Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


Equal Temperament


    a system of tuning based on a scale whose "steps" or degrees have logarithmically equal intervals between them, in contrast to the differently-spaced degrees of just intonation, meantone, well-temperament, or other tunings.

    Usually, but not always, equal temperaments assume octave-equivalence, of which the usual 12-EQ is the most obvious example.

    Examples of non-octave equal temperaments are Gary Morrison's 88CET (88 cents between degrees), and Wendy Carlos's alpha, beta, and gamma scales [listen to them here].

    Generally abbreviated ET.

    Below is a table showing advocates of various ETs with approximate dates. It does not claim to be complete. (click on the highlighted numbers to show more detail about those ETs)

    ET Date and Theorist/composer
    7 traditional Thai music
    9 Sunda: mapping of 3 pathet onto 7-out-of-9-equal
    10 ?? Elaine Walker
    12 1596 Prince Chu Tsai-yü
    1597 Simon Stevin
    1802 Georg Joseph Vogler
    1817 Gottfried Weber
    1900-1999 the predominant tuning of the 'developed' world
    15 1951 Augusto Novarro
    1991 Easley Blackwood
    16 1971 David Goldsmith
    17 1809 Villoteau (describing Arabic tuning)
    1929 Malherbe
    1935 Karapetyan
    1960s Ivor Darreg
    1999 Margo Schulter
    18 1960s Ivor Darreg
    19 1558 Guillaume Costeley
    1577 Salinas (19 notes of '1/3-comma meantone', almost identical to 19-ET)
    <1633 Jean Titelouze ('third-tones' may describe 19-ET)
    1835 Wesley Woolhouse (the most practical approximation of his 'optimal meantone')
    1852 Friedrich Opelt
    1911 Melchiorre Sachs
    1921 José Würschmidt
    1925 Ariel
    1930 Kornerup
    1932 Joseph Yasser
    1961 M. Joel Mandelbaum
    1960s Ivor Darreg
    1979 Yunik & Swift
    1996 Neil Haverstick
    1990s Elaine Walker
    1990s Jonathan Glasier
    1998 Joe Monzo
    20 1980 Gerald Balzano
    ?? Paul Zweifel
    22 (some older theories describe the Indian sruti system as this)
    1877 Bosanquet
    1921 José Würschmidt (for the future, after 19 runs its course)
    (1960s Erv Wilson used modulus-22, not necessarily ET)
    1960s Ivor Darreg
    1993 Paul Erlich
    24 1906 Richard H. Stein (first published 24-tET scores)
    1916 Charles Ives
    1917 Willi von Möllendorff
    1920 Alois Hába (and subsequently many of his students)
    1924 Julian Carrillo
    1933 Ivan Wyschnegradsky
    1950s Giacinto Scelsi (very loosely-conceived intonation)
    1980s Brian Ferneyhough (very loosely-conceived intonation)
    1983 Leo de Vries
    1994 Joe Monzo
    26 1998 Paul Erlich
    31 1555 Nicola Vicentino (31 notes of extended meantone - 1/4-comma? nearly identical to 31-ET)
    1606 Gonzaga (31 notes of extended meantone - 1/4-comma? nearly identical to 31-ET)
    <1618 Scipione Stella (31 notes of extended meantone - 1/4-comma? nearly identical to 31-ET)
    1618 Fabio Colonna (31 notes of extended meantone - 1/4-comma? nearly identical to 31-ET)
    1623 Daniel Hizler (used only 13 out of 31-ET in practice)
    1666 Lemme Rossi
    1691 Christiaan Huygens
    1722 Friedrich Suppig
    1725 Ambrose Warren
    1739 Quirinus van Blankenburg (as a system of measurement)
    1754 J. E. Gallimard
    1818 Pierre Galin
    1917-19 P. S. Wedell (quoted by Kornerup)
    1930 Thorvald Kornerup
    1932 Joseph Yasser (for the future, after 19 runs its course)
    1941 Adriaan Fokker
    1947 Mart. J. Lursen
    1950s Henk Badings (and many other Dutch composers)
    (1960s Erv Wilson used modulus-31, not necessarily ET)
    1967 Alois Hába
    1974 Sebastian von Hörner
    1980s Brian Ferneyhough (very loosely-conceived intonation)
    1989 John Bischoff and Tim Perkis
    1999 Paul Erlich
    34 Larry Hanson
    1997 Niel Haverstick
    36 1907 Ferrucio Busoni
    1923 Alois Hába
    41 1901 von Janko
    (1960s Erv Wilson claims that Partch was intuitively feeling out 41-ET)
    1998 Patrick Ozzard-Low
    43 1701 Joseph Sauveur (nearly identical to 1/5-comma meantone)
    46 1998 Graham Breed
    48 1924 Julian Carillo
    ?? Barbieri
    50 (nearly identical to 2/7-comma meantone, the first meantone to be described with mathematical exactitude, in 1562 by Zarlino)
    1710 Konrad Henfling
    1759 Robert Smith (as an approximation to his ideal 5/18-comma meantone system)
    1835 Wesley Woolhouse (almost identical to his 7/26-comma 'optimal meantone')
    53 400s BC Implied by Philolaus (disciple of Pythagoras)
    200s BC King Fang
    1608 Nicolaus Mercator (only as a system of measurement, not intended to be used on an instrument)
    1650 Athanasius Kircher
    1713 Confirmed as official scale in China
    1874-75 R. H. M. Bosanquet
    1890 Shohé Tanaka
    55 <1722 Johann Beer
    <1748 Georg Philip Telemann (nearly identical to 1/6-comma meantone)
    1755 Joseph Saveur, and Estève (nearly identical to 1/6-comma meantone)
    58 1770 Dom François Bedos de Celles
    68 1847 Meshaqah (describing modern Greek tuning)
    1989? John Chalmers (describing Byzantine tuning)
    72 1927 Alois Hába
    1951 Augusto Novarro
    1953 Ivan Wyschnegradsky
    1970 Ezra Sims
    1970 Franz Richter Herf
    1970s? Joe Maneri (and subsequently many of his students)
    1980s? James Tenney
    1999 Paul Erlich
    1999 Joe Monzo
    74 1762 Riccati (approximation to 3/14-comma meantone)
    1855 Drobisch (approximation to 2/9-comma meantone)
    76 1998 Paul Erlich (as a unified tuning for various tonal systems)
    96 1924 Julian Carillo
    118 1874-5 Bosanquet
    144 1946 Joseph Schillinger
    1999 Dan Stearns (chiefly for its value in notating several ET systems accurately)
    1999 Joe Monzo (chiefly for its value in notating several ET systems accurately)
    171 (date Groven - approximated by his 1/8-skhisma temperament)
    1926 Perrett
    1975 Martin Vogel
    301 1701 Joseph Sauveur (for ease of calculation with logs before calculators and computers were avaiable: log(2)~=0.301; and because 301 is divisible by 43)
    <1835 J. W. F. Herschel
    318 1999 Joe Monzo (in analyzing Aristoxenus)
    612 <1875 Captain J. Herschel (cited by Bosanquet)
    730 1835 Wesley Woolhouse (his unit of measurement)
    1200 1875 Alexander Ellis (his unit of measurement, called cents, 100 per 12-tET semitone)
    49152 1980s MIDI pitch-bend unit of increment, 213 (= 4096) units per 12-tET semitone

