Musical tuning

Musical tuning is the system used to define which tones, or pitcheses, to use when playing music. In other words, it is the choice of level and spacing of frequency values which are used. The tuning systems are usually defined in such a way that a listener perceives it as "nice".

The history of tuning is much more complex than it at first seems; this index page can be used as a starting point.

Table of contents

1 Subjects in general 2 Ways of tuning the twelve note chromatic scale 3 Tunings of other scale systems 4 Comparisons and controversies between tunings 5 External links

Subjects in general

• Musical theory
• Mathematics of the Western music scale
• Other musical tunings: Chinese, Indian etc.
• Microtonal music
• Psychoacoustics
• Physics of music
• MIDI

Ways of tuning the twelve note chromatic scale

• Pythagorean tuning, in which the ratios of the frequencies between all notes are all multiples of 3:2 - (.ogg format, 93.8KB)
• Just intonation, in which the ratios of the frequencies between all notes are based on relatively low whole numbers, such as 3:2, 5:4 or 7:4; or in which all pitches are based on the harmonic series, which are all whole number multiples of a single tone. Such a system may use two different ratios for what is the same interval in equal temperment depending on context; for instance, a major second may be either in the ratio 9:8 or 10:9. For this reason, just intonation may be less a suitable system for use on keyboard instruments or other instruments where the pitch of individual notes is not flexible. (On fretted instruments like guitars and lutes, multiple frets for one interval is practical.)
• Meantone temperament, a system of tuning which averages out pairs of ratios used for the same interval (such as 9:8 and 10:9), thus making it possible to tune keyboard instruments. The most common form of meantone is quarter comma meantone, which tunes major thirds justly in the ratio of 5:4 and divides them into two whole tones of equal size. To do this, eleven perfect fifths in each octave are flattened by a quarter of a syntonic comma, with the remaining fifth being left very sharp (such an unnacceptably out-of-tune fifth is known as a wolf interval).
• Well temperament, any one of a number of systems where the ratios between intervals are unequal, but approximate to ratios used in just intonation. Unlike meantone temperament, the amount of divergence from just ratios varies according to the exact notes being tuned, so that C-G will probably be tuned closer to a 3:2 ratio than, say, F#-C#. Because of this, well temperaments have no wolf intervals. A well temperament system is usually named after whoever first came up with it.
• Equal temperament, in which adjacent notes of the scale are all separated by logarithmically equal distances - (.ogg format, 96.9KB)

Tunings of other scale systems

• Slendro, a scale used in Indonesian gamelan music with five notes to the octave
• Pelog, the other main gamelan scale, with seven notes to the octave
• Harry Partch, a composer who wrote many pieces using a justly tuned 43-note scale of his own devising
• Lucy Tuning
• Xenharmonic
• Bohlen-Pierce scale

Comparisons and controversies between tunings

All musical tuning have advantages and disadvantages. Twelve tone equal temperament is the standard and most usual tuning system used in western music today because it gives the advantage of modulation to any key without dramatically going out of tune, as all keys are equally and slightly out of tune. However, just intonation provides the advantage of being entirely in tune, with at least some, and possible a great deal, loss in ease of modulation. Referring to 12-tet the composer Terry Riley, who has written music for both tuning systems, has been quoted as saying "Western music is fast because it's not in tune". Twelve tone equal temperament also, currently, has an advantage over just intonation in that most musicians will have instruments that can only play in equal temperament, since these are readily available. Other tuning systems have other advantages and disadvantages and are chosen for these qualities. It must be realized however, that just as many people who play music today in equal temperament without having heard of it, many musicians throughout the world and the past used just intonation without "knowing" it.

Equal temperament provides advantages outside of twelve tone, for instance the Bohlen-Pierce scale may consist of thirteen equal divisions of the tritave (an octave and a fifth, or 1902 cents), which can prove more practical, given the already counterintuitive nature of the scale. It is also a closer approximation of just intonation based on only low odd number ratios than regular twelve tone equal temperament is of just intonation based on all low whole number ratios.

Non-equal and non-just tunings also provide advantages. For instance, William Sethares shows that the tunings of Balinese gamelans are related to the inharmonic spectra or timbre of their metallophones and the harmonic spectra of stringed instruments such as the rebab, just as just intonation and twelve tone equal temperament are related to the spectra or timbre of harmonic instruments alone.

Some instruments, such as the violin, don't limit the musician to particular pitches, allowing to choose the tuning system "on the fly". Many performers on such instruments adjust the notes to be more in tune than the equal temperament system allows, perhaps even without realizing it.

External links

• The History of Musical Temperament and Pitch Before 1750
• The History of Musical Tuning and Temperament during the Classical and Romantic Periods
• Article on the history of absolute pitch to today's A=440 standard
• Tuning for beginners
• Understanding Temperaments - a concise, but comprehensive explanation of tempraments
• How to tune a few popular historic temperaments - instructions on how to tune a keyboard in various temperaments
• Temperament resources on the Web - lots more links
• Albert Gräf: Musical scale rationalization - a graph-theoretic approach (the paper is number 45 on the list, and is in .pdf format)