I’ve been talking about the marvelous tuning system that dominated western music from about 1550 to 1690: quarter-comma meantone. Above I’ve drawn it in its most mathematically beautiful form. For convenience I’m drawing it in the key of C, but you could use any key.
This scale has lots of ‘just major thirds’, which are pairs of notes with a frequency ratio of 5/4. I’ve drawn them as dark blue arrows.
This scale also has a visible symmetry under reflection across the vertical axis! What does this mean musically? It means that if you can reach some note in the scale by starting at C and multiplying the frequency by some number, you can reach some other note in the scale by starting at C and dividing by that number. Musicians call this symmetry ‘inversion’.
Unfortunately this scale has 13 notes, not 12. But the real problem is not that 13 is an unlucky number. It’s that two of the notes are absurdly close together! Their frequency ratio is a number very close to 1, called the ‘lesser diesis’:
So in practice, musicians usually leave out one of these two notes. This breaks the symmetry but gives a scale with 12 notes that are close to equally spaced.
Usually musicians leave out the higher of these two nearby notes, called the diminished fifth. When we’re in the key of C, as above, this is the one called G♭. With G♭ removed, we’re left with the following scale:
Now there’s one fewer of those dark blue arrows: one just major third is gone. Also, our scale now has a ‘wolf fifth’ with frequency ratio audibly bigger than 1.5. So, we pay a price for breaking the symmetry and leaving out the diminished fifth.
Alternatively, we could leave out the lower of the two nearby notes, called the augmented fourth. This is the one called F♯. Then we get this scale:
It’s just a reflected version of the scale above.
Is there any good reason to prefer the choice most people make, namely leaving out the diminished fifth? As a mathematician you might say “no, the two choices are related by a symmetry, so the choice we make is completely arbitrary”. However, the symmetry we’re talking about here—inversion symmetry, where we replace frequency ratios by their reciprocals—is not some god-given law of nature. After all, when you pluck a guitar string you hear an ‘overtone series’: not just the fundamental frequency but also the frequencies 2, 3, 4, etc. times that. You don’t hear an ‘undertone series’ with frequencies 1/2, 1/3, 1/4 etc. times the fundamental!
People do try to do music with undertones, and it’s fun—it’s called ‘negative harmony’. But undertones are physically not on an equal footing with overtones. And while a major triad and a minor triad are related by inversion symmetry, they sound very different. So there could be a reason to prefer the scale with an augmented fourth to the one with the diminished fifth. But I don’t know it.
The lesser diesis, syntonic comma and Pythagorean comma
As I keep emphasizing, trying to create a nice tuning system is like trying to make your carpet look good when it’s too big for your room. There are bound to be lumps in the carpet; all you can do is try to deal with them as best you can. You can try to spread them out evenly, try to gather them into one big lump and hide it under the couch, etc.
So, the math of tuning systems is the math of lumps. We can learn more about this by comparing quarter-comma meantone with Pythagorean tuning. Le’s compare the versions that are missing the diminished fifth. We’ll find a nice relation between three pesky numbers that are quite close to 1:
• the lesser diesis, 128/125,
• the syntonic comma, 81/80,
• the Pythagorean comma, 531441/524288.
First let’s take our quarter-comma meantone scale:
and write it using letter names for the lesser diesis and syntonic comma. I’ll leave out the blue arrows, because we won’t need them:
Here I’m writing the quarter-comma fifth, which is just Image may be NSFW.
Clik here to view.![\sqrt[4]{5},](http://s0.wp.com/latex.php?latex=%5Csqrt%5B4%5D%7B5%7D%2C&bg=ffffff&fg=333333&s=0&c=20201002)
as (3/2)σ-1/4 where σ is the syntonic comma. And I’m writing the wolf fifth as δ times the quarter-comma fifth, where δ is the lesser diesis.
As we go all the way around this circle clockwise, the frequency goes up by 7 octaves. So, multiplying all the numbers labeling the arrows, we must have
Clik here to view.
As you can see, here the lumps in the carpet are the cube of the reciprocal of the syntonic comma, which we have smeared out over the whole circle of fifths, together with the lesser diesis, which we’ve concentrated in the wolf fifth.
On the other hand Pythagorean tuning is based on fact that going around this circle, we go up 7 octaves:
Mathematically this says
Clik here to view.
where p is the Pythagorean comma. So here the lump in the carpet is the reciprocal of the Pythagorean comma, which we’ve packed into the wolf fifth.
Putting these two equations together we see
Clik here to view.
or in other words
Clik here to view.
The Pythagorean comma times the lesser diesis is the syntonic comma cubed! Knowing this may help us later, when I discuss the fancier methods of dealing with lumps in the carpet used in well-tempered scales.
First, however, I want to explain that word ‘meantone’. What’s mean about the tones in quarter-comma meantone? I’ll tell you next time!
For more on Pythagorean tuning, read this series:
For more on just intonation, read this series:
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: