- Ivor Darreg
- Easley Blackwood
- Brian McLaren
- Marc Jones
- Dan Stearns
a system of tuning based on a scale whose "steps" or degrees have logarithmically equal intervals between them, in contrast to the differently-spaced degrees of just intonation, meantone, well-temperament, or other tunings. Generally abbreviated as ET.
Usually, but not always, equal temperaments assume octave-equivalence, of which the usual 12-EQ is the most obvious example. My preferred abbreviation for 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:
-
The yellow lines (barely legible) represent
the just "perfect 5th"
and its "8ve"-complement the
"perfect 4th" (ratios 3:2 and 4:3, respectively),
and the just
"major 3rd / minor 6th"
(ratios 5:4 and 8:5, respectively) and
"minor 3rd / major 6th"
(ratios 6:5 and 5:3, respectively), and the
cents deviation from
those ratios (the quantization in cents varies according to the
zoom factor).
The green lines connect EDOs which are collinear on this graph. When this happens, it indicates that a certain important small interval acting as a unison-vector is being tempered out, which is also known as "vanishing".
Most of these green lines are labeled with the vanishing unison-vectors; some of them are labeled with names which may be consulted in this Dictionary for more information (meantone, syntonic comma, schismic, kleisma, MIRACLE, diaschisma, diesis).
I find it easier to view this using the negatives of paul's original graphics, and so i present both below. In the negatives, the blue lines represent the just ratios and the purple lines represent collinear EDOs. I have also added to the negative versions a small red square at the center which represents exact 5-limit JI.
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 vanishing commas depicted in
the diagram above, and their associated temperaments. (Thanks
to Carl Lumma for the original version of this table.) Where the
name has "--" the temperament family has not yet been assigned a name,
and where it has "x" I also did not draw the red line on the diagram
because of lack of space.
comma name(s) {2 3 5}-vector numerator:denominator cents temperament name(s) ETs heuristic complexity heuristic error
large limma [ 0 3 -2] 27:25 133.2375749 beep 4, 5, 9 3.295836866 38.90955906
classic chromatic semitone [ -3 -1 2] 25:24 70.67242686 dicot 3, 4, 7, 10 3.218875825 21.51352389
major limma, limma ascendant [ -7 3 1] 135:128 92.17871646 pelogic 7, 9, 16, 23 4.905274778 18.30023709
limma [ 8 -5 0] 256:243 90.22499567 blackwood 10, 15, 25 5.493061443 16.86080893
major diesis [ 3 4 -4] 648:625 62.565148 diminished, 'octatonic' 4, 12, 16, 28 6.43775165 9.89622099
maximal diesis [ 1 -5 3] 250:243 49.16613727 porcupine 7, 15, 22, 29, 37, 59 5.493061443 9.078897114
great diesis [ 7 0 -3] 128:125 41.05885841 augmented, diesic 3, 9, 12, 15, 18, 27, 39, 42 4.828313737 8.605409557
-- [ -14 3 4] 16875:16384 51.11985806 negri 9, 10, 19, 28, 29 9.733588516 5.175121053
syntonic comma [ -4 4 -1] 81:80 21.5062896 meantone 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 69, 74, 81, 88 4.394449155 4.863694883
small diesis [ -10 -1 5] 3125:3072 29.61356846 magic 3, 16, 19, 22, 25, 35, 41, 60, 63, 79 8.047189562 3.648693652
minimal diesis [ 5 -9 4] 20000:19683 27.65984767 tetracot 7, 27, 34 41, 48, 61, 75 9.887510598 2.819920032
diaschisma [ 11 -4 -2] 2048:2025 19.55256881 diaschismic, 5-limit pajara 12, 22, 34, 46, 56, 58, 70, 78, 80, 90 7.61332498 2.582761031
pythagorean comma [ -19 12 0] 531441:524288 23.46001038 aristoxenean 12, 48, 60, 72, 84, 96 13.18334746 1.767515597
-- [ 2 9 -7] 78732:78125 13.39901073 semisixths 19, 27, 46, 65, 73, 84 11.26606539 1.193939109
-- [ 26 -12 -3] 67108864:66430125 17.59884802 misty 12, 63, 75, 87, 99 18.0116612 0.982063908
