Musical tuning : Temperament

Musical tuning is the system used to define which tones, or pitches, 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.

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 - A harmonized C major scale in Pythagorean tuning (.ogg format, 93.8KB)
• Just intonation, in which the ratios of the frequencies between all notes are relatively simple, such as 3:2, 5:4 or 7:4. Such a system may use two different ratios for the same interval depending on context; for instance, a major second may be either in the ratio 9:8 or 10:9. For this reason, just intonation isn't a suitable system for use on keyboard instruments or other instruments where the pitch of individual notes isn't flexible.
• 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 - A harmonized C major scale in equal temperament (.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 43-note scale of his own devising
• Lucy Tuning

External links

• The History of Musical Temperament and Pitch Before 1750 (http://www.casaninja.com/christi/academic/tuningpre1750.html)
• The History of Musical Tuning and Temperament during the Classical and Romantic Periods (http://www.casaninja.com/christi/academic/classicaltuning.html)
• Article on the history of absolute pitch to today's A=440 standard (http://www.uk-piano.org/history/pitch.html)
• Tuning for beginners (http://www.microtonal.co.uk/start.htm)
• A beginner's guide to temperament (http://www.users.dircon.co.uk/~oneskull/3.6.04.htm)
• Understanding Temperaments (http://pages.globetrotter.net/roule/temper.htm) - a concise, but comprehensive explanation of tempraments
• How to tune a few popular historic temperaments (http://www.users.bigpond.com/hpschd/tech/tech09.html) - instructions on how to tune a keyboard in various temperaments
• Temperament: The Idea That Solved Music's Greatest Riddle (http://www.thenewrepublic.com/121001/franklin121001_print.html)
• Temperament resources on the Web (http://pages.globetrotter.net/roule/stimm.htm) - lots more links
• Albert Gräf: Musical scale rationalization - a graph-theoretic approach (http://www.musikwissenschaft.uni-mainz.de/Musikinformatik/schriftenreihe/) (the paper is number 45 on the list, and is in .pdf format)

Common misspelling and questions (FAQ)

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