Math Detours · Grade 6 · Connections · 1

Why there are twelve notes

Harmony is arithmetic. An octave is 2:1, a fifth is 3:2, and the reason a piano has twelve notes is that 1.5 to the twelfth power is almost, but not quite, 128.

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Tonight's idea

A musical note is a vibration. Pluck a guitar string and it wobbles back and forth hundreds of times a second; the number of wobbles per second is the frequency, measured in hertz (Hz). The A that orchestras tune to is 440 Hz. Higher frequency, higher note.

Two notes sound good together when their frequencies are in a simple ratio. The simplest ratio is 2:1. Play 440 Hz and 880 Hz together and they blend so completely that we give them the same name: both are A. That gap is called an octave. Halve the length of a string and it plays the octave above.

The next simplest ratio is 3:2. Play 440 Hz with \(440 \times \tfrac32 = 660\) Hz and you get the bright, open sound at the start of "Twinkle, Twinkle" (each "twinkle" repeats one note; the fifth is the leap from the first "twinkle" to the second). Musicians call it a fifth, because A up to E spans five letter names: A B C D E. The ratio 4:3 gives a fourth (A up to D, four letters), the opening of "Here Comes the Bride". The ratio 5:4 gives a major third (A up to C♯, three letters).

A B 2 C 3 D 5 E 7 F G A 1 4 6 C♯
One octave, A to A: 12 steps, one per key. A to E is 7 steps (a fifth), A to D is 5 (a fourth). Shaded: the chord A, C♯, E.

Why twelve notes in an octave? Try this. Start at some note and go up a fifth, multiplying the frequency by \(\tfrac32\). Do it again, and again. After twelve fifths you have multiplied by \(\left(\tfrac32\right)^{12}\), which is about 129.7. Seven octaves is \(2^7 = 128\), almost the same. So twelve fifths land almost exactly where seven octaves do, and the circle of fifths closes. That near-miss is why there are twelve notes.

The gap, a ratio of \(129.7 / 128 \approx 1.0136\), is called the Pythagorean comma.

The modern fix, equal temperament, spreads the error evenly: every one of the twelve steps in an octave multiplies the frequency by exactly the same amount, \(2^{1/12} \approx 1.0595\). Twelve of those multiply to exactly 2. The price is that a piano's fifth is \(2^{7/12} \approx 1.4983\), not \(1.5\). Every note on every piano is very slightly out of tune, on purpose, so that every key sounds equally good.

Watch one: building a chord from ratios

Start from A = 440 Hz. A major chord uses ratios 4 : 5 : 6 for its three notes. So the notes are \(440\), \(440 \times \tfrac54 = 550\) and \(440 \times \tfrac64 = 660\) Hz. Those are A, C♯ and E: an A major chord.

Now compare with the piano's equal-tempered versions. C♯ is 4 steps above A: \(440 \times 2^{4/12} \approx 554.4\) Hz. E is 7 steps above: \(440 \times 2^{7/12} \approx 659.3\) Hz. Close, but not identical. A choir singing without a piano will naturally drift to the pure ratios, 550 and 660, because they sound smoother. Hear the pure ratios in the tones demo.

Your turn

  1. Middle C is about 261.6 Hz. What is the frequency of the C one octave above? Two octaves above? One octave below?
  2. A fifth above 440 Hz is 660 Hz. A fourth above 660 Hz uses the ratio 4:3. What frequency is that, and what do you notice?
  3. Compute \(\left(\tfrac32\right)^{12}\) step by step (multiply by 1.5 twelve times; keep all the decimals as you go, or the rounding drifts), and compare with \(2^7\). How far off is the circle of fifths, as a percentage?
  4. A guitar string is 65 cm long. Measuring from the nut (the end near the tuning pegs), how far along do you press to play the octave? The fifth? Pressing leaves the part between your finger and the bridge vibrating, and half the length gives double the frequency.

Stretch: why a fifth and a fourth make an octave

Multiply the fifth's ratio \(\tfrac32\) by the fourth's \(\tfrac43\). What do you get, and what does it mean musically? Now look at the keyboard picture: a fifth is 7 steps and a fourth is 5. Why does that fit?

Then compute the piano's versions: \(2^{7/12}\) and \(2^{5/12}\), to four decimal places (a calculator is allowed). Multiply them. Why is the answer exactly 2 this time, with no comma left over?

Answers

  1. 523.2 Hz, 1046.4 Hz, 130.8 Hz.
  2. \(660 \times \tfrac43 = 880\) Hz, which is exactly an octave above 440. A fifth and then a fourth is an octave.
  3. 1.5, 2.25, 3.375, 5.0625, 7.59375, 11.390625, 17.0859375, 25.62890625, 38.443359375, 57.6650390625, 86.49755859375, 129.746337890625. (If you round to two places at each step you end near 130.1, which is why the instruction says not to.) Compared with 128 it overshoots by about 1.36 percent.
  4. Octave: press halfway, 32.5 cm from the nut, leaving half the string. Fifth: leave two thirds of the string vibrating, \(65 \times \tfrac23 \approx 43.3\) cm, so press one third of the way along, 21.7 cm from the nut.

Stretch. \(\tfrac32 \times \tfrac43 = 2\), an octave. A fifth up then a fourth up lands on the same note name an octave higher. On a piano 7 + 5 = 12 steps, the whole octave. \(2^{7/12} \approx 1.4983\) and \(2^{5/12} \approx 1.3348\); their product is \(2^{12/12} = 2\) exactly, because the exponents add to 1. Equal temperament has no comma because it is built from powers of a single number, and powers always multiply cleanly.

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