# teaching machines

## Why 12?

April 14, 2019 by . Filed under deltaphone, public.

We looked previously at how an octave—or doubling, as we called it—is partitioned non-linearly into intermediate tones. But we didn’t ultimately decide how many intermediate tones there should be. The pioneers of Western music converged on a 12-partition. But why not 8? Or 10? Or 11? Or 13?

Ultimately, we want our instruments to sound pleasing to our ears. Octaves are the prime way to achieve this consonance. The doublings in our instruments’ wires and passageways always sound good together. But we want other combinations to sound good together too.

What does it mean for two notes to sound good together? Physically, it means that the notes’ pressure waves oscillate in such a way that they coincide at regular intervals. In a doubling, for example, two oscillations of the faster wave coincide with one oscillation of the slower wave. The ratio of the frequencies is 2:1.

Of all the possible partitioning schemes, a 12-partition produces the largest proportion of simple ratios. See how nicely the waves for notes 0 and 7 coincide?

In the time that it takes note 0 in red to oscillate twice, note 7 in blue oscillates three times. Our ears will like that!

Meanwhile, as note 0 in red oscillates three times, note 5 in blue oscillates four times:

A whopping six of the intermediate tones can be expressed with simple ratios:

Note Ratio to Root Approximate Ratio
0 $2^\frac{0}{12}$ $1$
1 $2^\frac{1}{12}$ $-$
2 $2^\frac{2}{12}$ $-$
3 $2^\frac{3}{12}$ $\frac{6}{5}$
4 $2^\frac{4}{12}$ $\frac{5}{4}$
5 $2^\frac{5}{12}$ $\frac{4}{3}$
6 $2^\frac{6}{12}$ $-$
7 $2^\frac{7}{12}$ $\frac{3}{2}$
8 $2^\frac{8}{12}$ $\frac{8}{5}$
9 $2^\frac{9}{12}$ $\frac{5}{3}$
10 $2^\frac{10}{12}$ $-$
11 $2^\frac{11}{12}$ $-$
12 $2^\frac{12}{12}$ $2$

The ratios are not exact, it should be noted. For example, note 7 is $2^\frac{7}{12} \approx 1.498307 \neq \frac{3}{2}$. Our ears probably won’t notice this slight deviation.

Given its high probability of producing consonant notes, the 12-partition came to dominate our musical systems. The sequence of all 12 steps is called the chromatic scale, which can be programmed in Deltaphone:

It’s rare that we find all twelve notes in a piece of music. Instead we restrict ourselves to subsets of these notes to achieve an even greater likelihood of consonance.