Pythagorean Tuning Frequency Calculator

Enter a base frequency and an integer ratio to calculate the Pythagorean-tuned pitch, cents deviation, semitone approximation, and octave-reduced equivalent.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Enter the base frequency (Hz)
Input the reference pitch in Hertz, such as 261.63 Hz for middle C. All other frequencies will be calculated relative to this.
2
Specify the ratio numerator
Provide the top number of the Pythagorean interval ratio (e.g., 3 for a perfect fifth, which is 3:2).
3
Input the ratio denominator
Enter the bottom number of the Pythagorean interval ratio (e.g., 2 for a perfect fifth, 3:2).
4
Review the tuned frequency and intervals
The calculator will display the resulting tuned frequency, cents deviation, semitone approximation, and octave-reduced frequency.

Example Calculation

A musician wants to calculate the Pythagorean-tuned perfect fifth from a base frequency of 261.63 Hz (middle C) using a 3:2 ratio.

Base Frequency (Hz)

261.63

Ratio Numerator

Ratio Denominator

Results

392.45 Hz

Tips

Compare to Equal Temperament

Calculate frequencies for the same interval in both Pythagorean and equal temperament (e.g., A4 at 440 Hz) to understand the cents deviation and 'purity' differences.

Explore Enharmonic Differences

In Pythagorean tuning, enharmonic equivalents (e.g., C# vs. Db) are not the same frequency. Calculate both to hear the subtle differences that equal temperament eliminates.

Understand the 'Comma'

Repeatedly apply perfect fifths (3:2) and octave reductions (1:2) to a base frequency. After 12 fifths, you will find a small difference known as the Pythagorean comma (approx. 23.46 cents).

The Mathematics of Harmony: The Pythagorean Tuning Frequency Calculator

Pythagorean tuning, one of the oldest and most historically significant tuning systems, constructs musical scales using simple, pure integer ratios, primarily based on the perfect fifth.

The Pythagorean Tuning Frequency Calculator allows musicians and theorists to explore this ancient system, deriving specific note frequencies and understanding their deviation from modern standards.

For a base frequency of 261.63 Hz (middle C) and a 3:2 ratio for a perfect fifth, the calculator yields a tuned frequency of 392.45 Hz.

This tool offers deep insight into the mathematical underpinnings of musical intervals in 2025.

The Mathematical Basis of Musical Harmony

The concept of musical harmony is deeply rooted in mathematics, particularly in the relationships between vibrating strings and their resulting frequencies.

Ancient Greek mathematicians, most notably Pythagoras, discovered that simple integer ratios (like 2:1 for an octave, 3:2 for a perfect fifth, and 4:3 for a perfect fourth) produced consonant, pleasing musical intervals.

This observation formed the basis of Pythagorean tuning, where all notes are derived from stacking perfect fifths.

The overtone series further illustrates this natural phenomenon, where a vibrating string produces not only its fundamental frequency but also a series of higher, quieter frequencies (overtones) that correspond to these same simple integer ratios, intrinsically linking physics, mathematics, and musical aesthetics.

Calculating Tuned Frequencies with Pythagorean Ratios

The Pythagorean Tuning Frequency Calculator applies a simple ratio to a base frequency to determine the frequency of a new note within the Pythagorean system.

It also calculates the interval in cents, a logarithmic unit for measuring pitch differences, and approximates the semitone shift.

tuned frequency (Hz) = base frequency (Hz) × (ratio numerator / ratio denominator)
cents difference = 1200 × log2(tuned frequency / base frequency)
semitone approximation = cents difference / 100
octave equivalent = tuned frequency / (2 ^ floor(log2(ratio decimal)))

For a base frequency of 261.63 Hz, a numerator of 3, and a denominator of 2: Ratio decimal = 3 / 2 = 1.5 Tuned Frequency = 261.63 × 1.5 = 392.445 Hz Cents Difference = 1200 × log2(392.445 / 261.63) = 1200 × log2(1.5) ≈ 1200 × 0.58496 ≈ 701.96 cents.

Semitone Approximation = 701.96 / 100 = 7.02 semitones (a perfect fifth).

