a system of tuning based on a scale whose "steps" or degrees have logarithmically equal intervals between them, in contrast to the differentlyspaced degrees of just intonation, fractionofacommameantone, welltemperament, or other tunings. Generally abbreviated as ET.
Usually, but not always, equal temperaments assume octaveequivalence, of which the usual 12edo is the most obvious example. For many theorists the preferred abbreviation or these types of temperaments is EDO, for which some other theorists substitute ED2; both of these specify that it is the 2:1 ratio which is to be equally divided.
Below are some graphics by Paul Erlich, which show the amount of error for various EDOs for the basic concordant intervals in the 5limit. The farther a point is from a given axis, the larger the errors in the tuning corresponding to the point, of the intervals corresponding to the axis. The basic concept is the same as that of Dalitz plots in physics, and the Chalmers tetrachord plots (see diagrams #3 and #4 on that page).
LEGEND:
mouseover the following links to zoom in to the desired scaling:
paul's originals zoom: 1 zoom: 10 zoom: 100 zoom: 1000 zoom: 10000 
negatives zoom: 1 zoom: 10 zoom: 100 zoom: 1000 zoom: 10000 
Below is a table listing each of the linear temperaments depicted in the diagrams above (in order of decreasing distance from the origin), and their associated vanishing commas. (Thanks to Carl Lumma for the original version of this table, and Paul Erlich for the current version)
temperament name(s) 
sample realizations {ET\per.\gen.} 
[2 3 5] map in terms of [per., gen.] 
[period, optimal gen.] (cents) 
optimum RMS error (cents) 
comma name(s) 
comma [2 3, 5> monzo 
comma ratio 
comma ~cents 
father 
{3\3\1}, {5\5\2}, {8\8\3} 
[1,0] [2,1] [2,1] 
[1200, 442.179356] 
45.614107 
diatonic semitone 
[4 1, 1>  16 / 15  111.7 
beep 
{4\4\1}, {5\5\1}, {9\9\2} 
[1,0] [2,2] [3,3] 
[1200, 268.056439] 
35.60924  large limma  [0 3, 2>  27 / 25  133.2 
dicot 
{3\3\1}, {4\4\1}, {7\7\2}, {10\10\3} 
[1,0] [1,2] [2,1] 
[1200, 350.977500] 
28.851897 
minor chroma, classic chromatic semitone 
[3 1, 2>  25 / 24  70.67 
pelogic 
{7\7\3}, {9\9\4}, {16\16\7}, {23\23\10} 
[1,0] [2,1] [1,3] 
[1200, 522.862346] 
18.077734 
major chroma, major limma, limma ascendant 
[7 3, 1>  135 / 128  92.18 
blackwood 
{5\1\0}, {10\2\1}, {15\3\1}, {25\5\2} 
[5,0] [8,0] [12,1] 
[240, 84.663787] 
12.759741 
limma, pythagorean minor2nd 
[8 5, 0>  256 / 243  90.22 
diminished, 'octatonic' 
{4\1\0}, {8\2\1}, {12\3\1}, {16\4\1}, {28\7\2} 
[4,0] [6,1] [9,1] 
[300, 94.134357] 
11.06006  major diesis  [3 4, 4>  648 / 625  62.57 
augmented, diesic 
{3\1\0}, {9\3\1}, {12\4\1}, {15\5\1}, {18\6\1}, {27\9\2}, {39\13\3}, {42\14\3} 
[3,0] [5,1] [7,0] 
[400, 91.201856] 
9.677666 
diesis, great diesis, minor diesis 
[7 0, 3>  128 / 125  41.06 
porcupine 
{7\7\1}, {8\8\1}, {15\15\2}, {22\22\3}, {29\29\4}, {37\37\5}, {59\59\8} 
[1,0] [2,3] [3,5] 
[1200, 162.996026] 
7.975801 
maximal diesis 
[1 5, 3>  250 / 243  49.17 
negri 
{9\9\1}, {10\10\1}, {19\19\2}, {28\28\3}, {29\29\3} 
[1,0] [2,4] [2,3] 
[1200, 126.238272] 
5.942563    [14 3, 4>  16875 / 16384  51.12 
magic (5limit) 
{3\3\1}, {16\16\5}, {19\19\6}, {22\22\7}, {25\25\8}, {35\35\11}, {41\41\13}, {60\60\19}, {63\63\20}, {79\79\25} 
[1,0] [0,5] [2,1] 
[1200, 379.967949] 
