Well Temperaments (Part 2)

Last time I ended with a question: why are certain numbers close to 1 so important in tuning systems? It helps to understand a bit about this before we plunge into the study of well temperaments. It turns out that in some sense western harmony evolved one prime at a time, so let’s look at the subject that way.

The prime 2

If all the frequency ratios in our tuning system were powers of 2:

2ⁱ

life would be very simple. Multiplying a frequency by 2 raises its pitch by an octave, so the only chords we could play are those built out of octaves. Not much music could be made! But there’d be no difficult decisions, either.

The primes 2 and 3

In Pythagorean tuning, also called 3-limit tuning, we generate all our frequency ratios by multiplying powers of 2 and powers of 3:

2ⁱ · 3^j

This is more exciting. While the frequency ratio of 2 is an octave, that of 3/2 is called a just perfect fifth. So now we can use octaves and fifths to build other intervals (that is, frequency ratios).

But in fact, any positive real number can be approximated arbitrarily well by numbers of the form 2ⁱ · 3^j, so we have an embarrassment of riches: more intervals than we really want! To bring the system under control, we take some number of the form 2ⁱ · 3^j that’s really close to 1 and act like it is 1.

I examined the options in an earlier post and got a list of ‘winners’ according to some precise criterion. A couple of early winners are

2⁸ · 3^-5 = 256/243 ≈ 1.053

and

2^-11 · 3⁷ = 2143/2048 ≈ 1.068

These would be important in scales with 5 or 7 notes, but western music holds out for a much better one, called the Pythagorean comma:

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

This is important for a 12-tone scale, because it means that if we go up 12 fifths, multiplying the frequency by 3/2 each time, it’s almost the same as going up 7 octaves.

But not quite! There are many ways of dealing with this problem. In Pythagorean tuning we absorb the problem by dividing one of our fifths by the Pythagorean comma, turning it into an unpleasant ‘wolf fifth’:

Pythagorean wolf fifth = 3/2p = 2¹⁸ · 3^-11 = 262144/177147 ≈ 1.479811

For example:

But we can spread the inverse of the Pythagorean comma around the circle of fifths any way we like, and different ways give different tuning systems.

For example, in equal temperament we spread it completely evenly around the circle of fifths, using the equal tempered fifth everywhere:

equal tempered fifth = 3/2p^1/12 = 2^7/12 ≈ 1.498307

This is not an example of 3-limit tuning because it uses irrational numbers! But it’s an obvious way to solve the problem of the Pythagorean comma which emerges in 3-limit tuning. More interesting solutions tend to involve the next prime number.

The primes 2, 3, and 5

In 5-limit tuning we generate all our frequency ratios by multiplying powers of 2, 3 and 5:

2ⁱ · 3^j · 5^k

Equivalently, we build them using the octave (2), the just perfect fifth (3/2) and the just major third: 5/4.

There are some new simple fractions close to 1 that you can build with 2, 3 and also 5. The most important is the syntonic comma:

syntonic comma = σ = 2^-4 · 3⁴ · 5^-1 = 81/80 = 1.0125

This shows up when you try to reconcile the perfect fifth and the major third. If you go up four just perfect fifths, you boost the frequency by a factor of (3/2)⁴ = 81/16 = 5.0625, which is a bit more than a just major third and two octaves, namely 5/4 × 2² = 5. The ratio is the syntonic comma.

As we’ll see in future episodes, this realization is fundamental to many well tempered tuning systems. We’ve already seen the grand-daddy of these systems: quarter-comma meantone. It’s not a well tempered itself, but fixing its main flaw leads to well tempered systems. In quarter-comma meantone, we divide most of our fifths by the fourth root of the syntonic comma, which gives lots of just major thirds, shown in blue below:

So, this scale has many ‘quarter-comma fifths’ with a frequency ratio of (3/2)σ^-1/4. Going around the whole circle and multiplying 12 of these quarter-comma fifths would give 125, which is not quite the 128 we need to go up 7 octaves. So we need to take one of these quarter-comma fifths and multiply it by 128/125. The resulting ‘wolf fifth’ sounds terrible—and this is what well temperaments seek to cure.

The number 128/125 is an important fraction close to 1 built from just the primes 2 and 5. It’s called the lesser diesis:

lesser diesis = δ = 2⁷ · 5^-3 = 128/125 = 1.024

It’s not only a power of 2 divided by a power of 5, but also a power of 2 divided by a power of 10. You’ve bumped into it if you’ve ever wondered why people often use ‘kilobyte’ to mean 1024 bytes, not 1000.

From the Pythagorean comma, syntonic comma and lesser diesis we can generate other fractions close to 1 built from the primes 2, 3 and 5. For example, I’ve already discussed the product of the syntonic comma and lesser diesis, and also their ratio.

But when we study well temperaments, more important will be the Pythagorean comma divided by the syntonic comma. Called the schisma, this fraction is very close to 1:

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

I’ll talk about it more next time.

It’s also important to note that the lesser diesis is not independent from the Pythagorean comma and syntonic comma. We’ve already seen today that going up a fifth twelve times is the same as going up 7 octaves divided by the Pythagorean comma. Now we’re seeing that going up (3/2)σ^-1/4 twelve times is the same as going up 7 octaves times the lesser diesis. So, we have

(syntonic comma)^-3 · lesser diesis = (Pythagorean comma)^-1

or

p δ = σ³

We’ve already this in a slightly different way before.

Due to this relation there must be other fractions close to 1, built only from powers of the primes 2, 3, and 5, that are independent from p, σ and δ. In fact, we’ve already seen four such fractions appearing as the sizes of semitones in just intonation:

The ratios of these semitones include the syntonic comma, the lesser diesis, and also their product the greater diesis and their ratio the diaschisma!

But these semitones are not extremely close to 1. The smallest, the lesser chromatic semitone, is 25/24 ≈ 1.041666. So there must be interesting examples of fractions built from 2, 3 and 5, independent of the syntonic and Pythagorean commas, and much closer to 1. On Mastodon I asked for examples built solely from the primes 3 and 5, and a bunch of people helped me out. Here are some of the first few winners:

3^-1 · 5¹ = 1.666…
3³ · 5^-2 = 1.08
3^-19 · 5¹³ ≈ 1.050283
3²² · 5^-15 ≈ 1.028295
3^-41 5²⁸ ≈ 1.021383
3⁶³ · 5⁴³ ≈ 1.006767

The main thing to notice here is that we need fractions with impractically large numerators and denominators to get closer than the large diatonic semitone, 27/25 = 1.08. These fractions won’t play a role in well temperaments.

Higher primes

I won’t say much about primes after 5 now. But they’ve been studied in music theory at least since Ptolemy, and the compositions of Ben Johnston really run wild with them. For a tiny bit about the virtues of the prime 7, read my post on the harmonic seventh chord.

The facts I’ve crudely laid out above must be part of an elegant general theory of approximating the number 1 by fractions built from powers of a specified set of primes, and how to build scales from these fractions. Done systematically, this could be of interest not just to music theorists but even pure mathematicians. But I will not explore this now, since my goal was merely to recall some facts needed to understand the explosion of well temperaments starting around 1690!

Next time I’ll digress slightly into Kirnberger’s discovery of a tuning system with frequency ratios built only from the primes 2, 3, and 5 that comes extremely close to equal temperament. This is not a practical system, but it relies on an utterly astounding coincidence, and more importantly it highlights the role of the schisma, which will keep showing up in other systems.

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

---

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 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.