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Frequency Ratio Calculator

Enter two frequencies and get the simplified ratio, the cents value, the nearest just-intonation interval, and the nearest 12-TET interval — with cents deviation for both, so you can see which tuning system your interval is closer to.

Input

Frequency 1 (f₁)

Frequency 2 (f₂)

Common intervals

Result

Simplified ratio

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Cents (full)

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Decimal ratio

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Reduced to one octave

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Nearest just interval

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Nearest 12-TET interval

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Which is closer?

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Formulas

Ratio: r = f₂ / f₁

Cents: c = 1200 · log₂(r)

12-TET semitone: 2^1/12 ≈ 1.05946 = 100 cents

Just intonation: small integer ratios (3:2, 4:3, 5:4, …)

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Cents axis (0–1200, reduced to one octave)

12-TET semitones (below baseline) Just intervals (above baseline) Your ratio

About Frequency Ratios

A frequency ratio expresses one tone in terms of another. Musical intervals are perceived logarithmically: the brain hears a 440 → 880 Hz step (ratio 2:1) and a 220 → 440 Hz step as the "same" interval — both octaves. The two tuning systems below organize music's vocabulary of ratios in different ways.

Just intonation — small integer ratios

Just intonation (JI) defines intervals as small whole-number ratios: 2:1 octave, 3:2 fifth, 4:3 fourth, 5:4 major third, 6:5 minor third, etc. These ratios maximize "consonance" — the overtones of one note line up perfectly with those of the other, producing a beat-free sound. Pythagoras built tuning systems from just 2:1 and 3:2 (Pythagorean tuning); 5-limit just intonation adds 5:1 (the harmonic). Drawback: only one key sounds perfect; transposing requires re-tuning.

12-tone equal temperament — geometric spacing

12-TET divides the octave into 12 equal logarithmic steps. Each semitone is 2^1/12 ≈ 1.05946 (exactly 100 cents). All keys sound equally (slightly) out of tune. Drawback: the major third in 12-TET (400 cents) is 14 cents sharper than the just 5:4 (386 cents), giving piano thirds their characteristic beating. Benefit: any key sounds the same, transposition is free.

Cents — the universal interval unit

One cent is 1/100 of a 12-TET semitone, or 1/1200 of an octave. Cents convert any ratio to a linear measurement: cents = 1200 · log₂(ratio). Most listeners can detect pitch differences down to ~5 cents in isolation, ~10-20 cents in musical context. Beat rates also help: a 7 cent deviation at 440 Hz gives ~1.8 beats per second.

Continued-fraction approximation

This tool's "Best integer ratio" finds the simplest p:q with q ≤ 1000 closest to your decimal ratio, using continued fractions. For an exact 3:2 input (660/440 = 1.5), it returns 3:2 exactly. For a 12-TET fifth (440 × 2^7/12 ≈ 659.255 Hz), it returns 1.4983:1 — close to but not 3:2, illustrating the tuning gap.

Frequently Asked Questions

Why does the 12-TET major third sound "out of tune" compared to JI?

Because it IS — by about 14 cents. The just major third is 5:4 = 1.25 (386 cents); 12-TET is 2^4/12 = 1.2599 (400 cents). At 440 Hz, just 3rd lands at 550 Hz, 12-TET at 554.37 Hz. The overtone series of the lower note has its 5th harmonic at 5·f, which clashes with the 12-TET upper note's fundamental. Result: audible beating, mostly tolerable but noticeably "less locked in" than just thirds. Choirs and string quartets without fixed pitch tend to drift toward JI for harmonic clarity.

What does "small integer ratio" mean for consonance?

When two tones have a small-integer ratio like 3:2, their harmonic series align: the upper note's fundamental equals the lower note's 2nd harmonic; the upper's 2nd harmonic equals the lower's 3rd; etc. This creates many coincident harmonics → beat-free sound → maximum perceptual consonance (Helmholtz, Plomp). Complex ratios like 17:13 don't align nicely → multiple beat frequencies → dissonance. The brain seems to "prefer" simple ratios because they produce simpler combined waveforms.

What's the difference between just and Pythagorean intervals?

Pythagorean tuning uses only 2:1 (octave) and 3:2 (fifth) to derive everything. Major third in Pythagorean = (3:2)⁴ reduced by two octaves = 81:64 = 408 cents — actually sharper than 12-TET! The "just" 5:4 = 386 cents comes from adding 5 as a permitted prime. So just intonation = Pythagorean + 5-limit. Higher-limit JI (7, 11, 13…) brings in more exotic intervals; the Harmonic Seventh (7:4 = 969 cents) is a famous example, distinctly different from 12-TET's minor seventh at 1000 cents.

Can two different ratios produce the same cents value?

In principle no — each ratio maps to a unique cents value via 1200·log₂(r). In practice the cents value can be the same to within rounding for very close ratios (e.g., 16:9 vs 9:5 both round to ~1000 cents). The tool uses the actual decimal ratio for matching, not rounded cents. Different ratios with the same cents are mathematically impossible since the log is bijective on positive reals.

How does this relate to beat frequencies?

Beats are produced when two tones are close in frequency but not identical. Beat rate = |f₂ − f₁|. A 12-TET fifth (659.255 Hz) above 440 Hz beats against the 3rd harmonic of 440 (= 1320 Hz, but 2·659.255 = 1318.51 Hz, so 1.49 Hz beat). A just fifth (660 Hz) has 2·660 = 1320 Hz exactly — zero beats. This is the audible test for "in tune". Trained ears use beat-counting to tune to just intervals.

Why are octaves not slightly different in JI vs 12-TET?

Because both systems use 2:1 as the octave. The octave is the ONE interval that's identical in JI and 12-TET — by definition. The other 11 intervals diverge by various small amounts. Cuts both ways: pianos can be tuned with "stretched octaves" (slightly wider than 2:1) to compensate for inharmonicity of real strings, but that's a separate compromise involving physics of stiff strings.

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