equal-temperament

the smallest cardinality EDO which has any real musical usefulness, some theorists describe Indonesian slendro scale as this

2001 -- Herman Miller

the "whole-tone scale"

1787 -- Wolfgang A. Mozart (in his A Musical Joke)

1894 -- Claude Debussy

1902 -- Arnold Schönberg

traditional Thai music

1991 -- Clem Fortuna

1997 -- Randy Winchester

2001 -- Robert Walker

1980 -- Gordon Mumma (Octal Waltz for harpsichord)

1981 -- Daniel Wolf

1997 -- Randy Winchester

early 1900s -- Charles Ives (in Monzo, Ives "stretched" scales)

1930s-60s -- R. M. A. Kusumadinata (Sunda: mapping of 3 pathet onto 7-out-of-9-equal)

19?? -- James Tenney (piano part, The Road to Ubud)

1930s-60s -- R. M. A. Kusumadinata (Sunda)

1990s -- Elaine Walker

1978 -- Gary Morrison

1997 -- Randy Winchester

1998 -- William Sethares

1996 -- Daniel Wolf

semitone or "half-step"

before 3000 BC -- a possible Sumerian tuning (according to Monzo, Speculations on Sumerian Tuning)

1584 -- Prince Chu Tsai-yü (China)

1585 -- Simon Stevin (Netherlands)

1636 -- Marin Mersenne

c.1780-1828 -- Mozart, Beethoven, and Schubert compose many chord progressions which, by the use of enharmonic equivalence of the diesis, strongly imply 12-edo or a related 12-tone "circulating" well-temperament

1802 -- Georg Joseph Vogler

1817 -- Gottfried Weber

1900-1999 -- the nearly universal tuning of the 'developed' world

1911 -- Arnold Schönberg (along with his personal rejection of microtonality)

1962 -- Ernst Krenek (opera Ausgerechnet und Verspielt, op. 179)

1991 -- Paul Rapoport

1998 -- Herman Miller

1999 -- Dan Stearns

2001 -- X. J. Scott

1990 -- Ralph Jarzombek

2000 -- Herman Miller

1930s-60s -- R. M. A. Kusumadinata (Sunda)

1951 -- Augusto Novaro

1983 -- Joe Zawinul, on Molasses Run from Weather Report album Procession

1991 -- Easley Blackwood

1991 -- Clem Fortuna

1996 -- Herman Miller

1997 -- Randy Winchester

1998 -- Paul Erlich, with the group MAD DUXX (link to .ram audio file)

2001 -- Francesco Caratolozzo

1930s-60s -- R. M. A. Kusumadinata (Sunda)

1971 -- David Goldsmith

1993 -- Steve Vai

1997 -- Randy Winchester

1998 -- Herman Miller

2002 -- Victor Cerullo

1653 -- Brouncker

1809 -- Villoteau (describing Arabic tuning)

1929 -- Malherbe

1935 -- Karapetyan

1960s -- Ivor Darreg

1997 -- Herman Miller

1999 -- Margo Schulter (as a pseudo-Pythagorean tuning)

1907 -- Ferrucio Busoni (in his theory, but not used in his compositions)

1940s -- Julián Carrillo, 1/3rd-tone piano

1960s -- Ivor Darreg

third-tone (the tuning normally meant by that term)

1558 -- Guillaume Costeley

1577 -- Salinas (19 notes of '1/3-comma meantone', almost identical to 19-ET)

before 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

1922 -- Thorwald Kornerup

1925 -- Ariel

1926 -- Jacques Handschin

1932 -- Joseph Yasser

1940s -- Tillman Schafer

1961 -- M. Joel Mandelbaum

1960s -- Ivor Darreg

1976 -- Henri Pousseur: Racine 19e de 8/4, pour violoncelle seul

1979 -- Yunik & Swift

1979 -- Jon Catler

19?? -- Matthew Puzan

198? -- Erik Griswold

1987 -- Herman Miller

1996 -- Neil Haverstick

1990s -- Elaine Walker

1990s -- Jonathan Glasier

1990s -- William Casey Wesley

1998 -- Joseph Monzo (19-tone Samba)

1999 -- John Starrett

2004 -- Aaron Krister Johnson

1980 -- Gerald Balzano

1996 -- Paul Zweifel

1999 -- Herman Miller

2001 -- Herman Miller

(some older theories describe the Indian sruti system as this -- an interpretation now considered erroneous; it is now recognized that the steps are unequal)

