Last time we saw the importance of some tiny musical intervals: irritating but inevitable glitches in our search for perfectly beautiful harmonies. Today I want to talk about a truly microscopic interval called the ‘atom of Kirnberger’.
It was discovered by Bach’s student Johann Kirnberger, and it has a frequency ratio absurdly close to 1:
It arose naturally in Kirnberger’s attempt to find a tuning system close to equal temperament with only rational frequency ratios. But it relies on a mathematical miracle: a coincidence so eye-popping that a famous expert in black hole physics wrote a paper trying to explain it!
Throughout my discussion of tuning systems, we’ve repeatedly encountered two glitches in the fabric of music called ‘commas’:
The first shows up when you try to build a scale from fifths, while the second shows up when you try to have lots of nice fifths and major thirds.
They are quite close together, so their ratio is even closer to one! It has a cool name: it’s called the schisma. I’ll abbreviate it with the Greek letter chi:
The schisma is a kind of meta-glitch: a glitch between glitches! It may seem like a mere curiosity, since two pitches whose frequency ratio is a schisma sound the same to everyone. But precisely for this reason, it plays a role in some well tempered tuning systems.
You see, sometimes when you’re building a tuning system you need a Pythagorean comma to make your circle of fifths close up nicely, but to get a really nice major third you use the syntonic comma instead. When you do this, you’re off by a schisma! And like a lump in the carpet, this schisma has to go somewhere. It’s so small that it scarcely matters where you put it. If you’re not extremely careful in tuning, your notes are probably off by more than a schisma anyway. But mathematically, it’s there.
In future episodes, I’ll show you examples of how this happens in some well-known tuning systems. But today I want to show you a mind-bending, completely crazy way that Kirnberger used the schisma.
Let’s get started!
As we saw in our study of Pythagorean tuning, going up 12 just perfect fifths takes you up a bit more than 7 octaves. Their ratio is the Pythagorean comma:
As a result, if we divide the just perfect fifth (that is, 3/2) by the 12th root of the Pythagorean comma, we get the equal tempered fifth (that is, Image may be NSFW.
Clik here to view.
), whose 12th power is exactly 7 octaves. This correction, the 12th root of the Pythagorean comma,
is called a grad. I’ll call it γ for short:
So, what I’m saying is that if we divide 3/2 by the grad we get 27/12, which is the equal tempered fifth:
Now for the eye-popping coincidence. The grad
is remarkably close to the schisma:
Look at that! For no obvious reason, they match to almost seven decimal places!
Image may be NSFW.
Clik here to view.
But unlike the grad, the schisma is rational. This let Kirnberger create a tuning system very close to equal temperament but with rational frequency ratios. His idea was to use a circle of fifths where instead of using the equal tempered fifth
which is irrational, we use 3/2 divided by a schisma
which is rational. They are remarkably close!
The quantity 3/2χ is called the schismatic fifth:
We can try to build a circle of fifths using the schismatic fifth instead of the equal-tempered fifth. But there’s a slight problem. Actually, ‘slight’ is an overstatement: it’s a nearly infinitesimal problem.
If we go up 12 schismatic fifths we don’t go up exactly 7 octaves. We go up a microscopic amount more, since
= (27/12)12 (γ/χ)12
= 27 (γ/χ)12
and the number (γ/χ)12 is microscopically larger than one. Since this number was discussed by Kirnberger, it’s called the atom of Kirnberger. I’ll call it α for short:
Let’s work out what it equals! Remember that the grad is the 12th root of the Pythagorean comma, so
and turning the crank on the old calculator:
Using these ideas, Kirnberger created a tuning system called rational equal temperament. It’s very close to equal temperament, but with only rational numbers as frequency ratios. To do this, he used 11 schismatic fifths and one schismatic fifth divided by the atom of Kirnberger. Just for fun, I’ll call the last an atomic fifth:
I don’t know where Kirnberger put the atomic fifth, but I’ll follow the common tradition of putting problems right after the tritone, which is F♯ in the key of C:
Compare this to equal temperament:
Nobody can hear the difference, so Kirnberger’s rational equal temperament is not used in music. But it sheds light on the interaction between the Pythagorean comma and syntonic comma, and that is important for the well tempered scales we’ll be seeing next.
It also raises a math puzzle: why is the grad so close to the schisma? The physicist Don Page, famous for his work on black hole thermodynamics, has written a paper exploring this:
• Don N. Page, Why is the Kirnberger kernel so small?
He calls the 12th root of the atom of Kirnberger the Kirnberger kernel: it equals the grad divided by the schisma, or the equal tempered fifth divided by the schismatic fifth:
Since the schisma is already a meta-glitch, the Kirnberger kernel is a meta-meta-glitch! He shows that the closeness of this number to 1 is equivalent to a number of other coincidences, notably
He then wrestles this coincidence down to a fact involving only integers, which he tries to explain using properties of the hyperbolic tangent function! He is much better at these things than me, so if you enjoyed my article you should definitely take a look at his.
Next time I’ll get back to business and talk about well tempered tuning systems—starting with three more practical systems developed by Kirnberger.
For more on Pythagorean tuning, read this series:
For more on just intonation, read this series:
For more on quarter-comma meantone tuning, read this series:
For more on well-tempered scales, read this series:
• Part 1. An introduction to well temperaments.
• Part 2. How small intervals in music arise naturally from products of integral powers of primes that are close to 1. The Pythagorean comma, the syntonic comma and the lesser diesis.
• Part 3. Kirnberger’s rational equal temperament. The schisma, the grad and the atom of Kirnberger.
• Part 4. The music theorist Kirnberger: his life, his personality, and a brief introduction to his three well temperaments.
• Part 5. Kirnberger’s three well temperaments: Kirnberger I, Kirnberger II and Kirnberger III.
For more on equal temperament, read this series: