with now: 13 notes, with two separated by an absurdly tiny gap called the ‘lesser diesis’.

You can see this tiny gap between F♯ and G♭. Both these notes are versions of the ‘tritone’ in the scale of C major. F♯ is called the ‘augmented fourth’ and G♭ is called the ‘diminished fifth’. Unless we build a keyboard with split keys—as some people actually have, but they never caught on—one of these notes has got to go!

There are two options. The most popular is to remove the diminished fifth:

The other is to remove the augmented fourth:

Either approach gives a big bad ‘wolf fifth’, which must be avoided. But what does the resulting scale actually look like? What are the intervals between neighboring notes?

To understand this let’s go back to the start, our original ‘circle of fifths’ with 13 notes:

When we rearrange the notes, listing them in order, we get a ‘star of fifths’:

Now let’s add arrows showing the intervals between neighboring notes:

Except for the lesser diesis, the neighboring notes are all separated by two kinds of interval:

• the quarter-comma chromatic semitone:

• the quarter-comma diatonic semitone:

We worked out these wacky numbers last time. It may seem weird to have two sizes of semitone, but they sound fine. The problem, to repeat myself, is the lesser diesis between the augmented 4th and the diminished 5th. But when we remove either one of these notes, something nice happens!

This is the cool part. Last time we saw this relation:

So, when we remove either the augmented 4th or diminished 5th, the lesser diesis combines with one of the chromatic semitones adjacent to it to give an extra diatonic semitone!

If we remove the diminished 5th, we get this scale:

It’s not completely symmetrical, but it’s still quite pretty—and now the lesser diesis has been banished. Most of the fifths sound good, but there’s wolf fifth between the augmented fourth (F♯) and the minor second (C♯), so you should avoid this. The semitones alternate between chromatic and diatonic… except for two diatonic semitones in a row between F♯ and A♭, and between B and C♯.

Best of all, this scale has lots of just major thirds—though one fewer than in the mathematically beautiful 13-note version of the scale. Let’s figure out where they are. Last time we noticed this relation:

This implies that

But 5/4 is the frequency ratio of a just major third! So we get a just major third whenever we go up two chromatic semitones and two diatonic semitones. So the just major thirds are the blue arrows here:

It’s a bit random-looking, thanks to how we broke the symmetry. If there were a just major third from F♯ to B♭ the pattern of blue arrows would be symmetrical. But there’s not: there’s only a just major third from G♭ to B♭, and we’ve eliminated G♭ from this scale.

But still, this scale has lots of just major thirds! And that was the main goal.

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’.

For more on equal temperament, read this series:

• Equal temperament.