Stacking fifths

The most consonant interval that gets you somewhere new

The octave is the smoothest interval there is, but it is useless for building a scale: double a frequency and you arrive at something so similar we give it the same letter. To get new notes you want the next best thing, and lesson 3 was unambiguous about what that is. The fifth, 3:2, carves the deepest valley after the octave.

So: start at a note, go up a fifth. Go up another fifth. Whenever you climb past the top of your octave, halve the frequency to drop back inside it. Every note you generate is related to the last by a pure, beatless 3:2, which by lesson 3 is about as good as intervals get.

Keep going and something encouraging happens. After twelve fifths you have twelve distinct notes in the octave, and the twelfth lands almost exactly back where you started. Seven octaves up is 2^7 = 128. Twelve fifths is (3/2)^12 = 129.746. Almost.

Almost is not a number you can build an instrument on

The gap is 23.46 cents, about a quarter of a semitone. It has a name, the Pythagorean comma, and the demo above lets you hear it: played against the note it was supposed to be, it beats about three and a half times a second. This is not a rounding error that a careful instrument maker could sand away. It is arithmetic. Powers of three and powers of two never coincide, because 2 and 3 are both prime, so 3^a = 2^b has no solution in whole numbers other than the trivial one.

You can shove the problem around but you cannot remove it. Tune eleven pure fifths and the twelfth absorbs the whole comma, becoming so sour it was historically called the wolf: an interval you simply avoided, which meant avoiding whole keys. Real instruments were tuned this way for centuries, and the reason old music stays in a narrow range of keys is that the other ones were unusable.

The real constraint, stated properly

Here is the collision underneath all of it. You want two things:

Pure intervals. Notes related by small whole-number ratios, so their partials coincide and nothing beats.

Transposition. The ability to move a melody to a different starting note and have it still work, which requires the steps between notes to be the same wherever you start.

These are mutually exclusive. Not difficult to reconcile, not awaiting a cleverer tuner: mathematically incompatible. Any tuning built from pure ratios has intervals of unequal size, so moving a melody changes the intervals inside it. Any tuning with perfectly even steps cannot be built from whole-number ratios, because equal division of an octave into any number of parts requires irrational multipliers.

Worth noting where this stops being physics. That the comma exists is arithmetic and not up for debate. What anybody chose to do about it is culture, and different cultures did different things: various European temperaments each distributed the error differently, and plenty of musical traditions never accepted the trade at all and kept their pure intervals along with the restriction on modulation. The next lesson covers the compromise that won in the West, not the only one available.