4.569472  small diesis  [10 1, 5>  3125 / 3072  29.61 
meantone, 'diatonic' 
{5\5\2}, {7\7\3}, {12\12\5}, {19\19\8}, {26\26\11}, {31\31\13}, {43\43\18}, {50\50\21}, {55\55\23}, {74\74\31}, {81\81\34} {131\131\55} 
[1,0] [2,1] [4,4] 
[1200, 503.835154] 
4.217731 
comma, syntonic comma, comma of didymus 
[4 4, 1>  81 / 80  21.51 
diaschismic, 5limit pajara 
{10\5\1}, {12\6\1}, {22\11\2}, {34\17\3}, {46\23\4}, {56\28\5}, {58\29\5}, {70\35\6}, {78\39\7}, {80\40\7}, {90\45\8} 
[2,0] [3,1] [5,2] 
[600, 105.446531] 
2.612822  diaschisma  [11 4, 2>  2048 / 2025  19.55 
tetracot 
{7\7\1}, {27\27\4}, {34\34\5}, {41\41\6}, {48\48\7}, {61\61\9}, {75\75\11} 
[1,0] [1,4] [1,9] 
[1200, 176.282270] 
2.504205 
minimal diesis 
[5 9, 4>  20000 / 19683  27.66 
aristoxenean 
{12\1\0}, {48\4\1}, {60\5\1}, {72\6\1}, {84\7\1}, {96\8\1} 
[12,0] [19,0] [28,1] 
[100, 14.663787] 
1.382394 
pythagorean comma 
[19 12, 0>  531441 / 524288  23.46 
semisixths 
{19\19\7}, {27\27\10}, {46\46\17}, {65\65\24}, {73\73\27}, {84\84\31} 
[1,0] [1,7] [1,9] 
[1200, 442.979297] 
1.157498    [2 9, 7>  78732 / 78125  13.40 
wÃƒÂ¼rschmidt 
{28\28\9}, {31\31\10}, {34\34\11}, {37\37\12}, {65\65\21}, {71\71\23}, {96\96\31}, {99\99\32} 
[1,0] [1,8] [2,1] 
[1200, 387.819673] 
1.07195 
wÃƒÂ¼rschmidt's comma 
[17 1, 8>  393216 / 390625  11.45 
kleismic, hanson 
{15\15\4}, {19\19\5}, {23\23\6}, {34\34\9}, {53\53\14}, {72\72\19}, {83\83\22}, {87\87\23}, {91\91\24}, {125\125\33} 
[1,0] [0,6] [1,5] 
[1200, 317.079675] 
1.029625  kleisma  [6 5, 6>  15625 / 15552  8.107 
misty 
{12\4\1}, {63\21\5}, {75\25\6}, {87\29\7}, {99\33\8} 
[3,0] [5,1] [6,4] 
[400, 96.787939] 
0.905187    [26 12, 3>  67108864 / 66430125  17.60 
orwell (5limit) 
{9\9\2}, {22\22\5}, {31\31\7}, {53\53\12}, {75\75\17}, {84\84\19}, {97\97\22}, {128\128\29}, {243\243\55} 
[1,0] [0,7] [3,3] 
[1200, 271.589600] 
0.80041  semicomma  [21 3, 7>  2109375 / 2097152  10.06 
escapade 
{22\22\1}, {43\43\2}, {65\65\3}, {87\87\4}, {152\152\7}, {217\217\10} 
[1,0] [2,9] [2,7] 
[1200, 55.275493] 
0.483108    [32 7, 9>  4.2949E+9 / 4.2714E+9  9.492 
amity 
{7\7\2}, {39\39\11}, {46\46\13}, {53\53\15}, {60\60\17}, {99\99\28}, {152\152\43}, {205\205\58}, {311\311\88} 
[1,0] [3,5] [6,13] 
[1200, 339.508826] 
0.383104    [9 13, 5>  1600000 / 1594323  6.154 
parakleismic 
{19\19\5}, {42\42\11}, {61\61\16}, {80\80\21}, {99\99\26}, {118\118\31}, {217\217\57} 
[1,0] [5,13] [6,14] 
[1200, 315.250913] 
0.276603  parakleisma  [8 14, 13>  1.2244E+9 / 1.2207E+9  5.292 
semisuper 
{16\8\1}, {18\9\1}, {34\17\2}, {50\25\3}, {84\42\5}, {118\59\7}, {152\76\9}, {270\135\16}, {388\194\23} 
[2,0] [4,7] [5,3] 
[600, 71.146064] 
0.194018    [23 6, 14>  6.1152E+9 / 6.1035E+9  3.338 
schismic, helmholtz/groven 
{12\12\5}, {29\29\12}, {41\41\17}, {53\53\22}, {65\65\27}, {118\118\49}, {171\171\71}, {200\200\83}, {301\301\125} 
[1,0] [2,1] [1,8] 
[1200, 498.272487] 
0.161693  schisma  [15 8, 1>  32805 / 32768  1.954 
vulture 
{48\48\19}, {53\53\21}, {58\58\23}, {217\217\86}, {270\270\107}, {323\323\128} 
[1,0] [0,4] [6,21] 
[1200, 475.542233] 
0.153767    [24 21, 4>  1.0485E+10 / 1.0460E+10  4.200 
enneadecal 
{19\1\0}, {152\8\1}, {171\9\1}, {323\17\2}, {494\26\3}, {665\35\4} 
[19,0] [30,1] [44,1] 
[63.157894, 7.292252] 
0.104784 
'19tone comma' 
[14 19, 19>  1.9074E+13 / 1.9043E+13  2.816 
semithirds 
{118\118\19}, {205\205\33}, {323\323\52}, {441\441\71}, {559\559\90}, {1000\1000\161} 
