Interval & Tuning Calculator
Convert named intervals, ratios, and cents; compare them against 12-TET; and search cents input for constrained just-intonation ratios.
What the calculator does
Pick an interval by name, by frequency ratio, or by cent value, and the module reports its exact cent size and nearest 12-TET semitone, named by interval (Major 3rd, Minor 3rd). Exact grid matches are labeled as 12-TET; every off-grid interval also shows the bracketing semitone on the other side, so both signed distances are visible. Set a root pitch and the result also names the absolute 12-TET pitches (C♯4, D4) and reports the beat frequency against the nearest equal-tempered semitone at that root, so you can hear how the interval will wobble against equal temperament.
Finding a just-intonation approximation
In Cents input mode, the calculator searches for the simplest just-intonation ratio that lies within your configured error tolerance and prime limit. The search uses a Stern–Brocot tree walk between 1/1 and 2/1, collects ratios inside the tolerance band whose numerator and denominator primes fit under the limit, and then applies your tiebreaker. Simplest chooses the lowest numerator × denominator product, First keeps the first qualifying tree candidate, and Closest chooses the smallest cents error.
Anchoring at a root pitch
The Root section anchors the interval at a specific frequency. Toggle it on and enter the root as an exact frequency in Hz or as a chromatic pitch name; the mic button captures a pitch from the microphone. With a root set, the result names absolute pitches and reports the beat frequency against equal temperament. Toggle it off for cents-only analysis.
Auditioning the interval
Once an interval is set, play it back above the root pitch. Choose Together to sound both tones at once or Low→High to play them in sequence, then switch between Interval and 12-TET to compare the interval in its own tuning against the nearest equal-tempered equivalent. The 12-TET playback button is hidden when the interval is already on the equal-tempered grid. The beat glides in real time as you edit the interval or root, so A/B comparisons stay live.
Input modes
- By name: grouped by category in the dropdown: diatonic (minor 2nd through octave), septimal (7:6, 7:4, 9:7), undecimal (11:8, 11:9, 12:11), and tridecimal (13:8, 13:9, 16:13). Diatonic names use their equal-tempered semitone positions; septimal, undecimal, and tridecimal names use the catalog ratio cents.
- By ratio: e.g. 3:2, 7:4, 11:8, 13:8, 81:64. The ratio is reduced to cents and labeled by the nearest 12-TET interval slot; the signed distances to the two bracketing semitones show the microtonal offset.
- By cents: arbitrary cent values, including microtonal intervals. The simplest-ratio search runs in this mode.
Options
- Max error (1¢–50¢, default 5¢): the widest accepted deviation of the suggested ratio from the input cents value.
- Max prime limit (3, 5, 7, 11, 13, 17, 19, 23, 31; default 11): the highest prime allowed in the suggested ratio's numerator or denominator.
- Approximation tiebreaker: Simplest (lowest numerator × denominator product), First found (first qualifying Stern–Brocot candidate), or Closest (smallest cents error).
- Reference A4 (shared with Frequency & Pitch): sets the 12-TET grid used to name pitches. It only affects output while a root pitch is set.
Who this is for
Useful whenever tuning decisions have real musical weight. Composers working in just intonation or xenharmonic tunings can take a target cent value and find the simplest ratio that fits; theorists can show students that a "major third" covers a range of sizes; engravers and microtonalists can move between cent notation and ratio notation without leaving the app. For real-time pitch detection in historical temperaments, use Strobe Tuner.