Musical tuning

This page is about musical 'systems' of tuning, for the musical 'process' of tuning see tuning.

Ways of tuning the twelve-note chromatic scale

It is impossible to tune the twelve-note chromatic scale so that all intervals are "perfect"; many different methods with their own various compromises have thus been put forward. The main ones are:

Related Topics:
Chromatic scale - Interval

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• 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 (music), 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 temperament 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.)
• Pythagorean tuning, in which the ratios of the frequencies between all notes are all multiples of 3:2 - (.ogg format, 93.8KB) The Pythagorean system was further developed by Safi ad-Din al-Urmawi, who divided the octave into seventeen parts (limmas and commas) and used in the Turkish and Persian tone systems.
• 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. Next to the twelve-equal temperament, which some would not regard as a form of meantone, the best known form of this temperament 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 unacceptably out-of-tune fifth is known as a wolf interval). However, the fifth may be flattened to a greater or lesser degree than this and the tuning system will retain the essential qualities of meantone temperament; examples include the 31-equal fifth and Lucy tuning.
• Both just intonation and meantone temperament can be regarded as forms of regular temperament.
• 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-E will probably be tuned closer to a 5:4 ratio than, say, Db-F. 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 (100 cents) - (.ogg format, 96.9KB). Since this scale divides an octave into twelve equal-ratio steps, the frequency ratio between adjacent notes is then the twelfth root of two, 2^1/12, or ~1.05946309...