[1,0] [4,15] [2, 2] 
[1200, 193.199615] 
0.060822    [38 2, 15>  2.7488E+11 / 2.7466E+11  1.384 
vavoom 
{75\75\7}, {118\118\11}, {311\311\29}, {547\547\51}, {665\665\62}, {901\901\84} 
[1,0] [0,17] [4,18] 
[1200, 111.875426] 
0.058853    [68 18, 17>  2.9558E+20 / 2.9515E+20  2.523 
tricot 
{53\53\25}, {388\388\183}, {441\441\208}, {494\494\233}, {547\547\258}, {600\600\283} 
[1,0] [3,3] [16,29] 
[1200, 565.988015] 
0.0575    [39 29, 3>  6.8720E+13 / 6.8630E+13  2.246 
counterschismic 
{53\53\22}, {306\306\127}, {730\730\303} 
[1 0], [2,1], [21,45] 
[1200, 498.082318] 
0.026391    [69 45, 1>  2.9543E+21 / 2.9515E+21  1.661 
ennealimmal (5limit) 
({9\1\0},) {72\8\3), {99\11\4}, {171\19\7}, {243\27\10}, {270\30\11}, {441\49\18}, {612\69\25} 
[9,0] [15,2] [22,3] 
[133.333333, 49.008820] 
0.025593  ennealimma  [1 27, 18>  7.6294E+12 / 7.6256E+12  0.862 
minortone 
{46\46\7}, {125\125\19}, {171\171\26}, {217\217\33}, {388\388\59}, {559\559\85}, {730\730\111}, {901\901\137} 
[1,0] [1,17] [3,35] 
[1200, 182.466089] 
0.025466    [16 35, 17>  5.0032E+16 / 5.0000E+16  1.092 
kwazy 
{118\59\16}, {494\247\67}, {612\306\83}, {730\365\99}, {1342\671\182} 
[2,0] [1,8] [6,5] 
[600, 162.741892] 
0.017725    [53 10, 16>  9.0102E+15 / 9.0072E+15  0.569 
astro 
{118\118\13}, {1171\1171\129}, {2224\2224\245} 
[1,0] [5,31] [1,12] 
[1200, 132.194511] 
0.014993    [91 12, 31>  2.4759E+27 / 2.4747E+27  0.815 
whoosh 
{441\441\206}, {730\730\341}, {1171\1171\547}, {1901\1901\888}, {3072\3072\1435} 
[1,0] [17,33] [14,25] 
[1200, 560.546970] 
0.012388    [37 25, 33>  1.1645E+23 / 1.1642E+23  0.522 
monzismic 
{53\53\11}, {559\559\116}, {612\612\127}, {665\665\138}, {1171\1171\243}, {1783\1783\370} 
[1,0] [2,2] [10,37] 
[1200, 249.018448] 
0.005738  monzisma  [54 37, 2>  4.5036E+17 / 4.5028E+17  0.292 
egads 
{441\441\116}, {901\901\237}, {1342\1342\353}, {1783\1783\469}, {3125\3125\822} 
[1,0] [15,51] [16,52] 
[1200, 315.647874] 
0.00466    [36 52, 51>  4.4409E+35 / 4.4400E+35  0.339 
fortune 
{612\612\113}, {1901\1901\351}, {2513\2513\464}, {3125\3125\577} 
[1,0] [1,14] [11,47] 
[1200, 221.567865] 
0.003542    [107 47, 14>  1.6229E+32 / 1.6226E+32  0.277 
senior 
{171\171\46}, {1000\1000\269}, {1171\1171\315}, {1342\1342\361}, {2513\2513\676}, {3684\3684\991} 
[1,0] [11,35] [19,62] 
[1200, 322.801387] 
0.003022    [17 62, 35>  3.8152E+29 / 3.8147E+29  0.230 
gross 
{118\118\9}, {1783\1783\136}, {1901\1901\145}, {3684\3684\281} 
[1,0] [2,47] [4,22] 
[1200, 91.531021] 
0.002842    [144 22, 47>  2.2301E+43 / 2.2298E+43  0.245 
pirate 
{730\730/113}, {1783\1783\276}, {2513\2513\389}, {4296\4296\665} 
[1,0] [6,49] [0,15] 
[1200, 185.754179] 
0.000761    [90 15, 49>  1.7764E+34 / 1.7763E+34  0.047 
raider 
{1171\1171\335}, {3125\3125\894}, {4296\4296\1229} 
[1,0] [9,37] [26,99] 
[1200, 343.296099] 
0.000511    [71 99, 37>  1.71799E+47 / 1.71793E+47  0.062 
atomic 
({12\1\0},) {600\50\1}, {612\51\1}, {3072\256\5}, {3684\307\6}, {4296\358\7} 
[12,0] [19,1] [28,7] 
[100, 1.955169] 
0.00012 
atom of kirnberger 
[161 84, 12>  2.92300E+48 / 2.92298E+48  0.015 
Below is a lattice diagram of these "vanishing commas". Paul has included all the ones listed in the table above with numerator and denominator of seven digits or fewer. (Compare this diagram with those on Monzo, 5limit intervals, 100 cents and under.)
