Where do the frequencies of musical notes come from? Why are piano keys and guitar frets placed where they are?

The answer is a mathematical system called 12-Tone Equal Temperament.

To understand it, we start with three fundamental principles:

1. The Octave: When you double a sound's frequency ($2 \times f$), the human ear hears the two tones as the exact same note class, just higher in register. For example, A3 is $220\text{ Hz}$, A4 is $440\text{ Hz}$, and A5 is $880\text{ Hz}$.
2. 12 Semitones: In Western music, the span of one octave is split into $12$ steps called semitones (or half-steps): $$\text{C, C}\sharp\text{, D, D}\sharp\text{, E, F, F}\sharp\text{, G, G}\sharp\text{, A, A}\sharp\text{, B, C}$$
3. Equal Temperament: To allow musicians to play in any musical key without sounding out of tune, the frequency ratio $r$ between every pair of adjacent semitones must be identical. Since multiplying by $r$ twelve times in a row must double the frequency ($r^{12} = 2$), the ratio must be the twelfth root of two: $$r = 2^{1/12} = \sqrt[12]{2} \approx 1.059463094$$

Taking international standard concert pitch A4 ($440\text{ Hz}$) as our benchmark, the frequency of any note $n$ semitones away is given by the exponential formula: $$f(n) = 440 \cdot 2^{n/12}$$

• $n = 0$: $440 \cdot 2^0 = 440\text{ Hz}$ (Concert A4)
• $n = 1$: $440 \cdot 2^{1/12} \approx 466.16\text{ Hz}$ (A#4 / Bb4)
• $n = 12$: $440 \cdot 2^{12/12} = 440 \cdot 2 = 880\text{ Hz}$ (A5, one octave up)
• $n = -9$: $440 \cdot 2^{-9/12} \approx 261.63\text{ Hz}$ (Middle C, C4)
• $n = -12$: $440 \cdot 2^{-12/12} = 440 / 2 = 220\text{ Hz}$ (A3, one octave down)

Using Lua's exponentiation operator ^ along with sound(freq, duration) and play(), you can program any musical scale or melody purely from mathematical laws!

Move the mouse over a dotted box for more information.

• The Major Scale ("Do Re Mi Fa Sol La Ti Do"): A major scale is created by selecting 7 notes out of the 12 chromatic semitones using the interval pattern: Whole, Whole, Half, Whole, Whole, Whole, Half. That corresponds to semitone offsets: $$[0, 2, 4, 5, 7, 9, 11, 12]$$
• Why Guitar Frets Get Closer: On a stringed instrument, frequency is inversely proportional to vibrating length ($L \propto 1/f$). Each fret shortens the string by a factor of $2^{-1/12} \approx 0.9439$. Because this reduction is geometric, the frets get progressively closer together as you move down the neck!