wuerschmidt's comma [ 17 1 -8] 393216:390625 11.44528995 wuerschmidt 28, 31, 34, 37, 65, 71, 96, 99 12.8755033 0.891864646
kleisma [ -6 -5 6] 15625:15552 8.107278862 kleismic, hanson 15, 19, 23, 34, 53, 72, 83, 87, 91 9.656627475 0.837593197
semicomma [ -21 3 7] 2109375:2097152 10.06099965 orwell 9, 22, 31, 53, 75, 84, 97 14.56190225 0.688908734
-- [ 9 -13 5] 1600000:1594323 6.153558074 amity 39, 46, 53, 60, 99, 152 14.28195975 0.431628947
-- [ 32 -7 -9] 4294967296:4271484375 9.491569159 escapade 22, 43, 65, 87, 152 22.17522723 0.429201273
-- [ 8 14 -13] 1224440064:1220703125 5.291731873 parakleismic 19, 42, 61, 80, 99, 118, 217 20.92269286 0.25330523
schisma [ -15 8 1] 32805:32768 1.953720788 schismic, helmholtz/groven 29, 41, 53, 65, 77, 89, 118, 171 10.39833622 0.187781849
-- [ 24 -21 4] 10485760000:10460353203 4.199837286 vulture 48, 53, 58, 217, 323 23.07085806 0.18226178
-- [ 23 6 -14] 6115295232:6103515625 3.338011085 semisuper 16, 18, 34, 50, 84, 152, 270, 388 22.53213077 0.148287404
'19-tone' comma [ -14 -19 19] 1.90735E+13:1.90425E+13 2.81554699 enneadecal 19, 152, 171, 323, 494, 665 30.57932034 0.091998733
-- [ 39 -29 3] 6.87195E+13:6.86304E+13 2.246116498 tricot 53, 388, 441, 494, 547 31.85975637 0.070545869
-- [ -68 18 17] 2.96E+20:2.95E+20 2.523151279 vavoom 118, 547, 665 47.13546571 0.053490795
-- [ 38 -2 -15] 2.74878E+11:2.74658E+11 1.384290297 semithirds 118, 323, 441, 559 26.33879326 0.052578107
ennealimmal comma [ 1 -27 18] 7.62939E+12:7.6256E+12 0.861826202 ennealimmal 171, 441, 612 29.66253179 0.029061604
-- [ -16 35 -17] 5.00315E+16:5.0E+16 1.091894586 minortone 171, 388, 559, 730, 901 38.4514301 0.028387769
-- [ -53 10 16] 9.01016E+15:9.0072E+15 0.569430491 kwazy 118, 494, 612, 1342 36.73712949 0.015497587
-- [ 91 -12 -31] 2.47588E+27:2.47472E+27 0.814859805 astro 118, 1171, 2224 63.07639343 0.012915578
-- [ 37 25 -33] 1.1645E+23:1.16415E+23 0.522464095 whoosh 441, 730, 1171 53.1117529 0.009835587
monzisma [ 54 -37 2] 4.5036E+17:4.50284E+17 0.29239571 monzismic 53, 559, 612, 665, 1171, 1783 40.64882358 0.007192607
-- [ -36 -52 51] 4.44089E+35:4.44002E+35 0.339362106 egads 1342, 1783, 3125 82.08133353 0.004134056
-- [-107 47 14] 1.62285E+32:1.62259E+32 0.277034781 fortune 612, 1901, 2513, 3125 74.16690834 0.003734989
-- [ -17 62 -35] 3.8152E+29:3.8147E+29 0.230068385 senior 1171, 1342, 2513, 3684 68.1139619 0.003377473
-- [ 144 -22 -47] 2.23007E+43:2.22976E+43 0.245429314 gross 118, 1783, 1901, 3684 99.813194 0.002458712
-- [ -90 -15 49] 1.77636E+34:1.77631E+34 0.046966396 pirate 1783, 2513, 4296 78.86245771 0.00059554
-- [ 71 -99 37] 1.71799E+47:1.71793E+47 0.062327326 raider 1171, 4296 108.7626526 0.000573048
-- [ 161 -84 -12] 2.923E+48:2.92298E+48 0.015360929 atomic 3684, 4296 111.5966961 0.000137646
Below is a lattice diagram of these "vanishing commas". I have included all the ones listed in the table above except the two that plot the furthest away from the central 1/1 reference pitch: [10 -40 23] and [-11 26 -13]. I have labeled the ones that have names in current use among tuning theorists, and drawn vectors for a few of the others. (Compare this diagram with those on my webpage 5-limit intervals, 100 cents and under: they are essentially identical, except for the reversed orientation of prime-factors 3 and 5. Also see the lattice diagrams on the individual pages linked in the table.)
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.
Examples of non-octave equal temperaments are Gary Morrison's 88CET (88 cents between degrees), the Bohlen-Pierce scale, and Wendy Carlos's alpha, beta, and gamma scales [listen to them here].
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 12edo, and unless the context specifically indicates that a well-temperament or meantone is under discussion, "gleichschwebende Temperatur" in German treatises generally refers to 12edo.