It's my belief that the vectors of these intervals play a role in the patterns of shading and coloring in my gallery of EDO 5limit error lattices. Those lattices have the 3 and 5 axes oriented exactly as here.
This analysis only concerns the representations of various EDOs to the 5limit. See Monzo, EDO primeerror to get an idea of how different EDOs represent all of the prime factors from 3 to 43.
Examples of nonoctave equal temperaments are Gary Morrison's 88CET (88 cents between degrees), the BohlenPierce scale, and Wendy Carlos's alpha, beta, and gamma scales [audio examples on the Wendy Carlos site].
In a post to the Early Music list, Aleksander Frosztega wrote:
P.S. [quoting] >The phrase "equal temperament" has existed in print since 1781. French used the term "temperament egal" long before 1781.
German writers used the phrase gleichschwebende Temperatur to denote equalbeating temperament since the beginning of the 18th century. This is not to be confused with equaltemperament, and instead actually denotes certain meantones, welltemperament, and other tunings where the varying temperings of different intervals results in them having equal numbers of beats per second. However, most German writers have in fact used the term (and its variant spellings gleich schwebende Temperatur and gleichschwebende Temperatur) to designate regular 12edo, and unless the context specifically indicates that a welltemperament or meantone is under discussion, gleichschwebende Temperatur in German treatises generally refers to 12edo.
(Note also Schoenberg's frequent use, in his Harmonielehre, of the term schwebend to refer to a method of composition in which the sense of tonality is "suspended, floating", thus leading the way to his style of pantonality beginning around 1908.)
Below is a table showing advocates of various "octave"based ETs, with approximate dates. It does not claim to be complete, and keeps growing. (click on the highlighted numbers to show more detail about those ETs)
ET  Date and Theorist/composer 
5 
the smallest cardinality EDO which has any real musical usefulness, some theorists describe Indonesian slendro scale as this 2001  Herman Miller 
6 
the "wholetone scale" 1787  Wolfgang A. Mozart (in his A Musical Joke) 1894  Claude Debussy 1902  Arnold Schönberg 
7 
traditional Thai music 1991  Clem Fortuna 1997  Randy Winchester 2001  Robert Walker 
8 
1980  Gordon Mumma (Octal Waltz for harpsichord) 1981  Daniel Wolf 1997  Randy Winchester 
9 
early 1900s  Charles Ives (in Monzo, Ives "stretched" scales) 1930s60s  R. M. A. Kusumadinata (Sunda: mapping of 3 pathet onto 7outof9equal) 19??  James Tenney (piano part, The Road to Ubud) 
10 
1930s60s  R. M. A. Kusumadinata (Sunda) 1990s  Elaine Walker 1978  Gary Morrison 1997  Randy Winchester 1998  William Sethares 
11 
1996  Daniel Wolf 
12 
semitone or "halfstep" before 3000 BC  a possible Sumerian tuning (according to Monzo, Speculations on Sumerian Tuning) 1584  Prince Chu Tsaiyü (China) 1585  Simon Stevin (Netherlands) 1636  Marin Mersenne c.17801828  Mozart, Beethoven, and Schubert compose many chord progressions which, by the use of enharmonic equivalence of the diesis, strongly imply 12edo or a related 12tone "circulating" welltemperament 1802  Georg Joseph Vogler 1817  Gottfried Weber 19001999  the nearly universal tuning of the 'developed' world 1911  Arnold Schönberg (along with his personal rejection of microtonality) 
13 
1962  Ernst Krenek (opera Ausgerechnet und Verspielt, op. 179) 1991  Paul Rapoport 1998  Herman Miller 1999  Dan Stearns 2001  X. J. Scott 
14 
1990  Ralph Jarzombek 2000  Herman Miller 
15 
1930s60s  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 
16 
1930s60s  R. M. A. Kusumadinata (Sunda) 1971  David Goldsmith 1993  Steve Vai 1997  Randy Winchester 1998  Herman Miller 2002  Victor Cerullo 
17 
