Musical Intervals: Frequency Ratios, Cents, and Why Tuning Is a Compromise
Every musical note is a frequency — a number of cycles per second measured in hertz. The relationship between two notes is a ratio. The octave is exactly 2:1. The perfect fifth is close to 3:2. And there lies the problem: these pure ratios, stacked repeatedly, never quite close into a complete scale. The solution used by virtually all Western music for the last 300 years is a deliberate compromise called equal temperament.
The Two Systems: Just Intonation vs 12-TET
Just intonation uses frequency ratios expressed as small integers — 3:2 for the perfect fifth, 5:4 for the major third. These ratios produce intervals that are acoustically pure: the overtone series of the two notes aligns exactly, producing no audible beating.
12-tone equal temperament (12-TET) divides the octave into 12 equal semitones, each with a ratio of 2^(1/12) ≈ 1.05946. The only interval that remains acoustically pure is the octave (exactly 2:1). Every other interval is slightly "off" from the just ratio — but the error is distributed evenly, so every key sounds equally in tune.
The Unit of Comparison: Cents
A cent is 1/100th of an equal-tempered semitone, or 1/1200th of an octave. One semitone = 100 cents. One octave = 1200 cents.
The formula: cents = 1200 × log₂(f₂/f₁)
Human pitch perception: most trained musicians can detect a difference of about 5–10 cents. Untrained listeners typically need 15–25 cents before they hear something "wrong."
Interval Reference Table
| Interval | Semitones | Just ratio | Cents (just) | Cents (ET) | Δ cents |
|---|---|---|---|---|---|
| Unison | 0 | 1:1 | 0 | 0 | 0 |
| Minor 2nd | 1 | 16:15 | 111.7 | 100 | +11.7 |
| Major 2nd | 2 | 9:8 | 203.9 | 200 | +3.9 |
| Minor 3rd | 3 | 6:5 | 315.6 | 300 | +15.6 |
| Major 3rd | 4 | 5:4 | 386.3 | 400 | −13.7 |
| Perfect 4th | 5 | 4:3 | 498.0 | 500 | −2.0 |
| Tritone | 6 | 45:32 | 590.2 | 600 | −9.8 |
| Perfect 5th | 7 | 3:2 | 702.0 | 700 | +2.0 |
| Major 6th | 9 | 5:3 | 884.4 | 900 | −15.6 |
| Major 7th | 11 | 15:8 | 1088.3 | 1100 | −11.7 |
| Octave | 12 | 2:1 | 1200.0 | 1200 | 0 |
The Pythagorean Comma — Why Perfect Tuning Is Impossible
Tune 12 perfect fifths up from C: C → G → D → A → E → B → F# → C# → G# → D# → A# → F → C. Each fifth multiplies frequency by 3/2. After 12 fifths: (3/2)^12 = 129.746…
Seven octaves up from the same C: 2^7 = 128.
These should arrive at the same note — but they don't. The gap is (3/2)^12 / 2^7 ≈ 1.01364, or about 23.46 cents. This is the Pythagorean comma. Equal temperament "spreads" this comma across all 12 fifths, making each one 2 cents flat. The result: every fifth is slightly impure, but none are catastrophically wrong.
The Syntonic Comma — Why Major Thirds Sound Harsh on Old Organs
The equal-tempered major third is 400 cents. The just major third is 386.3 cents — a difference of 13.7 cents called the syntonic comma. That's well within hearing range, and it's why equal-tempered major thirds beat slightly while just-intoned ones don't. On a harpsichord or church organ, this difference is very audible. On electric guitar with distortion, it's inaudible — distortion generates so many overtones that precise tuning stops mattering.
The 440 Hz Standard
A440 — concert pitch A above middle C at 440 Hz — was standardized by the ISO in 1955. Before standardization, "A" ranged from 400 Hz to 480 Hz depending on country and era. Baroque instruments were often pitched at A415 (about a semitone lower), which is why some early music groups still tune to 415. Some orchestras use A442 or A443 for a slightly brighter sound.
Middle C (C4) at A440 is approximately 261.63 Hz. The lowest note on a standard piano (A0) is 27.5 Hz; the highest (C8) is 4186 Hz.
Using the Calculator
Enter any base frequency (or select a note from the dropdown), pick an interval, and optionally shift by octaves. Results show both the 12-TET frequency and the just intonation frequency, the ratio between them, and the deviation in cents. The interval table at the bottom of the page gives a full reference for all 15 standard intervals.
Frequently Asked Questions
How do I know if I can hear the difference between just intonation and equal temperament?
Most trained musicians can detect tuning differences of 5–10 cents, while untrained listeners typically need 15–25 cents to notice something sounds off. Use the cents calculator to find the difference between two frequency ratios — if it's under 10 cents, most ears won't catch it without careful listening. For example, the perfect fifth in equal temperament is only 2 cents flat compared to the just ratio of 3:2, which is why it sounds acceptably pure to most people.
Why does my synthesizer sound better in some keys than others if I'm using equal temperament?
Equal temperament distributes tuning errors evenly across all 12 notes, so theoretically every key should sound equally "in tune." However, you may be noticing that major thirds in some keys sound harsher because they're 13.7 cents sharp compared to the pure 5:4 just ratio. If you're composing in a key that features major chords heavily, you could use the frequency ratio calculator to check which intervals dominate your harmonic progression.
What's the Pythagorean comma and why should I care?
The Pythagorean comma is the 23.46-cent gap that appears when you stack 12 perfect fifths (3:2 ratio) — they should return to the same note seven octaves higher, but they overshoot by about 1.36%. Equal temperament solves this by making each fifth 2 cents flat, so the error doesn't accumulate into an audible discrepancy. Understanding this explains why Western music had to adopt equal temperament instead of just intonation: perfect ratios simply don't circle back.
How do I calculate cents from a frequency ratio?
Use the formula: cents = 1200 × log₂(f₂/f₁), where f₂ is the higher frequency and f₁ is the lower frequency. For example, a perfect fifth with the ratio 3:2 gives you 1200 × log₂(1.5) ≈ 702 cents, which you can compare to the equal-temperament value of 700 cents. The calculator handles this conversion automatically — just enter two frequencies or ratios and it will show you the difference in cents.
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