1877 -- Bosanquet

1921 -- José Würschmidt (for the future, after 19 runs its course)

(1960s -- Erv Wilson -- used modulus-22, not necessarily EDO)

1960s -- Ivor Darreg

1980 -- Morris Moshe Cotel

1993 -- Paul Erlich

1997 -- Steve Rezsutek -- customized guitars and a keyboard for Paul Erlich's 22edo scales

1997 -- Randy Winchester

1999 -- Herman Miller

1999 -- Peter Blasser

2000 -- Alison Monteith

some theorists describe Indonesian pelog scale as subset of this

1920s -- Hornbostel (describing Burmese music)

quarter-tone: 2¹ (= 2) quarter-tones per Semitone; 12 * 2¹ = 24 quarter-tones per octave. Also called enamu (1mu), a MIDI pitch-bend unit.

1760 - Charles Delusse: Air a la Greque (earliest notation of microtonal pitches in the "common-practice" era)

1849 - Fromental Halévy - his cantata Prométhée enchaîné uses quarter-tones between B:C and E:F, to revive the ancient enharmonic genus.

1890s -- John Foulds

1895 -- Julián Carrillo: String Quartet

1906 -- Richard H. Stein (first published 24-tET scores)

1906 -- Arnold Schönberg (schematic sketch, no extant compositions)

1908 -- Anton Webern (early drafts of two songs)

1916 -- Charles Ives

1917 -- Willi von Möllendorff

1918 -- Jörg Mager

1920 -- Alois Hába (and subsequently many of his students)

1924 -- Julián Carrillo

1932 -- adopted as standard tuning in Egypt and elsewhere in Arabic world

1933 -- Ivan Wyschnegradsky (Treatise on Quartertone Harmony)

1940s -- Julián Carrillo, 1/4th-tone piano

1941 -- Mildred Couper

1948 -- Pierre Boulez (original version of Le Soleil des Eaux)

1950s -- Giacinto Scelsi (very loosely-conceived intonation)

1967 -- Tui St. George Tucker

1969 -- Györgi Ligeti (Ramifications)

1960s-2000s -- John Eaton

1980s -- Brian Ferneyhough (very loosely-conceived intonation)

1983 -- Leo de Vries

1994 -- Joseph Monzo, 24-eq tune

1994 -- Paul Rapoport

1998 -- Paul Erlich

1998 -- Herman Miller

2001 -- Gene Ward Smith

2001 -- Herman Miller

1997 -- Paul Erlich (for music based on the diminished scale)

by 1875 -- Émile Chevé (by mistake)

1940s -- Julián Carrillo, 1/5th-tone piano

diesis (one of several meanings of that term)

1555 -- Nicola Vicentino (31 notes of extended meantone nearly identical to 31-ET)

1606 -- Vito Trasuntino (31 notes of extended meantone nearly identical to 31-ET)

1606 -- Gonzaga (31 notes of extended meantone nearly identical to 31-ET)

before 1618 -- Scipione Stella (31 notes of extended meantone nearly identical to 31-ET)

1618 -- Fabio Colonna (31 notes of extended meantone 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

1860s -- Josef Petzval

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. Lürsen

1950s -- Henk Badings (and many other Dutch composers)

(1960s -- Erv Wilson used modulus-31, not necessarily ET)

1962 -- Joel Mandelbaum (opera Dybbuk)

1967 -- Alois Hába

1970s -- Dr. Abram M. Plum

1973 -- Leigh Gerdine

1974 -- Sebastian von Hörner

1975 -- George Secor

1979 -- Jon Catler

1980s -- Brian Ferneyhough (very loosely-conceived intonation)

1989 -- John Bischoff and Tim Perkis

1999 -- Paul Erlich

1979 -- Dirk de Klerk

before 1998 -- Larry Hanson

1997 -- Neil Haverstick

sixth-tone; 3 units per Semitone = 12 * 3 units per "octave"

1907 -- Ferrucio Busoni (in his theory, but not used in compositions)

1923-1960s -- Alois Hába

1940s -- Julián Carrillo, 1/6th-tone piano

1952 -- Henri Pousseur: Prospection, pour un piano-triple à sixièmes de ton

by 1997 -- Tomasz Liese -- 19-out-of-36-edo subset scale

2010 -- Joseph Monzo -- using 83 degrees to map prime-factor 5 (best mapping of 5 is to 84 degrees - exactly the same size as 12-edo).

2012 -- Joseph Monzo -- very strong approximation of factors 5,7,11,13 in 13-limit JI.