1653  Brouncker 1809  Villoteau (describing Arabic tuning) 1929  Malherbe 1935  Karapetyan 1960s  Ivor Darreg 1997  Herman Miller 1999  Margo Schulter (as a pseudoPythagorean tuning) 
18 
1907  Ferrucio Busoni (in his theory, but not used in his compositions) 1940s  Julián Carrillo, 1/3rdtone piano 1960s  Ivor Darreg 
19 
thirdtone (the tuning normally meant by that term) 1558  Guillaume Costeley 1577  Salinas (19 notes of '1/3comma meantone', almost identical to 19ET) before 1633  Jean Titelouze ('thirdtones' may describe 19ET) 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  Joe Monzo (19tone Samba) 1999  John Starrett 2004  Aaron Krister Johnson 
20 
1980  Gerald Balzano 1990s  Dan Stearns 1996  Paul Zweifel 1999  Herman Miller 2022  Petr Chernobrivets (book: "TwentyTone Equal Temperament", and numerous compositions) 
21 
2001  Herman Miller 
22 
(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 modulus22, not necessarily EDO) 1960s  Ivor Darreg 1980  Morris Moshe Cotel 1993  Paul Erlich (notably, his piece Tibia) 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 2021  Joseph Monzo (demonstration of scale generation using Tonescape, and several short pieces for PNP2022) 
23 
some theorists describe Indonesian pelog scale as subset of this 1920s  Hornbostel (describing Burmese music) 
24 
quartertone: 2^{1} (= 2) quartertones per Semitone; 12 * 2^{1} = 24 quartertones per octave. Also called enamu (1mu), a MIDI pitchbend unit. 1760  Charles Delusse: Air a la Greque (earliest notation of microtonal pitches in the "commonpractice" era) 1849  Fromental Halévy  his cantata Prométhée enchaîné uses quartertones 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 24tET 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, 1/4thtone piano 1941  Mildred Couper 1948  Pierre Boulez (original version of Le Soleil des Eaux) 1950s  Giacinto Scelsi (very looselyconceived intonation) 1967  Tui St. George Tucker 1969  Györgi Ligeti (Ramifications) 1960s2000s  John Eaton 1980s  Brian Ferneyhough (very looselyconceived intonation) 1983  Leo de Vries 1994  Joe Monzo, 24eq tune 
25 
1994  Paul Rapoport 
26 
1998  Paul Erlich 1998  Herman Miller 
27 
2001  Gene Ward Smith 2001  Herman Miller 
28 
1997  Paul Erlich (for music based on the diminished scale) 
29 
by 1875  Émile Chevé (by mistake) 
30 
1940s  Julián Carrillo, 1/5thtone piano 
31 
diesis (one of several meanings of that term) 1555  Nicola Vicentino (31 notes of extended meantone nearly identical to 31ET) 1606  Vito Trasuntino (31 notes of extended meantone nearly identical to 31ET) 1606  Gonzaga (31 notes of extended meantone nearly identical to 31ET) before 1618  Scipione Stella (31 notes of extended meantone nearly identical to 31ET) 1618  Fabio Colonna (31 notes of extended meantone nearly identical to 31ET) 1623  Daniel Hizler (used only 13 out of 31ET 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 191719  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 modulus31, 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 looselyconceived intonation) 1989  John Bischoff and Tim Perkis 1999  Paul Erlich 
34 
1979  Dirk de Klerk before 1998  Larry Hanson 1997  Neil Haverstick 
36 
sixthtone; 3 units per Semitone = 12 * 3 units per "octave" 1907  Ferrucio Busoni (in his theory, but not used in compositions) 19231960s  Alois Hába 1940s  Julián Carrillo, 1/6thtone piano 1952  Henri Pousseur: Prospection, pour un pianotriple à sixièmes de ton by 1997  Tomasz Liese  19outof36edo subset scale 2010  Joe Monzo  using 83 degrees to map primefactor 5 (best mapping of 5 is to 84 degrees  exactly the same size as 12edo). 
37 
2012  Joe Monzo  very strong approximation of factors 5,7,11,13 in 13limit JI. 
38 
2004  tuningmath group, for situations where introducing a period of 1/2 or splitting the fifth into two over 19equal might be useful. 2^(11/38) is almost precisely an 11/9. 
