# Meantone temperament

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Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in a meantone, each fifth (except for the wolf fifth) is narrowed by the same amount relative to its width in Just Intonation. The meantone temperament:

• generates all non-octave intervals from a stack of tempered perfect fifths; and
• by choosing an appropriate size for the perfect fifths, tempers the syntonic comma to unison.

Quarter-comma meantone is the best known type of meantone temperament, and the term meantone temperament is often used to refer to it specifically.

## Contents

### Meantone temperaments

Though quarter-comma meantone is the most common, other systems which flatten the fifth by differing amounts but which still equate the major whole tone, which in just intonation is 9/8, with the minor whole tone, tuned justly to 10/9, are also called meantone systems. Since (9/8) / (10/9) = (81/80), the syntonic comma, the fundamental character of a meantone tuning is that all intervals are generated from fifths, and the syntonic comma is tempered to a unison.

All meantone temperaments fall on the syntonic temperament's tuning continuum,[1] and as such are "syntonic tunings." The distinguishing feature of each unique syntonic tuning is the width of its generator in cents, as shown in the central column of Figure 1. Historically-notable meantone temperaments, discussed below, occupy a narrow portion of the syntonic temperament's tuning continuum, ranging from approximately 695 to 699 cents. The criteria which define the limits (if any) of the meantone range of tunings within the syntonic temperament's tuning continuum are not yet well-defined.

While the term meantone temperament refers primarily to the tempering of 5-limit musical intervals, optimum values for the 5-limit also work well for the 7-limit, defining septimal meantone temperament. In Figure 1, the valid tuning ranges of 5-limit, 7-limit, and 11-limit syntonic tunings are shown, and can be seen to include many notable meantone tunings.

Meantone temperaments can be specified in various ways. We can, as above, specify by what fraction (logarithmically) of a syntonic comma the fifth is being flattened, what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or what the ratio of the whole tone to the diatonic semitone is. This ratio was termed "R" by American composer, pianist and theoretician Easley Blackwood, but in effect has been in use for much longer than that. It is useful because it gives us an idea of the melodic qualities of the tuning, and because if R is a rational number N/D, so is (3R+1)/(5R+2) or (3N+D)/(5N+2D), which is the size of fifth in terms of logarithms base 2, and which immediately tells us what division of the octave we will have. If we multiply by 1200, we have the size of fifth in cents.