| Interval | ET (Hz) | JI Ratio | JI (Hz) | Cents Δ |
|---|
You choose a root note and octave. The tool calculates the root frequency from a lookup table (based on A4 = 440 Hz), then builds a comparison table for all 13 intervals in the octave. Equal temperament frequencies use root × 2^(n/12). Just intonation frequencies use the classic 5-limit integer ratios. The cents difference is computed as 1200 × log₂(JI/ET). The JI ratios shown are one common set; alternative tuning systems use different ratios for some intervals.
Just intonation sounds beautiful in one key but falls apart when you modulate to distant keys — some intervals become unusably out of tune. Equal temperament compromises slightly on every interval so that all 12 keys are equally usable, which is essential for keyboards, fretted instruments, and ensemble music.
It means the ratios only use prime factors up to 5 (i.e., 2, 3, and 5). This is the most common system for Western JI. Higher-limit systems (7-limit, 11-limit) introduce additional ratios for intervals like the harmonic seventh.
For intervals like the major third (about 14 cents off), trained musicians can usually hear the difference, especially on sustained chords. For intervals like the perfect fifth (about 2 cents off), the difference is extremely subtle and rarely noticeable in practice.
Barbershop quartets, some a cappella choirs, and fretless string ensembles naturally gravitate toward JI. Most fixed-pitch instruments (piano, guitar, organ) are tuned to equal temperament or a historical temperament like Werckmeister III.
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.