## Definitions of tuning terms © 1998 by Joseph L. Monzo All definitions by Joe Monzo unless otherwise cited

Equal Temperament

a system of tuning based on a scale whose "steps" or degrees have logarithmically equal intervals between them, in contrast to the differently-spaced degrees of just intonation, meantone, well-temperament, or other tunings. 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:
• Ivor Darreg
• Easley Blackwood
• Brian McLaren
• Marc Jones
• Dan Stearns

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

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

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

[Note 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]

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