# Christiaan Huygens

Christiaan Huygens described the 31-tone system in his Lettre touchant le cycle harmonique (Rotterdam 1691) and in Novus cyclus harmonicus (Leiden 1724). Earlier in 1661 he had already made notes in which he accomplished the following:

• He developed an elegant method to calculate string lengths for any regular tuning system, which he applied exclusively to meantone tuning.
• He worked out the use of logarithms in the calculation of string lengths and interval sizes.
• He demonstrated the close relationship between meantone tuning and 31-tone equal temperament.
• He recognised the possible consonant nature of intervals with septimal ratios like 4:7 and 5:7, and showed that they are approximated by the augmented sixth and augmented fourth, respectively, in meantone temperament and 31-tone equal temperament.

Engraving by Frederik Ottens based on the portrait of Edelinck for the publication by 's-Gravesande of the Opera varia (1724)

Engraving by Gerard Edelinck (1687)

Huygens' name is invariably associated with the division of the octave into 31 equal parts. He was not the first one however to describe the 31-tone octave division. A similar division is implicit in the works of authors such as Nicola Vicentino (1555) and Fabio Colonna (1618), while several 17th-century 31-tone keyboards were built, all following more or less the design of Vicentino's archicembalo. The first explicit description of the 31-tone scale, together with string lengths, is in Lemme Rossi's Sistema musico (1666). Huygens didn't know about any of these writings and instruments; he only knew of the existence of Vicentino's archicembalo via Salinas. However, it was not his intention to provide a system with 31 tones available per octave but his main thesis was merely that meantone tuning - the traditional tuning system - could be described by a selection from the 31-tone scale, in his eyes a much nicer and more general way to describe pitches than the original meantone tuning, which he found to be the best one: "Optimum est Temperamentum in chordarum systemate, cum ex diapente quarta pars commatis ubique deciditur".
Huygens, being an inventor of countless appliances, also has designed a keyboard instrument with 31 strings per octave. The keyboard with twelve normal keys per octave was positioned above the strings and movable. Through a system of pins, the keys had to control twelve of the 31 strings at any time. Transposing amounted to shifting the keyboard. This design was probably never realised.

Division of the whole tone VT (= C) - Re (= D) in five steps according to Colonna (1618)

Huygens has also corresponded with Joan Albert Ban (1597/8-1644).

### Biographies of Huygens

#### In print

• C.D. Andriesse, Titan kan niet slapen. Een biografie van Christiaan Huygens (Uitgeverij Contact, Amsterdam, 1994)

### Publications of his work

• Huygens, Christiaan. Le cycle harmonique. Rotterdam, 1691; Novus cyclus harmonicus, Leiden, 1724. 2nd edition with English and Dutch translations, Rudolf Rasch (ed.), Tuning and temperament library vol. 6, Diapason Press, Utrecht, 1986.
• Huygens, Christiaan. Oeuvres complètes. Tome premier, deuxième, troisième, quatrième, cinquième, sixième, septième, huitième, neuvième, dixième, dix-neuvième, vingtième, vingt-et-unième, vingt-et-deuxième. The Hague, 1888-1950.