41 
1901  Paul von Jankó (1960s  Erv Wilson claims that Partch was intuitively feeling out 41ET) 1975  George Secor 1989  Helen Fowler 1993  Joe Monzo 1998  Carl Lumma 1998  Patrick OzzardLow 2002  Gene Ward Smith 
42 
1940s  Julián Carrillo, 1/7thtone piano 
43 
méride 1701  Joseph Sauveur  a unit of interval measurement, nearly identical to 1/5comma meantone 
46 
1989  R. Fuller 1998  Graham Breed 2000  Dave Keenan and Paul Erlich 2002  Gene Ward Smith 
48 
doamu (2mu), a MIDI pitchbend unit: 2^{2} (= 4) 2mus per Semitone = 12 * 2^{2} = 48 2mus per octave. Also called "eighthtone". early 1900s  Charles Ives (according to Monzo, Ives stretchedoctave scales) 1915  N. Kulbin 1924  Julián Carrillo 1940s  Julián Carrillo, 1/8thtone piano 19??  Patrizio Barbieri 19??  ClausSteffen Mahnkopf 19??  Volker Staub 1998  Joseph Pehrson 
50 
(1558  Zarlino  fair approximation to 2/7comma meantone, the first meantone to be described with mathematical exactitude) 1710  Konrad Henfling 1759  Robert Smith (as an approximation to his ideal 5/18comma meantone system) 1835  Wesley Woolhouse (as practical approximation to his 7/26comma 'optimal meantone') 1940s  Tillman Schafer 
53 
mercator; nearly identical to pythagorean tuning and a very good approximation to 5limit 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  53tone Pythagorean tuning became official scale in China) 187475  R. H. M. Bosanquet 1875  Alexander J. Ellis (appendix to Helmholtz, On the Sensations of Tone) 1890  Shohé Tanaka c.1900  Standard Turkish musictheory 1911  Robert Neumann (quoted by Schönberg in Harmonielehre) 1927  Augusto Novaro 1978  Larry Hanson 2002  Gene Ward Smith 2005  Chris Mohr 
54 
1940s  Julián Carrillo, 1/9thtone piano 
55 
(good approximation to 1/6comma 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 nonkeyboard instruments (according to Monzo, Mozart's Tuning) 
58 
(1770  Dom FranÃƒÂ§ois Bedos de Celles  according to Barbour, but this may be erroneous. See Yahoo tuning message 63618) 2002  Gene Ward Smith 
60 
5 units per Semitone = 12 * 5 units per "octave". 1940s  Julián Carrillo, 1/10thtone piano 1980s?  Richard Boulanger 
65 
1927  Augusto Novaro 1951  J. Murray Barbour, as a 5limit system. 2004  Gene Ward Smith, it is simultaneously a schismatic system and a semisixths (78732/78125 comma) system. 
66 
1940s  Julián Carrillo, 1/11thtone piano 
68 
1847  Meshaqah (describing modern Greek tuning) 1989?  John Chalmers (describing Byzantine tuning) 
72 
twelfthtone / 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 193858  Evgeny Alexandrovich Murzin created a 72tET 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, 1/12thtone piano 1953  Ivan Wyschnegradsky 1963  Iannis Xenakis (cf. his book Musiques formelles) 1970  Ezra Sims 1970  Franz Richter Herf 1970  Rolf Maedel 1970s2000s  Joe Maneri (and subsequently many of his students) 1980s?  James Tenney 1990s  Ted Mook 1999  Paul Erlich 1999  Joe Monzo (as basis of simplified HEWM notation) 1999  Rick Tagawa 2001  Dave Keenan, Graham Breed, Joseph Pehrson, Paul Erlich, Joe Monzo (for notation of miracle family scales) 2001  Julia Werntz 2002  Gene Ward Smith 
74 
1762  Riccati (approximation to 3/14comma meantone) 1855  Drobisch (approximation to 2/9comma meantone) 1991  John Cage, in "Ten" for chamber ensemble 
76 
1998  Paul Erlich (as a unified tuning for various tonal systems) 
78 
1940s  Julián Carrillo, 1/13thtone piano 
80 
2004  Gene Ward Smith  a strong 19limit system. Chains of 80edo fifths have been proposed for neoGothic or Arabic inspired music. 
