Lesson 5
Equal temperament
Lesson 4 ended at an impossibility: pure intervals and free transposition cannot coexist. This is the lesson where somebody picks. The choice made in Western music was to abandon purity entirely, on purpose, everywhere, and it is the single most consequential engineering decision in the history of the field.
Give up, deliberately and evenly
If the steps cannot be both pure and equal, make them exactly equal and accept that none of them are pure. Divide the octave into twelve identical multiplicative steps. Each step is the twelfth root of two, about 1.05946, and twelve of them multiply back to exactly 2.
Now every interval is the same size wherever you start. Transpose anything anywhere and it survives intact. There is no wolf, because the error has been smeared uniformly across all twelve keys instead of being dumped on one. That is what the evenly spaced rows in your piano roll actually buy you.
The price is that not one interval except the octave is in tune any more. Every third, fifth and sixth you have ever played on a keyboard is slightly wrong, on purpose.
The same triad, tuned two ways
Start it playing, then switch. Listen to the middle note.
- Root
- 261.63 Hz
- Third
- 327.04 Hz
- Fifth
- 392.44 Hz
- Third is off by
- 0.00c
How far each tempered interval sits from its pure ratio
- Minor second16:15-11.73c
- Major second9:8-3.91c
- Minor third6:5-15.64c
- Major third5:4+13.69c
- Fourth4:3+1.96c
- Tritone45:32+9.78c
- Fifth3:2-1.96c
- Minor sixth8:5-13.69c
- Major sixth5:3+15.64c
- Minor seventh9:5-17.60c
- Major seventh15:8+11.73c
- Octave2:1+0.00c
The fifth is off by -1.96c, which is below what anyone can hear. The major third is off by +13.69c, which is not. Equal temperament did not share the damage out evenly. It protected the fifth and sent the bill to the third, and the shimmer you hear on the tempered chord is almost entirely that one note.
The damage is not shared out evenly
This is the part worth sitting with. The tempered fifth misses its pure 3:2 by two cents, which is comfortably below what anyone can detect. The tempered major third misses 5:4 by nearly fourteen, which is not subtle once you know what to listen for. Play the chord above and switch between tunings: the shimmer that appears is almost entirely that third.
That asymmetry is the point of the maths rather than an accident of it. Twelve equal steps happen to approximate the fifth extremely well and the third only tolerably, and the whole system was adopted because the fifth is the interval that matters most structurally. We did not solve the problem. We chose where to put it.
Why twelve
Nothing so far explains the number. Twelve is not sacred; it is the answer to an approximation problem, and you can watch it fall out.
To divide the octave into n equal steps and still have a good fifth, you need some whole number of steps to land near a 3:2. In other words you want a fraction k/n close to log2(3/2) = 0.5849625. The best fractions for a given size of denominator are its continued-fraction convergents:
1/2 is 0.5, badly off. 3/5 is 0.6, closer. 7/12 is 0.58333, off by only 0.0016. 24/41 and 31/53 are better still.
Twelve is where the accuracy becomes good enough that further improvement is not worth the extra notes. Nineteen, 31 and 53 tone equal temperaments all exist, all have better thirds than ours, and all have been built as real instruments. They lost because twelve was already good enough and had far fewer keys.
Which is the honest summary of the whole thing: twelve is an engineering optimum for a particular set of priorities, not a law of nature. Change the priorities, or change the timbres as lesson 3 did, and a different number wins. Gamelan tunings divide the octave into five or seven uneven steps and are perfectly coherent, because the instruments have inharmonic partials and the consonances sit somewhere else entirely.