exupero's blog

Musical temperament

Western music uses a system of 12 notes, the frequencies of which are typically spaced by equal temperament so each half-step interval has the same ratio of frequencies as every other half-step interval. Choosing octaves to double the frequency of the octave below, the 11 notes between octaves are each tuned to be a twelfth root of two greater in pitch than the note below. Increasing the frequency by 12 twelfth roots of two produces a doubled frequency at the other end of the octave.

That's not the only way to do it, though. Notes can also be tuned to just intonation, which instead prefers whole-number ratios of frequencies to the ratio of a twelfth roots of two (an irrational ratio). Just intonation is less common because it narrows the range of musical expression. Whole-number ratios of frequencies depend on the choice of basis note, which leads to different keys having different intervals between what on paper are the same notes. The scale starting on G ends up with notes at different frequencies than the scale starting on C, and a piece of music couldn't change keys midway through without retuning the instrument. Same for a symphony and an entire performance. Western music would be much less dynamic under just intonation than musical temperament.

That's true historically, but modern electronic devices don't need to be retuned to produce different sets of frequencies, so the possibilities are open. To explore them, let's start by visualizing equal temperament and how frequency changes between notes:


Increasing by a mere twelfth root of two makes the line appear almost straight.

Just intonation not only prefers whole-number ratios, but also ratios of smaller numbers, the smallest being 2:1, which defines the ratio of one octave to the next. While the next simplest ratio is 3:1, for notes in the same octave we have to pick ratios less than 2:1. In that case, the next simplest ratio is 3:2. Here it is on the above equal temperament curve, choosing the note such that it falls closest to the curve:


Visually it's a good match, right on the curve. Under equal temperament, the seventh half-note from the basis note has a frequency 27/12 higher than the basis note, or 1.498:1, very close to 3:2.

The next simplest ratios are 4:3 and 5:3:


Those are also close, though 5:3 falls a little below the curve.

Here's the full set of notes, adding the ratios 5:4, 6:5, 7:4, 7:5, 7:6, 8:5, 8:7, and 9:5:


Several fall close to the curve, but four deviate noticeably, specifically the first, second, tenth, and eleventh notes. Let's see how far off all the notes are:

Equal temperamentJust intonationDifference
21/12 ≅ 1.0598/7 ≅ 1.1437.9%
22/12 ≅ 1.1227/6 ≅ 1.1673.9%
23/12 ≅ 1.1896/5 = 1.20.9%
24/12 ≅ 1.2605/4 = 1.250.8%
25/12 ≅ 1.3354/3 ≅ 1.3330.1%
26/12 ≅ 1.4147/5 = 1.41.0%
27/12 ≅ 1.4983/2 = 1.50.1%
28/12 ≅ 1.5878/5 = 1.60.8%
29/12 ≅ 1.6825/3 ≅ 1.6670.9%
210/12 ≅ 1.7827/4 = 1.751.8%
211/12 ≅ 1.8889/5 = 1.84.6%

The first two notes and last two notes are the only ones that deviate from equal temperament by more than 1%.

It's hard for me to determine if these just intonation intervals sound objectively better or worse. I'm not particularly sensitive to dissonance, so judge for yourself:

Equal temperament certainly seems more familiar, which makes it difficult to know if these simplest-possible just intonation intervals are unpleasant or just unfamiliar. In a piece of music that knows how to use them, they might be fine.