Why Does a Scale Have Twelve Notes?

A string’s pitch is how many times a second it vibrates, its frequency, measured in hertz (Hz). For the same string at the same tension, the frequency goes as one over the length: half the length vibrates twice as fast. So the octave, half the string, is a frequency ratio of 2 : 1. Two thirds of the string gives Pa, 3 : 2 above Sa, the step called a perfect fifth. Three quarters gives Ma, 4 : 3 above Sa, a perfect fourth.

What makes these simple ratios sound sweet? A plucked string or a harmonium reed doesn’t sound only its pitch. It also sounds, more quietly, 2, 3, 4, … times that frequency, its harmonics. Sa at 200 Hz sounds 200, 400, 600, 800 Hz and so on. When two notes are played together, their harmonics can meet or nearly meet. Where two nearly meet, the sound wobbles in loudness, a beat, as many times a second as their frequencies differ. Hermann von Helmholtz argued in 1863 that this is why some pairs sound smooth and others rough.

Try these with the widget.

• a.Which ratio makes the most harmonics meet? Which makes none meet?
• b.For 3 : 2, which harmonic of Sa meets the 2nd harmonic of Pa? Why must it?
• c.Set 3 : 2 and detune Pa by +2 Hz. How fast does the fastest pair beat, and why that number?

Go up a fifth from Sa (× 3/2), then up a fourth from Pa (× 4/3). You land on the next Sa, because

$$\tfrac{3}{2} \times \tfrac{4}{3} = 2.$$

So stacking intervals multiplies their ratios. Yet to the ear, a fifth plus a fourth adds up to an octave, step on step. Our ears hear pitch roughly by ratios: 110, 220, 440 and 880 Hz sound equally spaced, each an octave above the last, though the gaps in hertz are 110, 220 and 440.

There is a tool that turns multiplying into adding: the logarithm. Measure an interval by how many octaves it is, that is, how many times you must multiply by 2 to get its ratio:

$$\text{size in octaves} = \log_2(\text{ratio}).$$

An octave is $\log_2 2 = 1$; two octaves, a ratio of 4, are $\log_2 4 = 2$. And because $\log_2(ab) = \log_2 a + \log_2 b$, sizes add when ratios multiply. Musicians split the octave into 1,200 small parts called cents:

$$\text{cents} = 1200 \times \log_2(\text{ratio}).$$

Most calculators have no $\log_2$ key, but any logarithm will do: $\log_2 x = \dfrac{\log x}{\log 2}$, with $\log$ to base 10 (the change-of-base rule).

On the widget’s circle, a note k cents above Sa sits $\tfrac{k}{1200}$ of the way round. That is why every fifth turns the circle by the same angle, 701.96 cents’ worth, though in hertz each fifth is a bigger jump than the last.

Climbing n fifths multiplies Sa’s frequency by $\left(\tfrac{3}{2}\right)^n$. Landing exactly on a Sa, m octaves up, would need

$$\left(\tfrac{3}{2}\right)^n = 2^m, \quad\text{that is,}\quad 3^n = 2^{m+n}.$$

Look at the two sides. $3^n$ is 3 × 3 × … × 3, a product of odd numbers, so it is odd. $2^{m+n}$, for n ≥ 1, is a product of 2s, so it is even. An odd number can’t equal an even one. No number of fifths ever comes back exactly to Sa, and no computer search could ever find one: the proof covers every n at once.

The same argument proves something about logarithms. If $\log_2 1.5$ were a fraction $\tfrac{m}{n}$, then $1.5^n = 2^m$, which we have just seen is impossible. So $\log_2 1.5 = 0.58496\ldots$ is an irrational number: its decimals never end and never repeat.

The widget showed how near twelve fifths come. They climb a ratio of

$$\left(\tfrac{3}{2}\right)^{12} = \frac{531{,}441}{4{,}096} = 129.746\ldots$$

while seven octaves are $2^7 = 128$. The leftover, $\tfrac{531{,}441}{524{,}288} = 1.01364\ldots$, about 23.46 cents, is the Pythagorean comma: the amount by which twelve pure fifths overshoot seven octaves. It is about a quarter of the smallest step on a harmonium, and it never goes away.

The circle never closes, but it comes close at some steps. From five fifths on, here are the closest returns, each closer than every one before it:

Fifths	Octaves	Fifths overshoot (+) or fall short (−), cents
5	3	−90.2
12	7	+23.5
41	24	−19.8
53	31	+3.6
306	179	−1.8
665	389	+0.08

Near misses make scales. Stop the climb after four fifths, Sa, Pa, Re, Dha and Ga, and put them in order: Sa Re Ga Pa Dha, the five notes of Raag Bhupali, the same five-note scale (the major pentatonic) heard in folk tunes across East Asia and far beyond. The next fifth, Ni, would land only 90 cents below Sa, crowding it; stopped at five notes, the scale has no gap smaller than a whole step. Twelve fifths give the twelve notes of the harmonium. And 53 fifths come within 3.6 cents of 31 octaves, a near miss noticed in China over two thousand years ago (the story below).

