Pythagorean tuning interval ratio calculator

Enter a positive count to travel upward around the circle of fifths or a negative count to travel downward. The calculator can move the result into one octave without changing its pitch class, then reports the exact fraction, decimal ratio, octave shift, and size in cents. It is useful for studying tuning theory, checking interval tables, and comparing Pythagorean spellings without rounding away the underlying arithmetic.

Build an interval from pure fifths

Pythagorean tuning begins with the pure fifth, whose frequency ratio is exactly 3:2. Stacking fifths means multiplying that ratio once for every upward step, so seven fifths produce (3/2)^7. A negative count travels in the opposite direction and uses the reciprocal, (2/3) for each step. Enter zero to obtain unison, one for a pure fifth, two for the Pythagorean major second after octave reduction, or any signed integer within the stated limit. The calculator performs the rational arithmetic with integers rather than repeatedly rounding decimal values. Its numerator and denominator therefore preserve the exact theoretical ratio. This distinction matters when many fifths are stacked: a displayed decimal is convenient for measurement, but it is only an approximation, while the fraction records precisely how the interval was constructed. Positive and negative counts are handled symmetrically, making the tool suitable for moving either direction through a chain of fifths and for checking inverse relationships between intervals.

Understand octave normalization and the reported shift

Intervals produced by several fifths quickly extend beyond a single octave. Octave normalization multiplies or divides the raw ratio by powers of two until the result lies from 1:1 inclusive to 2:1 exclusive. Because an octave identifies pitches of the same pitch class, this reduction keeps the Pythagorean identity of the interval while presenting it in the range most tuning tables use. The `octave_shift` field explains exactly what happened: a negative value means the raw stacked ratio was divided by that many octaves, while a positive value means it was raised by octaves after descending fifths. Disable normalization when you need the literal compound interval formed by the complete stack. In that mode the octave shift is zero and the fraction remains (3/2)^n or its reciprocal. Comparing normalized and unnormalized calls is a direct way to separate two ideas that are sometimes conflated: the physical frequency multiplication accumulated by the stack and the octave-equivalent simple interval normally assigned a familiar musical name.

Use exact ratios, decimals, and cents together

The result offers several views of the same interval because each serves a different task. Use `ratio`, `numerator`, and `denominator` when copying exact values into a tuning table, proving a relationship, or calculating a target frequency with rational arithmetic. Use `decimal_ratio` when multiplying a reference frequency in a synthesizer script or spreadsheet. Use `cents` when comparing the result with twelve-tone equal temperament or another tuning system, since cents convert multiplicative frequency ratios into an additive scale. For example, you can calculate a Pythagorean pitch, compare its cents value with the nearest 100-cent equal-tempered step, and interpret the difference as a tuning offset. Remember that the calculator describes intervals rather than naming notes: enharmonic spelling depends on the chosen starting note and direction through the fifth chain. The exact fraction remains unambiguous even when two spellings look similar on a keyboard, and it exposes small discrepancies such as the Pythagorean comma that rounded pitch labels can conceal.