💡 Understanding precise numerical relationships, like those in Pythagorean tuning, is fundamental in many fields. For presenting these values clearly, our Engineering Notation Formatter can be a helpful tool.

Deriving a Pythagorean Perfect Fifth

Let's calculate the frequency of a perfect fifth above middle C (C4) in Pythagorean tuning.

The base frequency for middle C is 261.63 Hz, and a perfect fifth has a frequency ratio of 3:2.

Identify base frequency: C4 = 261.63 Hz.
Apply the ratio: Multiply the base frequency by the ratio of the perfect fifth (3/2): 261.63 Hz × (3 / 2) = 261.63 Hz × 1.5 = 392.445 Hz.
Resulting frequency: The Pythagorean-tuned G4 (a perfect fifth above C4) is approximately 392.45 Hz.
Calculate cents deviation: This interval is 701.96 cents, very close to the 700 cents of an equal-tempered perfect fifth, but slightly wider, creating a "purer" sound.

This demonstrates how simple ratios yield specific frequencies in this historical tuning system.

💡 Just as Pythagorean tuning relies on precise mathematical relationships, other areas of math also involve fundamental concepts like ratios and linearity. Our Equation of a Line (Two Points) Calculator can help explore linear relationships.

Common Intervals and Their Pythagorean Ratios

Pythagorean tuning is built upon simple integer ratios derived primarily from the perfect fifth (3:2) and the octave (2:1).

These ratios define the "purity" of intervals within this system.

Here are some common musical intervals and their corresponding Pythagorean ratios:

Unison: 1:1 ratio. If the base frequency is 261.63 Hz (C), the unison is also 261.63 Hz.
Perfect Octave: 2:1 ratio. From C (261.63 Hz), the octave up is 523.26 Hz.
Perfect Fifth: 3:2 ratio. From C (261.63 Hz), the perfect fifth (G) is 392.45 Hz. This is the foundational interval.
Perfect Fourth: 4:3 ratio. From C (261.63 Hz), the perfect fourth (F) is 348.84 Hz. This is derived as an inverted perfect fifth (2:3) plus an octave.
Major Second (Tone): 9:8 ratio. From C (261.63 Hz), the major second (D) is 294.33 Hz. This is derived from two stacked perfect fifths, reduced by an octave.
Major Third: 81:64 ratio. From C (261.63 Hz), the major third (E) is approximately 333.15 Hz. This is a relatively complex ratio compared to those found in just intonation, and often sounds noticeably sharper in Pythagorean tuning.

Frequently Asked Questions

What is Pythagorean tuning?

Pythagorean tuning is a system of musical tuning where all intervals are derived from a series of perfectly tuned perfect fifths (a frequency ratio of 3:2) and octaves (2:1). This system creates very pure-sounding fifths and fourths but results in other intervals, particularly major and minor thirds, that can sound dissonant compared to modern equal temperament, due to the accumulation of small discrepancies.

How does Pythagorean tuning differ from equal temperament?

Pythagorean tuning creates intervals based on pure 3:2 and 2:1 ratios, resulting in perfectly consonant fifths and octaves, but 'wolf intervals' for other notes. Equal temperament, conversely, divides the octave into 12 exactly equal semitones, making all intervals slightly impure but allowing music to be played in any key without sounding significantly out of tune, which is standard for modern Western music.

What is a 'cent' in music theory?

A cent is a logarithmic unit of measure used for musical intervals, where one octave is divided into 1200 cents. Each semitone in equal temperament is exactly 100 cents. This unit allows musicians and acousticians to precisely quantify the difference between various tuning systems and the 'purity' or deviation of intervals from their theoretical values.

What is the Pythagorean comma?

The Pythagorean comma is a small, dissonant interval (approximately 23.46 cents) that arises in Pythagorean tuning when a series of 12 perfect fifths (3:2 ratio) does not perfectly align with a series of 7 octaves (2:1 ratio). This discrepancy means that a note reached by stacking 12 fifths will be slightly higher than the same enharmonic note reached by stacking 7 octaves, creating a noticeable 'out of tune' effect.