Frequency Ratios

A frequency ratio is the comparison of two frequencies by division: f₂/f₁. It is the natural way to measure the relationship between two pitches, because the human auditory system perceives pitch in terms of ratios, not differences (see Sound Waves — Logarithmic Perception).

🎯 Simple version: To compare two pitches, divide one frequency by the other. A ratio of 2:1 is an octave — the biggest musical “step.” A ratio of 3:2 is the next most natural step. Simple ratios (small numbers) sound smooth together. Complex ratios (big numbers) sound rough. The cent is a ruler for measuring these ratios: 1200 cents = 1 octave, 100 cents = 1 chromatic step.

Why Ratios, Not Differences

Two demonstrations:

Same Hz difference (100 Hz) → completely different musical intervals. Same ratio (2:1) → same musical interval regardless of starting frequency.

This ratio-based perception is a consequence of the cochlea’s logarithmic frequency mapping (see ear-cochlea.md). Equal distances along the basilar membrane correspond to equal frequency ratios, not equal Hz differences. Music is built on multiplication, not addition.

Simple Ratios and the Harmonic Series

The simplest frequency ratios emerge directly from the harmonic series:

Pattern: ratios with smaller integers produce greater perceptual fusion. This is not subjective — it is a measurable consequence of harmonic overlap. When two tones have a 3:2 ratio, every 3rd harmonic of the lower tone aligns with every 2nd harmonic of the upper tone, producing reinforcement instead of roughness within critical bandwidth (see consonance-dissonance.md).

Perceptual Fusion vs. Aesthetic Preference

A critical distinction, established by McDermott et al.’s studies with the Tsimané people of Bolivia (who have minimal exposure to Western music):

• Perceptual fusion — the tendency for tones in simple ratios to be heard as a single blended sound — is universal biology. It arises from harmonic template matching in the auditory system.
• Aesthetic preference — whether people like consonant intervals more than dissonant ones — is culturally learned.

The Tsimané can hear the difference between consonant and dissonant intervals, but they don’t rate consonant ones as more pleasant. Western listeners strongly prefer consonance, but this is learned, not innate. PhizMusic describes the physics of fusion without prescribing aesthetic judgments.

The Cent System

Ratios are powerful for understanding physics, but awkward for arithmetic. Is 5:4 plus 6:5 equal to 3:2? (It is, but proving it requires multiplication, not addition.) The cent system solves this by converting ratios to a logarithmic scale where intervals add:

Cents = 1200 × log₂(f₂ / f₁)

By definition:

• 1 octave (2:1) = 1200 cents
• 1 chromatic step in 12-TET = exactly 100 cents
• 1 cent = the ratio 2^(1/1200) ≈ 1.000578

The cent scale is logarithmic, so:

Examples

Verify: 3:2 = 5:4 × 6:5. In cents: 702.0 ≈ 386.3 + 315.6 = 701.9 (rounding).

Just Ratios vs. Equal Temperament

The simple ratios above are called just intonation — intervals tuned to exact harmonic-series ratios. In 12-TET (see twelve-tet.md), every chromatic step is an equal ratio of 2^(1/12), producing intervals that are close to but never exactly match just ratios (except the octave):

The 3:2 and 4:3 ratios are approximated very well by 12-TET (within 2 cents). The 5:4 and 6:5 ratios have larger errors (~14-16 cents) — perceptible to trained ears, especially in sustained chords. This is the central trade-off of 12-TET: universal transposability at the cost of tuning purity.

Prime-Limit Classification

Different cultures’ tuning systems can be classified by the largest prime factor appearing in their frequency ratios:

This classification reveals that different tuning systems are different engineering solutions to the same mathematical constraints — not progressive stages of development. 3-limit tuning isn’t “primitive” — it optimizes for maximal consonance of fifths. 5-limit adds sweeter thirds at the cost of complexity. Each choice reflects different aesthetic priorities.

The mathematical constraint underlying all ratio-based tuning: log₂(3) is irrational. You cannot stack any number of 7-step-intervals (3:2 ratio) and arrive at an exact octave (2:1). Twelve 7-step-intervals = 84 chromatic steps = 7 octaves in integer arithmetic, but (3/2)^12 ≈ 129.75 while 2^7 = 128 — the frequencies don’t close. This impossibility forces every culture to choose how to distribute the inevitable error. 12-TET distributes it equally across all steps.

Translation Table

Connections

• Harmonic Series — the physical source of simple integer ratios
• Intervals — how ratios map to step-intervals in the chromatic system
• Scales — how ratio-based reasoning explains different scale choices across cultures
• Twelve-TET — the equal-temperament compromise that approximates just ratios
• Consonance & Dissonance — why simple ratios produce perceptual fusion
• Sound Waves — logarithmic perception as the basis for ratio-based hearing

Suggested References

• Sethares Dissonance Curves (aatishb.com) — Click anywhere on the dissonance curve to hear frequency ratios and their perceived consonance/dissonance