**Meantone Temperaments**

Though quarter-comma meantone is the most common type, other systems that flatten the fifth by some amount, but that still equate the major whole tone (9/8 in just intonation) with the minor whole tone (10/9 in just intonation), are also called meantone systems. Since (9/8) / (10/9) = (81/80)—the syntonic comma—the fundamental characteristics of meantone systems are 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, 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: by what fraction (logarithmically) of a syntonic comma the fifth is being flattened (as above), what equal temperament has the meantone fifth in question, the width of the tempered perfect fifth in cents, or the ratio of the whole tone to the diatonic semitone. This last 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.

In these terms, some historically notable meantone tunings are listed below. The relationship between the first two columns is exact, while that between them and the third is closely approximate.

R | Size of the fifth in octaves | Fraction of a (syntonic) comma |
---|---|---|

9/4 | 31/53 | 1/315 (nearly Pythagorean Tuning) |

2/1 | 7/12 | 1/11 (1/12 Pythagorean comma) |

9/5 | 32/55 | 1/6 |

7/4 | 25/43 | 1/5 |

5/3 | 18/31 | 1/4 |

8/5 | 29/50 | 2/7 |

3/2 | 11/19 | 1/3 |

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