81 
2002  Joe Monzo, good approximation to Kornerup's "golden meantone" 2005  Gene Ward Smith, approximation to 5/19comma meantone 
84 
7 units per Semitone = 12 * 7 units per "octave". 1940s  Julián Carrillo, 1/14thtone piano 1985  Harald Waage (for 5limit just intonation) 
87 
1951  J. Murray Barbour, as a 5limit system 1998  Paul Erlich  it is consistent and unique in the 13oddlimit 2004  Gene Ward Smith, it supports kleismic tempering 2020  Jeff Brown 
88 
for most purposes, essentially the same as LucyTuning. 1775  John "Longitude" Harrison 1987  Charles Lucy 
90 
1940s  Julián Carrillo, 1/15thtone piano 
94 
unique up to 13oddlimit, and consistent up to 23oddlimit 1998  Paul Erlich 2015  Cam Taylor 
96 
triamu (3mu), a MIDI pitchbend unit: 2^{3} (= 8) 3mus per Semitone = 12 * 2^{3} = 96 3mus per "octave". Also called "1/16tone". 1924  Julián Carrillo 1940s  Julián Carrillo, 1/16thtone piano 1980  Pascale Criton 2001  VincentOlivier Gagnon 
99 
2004  Gene Ward Smith, 99edo is significant as a 7limit 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. 
100 
1980s  Barry Vercoe  built into CSound software 
106 
2 * 53 degrees per "octave" 2004  Joe Monzo (in analyzing Philolaus's small intervals) 
111 
2004  Gene Ward Smith  strong 1321limit system 
118 
18745  Bosanquet 
130 
2004  Gene Ward Smith  twice 65, and important in the 7, 13 (and 15) limits in particular. 
140 
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 7limit division. 
144 
farab; 12 units per Semitone = 12 * 12 units per "octave". 300s BC  Aristoxenus (most likely interpretation of his theories) early 900s  Abu Nasr alFarabi 1946  Joseph Schillinger 1999  Dan Stearns and Joe Monzo (chiefly for its value as a unified notation for mixed EDOs and/or complex just intonation tunings) 
152 
1999(?)  Paul Erlich, "Universal Tuning" 
171 
(19??  Eivend Groven  approximated by his 1/8skhisma temperament) 1926  Perrett 1975  Martin Vogel 2002  Gene Ward Smith 
175 
2002  Gene Ward Smith 
192 
tetramu (4mu), a MIDI pitchbend unit: 2^{4} (= 16) 4mus per Semitone = 12 * 2^{4} = 192 4mus per octave. 
200 
16 ^{2}/_{3} degrees per Semitone 2002  Joe Monzo (in analyzing Werckmeister III) 
205 
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 
217 
7 * 31 degrees per octave = 18 ^{1}/_{12} degrees per Semitone 2002  Joe Monzo (proposed for adaptiveJI 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 multiEDOs) 
224 
2004  Gene Ward Smith, it is important in the 13limit in particular. 2004  George Secor, as an important part of Sagittal notation 
270 
tredek; named by Joe Monzo in accordance with its excellent approximation of 13limit JI 1970s?  Erv Wilson and John Chalmers 1997  Paul Hahn 2013  Joe Monzo  for use as a unit of interval measurement without need for decimal places, strongest 3digit EDO for 13limit JI. 
288 
early 1900s  Charles Ives (as analyzed in Monzo, Ives stretched octave scales) 
300 
25 units per Semitone = 12 * 25 units per "octave". 1800s  system of savarts 
301 
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) 
311 
gene; named by Joe 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  Joe Monzo  for use as a unit of interval measurement without need for decimal places, very strong thru 41limit JI. 
318 
1999  Joe Monzo (in analyzing Aristoxenus: 318 = 53*6) 
384 
pentamu (5mu), a MIDI pitchbend unit: 2^{5} (= 32) 5mus per semitone, 2^{5} * 12 = 384 5mus per octave. 
441 
2004  Gene Ward Smith  a very strong 5 or 7 limit system, along with 612 a good way to tune ennealimmal. 
494 
2004  Gene Ward Smith  strong 1115 limit system, and still good as a 17limit system. 
512 
2^{9} units per octave. 1980s  tuning resolution of some electronic instruments, notably Ensoniq VFX and VFXSD. 
581 
2013  Scott Dakota 2015  Cam Taylor 2016  Joe Monzo  for use as a unit of interval measurement without need for decimal places, strongest 3digit EDO for 23limit JI. 
600 
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 
612 
51 (= 3 * 17) units per semitone = 2^{2} * 3^{2} * 17 units per octave; an excellent unit of interval measurement for 11limit 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  Joe Monzo (in analyzing Werckmeister III) 
665 
(a remarkably close approximation to pythagorean tuning) before 1975  Jacques Dudon 1980s?  Marc Jones  see satanic comma 
730 
1835  Wesley Woolhouse  his unit of measurement for 5limit JI, and an analogue of cents; 60 ^{5}/_{6} degrees per Semitone. 