2004 -- tuning-math group, for situations where introducing a period of 1/2 or splitting the fifth into two over 19-equal might be useful. 2^(11/38) is almost precisely an 11/9.

1901 -- Paul von Jankó

(1960s -- Erv Wilson claims that Partch was intuitively feeling out 41-ET)

1975 -- George Secor

1989 -- Helen Fowler

1993 -- Joseph Monzo

1998 -- Carl Lumma

1998 -- Patrick Ozzard-Low

2002 -- Gene Ward Smith

1940s -- Julián Carrillo, 1/7th-tone piano

méride

1701 -- Joseph Sauveur - a unit of interval measurement, nearly identical to 1/5-comma meantone

1989 -- R. Fuller

1998 -- Graham Breed

2000 -- Dave Keenan and Paul Erlich

2002 -- Gene Ward Smith

doamu (2mu), a MIDI pitch-bend unit: 2² (= 4) 2mus per Semitone = 12 * 2² = 48 2mus per octave. Also called "eighth-tone".

early 1900s -- Charles Ives (according to Monzo, Ives stretched-octave scales)

1915 -- N. Kulbin

1924 -- Julián Carrillo

1940s -- Julián Carrillo, 1/8th-tone piano

19?? -- Patrizio Barbieri

19?? -- Claus-Steffen Mahnkopf

19?? -- Volker Staub

1998 -- Joseph Pehrson

(1558 -- Zarlino - fair approximation to 2/7-comma meantone, the first meantone to be described with mathematical exactitude)

1710 -- Konrad Henfling

1759 -- Robert Smith (as an approximation to his ideal 5/18-comma meantone system)

1835 -- Wesley Woolhouse (as practical approximation to his 7/26-comma 'optimal meantone')

1940s -- Tillman Schafer

mercator; nearly identical to pythagorean tuning and a very good approximation to 5-limit just intonation

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 -- 53-tone Pythagorean tuning became official scale in China)

1874-75 -- R. H. M. Bosanquet

1875 -- Alexander J. Ellis (appendix to Helmholtz, On the Sensations of Tone)

1890 -- Shohé Tanaka

c.1900 -- Standard Turkish music-theory

1911 -- Robert Neumann (quoted by Schönberg in Harmonielehre)

1927 -- Augusto Novaro

1978 -- Larry Hanson

2002 -- Gene Ward Smith

2005 -- Chris Mohr

1940s -- Julián Carrillo, 1/9th-tone piano

(good approximation to 1/6-comma meantone)

1711 -- Joseph Sauveur, "the system which ordinary musicians use"

before 1722 -- Johann Beer

1723 -- Pier Francesco Tosi

before 1748 -- Georg Philip Telemann

1748 -- Georg Andreas Sorge

1752 -- Johann Joachim Quantz

1755 -- Estève

1780s -- W. A. Mozart, subsets of up to 20 tones, for non-keyboard instruments (according to Monzo, Mozart's Tuning)

(1770 -- Dom François Bedos de Celles - according to Barbour, but this may be erroneous. See Yahoo tuning message 63618)

2002 -- Gene Ward Smith

5 units per Semitone = 12 * 5 units per "octave".

1940s -- Julián Carrillo, 1/10th-tone piano

1980s? -- Richard Boulanger

1927 -- Augusto Novaro

1951 -- J. Murray Barbour, as a 5-limit system.

2004 -- Gene Ward Smith, it is simultaneously a schismatic system and a semisixths (78732/78125 comma) system.

1940s -- Julián Carrillo, 1/11th-tone piano

1847 -- Meshaqah (describing modern Greek tuning)

1989? -- John Chalmers (describing Byzantine tuning)

twelfth-tone / moria; 6 units per Semitone = 12 * 6 units per "octave".

1800s -- standard quantization for Byzantine Chant

1927 -- Alois Hába (in his book Neue Harmonielehre)

1927 -- Augusto Novaro

1938-58 -- Evgeny Alexandrovich Murzin created a 72-tET synthesizer. Among composers to write for it: Andrei Volkonsky, Nikolai Nikolsky, Eduard Artemiev, Alexander Nemtin, Andrei Eshpai, Gennady Gladkov, Pyotr Meshchianinov, Stanislav Kreichi (see Anton Rovner's article in TMA).

