explainer
Why Every Octave Doubles: The Math Behind Musical Notes
By the LazyTools team · Published 2026-07-12 · Updated 2026-07-12 · 3 min read
Every musical note is just a frequency — a number of vibrations per second — and the whole system runs on two simple rules. Rule one: an octave doubles the frequency. Rule two: that octave is split into twelve equal steps, each multiplying the frequency by the same small factor. From those two ideas you can calculate the pitch of every note on every instrument.
Pitch is a frequency, and octaves double it
A musical note is a sound wave vibrating at a steady rate, measured in hertz (Hz). The magic of the octave is that two notes an octave apart — say the low and high A of a scale — sound like “the same note,” and that perception always maps to an exact doubling of frequency. A4 is 440 Hz; the A an octave up is 880 Hz; an octave down is 220 Hz. Our hearing is logarithmic, so equal musical steps are equal ratios, not equal differences.
Twelve equal steps: the twelfth root of two
To divide that doubling into the twelve semitones of Western music equally, each step must multiply the frequency by the same factor — and twelve of those factors must multiply to 2. That factor is the twelfth root of two, 2^(1/12) ≈ 1.05946. Go up one semitone, multiply by 1.0595; go up twelve, and you’ve multiplied by 2 — a full octave. This is equal temperament, and it gives the master formula for any note:
f = 440 × 2^((n − 69) / 12)
where n is the note’s MIDI number and 69 is A4 (440 Hz). Middle C is C4, MIDI 60, which comes out to 440 × 2^((60−69)/12) ≈ 261.63 Hz. The note frequency calculator runs this for any note — and lets you change the 440 reference for A=432 or historical pitches.
Why A = 440 Hz?
There’s nothing physically special about 440 Hz — it’s a standard, agreed so that instruments built and played around the world are in tune with each other. It was fixed internationally as ISO 16. Before standardisation, “A” varied widely between cities and eras, which made travelling musicians’ lives difficult. Some orchestras still tune slightly higher (442–443 Hz) for brilliance, and A = 432 Hz has a following as a warmer alternative — the calculator retunes every note if you change the reference.
Cents, and the piano’s tiny white lie
To measure tuning finer than a semitone, musicians use cents: 100 cents to a semitone, 1,200 to an octave. They’re how you say a note is “5 cents flat” — just barely noticeable — versus “20 cents flat,” which sounds wrong.
Cents also reveal equal temperament’s clever compromise. A pure perfect fifth is a simple 3:2 frequency ratio (exactly 1.5) — the interval that sounds most consonant. But equal temperament makes the fifth 2^(7/12) ≈ 1.4983, about 2 cents flat of pure. Spread across all twelve keys, those tiny errors are the price of being able to play in every key on one fixed set of strings or frets. Tune a piano to make one key perfect and the distant keys would sound painfully out; equal temperament makes them all equally, imperceptibly imperfect. The interval calculator shows the exact ratio and cents for any two notes.
All figures use equal temperament with A4 = 440 Hz (ISO 16) and middle C = C4 = MIDI 60 (scientific pitch notation) — the standard conventions. These are exact mathematical relationships, computed in your browser.
Frequently asked questions
Why is A tuned to 440 Hz?
440 Hz for the A above middle C (A4) is the modern international standard, ISO 16, adopted so instruments everywhere agree on pitch. It's a convention, not a law of physics — some orchestras use 442 or 443 Hz, and 432 Hz is a popular alternative tuning.
Why does an octave double the frequency?
Because our ears perceive pitch logarithmically: notes an octave apart sound like 'the same note, higher,' and that always corresponds to a 2:1 frequency ratio. A4 is 440 Hz, so A5 is 880 Hz and A3 is 220 Hz.
What is equal temperament?
A tuning system that splits the octave into twelve equal steps, each multiplying the frequency by 2^(1/12) ≈ 1.0595. It lets instruments play in tune in every key, at the cost of making most intervals very slightly off the pure whole-number ratios.
How do I calculate a note's frequency?
Use f = 440 × 2^((n − 69)/12), where n is the note's MIDI number (A4 = 69). Middle C (C4, MIDI 60) works out to about 261.63 Hz. A note frequency calculator does it instantly for any note.
What is a cent in music?
One hundredth of a semitone, so 1,200 cents to an octave. Cents measure tiny tuning differences — a note 5 cents flat is only just detectable, while 20 cents is clearly out of tune.
Why aren't a piano's intervals perfectly in tune?
Equal temperament is a compromise. A pure perfect fifth is a 3:2 frequency ratio (1.5), but equal temperament makes it 2^(7/12) ≈ 1.4983 — about 2 cents flat. That tiny error is what lets a piano sound acceptable in all 12 keys instead of perfect in one.
What is A = 432 Hz tuning?
An alternative reference that tunes A4 to 432 Hz instead of 440, shifting every note down slightly. Some listeners prefer it, though claims of special properties aren't scientifically supported. A note frequency calculator can retune every note to a 432 reference.