Foundations

Tuning and Temperament: A Short Guide

Pitch is agreed rather than fixed, and every scale involves compromise, from the A that an orchestra tunes to, to the frets on a guitar.

Musicians do not simply tune to a natural law. They agree a reference pitch, choose how to divide the octave, and accept compromises that suit their instruments. This guide explains standard pitch and A4, how concert pitch has varied, what just intonation and equal temperament mean, why keyboards and fretted instruments need compromise, and how flexible instruments, drones and orchestras handle tuning.

Naming the pitch

Pitch is the highness or lowness of a note, and for a steady note it corresponds to a frequency, the number of vibrations per second, measured in hertz. Musicians label notes by letter and octave, and the A above middle C, which is written A4 in the standard notation used here, is the usual reference for tuning. Today that note is most commonly set to 440 hertz, a convention that was later confirmed by an international standard.

It is a convention, not a fact of nature. Nothing in acoustics requires that particular number, and some ensembles choose a slightly higher A, commonly a few hertz above 440, because a brighter sound is sometimes preferred. What matters in practice is that everyone playing together agrees on the same number, since two instruments a few hertz apart sound audibly out of tune when they play the same note.

A pitch that moved

Historians generally agree that before the nineteenth century there was no single pitch standard. Reference pitch varied between places, between institutions and between types of instrument, and a church organ in one town might sit noticeably higher or lower than an organ in the next. Musicians had to adapt to the local pitch, and instrument makers built for the place where an instrument would be used.

Efforts to standardise gathered pace in the nineteenth century, and a French standard near 435 hertz was influential before 440 became the dominant reference. Early-music players today often use lower pitches for older repertoire, with a level roughly a semitone below 440 being a common choice for baroque music, although the exact practice varies with region and period. The University of Edinburgh’s Collection of Historic Musical Instruments has mounted a display on these topics, which examined methods for finding the right pitch and keeping an instrument in tune.

Pure intervals and their arithmetic

Just intonation builds a scale from simple whole-number ratios. A pure fifth has two notes whose frequencies stand in the ratio three to two, and a pure major third has the ratio five to four. Such intervals sound smooth and steady, because the overtones of the two notes line up and produce no slow pulsing, called beating, between them.

The difficulty is that pure intervals do not fit together. If one climbs twelve pure fifths, the result is slightly higher than seven octaves above the starting note, and the small discrepancy is known as a comma. As a result, a scale tuned in pure intervals for one key sounds excellent there and increasingly out of tune in distant keys. Singers, string players and ensembles can adjust as they go, but an instrument with fixed pitches cannot, so its maker has to decide where the compromise falls.

Compromise on the keyboard

That decision is called temperament. On a Piano or a Harp, each note has one fixed pitch, and a single key has to serve as, for instance, both D-sharp and E-flat. One essay from the Sound and Science database describes the twelve-note octave as a compromise by design. In quarter-comma meantone, widely used for keyboards in the sixteenth and seventeenth centuries, the fifths are narrowed slightly to make the thirds pure. Keys close to the starting point sound sweet, but distant keys can sound harsh, and some instruments added split keys so that two different pitches were available for notes that would otherwise have to be merged.

Later systems, often grouped as well temperaments, spread the discomfort unevenly so that every key is usable but each has a different character. Equal temperament, which divides the octave into twelve identical semitones, removes the differences altogether. Its fifths are very slightly narrow and its thirds noticeably wider than pure, but every key sounds the same. It spread across Europe gradually, and it is now the usual basis for the piano, although the dates and the speed of adoption differed from place to place.

Frets, slides and flexible players

Fretted instruments carry a version of the same problem. On an acoustic guitar, a Ukulele or a Mandolin, a metal fret fixes each pitch along the string, and the placement of the frets usually follows equal temperament. Players and makers compensate in small ways, for instance by adjusting bridge position, and skilled players often pull a string slightly to sharpen a note.

Instruments without frets or keys can do much more. The Violin, Viola and Cello have a smooth fingerboard, the Trombone has a slide, and brass and woodwind players can bend pitch with lips and breath, as can singers. Many players of these instruments adjust pitch to suit the harmony they are in, rather than following one fixed scale. The strings themselves are tuned in perfect fifths, with the violin commonly tuned G3, D4, A4 and E5 and the viola a fifth lower, according to the Philharmonia Orchestra.

Drones and a fixed reference

In many musical traditions a sustained note, called a drone, supplies the reference against which all other pitches are judged. In Hindustani music the tanpura, a usually four-stringed instrument, sounds the tonic continuously, and the Sitar itself carries drone strings as well as melody strings. Bagpipes carry drone pipes for the same purpose. When a drone sounds, the ear readily hears whether a note stands in a pure relationship to it, and many traditions tune their scales by such relationships, not by equal division of the octave.

The same idea appears in instruments designed for flexibility. The Koto has a movable bridge under each string, so the player can reposition the bridges to set the scale a piece requires. Tuning is here a skilled and creative part of performance, rather than a preliminary chore, and the range of scales in use around the world is far wider than the twelve notes of the piano.

How an orchestra tunes

Before a concert the oboe sounds an A, and the rest of the orchestra tunes to it, according to the Philharmonia Orchestra. A few practical reasons are commonly given. The oboe’s pitch is difficult to change quickly, so it makes a steady reference, and its penetrating tone carries across a stage. A is also the pitch of an open string on the violin, viola and cello, which gives string players a convenient starting point from which to tune the other strings.

Some instruments cannot be retuned on stage at all. A piano is tuned beforehand by a technician, and the Philharmonia notes that harpists often spend about an hour before a concert tuning their forty-seven strings. The Timpani are the reverse case, since pedals let a player change the pitch of a drum within a piece. The orchestra’s brief ritual is therefore a meeting of several different answers to the same question, namely how to make many instruments agree.

Terms used in this article

beating: The slow pulsing heard when two notes are almost, but not exactly, in tune.
comma: The small gap that appears when pure intervals are stacked and fail to meet exactly, such as twelve fifths against seven octaves.
drone: A sustained note, usually the tonic, that provides a constant reference for the melody.
equal temperament: A temperament dividing the octave into twelve identical semitones.
hertz: The unit of frequency, one cycle of vibration per second.
just intonation: A tuning built from pure whole-number frequency ratios, such as three to two for a fifth.
temperament: A system of slightly adjusting intervals so that an instrument with fixed pitches can play in many keys.