    Note that Easley Blackwood's Microtonal Etudes contain one etude for each ET from 13 thru 24, and that Ivor Darreg (in the 1970s and 80s) and Brian McLaren (in the 1990s) composed pieces for every ET between 5 and 53.

    [from Joe Monzo, JustMusic: A New Harmony. Thanks to John Chalmers, Manuel op de Coul, Margo Schulter, and especially Paul Erlich, for helpful criticism and additional info.]

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

    Any tuning system which divides the octave (2/1) into n aliquot parts is termed an n-tone Equal Temperament. Mathematically, an Equal Temperament is a geometric series and each degree is a logarithm to the base 2n.

    [Note: the base is 2 only in 'octave'-equivalent equal-temperaments. It is possible to construct an equal temperament using any number as a base, as noted below. An example would be to divide the 'perfect 12th', which has the ratio 3:1, into equal 'steps'; this is a geometric series where each degree is a logarithm to the base 3n. -Monzo]

    Because of the physiology of the human auditory system, the successive intervals of Equal Temperaments sound perceptually equal over most of the audible range.

    It is also possible to divide intervals other than the octave as in the recent work of Wendy Carlos (Carlos,1986), but musical examples are still rather uncommon.

    [from John Chalmers, Divisions of the Tetrachord]


(to download a zip file of the entire Dictionary, click here)

  • For many more diagrams and explanations of historical tunings, see my book.
  • If you don't understand my theory or the terms I've used, start here
  • I welcome feedback about this webpage:
    corrections, improvements, good links.
    Let me know if you don't understand something.


    return to the Microtonal Dictionary index
    return to my home page
    return to the Sonic Arts home page