1940s -- Julián Carrillo, 1/12th-tone piano

1953 -- Ivan Wyschnegradsky

1963 -- Iannis Xenakis (cf. his book Musiques formelles)

1970 -- Ezra Sims

1970 -- Franz Richter Herf

1970 -- Rolf Maedel

1970s-2000s -- Joe Maneri (and subsequently many of his students)

1980s? -- James Tenney

1990s -- Ted Mook

1999 -- Paul Erlich

1999 -- Joseph Monzo (as basis of simplified HEWM notation)

1999 -- Rick Tagawa

2001 -- Dave Keenan, Graham Breed, Joseph Pehrson, Paul Erlich, Joseph Monzo (for notation of miracle family scales)

2001 -- Julia Werntz

2002 -- Gene Ward Smith

1762 -- Riccati (approximation to 3/14-comma meantone)

1855 -- Drobisch (approximation to 2/9-comma meantone)

1991 -- John Cage, in "Ten" for chamber ensemble

1998 -- Paul Erlich (as a unified tuning for various tonal systems)

1940s -- Julián Carrillo, 1/13th-tone piano

2004 -- Gene Ward Smith - a strong 19-limit system. Chains of 80-edo fifths have been proposed for neo-Gothic or Arabic inspired music.

2002 -- Joseph Monzo, good approximation to Kornerup's "golden meantone"

2005 -- Gene Ward Smith, approximation to 5/19-comma meantone

7 units per Semitone = 12 * 7 units per "octave".

1940s -- Julián Carrillo, 1/14th-tone piano

1985 -- Harald Waage (for 5-limit just intonation)

1951 -- J. Murray Barbour, as a 5-limit system

1998 -- Paul Erlich - it is consistent and unique in the 13-odd-limit

2004 -- Gene Ward Smith, it supports kleismic tempering

2020 -- Jeff Brown

for most purposes, essentially the same as LucyTuning.

1775 -- John "Longitude" Harrison

1987 -- Charles Lucy

1940s -- Julián Carrillo, 1/15th-tone piano

unique up to 13-odd-limit, and consistent up to 23-odd-limit

1998 -- Paul Erlich

2015 -- Cam Taylor

triamu (3mu), a MIDI pitch-bend unit: 2³ (= 8) 3mus per Semitone = 12 * 2³ = 96 3mus per "octave". Also called "1/16-tone".

1924 -- Julián Carrillo

1940s -- Julián Carrillo, 1/16th-tone piano

1980 -- Pascale Criton

2001 -- Vincent-Olivier Gagnon

2004 -- Gene Ward Smith, 99-edo is significant as a 7-limit system, having commas of 2401/2400, 3136/3125 and 4375/4374. Its errors, well under 2 cents, are by some people's ears just enough to be pleasing.

1980s -- Barry Vercoe - built into CSound software

2 * 53 degrees per "octave"

2004 -- Joseph Monzo (in analyzing Philolaus's small intervals)

2004 -- Gene Ward Smith - strong 13-21-limit system

1874-5 -- Bosanquet

2004 -- Gene Ward Smith - twice 65, and important in the 7, 13 (and 15) limits in particular.

2004 -- Gene Ward Smith, mentioned in a manner of speaking by Edward Charles Titchmarsh in his book The Theory of the Riemann Zeta Function, where he discusses a high value of the Riemann zeta function which corresponds to the 140 division. It is an important 7-limit division.

farab; 12 units per Semitone = 12 * 12 units per "octave".

300s BC -- Aristoxenus (most likely interpretation of his theories)

early 900s -- Abu Nasr al-Farabi

1946 -- Joseph Schillinger

1999 -- Dan Stearns and Joseph Monzo (chiefly for its value as a unified notation for mixed EDOs and/or complex just intonation tunings)

1999(?) -- Paul Erlich, "Universal Tuning"

(19?? -- Eivend Groven - approximated by his 1/8-skhisma temperament)

1926 -- Perrett

1975 -- Martin Vogel

2002 -- Gene Ward Smith

2002 -- Gene Ward Smith

tetramu (4mu), a MIDI pitch-bend unit: 2⁴ (= 16) 4mus per Semitone = 12 * 2⁴ = 192 4mus per octave.

16 ²/₃ degrees per Semitone

2002 -- Joseph Monzo (in analyzing Werckmeister III)

meme

2001 -- Aaron Hunt: 205 = 41 x 5 = [(7 x 6) - 1] x 5 = (12 x 17) + 1; used as the basis tuning for his Tonal Plexus microtonal keyboard

7 * 31 degrees per octave = 18 ¹/₁₂ degrees per Semitone

2002 -- Joseph Monzo (proposed for adaptive-JI tuning of Mahler's compositions)

2002 -- Bob Wendell (for quantification of just intonation to facilitate composing in a polyphonic blues style)

2002 -- George Secor & Dave Keenan (as a basis for notation for JI and multi-EDOs)

2004 -- Gene Ward Smith, it is important in the 13-limit in particular.

