Well Temperaments (Part 5)

Okay, let’s study Kirnberger’s three well-tempered tuning systems! I introduced them last time, but now I’ve developed a new method for drawing tuning systems, which should help us understand them better.

As we’ve seen, tuning theory involves two numbers close to 1, called the Pythagorean comma (≈ 1.0136) and the syntonic comma (= 1.0125). While they’re not equal, they’re so close that practical musicians often don’t bother to distinguish them! They call both a comma.

So, my new drawing style won’t distinguish the two kinds of comma.

Being a mathematician, I would like to say a lot about why we can get away with this. But that would tend to undercut my claim that the relaxed approach makes things simpler! I don’t want to be like the teacher who prefaces the explanation of a cute labor-saving trick with a long and confusing theoretical discussion of when it’s justified. So let me start by just diving in and using this new approach.

First I’ll illustrate this new approach with some tuning systems I’ve already discussed. Then I’ll show you Kirnberger’s three well-tempered systems. At that point you should be in a good position to make up your own well temperaments!

Pythagorean tuning

Here is Pythagorean tuning in my new drawing style:

The circle here is the circle of fifths. Most of these fifths are black arrows labeled by +0. These go between notes that have a frequency ratio of exactly 3/2. This frequency ratio gives the nicest sounding fifth: the Pythagorean fifth.

But one arrow on the circle is red, and labeled by -1. This fifth is one comma flat compared to a Pythagorean fifth. In other words, the frequency ratio of this fifth is 3/2 divided by a comma. This arrow is red because it’s flat—and it’s a fairly bright red because one comma flat is actually quite a lot: this fifth sounds pretty bad!

(The comma here is a Pythagorean comma, but never mind.)

This illustrates a rule that holds for every tuning system we’ll consider:

Rule 1. The numbers labeling arrows on the circle of fifths must sum to -1.

Now let’s look at Pythagorean tuning again, this time focusing on the arrows inside the circle of fifths:

The arrows inside the circle are major thirds. A few of them are black and labeled +0. These go between notes that have a frequency ratio of exactly 5/4. That’s the nicest sounding major third: the just major third.

But a some the arrows inside the circle are green, and labeled by +1. These major thirds are one comma sharp compared to the just major third. In other words, the frequency ratio between notes connected by these arrows is 5/4 times a comma. These arrows are green because they’re sharp—and it’s a fairly bright green because one comma sharp is actually quite a lot.

(These commas are syntonic commas, but never mind.)

Why do the major thirds work this way? It’s forced by the other rule governing all the tuning systems we’ll talk about:

Rule 2. The sum of the numbers labeling arrows for any four consecutive fifths, plus 1, equals the number labeling the arrow for the corresponding major third.

This rule creates an inherent tension in tuning systems! To get major thirds that sound really nice, not too sharp, we need some fifths to be flat. Pythagorean tuning is one way this tension can play out.

Equal temperament

Now let’s look at another tuning system: equal temperament.

Pythagorean tuning had eleven fifths that are exactly right, and one that’s 1 comma flat. The flatness was as concentrated as possible! Equal temperament takes the opposite approach: the flatness is spread out equally among all twelve fifths. Rule 1 must still hold: the total flatness of all the fifths is still 1 comma. So each fifth is 1/12 of a comma flat.

How does this affect the major thirds? Rule 2 says that each major third must be 2/3 of a comma sharp, since

2/3 = – 1/12 – 1/12 – 1/12 – 1/12 + 1

My pictures follow some color rules that are too boring to explain in detail, but bright colors indicate danger: intervals that are extremely flat or extremely sharp. In equal temperament the fifths are all reddish because they’re all flat—but it’s a very dark red, almost black, because they’re only slightly flat. The major thirds are fairly sharp, so their blue-green color is more noticeable.

Quarter-comma meantone

Now let’s look at another important tuning system: quarter-comma meantone. This system was very popular from 1550 until around 1690. Then people started inventing well temperaments as a reaction to its defects. So we need to understand it well.

Here it is:

All but one of the fifths are slightly flat: 1/4 comma flat. This is done to create a lot of just major thirds, since Rule 2 says

0 = -1/4 – 1/4 – 1/4 – 1/4 + 1

This is the beauty of quarter-comma meantone! But it’s obtained at a heavy cost, as we can see from the glaring fluorescent green.

Because 11 of the fifths are 1/4 comma flat, the remaining one must be a whopping 7/4 commas sharp, by Rule 1:

7/4 + 11 × -1/4 = -1

This is the famous ‘wolf fifth’. And by Rule 2, this wolf fifth makes the major thirds near it 2 commas sharp, since

2 = 7/4 – 1/4 – 1/4 – 1/4 + 1

In my picture I wrote ‘8/4’ instead of 2 because I felt like keeping track of quarter commas.

The colors in the picture should vividly convey the ‘extreme’ nature of quarter-comma meantone. As long as you restrict yourself to playing the dark red fifths and black major thirds, it sounds magnificently sweet. But as soon as you enter the fluorescent green region, it sounds wretched! Well temperaments were created to smooth this system down… without going all the way to the bland homogeneity of equal temperament.

And now let’s look at Kirnberger’s three well tempered systems. Only the third was considered successful, and we’ll see why.

