Intervalo Musical (semitons)

Musical intervals as frequency ratios: rule and example

At its core, a musical interval is just a ratio between two frequencies. The octave sits at 2:1, the perfect fifth at 3:2 (the Pythagorean ratio), the perfect fourth at 4:3. The major third lands on 5:4 if you tune in just intonation, or on 2^(4/12) ≈ 1.260 under equal temperament. To get the frequency of a note n semitones above some base, you compute f₀ · 2^(n/12). Run that with A4 = 440 Hz and you get A5 = 880 Hz, while C8, the highest piano key, comes out around 4186 Hz.

Context and applications

Equal temperament cuts the octave into 12 identical semitones, each one a constant ratio of 2^(1/12) ≈ 1.0595. That's the bargain that lets a piano modulate freely into any key, paid for with thirds and fifths that come out a touch impure. The reference pitch A4 = 440 Hz was nailed down by ISO 16 in 1955. The same math shows up everywhere musicians work with pitch: tuning instruments, writing music, designing synth oscillators like Moog or Serum, exploring microtonal theory, and building audio plugins that shift pitch by semitones.

FAQ

Why does the just-intonation fifth (3:2) sound "better" than the tempered one? The 3:2 ratio gives you simple, slow beat patterns. The tempered fifth sits about 2 cents flat, and trained ears can hear it.

What is a "cent"? It's a hundredth of a tempered semitone, or 2^(1/1200), and it's the unit people reach for when they want to describe how far off a note is.

How do I calculate any note's frequency? Start from A4 = 440 Hz and multiply by 2^(n/12). Here n counts the semitones between your note and A4: positive if you're going up, negative if you're going down.

Why is A4 sometimes 432 Hz? A handful of musicians favor 432 Hz on esoteric grounds. In practice, modern orchestras stick to the ISO standard of 440 Hz, or bump it to 442 in parts of Europe.