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equal-temperament

[Joe Monzo, with Paul Erlich, John Chalmers, Manuel Op de Coul, Margo Schulter, Carl Lumma]

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, fraction-of-a-comma-meantone, well-temperament, or other tunings. Generally abbreviated as ET.

Usually, but not always, equal temperaments assume octave-equivalence, of which the usual 12-edo 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 5-limit. 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:

mouse-over 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
minor-2nd
[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 (5-limit) {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,
5-limit 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
würschmidt {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 würschmidt'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 (5-limit) {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 '19-tone
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
(5-limit)
({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, 5-limit intervals, 100 cents and under.)

lattice diagram of 5-limit EDO vanishing commas

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 5-limit 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 5-limit. See Monzo, EDO prime-error to get an idea of how different EDOs represent all of the prime factors from 3 to 43.

Examples of non-octave equal temperaments are Gary Morrison's 88-CET (88 cents between degrees), the Bohlen-Pierce 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 equal-beating temperament since the beginning of the 18th century. This is not to be confused with equal-temperament, and instead actually denotes certain meantones, well-temperament, 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 gleich-schwebende Temperatur) to designate regular 12-edo, and unless the context specifically indicates that a well-temperament or meantone is under discussion, gleichschwebende Temperatur in German treatises generally refers to 12-edo.

(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 "whole-tone 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)

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)

10

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

1990s -- Elaine Walker

1978 -- Gary Morrison

1997 -- Randy Winchester

1998 -- William Sethares

11

1996 -- Daniel Wolf

12

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)

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

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

16

1930s-60s -- 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 pseudo-Pythagorean tuning)

18

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

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

1960s -- Ivor Darreg

19

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 -- Joe Monzo (19-tone Samba)

1999 -- John Starrett

2004 -- Aaron Krister Johnson

20

1980 -- Gerald Balzano

1996 -- Paul Zweifel

1999 -- Herman Miller

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 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

23

some theorists describe Indonesian pelog scale as subset of this

1920s -- Hornbostel (describing Burmese music)

24

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 -- Joe Monzo, 24-eq 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, 1/5th-tone piano

31

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

34

1979 -- Dirk de Klerk

before 1998 -- Larry Hanson

1997 -- Neil Haverstick

36

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

37

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

38

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.

41

1901 -- Paul von Jankó

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

1975 -- George Secor

1989 -- Helen Fowler

1993 -- Joe Monzo

1998 -- Carl Lumma

1998 -- Patrick Ozzard-Low

2002 -- Gene Ward Smith

42

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

43

méride

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

46

1989 -- R. Fuller

1998 -- Graham Breed

2000 -- Dave Keenan and Paul Erlich

2002 -- Gene Ward Smith

48

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

50

(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

53

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

54

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

55

(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)

58

(1770 -- Dom François 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, 1/10th-tone piano

1980s? -- Richard Boulanger

65

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.

66

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

68

1847 -- Meshaqah (describing modern Greek tuning)

1989? -- John Chalmers (describing Byzantine tuning)

72

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 -- 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 3/14-comma meantone)

1855 -- Drobisch (approximation to 2/9-comma 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, 1/13th-tone piano

80

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

81

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

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

84

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

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

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

87

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

2004 -- Gene Ward Smith, it supports kleismic tempering

88

for most purposes, essentially the same as LucyTuning.

1775 -- John "Longitude" Harrison

1987 -- Charles Lucy

90

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

96

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

99

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.

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 13-21-limit system

118

1874-5 -- 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 7-limit 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 al-Farabi

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 1/8-skhisma temperament)

1926 -- Perrett

1975 -- Martin Vogel

2002 -- Gene Ward Smith

175

2002 -- Gene Ward Smith

192

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

217

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

2002 -- Joe 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)

224

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

270

tredek; named by Joe Monzo in accordance with its excellent approximation of 13-limit 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 3-digit EDO for 13-limit 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 41-limit JI.

318

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

384

pentamu (5mu), a MIDI pitch-bend unit: 25 (= 32) 5mus per semitone, 25 * 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 11-15 limit system, and still good as a 17-limit system.

512

29 units per octave.

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

581

2016 -- Joe Monzo -- for use as a unit of interval measurement without need for decimal places, strongest 3-digit EDO for 23-limit 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 = 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 -- 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 5-limit JI, and an analogue of cents; 60 5/6 degrees per Semitone.

768

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 -- Joe 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 -- Joe Monzo -- proposed as de facto hardware tuning standard

1000

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)

1024

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 -- Joe Monzo (tuning resolution of Yamaha TG-77)

1200

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.

1536

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

1700

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

1728

19?? -- Paul Beaver (rendered as 123)

2460

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

3072

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

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 7-limit JI.

4296

358 units per semitone

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

6144

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

8539

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

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

10600

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

12288

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

24576

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

30103

jot

1864 -- Augustus De Morgan -- his unit of measurement; chosen because of its closeness to log10(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 41-limit JI.

36829

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

46032

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.

49152

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 CakewalkTM and many other popular sequencer programs.

58973

5587-EDA (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 41-limit JI.

98304

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

196608

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.
. . . . . . . . .
[John Chalmers, Divisions of the Tetrachord]

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 from Monzo: 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 (as in the Bohlen-Pierce scale); this is a geometric series where each degree is a logarithm to the base 3n.]

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.

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

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