2004 -- George Secor, as an important part of Sagittal notation

tredek; named by Joseph Monzo in accordance with its excellent approximation of 13-limit JI

1970s? -- Erv Wilson and John Chalmers

1997 -- Paul Hahn

2013 -- Joseph Monzo -- for use as a unit of interval measurement without need for decimal places, strongest 3-digit EDO for 13-limit JI.

early 1900s -- Charles Ives (as analyzed in Monzo, Ives stretched octave scales)

25 units per Semitone = 12 * 25 units per "octave".

1800s -- system of savarts

1701 -- Joseph Sauveur - heptameride (for ease of calculation with logs: log(2)~=0.301; and because 301 is divisible by 43)

before 1835 -- Captain J. W. F. Herschel (cited by Woolhouse)

gene; named by Joseph Monzo both in honor of Gene Ward Smith and for its connotation of a basic biological unit

2004 -- Gene Ward Smith - this remarkable division is important in the 13 through 41 limits, in every one of those odd limits. As a generic way of representing what some might maintain is anything anyone could reasonably want to represent it is of interest.

2007 -- Joseph Monzo -- for use as a unit of interval measurement without need for decimal places, very strong thru 41-limit JI.

1999 -- Joseph Monzo (in analyzing Aristoxenus: 318 = 53*6)

pentamu (5mu), a MIDI pitch-bend unit: 2⁵ (= 32) 5mus per semitone, 2⁵ * 12 = 384 5mus per octave.

2004 -- Gene Ward Smith - a very strong 5 or 7 limit system, along with 612 a good way to tune ennealimmal.

2004 -- Gene Ward Smith - strong 11-15 limit system, and still good as a 17-limit system.

2⁹ units per octave.

1980s -- tuning resolution of some electronic instruments, notably Ensoniq VFX and VFX-SD.

dexl, named by Joseph Monzo after its Roman numeral (DXL).

2023 -- Joseph Monzo, as a replacement for cents to measure 43-limit JI

2013 -- Scott Dakota

2015 -- Cam Taylor

2016 -- Joseph Monzo -- for use as a unit of interval measurement without need for decimal places, strongest 3-digit EDO for 23-limit JI.

50 units per Semitone = 12 * 50 units per octave.

1898 -- Widogast Iring -- "iring" unit of interval measurement

1932 -- Joseph Yasser "centitone" unit of interval measurement

51 (= 3 * 17) units per semitone = 2² * 3² * 17 units per octave; an excellent unit of interval measurement for 11-limit JI.

before 1875 -- Captain J. W. F. Herschel (cited by Bosanquet)

1917 -- Josef Sumec

c.1970 -- Gene Ward Smith -- for interval measurment, an analogue of cents

2002 -- Joseph Monzo (in analyzing Werckmeister III)

(a remarkably close approximation to pythagorean tuning)

before 1975 -- Jacques Dudon

1980s? -- Marc Jones -- see satanic comma

1835 -- Wesley Woolhouse -- his unit of measurement for 5-limit JI, and an analogue of cents; 60 ⁵/₆ degrees per Semitone.

hexamu (6mu), a MIDI pitch-bend unit: 2⁶ (= 64) 6mus per semitone, 2⁶ * 12 = 768 6mus per octave.

1980s-2000s -- Tuning resolution of many electronic instruments, including several by Yamaha, Emu, and Ensoniq; also the resolution of some early sequencer software, including Texture.

1980s-2000s -- Joseph Monzo (using Texture software in 1980s, then using computer soundcards with 6mu resolution in 1990s and 2000s.)

1980s-2000s -- myriad artists using MIDI hardware.

2003 -- Joseph Monzo -- proposed as de facto hardware tuning standard

millioctave, an interval measurement, an analogue of cents: 1000 = 2³ * 5³ = 83 ¹/₃ units per Semitone.

1980s -- Csound software: its "oct" pitch format

1993 -- Mark Lindley (in his book Mathematical Models of Musical Scales)

2¹⁰ (= 1024) units per octave = 85 ¹/₃ units per Semitone; an analogue of cents.

