Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


meantone


    a system of tuning in which the two JI "whole tones" (with ratios of 9:8 and 10:9) are conflated into one "mean tone" which lies between the two, the objective being to produce or approximate JI "major 3rd"s (with a ratio of 5:4) but to eliminate the syntonic comma and its associated problem of commatic drift by having only one size of whole-tone.

    The "5th" is flatter (by a fraction of a syntonic comma) than the just 3:2, and this is normally how the tuning is described. Thus the names "Quarter-comma meantone", "2/9-comma meantone", etc.

    [from Joe Monzo, JustMusic: A New Harmony]

    [I had originally stated that the '5th' is flattened slightly in order to produce a closed system where a "cycle of 5ths" eventually returns to the origin. Paul Erlich corrected me, saying that ETs create a closed system, but most meantones do not.]

    (see also my paper explaining the derivation of W. S. B. Woolhouse's optimal meantone.)

    ...............................

    [from John Chalmers:]

    ...the definition of meantone could be tightened up a bit as it implies that all meantone-like tunings have 5/4 major thirds. I would describe meantone [that is, meantone proper] as the temperament whose fifth is equal to the fourth root of 5 and is thus 1/4 of a syntonic comma flatter than 3/2.

    Meantone-like temperaments are those cyclic systems which have fifths flatter than 3/2 by some fraction (rational or irrational) of the syntonic comma and which form their major thirds by four ascending fifths reduced by two octaves. Such tunings divide the ditone (which is here equivalent to the major third) into two equal "mean tones" Well-known examples include the 1/3-comma (just 6/5 and 5/3), the 1/5-comma (just 15/8, 16/15), and John Harrison's tuning whose major third equals 2(1/pi) and which has been revived and promoted in an extended form by Charles Lucy.

    Certain meantone-like temperaments are audibly equivalent to equal temperaments of the octave. For example, the 1/4-comma system corresponds to 31-tet and the 1/3 comma to 19-tet. Arbitrarily close equivalences may be found by using continued fractions or Brun's Algorithm.

    The upper limit of meantone-like fifths is the fifth of 12-tone ET (700 cents) and the upper limit of meantone-like major thirds is the major third of 12-tet (400 cents).

    (While Pythagorean tuning could be considered as a meantone-like system in which the tempering fraction is zero, temperaments whose fifths are larger than 700 cents are best thought of as positive systems, which make their major thirds by a chain of 8 descending fifths (or ascending fourths). These major thirds are formally diminished fourths (e.g., C-Fb). Other relations occur as the fifth become still sharper (9 fifths up, augmented second, C-D#).)

    [from John Chalmers, personal communication]

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[from Paul Erlich:]

From: Paul H. Erlich
Sent: Monday, December 20, 1999 7:24 PM
To: 'tuning@onelist.com'
Subject: Summary of optimal meantone tunings

Here is a summary of the meantone tunings (giving the size of fifth in cents, the fraction of the syntonic comma by which the fifth is reduced, and the first known advocate or reference to a TD posting by me from Brett Barbaro's e-mail) that are optimal under various error criteria for the three "classical" consonant interval classes: the one 3-limit interval, the p4/p5; and the two 5-limit intervals, the m3/M6 and the M3/m6.

* For some cautionary annotations by Margo Schulter concerning Aron and Vicentino in this table, see her posting in Onelist Tuning Digest # 532, message 12.


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