The record-breaking near misses can be found without trying every number. They come from writing $\log_2 1.5$ as a continued fraction, a method for the best fractions near any number; its best fractions here are $\tfrac{3}{5}, \tfrac{7}{12}, \tfrac{24}{41}, \tfrac{31}{53}, \ldots$, the octaves over the fifths in the table.

If twelve fifths overshoot seven octaves by a comma, share the comma out: make each fifth smaller by a twelfth of it, 23.46 ÷ 12 = 1.96 cents. Then twelve fifths close the circle exactly, and the twelve notes are evenly spaced, each 100 cents above the one before. This is twelve-tone equal temperament, the usual tuning of pianos, guitars and harmoniums today. Each small step, a semitone, is the same ratio:

$$r^{12} = 2, \quad\text{so}\quad r = \sqrt[12]{2} = 1.0594630943\ldots$$

The twelve notes of an octave are a geometric sequence with common ratio $\sqrt[12]{2}$, and the eleven between one Sa and the next are the eleven geometric means between, say, 440 and 880 Hz (Grade 10’s Sequence and Series).

And the twelfth root needs no special key. Twelve is 2 × 2 × 3, so

$$\sqrt[12]{2} = \sqrt{\sqrt{\sqrt[3]{2}}},$$

a cube root and two square roots (in any order), the roots of Grade 9’s Indices. The page of Zhu Zaiyu’s book in the story below shows the square roots, worked by hand over 400 years ago.

Why twelve and not some other number? Cut the octave into N equal steps and ask how near some step lands to the two sweetest intervals after the octave, the fifth (3 : 2) and the major third (5 : 4).

Equal temperament isn’t free. Every interval except the octave is slightly out of tune, and you can hear it as beats. Take A at 440 Hz and the tempered E at 659.26 Hz. A’s 3rd harmonic is 1,320 Hz; E’s 2nd is 1,318.51 Hz. They nearly meet, 1.49 Hz apart, so the fifth beats about one and a half times a second: a slow, gentle wobble. The major third is worse. A at 440 Hz and the tempered C♯ at 554.37 Hz: A’s 5th harmonic is 2,200 Hz, C♯’s 4th is 2,217.46 Hz, and they beat about 17 times a second, a buzz you can hear on any piano.

This is where the rule breaks, twice. First, much of the world’s music is not built on twelve equal steps. The Natyashastra, the oldest surviving Indian treatise on the performing arts, music among them (its first complete version from somewhere between 200 BCE and 200 CE), divides the octave into 22 shrutis, and a singer in a raag places a note by ear, a little higher or lower than the same note in a neighboring raag, and slides between notes. A harmonium can’t: its pitches are fixed, and standard Indian models are equal-tempered. All India Radio kept the harmonium off its broadcasts from 1940 to 1971, for reasons that included exactly this.

Second, real strings don’t quite obey the whole-number harmonics. A piano’s thick steel strings are stiff, and their high harmonics run slightly sharp. So piano tuners stretch the octaves: high notes are tuned a little higher, low notes a little lower, by a few tens of cents at the ends of the keyboard, so that the octaves sound pure with the harmonics the strings really make. The mathematics of perfect ratios is exact; the strings are not.

• Guitar frets. On a guitar, the ratio of the distances of two neighboring frets to the bridge is $\sqrt[12]{2}$, so the nth fret is a distance $L\left(1 - 2^{-n/12}\right)$ from the nut on a string of length L, and the 12th fret is halfway. That is why the frets get closer together towards the body: equal ratios, not equal lengths.
• Electronic instruments. The MIDI standard, which keyboards, phones and music software use to name notes, numbers them so that note 69 is A at 440 Hz; note N has frequency $f = 440 \times 2^{(N - 69)/12}$, and a frequency’s note number is $69 + 12 \log_2\!\left(\tfrac{f}{440}\right)$, the same logarithm a tuner works out.
• The standard A. A = 440 Hz was agreed at a meeting in London in May 1939 and became the international standard ISO 16 in 1975.

• Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7? Another logarithmic scale, where one step multiplies the energy by about 32.
• How Did a Book of Numbers Make Multiplying Easy? Logarithms as a calculating tool, from tables to the slide rule.
• Can You Prove √2 Is Not a Fraction? The same odd-against-even argument as this page’s proof.
• Computing, Going Further: How Does a Computer Make a Sitar-Like Note Out of Noise? Make plucked notes and Raag Bhupali’s scale in Python; How Does a Tuner Hear Which Note You Play? (Coming soon.); Can a Computer Compose a Tune That Follows a Raag? (Coming soon.)
• Grade 10, Sequence and Series: geometric means, the twelve notes of an octave among them.