1980s -- Tuning resolution for many synthesizers with tuning tables, including the popular Yamaha DX, SY and TG series

1990-95 -- Joseph Monzo (tuning resolution of Yamaha TG-77)

1875 -- Alexander Ellis (his unit of measurement, called cents, 100 per 12-tET semitone)

1980s-2000s -- many synthesizers and soundcards with 1-cent resolution give a 768-out-of-1200-edo subset tuning.

heptamu (7mu), a MIDI pitch-bend unit; 2⁷ = 128 7mus per Semitone; 12 * 2⁷ = 1536 7mus per octave

2002 -- Margo Schulter (for interval measurement, called "iota")

19?? -- Paul Beaver (rendered as 12³)

mina (short for "schismina"); 233-EDA (233 equal divisions of the apotome); quite close to ¹/₂ cent.

2004 -- The largest ET that can be notated in the Sagittal notation system.

2004 -- Gene Ward Smith, George Secor, Dave Keenan

oktamu (8mu), a MIDI pitch-bend unit: 2⁸ (= 256) 8mus per Semitone; 12 * 2⁸ = 3072 8mus per octave.

1990s -- Apple's QuickTime Musical Instruments tuning spec

2004 -- Gene Ward Smith - a strong 7 or 9 limit system, but mentioned here because it is 5^5, which might be useful for something.

2007 - Joseph Monzo - advocated as a unit of interval measurement for 7-limit JI.

358 units per semitone

1992 -- Marc Jones (used as most convenient UHT [ultra-high temperament] to measure 5-limit just intonation intervals)

enneamu (9mu), a MIDI pitch-bend unit: 2⁹ (= 512) 9mus per Semitone; 12 * 2⁹ = 6144 9mus per octave.

tina; 809-EDA (809 equal divisions of the apotome)

2007 -- Joseph Monzo -- for use as a unit of interval measurement without need for decimal places, strong thru 31-limit JI and also good for 41.

1965 -- M. Ekrem Karadeniz -- his unit of measurement, called türk-sents, 200 units per 53-edo comma.

dekamu (10mu), a MIDI pitch-bend unit: 2¹⁰ (= 1024) 10mus per Semitone; 12 * 2¹⁰ = 12288 10mus per octave.

endekamu (11mu), a MIDI pitch-bend unit: 2¹¹ (= 2048) 11mus per Semitone; 12 * 2¹¹ = 24576 11mus per octave.

jot

1864 -- Augustus De Morgan -- his unit of measurement; chosen because of its closeness to log₁₀(2) * 100,000.

2007 -- Gene Ward Smith, Joseph Monzo -- for use as a unit of interval measurement which is both strong (i.e., low logflat badness) and consistent thru 41-limit JI.

(198? -- approximation to John Brombaugh's scale of temperament units.)

flu -- useful for discussing 5-limit tempering.

2004 -- Gene Ward Smith - The "Diophantine clarity" division: pythagorean-comma = 900 flus, syntonic-comma ("Didymus comma") = 825 flus, therefore schisma = 75 flus. The flu system tempers the atom out of the discussion. Gene recommeds it as a replacement for Tuning Units.

dodekamu (12mu), a MIDI pitch-bend unit: 2¹² (= 4096) 12mus per Semitone; 12 * 2¹² = 49152 12mus per octave; formerly called cawapu.

1980s -- pitch-bend resolution of Cakewalk^TM and many other popular sequencer programs.

5587-EDA (5587 equal divisions of the apotome)

2007 -- Joseph Monzo -- for use as a unit of interval measurement without need for decimal places, strong and consistent thru 41-limit JI.

tridekamu (13mu), a MIDI pitch-bend unit: 2¹³ (= 8192) 13mus per Semitone; 12 * 2¹³ = 98304 13mus per octave.

1983 -- the maximum resolution possible in MIDI pitch-bend

tetradekamu (14mu), a MIDI pitch-bend unit: 2¹⁴ (= 16384) 14mus per Semitone; 12 * 2¹⁴ = 196608 14mus per octave; formerly called midipu.

1983 -- finest possible resolution in the MIDI tuning Spec.

1999 -- MTS (MIDI tuning standard)

Notes:

• Easley Blackwood's Microtonal Etudes contain one etude for each ET from 13 thru 24;
• Ivor Darreg (in the 1970s and 80s) and Brian McLaren (in the 1990s) composed pieces for every ET between 5 and 53;
• Dan Stearns and Marc Jones (in the 1990s and 2000s) have composed in numerous different ETs, often mixing several of them in the same piece;
• Gene Ward Smith has explored a wide variety of equal (and unequal) temperaments in his compositions.