Equal Temperament vs Just Intonation
| Interval | ET (Hz) | JI Ratio | JI (Hz) | Cents Δ |
|---|
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What Your Result Means
- Root Frequency: The frequency in Hz of the note and octave you selected. A4 = 440 Hz is the standard reference; all other notes derive from this.
- ET (Hz): The equal temperament frequency for each interval, calculated as root × 2^(n/12). Every semitone is exactly the same ratio apart.
- JI (Hz): The just intonation frequency using small integer ratios (5-limit tuning). These intervals are acoustically "pure" and produce no beating between harmonics.
- Cents Δ: The difference between ET and JI in cents (1/100 of a semitone). Positive means JI is sharper than ET; negative means JI is flatter. Differences under ~5 cents are generally imperceptible to most listeners.
How This Calculator Works
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.
Quick Questions
Why does equal temperament exist if just intonation sounds "purer"?
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.
What does "5-limit" mean for the JI ratios?
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.
Can I hear the difference between ET and JI?
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.
Which instruments use just intonation?
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.
Sources
- Wikipedia — Equal Temperament (history and mathematics of 12-TET)
- Wikipedia — Just Intonation (5-limit ratios and comparison tables)
- Michigan Tech — Note Frequencies (standard A440 frequency reference table)
Method & review
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.