Musical interval ratio calculator
Just intervals are simple whole-number ratios. Equal temperament approximates them so that every key sounds equally, slightly wrong.
Hz
Temperament error
1.9550cents
- Equal-tempered interval
- 700.0 cents
- Just interval
- 701.9550 cents
- Equal-tempered ratio
- 1.498 307 08
- Just ratio
- 1.500 000 00
- Tempered frequency
- 659.2551 Hz
- Just frequency
- 660.0000 Hz
- Beats per second between them
- 0.7449 Hz
- Error is likely audible
- 0
Show the working
- Interval in semitones = 7
- Just ratio numerator = 3
- Just ratio denominator = 2
- Base frequency = 440
- Equal-tempered ratio = 2^(7/12) = 1.498 31
- Just ratio = 3 / 2 = 1.5
- Equal-tempered interval = 7 * 100 = 700
- Just interval = 1200 * log2(1.5) = 701.955
- Temperament error = 701.955 - 700 = 1.955
- Tempered frequency = 440 * 1.498 31 = 659.255
- Just frequency = 440 * 1.5 = 660
- Beats per second between them = abs(660 - 659.255) = 0.744 886
- Error is likely audible = abs(1.955) > 5 ? 1 : 0 = 0
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- cents = 1200 log2(ratio)
- equal temperament: each semitone is exactly 100 cents
The tempered major third is 13.7 cents sharp of just, which is why triads on a piano beat noticeably.
Around 5 cents is the threshold where most listeners start to hear a mistuning on a sustained note.