MusicReasonPitch, Tuning, Intonation, and Fret Geometry
What frequency is this pure ratio, exactly?
Choose a fundamental frequency and a small-integer ratio such as 3:2. This tool multiplies the fundamental by that exact ratio, shows the resulting frequency and cents distance, and compares it against the nearest 12-TET equal-tempered equivalent. It does not claim real instruments actually maintain the ratio in practice.
Answer
Understand
Just intonation builds intervals from small whole-number frequency ratios rather than the equal division of the octave into 12 identical semitones. A 3:2 ratio is a genuinely pure perfect fifth, with no beating between its overtones; the equal-tempered fifth used on a piano is a close but deliberately imperfect approximation of that same ratio, chosen so every key sounds equally (im)pure.
Worked examples
A 220 Hz fundamental times a 3:2 ratio gives exactly 330 Hz, with the exact ratio text preserved rather than only a decimal approximation. That same interval, measured in cents using the standard logarithmic formula, sits about 2 cents wider than the equal-tempered fifth used on a piano, a small but measurable difference from pure.
Method and limitations
The upper frequency is the fundamental multiplied by the ratio, further multiplied by two raised to the requested octave placement to move the result into a convenient register. Cents distance from the fundamental uses the standard 1200 ยท log2(f1/f0)formula, and the deviation from the nearest 12-TET semitone is reported alongside it. Only the Locked small-integer ratio vocabulary is offered; this tool does not claim any real instrument, voice, or ensemble actually holds these ratios in practice, since intonation in real performance is influenced by countless factors this tool cannot observe.
The octave-placement input exists because a raw ratio like 3:2 applied directly to a low fundamental can land outside a convenient hearing or playback register; multiplying by an extra power of two shifts the result a whole octave without disturbing the ratio's own exact relationship to the fundamental. Moving the octave placement up or down by one always changes the reported cents distance by exactly 1200, never a fractional amount, since an octave shift is itself always an exact doubling or halving of frequency.