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)
NOTE:
-
The following composers have written in so many different
equal-temperaments that they will (eventually) each have their own page:
| ET | Date and Theorist/composer |
| 5 | some theorists describe Indonesian slendro scale as this
2001 Herman Miller |
| 6 | the "whole-tone scale"
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 analysis of 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 | before 3000 BC a possible Sumerian tuning (according to speculations by Monzo)
1584 Prince Chu Tsai-yü 1585 Simon Stevin 1636 Marin Mersenne 1802 Georg Joseph Vogler 1817 Gottfried Weber 1900-1999 the predominant 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 | 19?? Ralph Jarzombek
2000 Herman Miller |
| 15 | 1930s-60s R. M. A. Kusumadinata (Sunda)
1951 Augusto Novaro 1983 Joe Zawinul, with the group Weather Report 1991 Easley Blackwood 1991 Clem Fortuna 1996 Herman Miller 1997 Randy Winchester 1998 Paul Erlich, with the group MAD DUXX 2001 Francesco Caratozzolo |
| 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)
1960s Ivor Darreg |
| 19 | 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 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 Bill Wesley 1998 Joe Monzo |
| 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)
1877 Bosanquet 1921 José Würschmidt (for the future, after 19 runs its course) (1960s Erv Wilson -- used modulus-22, not necessarily ET) 1960s Ivor Darreg 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 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 8ve. Also called enamu,
a MIDI pitch-bend unit.
1906 Richard H. Stein (first published 24-tET scores) 1908 Arnold Schönberg (sketches, no longer extant) 1908 Anton Webern (sketches) 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) 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 |
| 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 | 18?? Émile Chevé (by mistake) |
| 31 | 1555 Nicola Vicentino (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 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 | 3 units per Semitone = 12 * 3 units per "octave".
1907 Ferrucio Busoni (in his theory, but not used in compositions) 1923-1960s Alois Hába 19?? Andrzej Gawel's 19-of-36-tET subset scale |
| 41 | 1901 Paul von Jankó
(1960s Erv Wilson claims that Partch was intuitively feeling out 41-ET) 1975 George Secor 1989 Helen Fowler 1998 Patrick Ozzard-Low |
| 43 | 1701 Joseph Sauveur meride (nearly identical to 1/5-comma meantone) |
| 46 | 1989 R. Fuller
1998 Graham Breed 2000 Dave Keenan and Paul Erlich |
| 48 | doamu, a MIDI pitch-bend unit:
22 (= 4) doamus per Semitone = 12 * 22 = 48 doamus per 8ve.
Also called "eighth-tones".
early 1900s Charles Ives (in Monzo analysis of Ives "stretched" scales 1915 N. Kulbin 1924 Julián Carrillo 19?? Patrizio Barbieri 19?? Claus-Steffen Mahnkopf 19?? Volker Staub 1998 Joseph Pehrson |
| 50 | (nearly identical to
2/7-comma meantone,
the first meantone to be
described with mathematical exactitude, in 1558 by Zarlino)
1710 Konrad Henfling 1759 Robert Smith (as an approximation to his ideal 5/18-comma meantone system) 1835 Wesley Woolhouse (almost identical to his 7/26-comma 'optimal meantone') 1940s Tillman Schafer |
| 53 | (nearly identical to both Pythagorean and 5-limit JI tuning)
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) |
| 55 | (nearly identical 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) |
| 58 | 1770 Dom François Bedos de Celles
2002 Gene Ward Smith |
| 60 | 5 units per Semitone = 12 * 5 units per "octave".
1980s? Richard Boulanger |
| 68 | 1847 Meshaqah (describing modern Greek tuning)
1989? John Chalmers (describing Byzantine tuning) |
| 72 | 6 units per Semitone = 12 * 6 units per "octave".
1800s standard quantization for Byzantine Chant 1927 Alois Hába (in his book Neue Harmonielehre) 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). 1951 Augusto Novaro 1953 Ivan Wyschnegradsky 1963 Iannis Xenakis (cf. his book Musiques formelles) 1970 Ezra Sims 1970 Franz Richter Herf 1970 Rolf Maedel 1970s? 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 (as superset and for notation of MIRACLE subset scales) 2001 Julia Werntz |
| 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) |
| 84 | 7 units per Semitone = 12 * 7 units per "octave".
1985 Harald Waage (for 5-limit JI) |
| 96 | triamu, a MIDI pitch-bend unit:
23 (= 8) triamus per Semitone = 12 * 23 = 96 triamus per "octave".
Also called "1/16-tones".
1924 Julián Carrillo 1980 Pascale Criton 2001 Vincent-Olivier Gagnon |
| 100 | 1980s Barry Vercoe - built into CSound software |
| 118 | 1874-5 Bosanquet |
| 144 | 12 units per Semitone = 12 * 12 units per "octave".