768 
hexamu (6mu), a MIDI pitchbend unit: 2^{6} (= 64) 6mus per semitone, 2^{6} * 12 = 768 6mus per octave. 1980s2000s  Tuning resolution of many electronic instruments, including several by Yamaha, Emu, and Ensoniq; also the resolution of some early sequencer software, including Texture. 1980s2000s  Joe Monzo (using Texture software in 1980s, then using computer soundcards with 6mu resolution in 1990s and 2000s.) 1980s2000s  myriad artists using MIDI hardware. 2003  Joe Monzo  proposed as de facto hardware tuning standard 
1000 
millioctave, an interval measurement, an analogue of cents: 1000 = 2^{3} * 5^{3} = 83 ^{1}/_{3} units per Semitone. 1980s  Csound software: its "oct" pitch format 1993  Mark Lindley (in his book Mathematical Models of Musical Scales) 
1024 
2^{10} (= 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 199095  Joe Monzo (tuning resolution of Yamaha TG77) 
1200 
1875  Alexander Ellis (his unit of measurement, called cents, 100 per 12tET semitone) 1980s2000s  many synthesizers and soundcards with 1cent resolution give a 768outof1200edo subset tuning. 
1536 
heptamu (7mu), a MIDI pitchbend unit; 2^{7} = 128 7mus per Semitone; 12 * 2^{7} = 1536 7mus per octave 
1700 
2002  Margo Schulter (for interval measurement, called "iota") 
1728 
19??  Paul Beaver (rendered as 12^{3}) 
2460 
mina (short for "schismina"); 233EDA (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 
3072 
oktamu (8mu), a MIDI pitchbend unit: 2^{8} (= 256) 8mus per Semitone; 12 * 2^{8} = 3072 8mus per octave. 1990s  Apple's QuickTime Musical Instruments tuning spec 
3125 
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  Joe Monzo  advocated as a unit of interval measurement for 7limit JI. 
4296 
358 units per semitone 1992  Marc Jones (used as most convenient UHT [ultrahigh temperament] to measure 5limit just intonation intervals) 
6144 
enneamu (9mu), a MIDI pitchbend unit: 2^{9} (= 512) 9mus per Semitone; 12 * 2^{9} = 6144 9mus per octave. 
8539 
tina; 809EDA (809 equal divisions of the apotome) 2007  Joe Monzo  for use as a unit of interval measurement without need for decimal places, strong thru 31limit JI and also good for 41. 
10600 
1965  M. Ekrem Karadeniz  his unit of measurement, called türksents, 200 units per 53edo comma. 
12288 
dekamu (10mu), a MIDI pitchbend unit: 2^{10} (= 1024) 10mus per Semitone; 12 * 2^{10} = 12288 10mus per octave. 
24576 
endekamu (11mu), a MIDI pitchbend unit: 2^{11} (= 2048) 11mus per Semitone; 12 * 2^{11} = 24576 11mus per octave. 
30103 
jot 1864  Augustus De Morgan  his unit of measurement; chosen because of its closeness to log_{10}(2) * 100,000. 
31920 
2007  Gene Ward Smith, Joe Monzo  for use as a unit of interval measurement which is both strong (i.e., low logflat badness) and consistent thru 41limit JI. 
36829 
(198?  approximation to John Brombaugh's scale of temperament units.) 
46032 
flu  useful for discussing 5limit tempering. 2004  Gene Ward Smith  The "Diophantine clarity" division: pythagoreancomma = 900 flus, syntoniccomma ("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. 
49152 
dodekamu (12mu), a MIDI pitchbend unit: 2^{12} (= 4096) 12mus per Semitone; 12 * 2^{12} = 49152 12mus per octave; formerly called cawapu. 1980s  pitchbend resolution of Cakewalk^{TM} and many other popular sequencer programs. 
58973 
5587EDA (5587 equal divisions of the apotome) 2007  Joe Monzo  for use as a unit of interval measurement without need for decimal places, strong and consistent thru 41limit JI. 
98304 
tridekamu (13mu), a MIDI pitchbend unit: 2^{13} (= 8192) 13mus per Semitone; 12 * 2^{13} = 98304 13mus per octave. 1983  the maximum resolution possible in MIDI pitchbend 
196608 
tetradekamu (14mu), a MIDI pitchbend unit: 2^{14} (= 16384) 14mus per Semitone; 12 * 2^{14} = 196608 14mus per octave; formerly called midipu. 1983  finest possible resolution in the MIDI tuning Spec. 1999  MTS (MIDI tuning standard) 
Notes:

Any tuning system which divides the octave (2/1) into n aliquot parts is termed an ntone Equal Temperament. Mathematically, an Equal Temperament is a geometric series and each degree is a logarithm to the base 2^{n}.
[Note from Monzo: the base is 2 only in octaveequivalent equaltemperaments. 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 (as in the BohlenPierce scale); this is a geometric series where each degree is a logarithm to the base 3^{n}.]
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.
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