A cent is 1/100 of a semitone; cents are logarithmic so intervals add while ratios multiply

The cent, introduced by Alexander Ellis, is the standard logarithmic unit for measuring musical intervals. There are 100 cents in one equal-tempered semitone and 1200 cents in one octave. The cent is logarithmic: perceived pitch intervals correspond to ratios of frequencies, and logarithms turn multiplication into addition. So a perfect fifth (3:2 = 702 cents) followed by a major third (5:4 = 386 cents) spans 702 + 386 = 1088 cents = a just major seventh. The conversion formula is: cents = 1200 × log₂(ratio) = 3986 × log₁₀(ratio). Cents make scale comparison easy: 12-tet semitone = 100 cents exactly; Pythagorean comma = 23.46 cents; just major third = 386.3 cents vs. 12-tet major third = 400 cents (difference of 13.7 cents).

Examples

Just perfect fifth: 3:2 → 702 cents. 12-tet perfect fifth: 700 cents. Difference: 2 cents (barely perceptible). Just major third: 5:4 → 386 cents. 12-tet major third: 400 cents. Difference: 14 cents (perceptible, especially sustained). Pythagorean comma: 531441/524288 → 23.46 cents.

Assessment

Convert the ratio 7:4 (harmonic seventh) to cents (use log₂(7/4) × 1200). Is this larger or smaller than the 12-tet minor seventh (1000 cents)? By how many cents?