1946 Joseph Schillinger 1999 Dan Stearns and Joe Monzo (chiefly for its value as a unified notation for mixed EDOs and/or complex JI tunings) |
| 152 | 1999(?) Paul Erlich, "Universal Tuning" |
| 171 | 1926 Perrett
1975 Martin Vogel |
| 192 | tetramu, a MIDI pitch-bend unit: 24 (= 16) tetramus per Semitone = 12 * 24 = 192 tetramus per "octave". |
| 200 | 16 2/3 degrees per Semitone
2002 Joe Monzo (in analyzing Werckmeister III) |
| 205 | 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 (for adaptive-JI tuning of Mahler's compositions) 2002 Bob Wendell (for quantification of JI to facilitate composing in a polyphonic blues style) 2002 George Secor & Dave Keenan (as a basis for notation for JI and multi-EDOs) |
| 270 | 1970s? Erv Wilson and John Chalmers
1997 Paul Hahn |
| 288 | early 1900s Charles Ives (in Monzo analysis of Ives "stretched" 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) |
| 318 | 1999 Joe Monzo (in analyzing Aristoxenus; 318 = 53*6) |
| 384 | pentamu, a MIDI pitch-bend unit: 25 (= 32) pentamus per semitone, 25 * 12 = 384 pentamus per 8ve. |
| 512 | 29 units per "octave".
1980s tuning resolution of some electronic instruments, notably Ensoniq VFX and VFX-SD. |
| 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"
before 1875 -- Captain J. W. F. Herschel (cited by Bosanquet) c.1970 -- Gene Ward Smith (for interval measurment, his 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 and an analogue of cents; 60 5/6 degrees per Semitone.) |
| 768 | hexamu, a MIDI pitch-bend unit:
26 (= 64) hexamus per semitone, 26 * 12 = 768 hexamus per 8ve.
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.) 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 my Yamaha TG-77) |
| 1200 | 1875 -- Alexander Ellis (his unit of measurement, called cents, 100 per 12-tET semitone) |
| 1536 | heptamu, a MIDI pitch-bend unit; 27 = 128 heptamus per Semitone; 12 * 27 = 1536 heptamus per 8ve |
| 1728 | 19?? -- Paul Beaver (rendered as 123) |
| 3072 | oktamu, a MIDI pitch-bend unit:
28 (= 256) oktamus per Semitone; 12 * 28 = 3072 oktamus per 8ve.
1990s -- Apple's QuickTime Musical Instruments spec |
| 4296 | 358 units per semitone
1992 -- Marc Jones (used as most convenient UHT [ultra-high temperament] to measure 5-limit intervals) |
| 6144 | enneamu, a MIDI pitch-bend unit: 29 (= 512) enneamus per Semitone; 12 * 29 = 6144 enneamus per 8ve. |
| 10600 | 1965 -- M. Ekrem Karadeniz -- his unit of measurement, called türk-sents, 200 units per 53edo-comma. |
| 12288 | dekamu, a MIDI pitch-bend unit: 210 (= 1024) dekamus per Semitone; 12 * 210 = 12288 dekamus per 8ve. |
| 24576 | endekamu, a MIDI pitch-bend unit: 211 (= 2048) endekamus per Semitone; 12 * 211 = 24576 endekamus per 8ve. |
| 30103 | 1864 -- Augustus De Morgan -- his unit of measurement, called jots. |
| 36829 | (19?? -- approximation to John Brombaugh's scale of tuning units.) |
| 49152 | dodekamu, a MIDI pitch-bend unit:
212 (= 4096) dodekamus per Semitone; 12 * 212 = 49152 dodekamus per 8ve. Also called cawapu.
1980s -- pitch-bend resolution of CakewalkTM and many other popular sequencer programs. |
| 98304 | tridekamu, a MIDI pitch-bend unit:
213 (= 8192) tridekamus per Semitone; 12 * 213 = 98304 tridekamus per 8ve.
1983 -- the maximum resolution possible in MIDI pitch-bend |
| 196608 | tetradekamu, a MIDI pitch-bend unit: 214 (= 16384) tetradekamus per Semitone; 12 * 214 = 196608 tetradekamus per 8ve. Also called midipu.
1983 -- finest possible resolution in the MIDI tuning Spec. 1999 -- MTS (MIDI tuning standard) |
|
[from Joe Monzo, JustMusic: A New Harmony.
Thanks to John Chalmers, Manuel Op de Coul, Margo Schulter,
and especially Paul Erlich,
for helpful criticism and additional info.]
[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.
[from 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.
Updated:
2003.08.02 - added 665edo to list, added central "JI" red dot to zoom graphs
2003.07.24 - clarified meaning of "gleichschwebende Temperatur"
2003.07.04 - added new Greek-like MIDI pitch-bend unit terms to list
2003.07.03 - added negatives of Paul's mouse-over graphics
2003.01.29 - mouse-over graphics by Paul Erlich added, gallery of lattices split off
into separate page of its own
2002.09.16 - links to Huygens-Fokker site, corrections from Paul Erlich on first diagram
2002.09.13 - many new names and links added to list
2002.06.20
2002.02.14-18
2002.01.25
