Savart interval: How Music Became a Ratio
In 1902, Auguste Guillemin proposed a musical measuring unit named after Felix Savart. The Savart interval expressed the relationship between the frequencies of notes through logarithms, joining a long effort to give musical distances a numerical form. An octave contains 301.03 savarts. Savart, who lived from 1791 to 1841, had never considered measuring intervals through logarithms. The unit bearing his name arrived after his lifetime.
The mathematical trail led back to the seventeenth century. John Napier introduced logarithms in 1614, and in 1647 Juan Caramuel y Lobkowitz wrote to Athanasius Kircher about using logarithms to measure music. With base-two logarithms, an octave has the value one, while a semitone has the value one-twelfth. This approach represents intervals through frequency ratios rather than the gap between frequencies on a linear scale. Octaves grow exponentially on that linear scale but stand at equal distances on a logarithmic scale. The same musical step can therefore retain the same numerical size at different pitches.
Joseph Sauveur took another route in his 1701 Principes d'acoustique et de musique. He used base-ten logarithms calculated to three decimal places. The logarithm of two was approximately 0.301; multiplying it by 1000 gave him 301 units per octave. He then arranged the octave into 43 parts called merides, each divided into seven heptamerides. Sauveur also considered splitting each heptameride into ten smaller parts, although he did not make practical use of those tiny divisions. His scheme introduced both a decimal calculation and a set of named subdivisions, long before Guillemin proposed the Savart interval.
Guillemin's original terminology differed from the later convention. At first, one savart represented the decimal logarithm of a ten-to-one frequency ratio, and a millisavart represented the base-ten logarithm multiplied by 1000. That smaller measure later became the savart itself. In the later definition, a Savart interval is calculated by multiplying the base-ten logarithm of the frequency ratio by 1000. The attribution of logarithmic interval measurement to Savart himself appeared later in Anglo-Saxon writings. Decimal precision affected the reported octave value, and practical versions sometimes rounded it to 300, alongside the more detailed figure.
Another family of interval measures grew around the equal-tempered semitone. Early in the nineteenth century, Gaspard de Prony proposed a logarithmic unit corresponding to that semitone. Alexander John Ellis later described pitch standards in pronys to two decimal places, comparing them with a theoretical reference pitch of 370 Hz. At Robert Holford Macdowall Bosanquet's suggestion, Ellis based the cent on a hundredth of a semitone. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Its complete octave therefore spans 1200 cents, a different numerical scale for the same frequency doubling measured by a Savart interval.
Ellis carried that framework into comparisons of instruments from around the world. He described and used cents in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone, defined the system in History of Musical Pitch, and formally introduced it in his 1885 paper On the Musical Scales of Various Nations. His comparisons ranged across Ancient Greece, modern Europe, Persia, Arabia, Syria, the Scottish Highlands, India, Singapore, Burma, and Siam. He also examined pentatonic scales from the South Pacific, Java, China, and Japan. Their variety led him to reject the idea of a single universal musical scale.
Further proposals gave musical intervals still more numerical forms. Widogast Iring proposed the centitone in 1898 as a division of the octave into 600 steps; each step equals two cents. Joseph Yasser later used that measure in A Theory of Evolving Tonality in 1932 and promoted decitones and millitones as well. An equal-tempered perfect fifth illustrates how these systems meet: it spans 700 cents, 175.6 savarts, 583.3 millioctaves, or 350 centitones. Each figure describes a relationship between notes, while the choice of unit changes the scale of the number used to express it. A major third provides a comparison between tunings: in just intonation its frequency ratio is 5:4, about 386 cents, while in equal temperament it spans 400 cents. That 14-cent difference is about a seventh of a half step and is clearly audible.
At the instrument, tiny numerical differences can become audible through interaction. A single cent is too small for most listeners to perceive between successive notes, but two notes played together can produce beating as their waves reinforce and cancel one another. Middle C sounded with a note 10.06 cents higher produces a beat frequency of 1.53 Hz; with a note 25 cents higher, the beat frequency is 3.81 Hz. A piano tuner can time those beats when two strings sound together, using the changing strength of their combined sound to check the tuning.