MUSICAL TEMPERAMENT CALCULATOR

a guide to musical tuning

Abstract

Research Question/Problem

Throughout the history of Western music, tuning systems applied to fixed-pitch instruments were developed on the basis of the auditory perception of experienced musicians, who manipulated the dilution of the Pythagorean and syntonic commas in the construction of temperaments (Barbour, 1951, pp. 177–178). These systems can be organized into four major categories: Pythagorean tuning, meantone temperaments, well temperaments, and equal temperament. Although each historical period favored particular intervals—pure perfect fifths, pure major thirds, or tonal mobility—all relied on empirical listening criteria, especially the perception of beats and the controlled redistribution of commas.

With the consolidation of electronic tuning devices, a paradigm grounded in mathematical models became widespread, often detached from perceptual-acoustic variables such as string inharmonicity (Railsback, 1938, pp. 274–276) and aesthetic choices involved in temperaments. Although such devices provide numerical precision, they do not allow musicians or researchers to consciously visualize and distribute the comma throughout the cycle of fifths, a structural element of any tempered system.

In light of this gap, the following research question arises: would it be possible to develop a tool capable of accurately measuring the dilution of the Pythagorean comma in perfect fifths while enabling the simulation and creation of different temperament systems, integrating mathematical, acoustic, and aesthetic criteria?

Objectives

The core of this research lies in the development of a musical temperament calculator, a tool capable of mathematically modeling and quantifying cumulative dilutions of the Pythagorean comma in perfect fifths (P5s), thereby enabling its redistribution throughout the cycle of fifths for the creation of a musical temperament.

Results and Main contributions

The calculator was implemented in a spreadsheet, applying logarithmic formulas to compute the difference, in cents, between two frequencies (1200 × log₂ f1/f2). The comparison of this result with 701.955 cents—the value corresponding to a pure P5—determines the deviation of each P5 within the constructed system.

The overall percentage of dilution of the Pythagorean comma is obtained by calculating the mean deviation of the first eleven fifths in the cycle (Lindley, 1980, p. 670), considering that the octave must remain pure in practice. This procedure prevents the calculation of all twelve fifths from inevitably resulting in the complete dilution of the comma within the 1200-cent octave. The tool also makes it possible to determine the total percentage of dilution and to transpose the system to different reference frequencies.

Conclusions and Implications

This research demonstrates that it is possible to integrate mathematical modeling and auditory perception in the conscious construction of temperament systems. The calculator does not replace the musician’s or tuner’s ear; rather, it expands structural understanding by making the redistribution of the Pythagorean comma visible.

As an implication, the study contributes both to tuning theory and to the experimental practice of tuners interested in the creation of new systems. As a further development, the methodology may incorporate variables such as instrumental inharmonicity and the dilution of the syntonic comma, thereby broadening its analytical scope and opening new perspectives for investigations in musical tuning.