Monochord

A monochord, also known as sonometer (see below), is an ancient musical and scientific laboratory instrument, involving one (mono) string (Chord). The term monochord is sometimes used as the class-name for any musical stringed instrument having only one string and a stick shaped body, also known as musical bows. According to the Hornbostel–Sachs system, string bows are bar zithers (311.1) while monochords are traditionally board zithers (314). The "harmonical canon", or monochord is, at its least, "merely a string having a board under it of exactly the same length, upon which may be delineated the points at which the string must be stopped to give certain notes," allowing comparison.[2]

A string is fixed at both ends and stretched over a sound box. One or more movable bridges are then manipulated to demonstrate mathematical relationships among the frequencies produced. "With its single string, movable bridge and graduated rule, the monochord (kanōn ) straddled the gap between notes and numbers, intervals and ratios, sense-perception and mathematical reason."[3] However, "music, mathematics, and astronomy were inexorably linked in the monochord."[4] As a pedagogical tool for demonstrating mathematical relationships between intervals, the monochord remained in use throughout the middle ages.[5]

The monochord can be used to illustrate the mathematical properties of musical pitch and to illustrate Mersenne's laws regarding string length and tension: "essentially a tool for measuring musical intervals".[4] For example, when a monochord's string is open it vibrates at a particular frequency and produces a pitch. When the length of the string is halved, and plucked, it produces a pitch an octave higher and the string vibrates at twice the frequency of the original (2:1)  . Half of this length will produce a pitch two octaves higher than the original—four times the initial frequency (4:1)—and so on. Standard diatonic Pythagorean tuning (Ptolemy's Diatonic Ditonic) is easily derived starting from superparticular ratios, (n+1)/n, constructed from the first four counting numbers, the tetractys, measured out on a monochord.[citation needed] The mathematics involved include the multiplication table, least common multiples, and prime and composite numbers.[4]

"As the name implies, only one string is needed to do the experiments; but, since ancient times, several strings were used, all tuned in exact unison, each with a moveable bridge, so that various intervals can be compared to each other [consonance and dissonance]."[4] A "bichord instrument" is one, "having two strings in unison for each note ," such as the mandolin.[6] With two strings one can easily demonstrate how various musical intervals sound. Both open strings are tuned to the same pitch, and then the movable bridge is put in a mathematical position on the second string to demonstrate, for instance, the major third (at 4/5th of the string length)   or the minor third (at 5/6th of the string length)  .

Many contemporary composers focused on microtonality and just intonation such as Harry Partch, Ivor Darreg, Tony Conrad, Glenn Branca, Bart Hopkin, and Yuri Landman constructed multistring variants of sonometers with movable bridges.

Parts of a monochord include a tuning peg, nut, string, moveable bridge, fixed bridge, calibration marks, belly or resonating box, and an end pin.[4]

Instruments derived from the monochord (or its moveable bridge) include the guqin, dan bau, koto, vina, hurdy-gurdy, and clavichord ("hence all keyboard instruments").[4] A monopipe is the wind instrument version of a monochord; a variable open pipe which can produce variable pitches, a sliding cylinder with the numbers of the monochord marked.[7] End correction must be used with this method, to achieve accuracy.

The monochord is mentioned in Sumerian writings, and, according to some, was reinvented by Pythagoras (sixth century BCE).[4] Dolge attributes the invention of the moveable bridge to Guido of Arezzo around 1000 CE.[8]