Kirnberger I

Here is Kirnberger I:

The flatness of the fifths is concentrated in a single fifth, just as in Pythagorean tuning. Indeed, from this picture Kirnberger I looks just like a rotated version of Pythagorean tuning! That’s a bit deceptive, because in Kirnberger I the flat fifth is flat by a syntonic rather than a Pythagorean comma. But this is precisely the sort of nuance my new drawing style ignores. And that’s okay, because the difference between the syntonic and Pythagorean comma is inaudible.

So the only noticeable difference between Kirnberger I and Pythagorean tuning is the location of flat fifth. And it’s hard to see any advantage of putting it so close to C as Kirnberger did, rather than putting it as far away as possible.

Thus, it’s not so suprising that I’ve never heard of anyone actually using Kirnberger I. Indeed it’s rare to even see a description of it: it’s very obscure compared to Kirnberger II and Kirnberger III. Luckily it’s on Wikipedia:

• Wikipedia, Kirnberger temperament.

Kirnberger II

Here is Kirnberger’s second attempt:

This time instead of a single fifth that’s 1 comma flat, he used two fifths that are 1/2 comma flat.

As a result, only 3 major thirds are just, as compared to 4 in Kirnberger I. But the number of major thirds that are 1 comma sharp has gone down from 8 to 7. The are also 2 major thirds that are 1/2 comma sharp—the bluish ones. So, this system is less ‘extreme’ than Kirnberger I: the pain of sharp major thirds is more evenly distributed. As a result, this system was more widely used. But it was never as popular as Kirnberger III.

For more, see:

• Carey Beebe, Technical Library: Kirnberger II.

Kirnberger III

Here is Kirnberger’s third and final try:

This time instead of a two fifths that are 1/2 comma flat, he used four fifths that are 1/4 comma flat! A very systematic fellow.

This system has only one just major third. It has 2 that are 1/4 comma sharp, 2 that are 2/4 comma sharp, 2 that are 3/4 comma sharp, and only 3 that are 1 comma sharp. So it’s noticeably less ‘extreme’ than Kirnberger II: fewer thirds that are just, but also fewer that are painfully sharp.

I think you really need to stare at the picture for a while, and think about how Rule 2 plays out, to see the beauty of Kirnberger III. But the patterns become a bit more visible if we rotate this tuning system to give it bilateral symmetry across the vertical axis, and write the numbers in a symmetrical way too:

Rotating a tuning system just means we’re starting it at a different note—‘transposing’ it, in music terminology.

The harpsichord tuning expert Cary Beebe writes:

One of the easiest—and most practical—temperaments to set dates from 1779 and is known as Kirnberger III. For a while, some people thought that this might be Bach’s temperament, seeing as Johann Philipp Kirnberger (1721–1783) learnt from the great JS himself. Despite what you might have been taught, Bach neither invented nor used Equal Temperament. He probably used many different tuning systems—and if he had one particular one in mind for any of his works, he never chose to write clear directions for setting it. Note that his great opus is called the Well-tempered Clavier in English, not the “Equal Tempered Clavichord”, as it has too often been mistranslated. You will find several other Bach temperaments discussed later in this series.

There are other commas to learn, and a whole load of other technical guff if you really want to get into this quagmire, but here you will forgive me if we regard the syntonic comma as for all practical purposes the same size as the Pythagorean. After all, don’t you just want to tune your harpsichord instead of go for a science degree?

Here’s how you go about setting Kirnberger III…

Then he explains how to tune a harpischord in this system:

• Carey Beebe, Technical Library: Kirnberger III.

Carey Beebe is my hero these days, because he explains well temperaments better than anyone else I’ve found. My new style of drawing tuning systems is inspired by his, though I’ve added some extra twists like drawing all the major thirds, and using colors.

Technical details

If you’re wondering what Beebe and I mean about Pythagorean versus syntonic commas, here you can see it. Here is Kirnberger I drawn in my old style, where I only drew major thirds that are just, and I drew them in dark blue:

Kirnberger I has one fifth that’s flat by a factor of the syntonic comma:

σ = 2^-4 · 3⁴ · 5^-1 = 81/80 = 1.0125

But as we go all the way around the circle of fifths the ‘total flatness’ must equal the Pythagorean comma:

p = 2^-19 · 3¹² = 531441/524288 ≈ 1.013643

That’s just a law of math. So Kirnberger compensated by having one fifth that’s flat by a factor of p/σ, which is called the ‘schisma’:

χ = p/σ = 2^-15 · 5 · 3⁸ = 32805/32768 ≈ 1.001129

He stuck this ‘schismatic fifth’ next to the tritone, since that’s a traditional dumping ground for annoying glitches in music. But it barely matters since the schisma is so small.

(That said, the schisma is almost precisely 1/12th of a Pythagorean comma, or more precisely p^1/12—a remarkable coincidence discovered by Kirnberger, which I explained in Part 3. And I did draw the 1/12th commas in equal temperament! So you may wonder why I didn’t draw the schisma in Kirnberger I. The answer is simply that in both cases my decision was forced by rules 1 and 2.)

Here’s Kirnberger II in a similar style:

Here the schismatic fifth compensates for using two 1/2 commas that are syntonic rather than Pythagorean.

And here’s Kirnberger III:

Now the schismatic fifth compensates for using four 1/4 commas that are syntonic rather than Pythagorean.

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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 this series:

• Quarter-comma meantone.

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:

• Equal temperament.