Just Intonation (Part 3)

Azimuth 2023-11-09

Last time I said a bit about ‘just intonation’: that is, tuning where the most important notes have frequency ratios that are simple fractions. I focused on the historically important case of ‘5-limit tuning’, where the frequency ratios are products of powers of primes ≤ 5.

I went a long way toward getting some popular scales in 5-limit tuning. I started by drawing a chart called a ‘Tonnetz’ where going east multiplies the frequency by 3/2, and going roughly northeast multiplies the frequency by 5/4:

Then, I kept only the numbers in a parallelogram whose corners are numbers very close to a tritone: that is, $\sqrt{2}$ or $1/\sqrt{2}$ .

Then I multiplied these numbers by appropriate powers of 2 to make them lie between 1 and 2:

If we curl this parallelogram up into a torus, this torus has exactly 12 notes on it! To curl it up, we need to glue each note on the parallelogram’s left edge to its partner on the right edge:

We also need to glue each note on the parallelogram’s top edge to its partner on the bottom:

The problem is that not all the notes we’re gluing together have the same frequency! Luckily, they’re close.

How close are these notes, exactly? That’s what I want to analyze today. Believe it or not, music theorists have thought about this for thousands of years and made up special terms to describe the answers.

First let’s compare the notes on the left edge of the parallelogram to their partners on the right edge:

If we work out their frequency ratios, we see the notes on the right are a bit higher than those on the left:

$\begin{array}{ccl} \frac{45/32}{25/18} &=& \frac{81}{80} \\ \\ \frac{9/8}{10/9} &=& \frac{81}{80} \\ \\ \frac{9/5}{16/9} &=& \frac{81}{80} \\ \\ \frac{36/25}{64/45} &=& \frac{81}{80} \end{array}$

We always get the same ratio! And it’s our friend the syntonic comma:

81/80 = 1.0125

This is no accident, of course: it’s built into the structure of the Tonnetz. In music terminology, when we go up 4 just perfect fifths and then go down a just major third and 2 octaves, the frequency gets multiplied by

3/2 × 3/2 × 3/2 × 3/2 × 4/5 × 1/2 × 1/2 = 81/80

which is the syntonic comma.

So: curling up the parallelogram means deciding whether to use tones on the its left edge and tones on its right edge—which forces us into the jaws of the syntonic comma.

Next let’s compare the notes on the top edge of the parallelogram to their partners on the bottom edge:

If we work out their frequency ratios, we see the notes on the bottom are a bit higher than their partners on the top:

$\begin{array}{ccl} \frac{64/45}{25/18} &=& \frac{128}{125} \\ \\ \frac{36/25}{45/32} &=& \frac{128}{125} \end{array}$

Yet another glitch! This number is called the lesser diesis:

128/125 = 1.024

‘Diesis’ is a Greek word meaning ‘leak’ or ‘escape’—though ‘glitch’ might be a more idiomatic translation. In music terminology, if you go up an octave and then go down 3 just major thirds, the frequency gets multiplied by

2 × 4/5 × 4/5 × 4/5 = 128/125

which is the lesser diesis. You can see how this works by staring at the Tonnetz.

In short, when we curl up the parallelogram we have some choices to make concerning the major second, the tritone and the major seventh:

$\begin{array}{ll} \textbf{tonic} & \textbf{1} \\ \textbf{minor 2nd} & \textbf{16/15} \\ \textbf{major 2nd} & \textbf{10/9 or 9/8} \\ \textbf{minor 3rd} & \textbf{6/5} \\ \textbf{major 3rd} & \textbf{5/4} \\ \textbf{perfect 4th} & \textbf{4/3} \\ \textbf{tritone} & \textbf{25/18 or 45/32 or 65/45 or 36/25} \\ \textbf{perfect 5th} & \textbf{3/2} \\ \textbf{minor 6th} & \textbf{8/5} \\ \textbf{major 6th} & \textbf{5/3} \\ \textbf{minor 7th} & \textbf{16/9 or 9/5} \\ \textbf{major 7th} & \textbf{15/8} \\ \textbf{octave} & \textbf{2} \\ \end{array}$

For the major second we have two choices whose ratio is the syntonic comma: 10/9 and 9/8. In a symmetrical way, for the minor seventh below the tonic we have two choices whose ratio is the syntonic comma: 8/9 and 9/10. But we multiply these choices by 2 to get choices for the minor seventh above the tonic: 16/9 and 9/5.

The biggest challenge involves the tritone, where we have four choices, coming from the corners of our parallelogram:

Here I’ve drawn arrows from lower tones to higher tones, and labeled each edge by the frequency ratio between the tones it connects.

The top and bottom edges have a frequency ratio of a syntonic comma, 81/80, while the left and right edges have a frequency ratio of a lesser diesis, 128/125. The diagonals also have standard names!

The diagonal from upper left to lower right gives a frequency ratio called the greater diesis:

$\frac{36/25}{25/18} = \frac{648}{625} = 1.0368$

In other words,

greater diesis = lesser diesis × syntonic comma

The diagonal from the upper right to the lower left gives a frequency ratio called the diaschisma:

$\frac{64/45}{45/32} = \frac{2048}{2025} \approx 1.011358\dots$

In other words,

diaschisma = lesser diesis / syntonic comma

Apparently the diaschisma got its name from the physicist Helmholtz, but it was already studied by Boethius, who wrote a book called De musica in 510 AD. ‘Schisma’ means something like ‘split’, so I guess ‘diaschisma’ means ‘the split between’.

Here you can see the syntonic comma, lesser and greater diesis, and diaschisma in decimals, to make it easier to compare how big they are:

Sometimes the tritone is called diabolus in musica—‘the devil in music’. The usual explanation is that this frequency ratio is so dissonant. But as we’ve seen, it’s also devilishly difficult to deal with in just intonation! We have four choices. What should we do?

Stay tuned for more!