41 equal temperament

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In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios). About this soundPlay  Each step represents a frequency ratio of 21/41, or 29.27 cents (About this soundPlay ), an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic,[1] magic and miracle[2] temperaments. It is the second smallest equal temperament, after 29-ET, whose perfect fifth is closer to just intonation than that of 12-ET. In other words, is a better approximation to the ratio than either or .

History and use[edit]

Although 41-ET has not seen as wide use as other temperaments such as 19-ET or 31-ET[citation needed], pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague.[3] 41-ET can also be seen as an octave-based approximation of the Bohlen–Pierce scale.

41-ET is also a subset of 205-ET, for which the keyboard layout of the Tonal Plexus is designed.

Interval size[edit]

Here are the sizes of some common intervals (shaded rows mark relatively poor matches):

interval name size (steps) size (cents) midi just ratio just (cents) midi error
octave 41 1200 2:1 1200 0
harmonic seventh 33 965.85 About this soundPlay  7:4 968.83 About this soundPlay  −2.97
perfect fifth 24 702.44 About this soundPlay  3:2 701.96 About this soundPlay  +0.48
septimal tritone 20 585.37 About this soundPlay  7:5 582.51 About this soundPlay  +2.85
11:8 wide fourth 19 556.10 About this soundPlay  11:8 551.32 About this soundPlay  +4.78
15:11 wide fourth 18 526.83 About this soundPlay  15:11 536.95 About this soundPlay  −10.12
27:20 wide fourth 18 526.83 About this soundPlay  27:20 519.55 About this soundPlay  +7.28
perfect fourth 17 497.56 About this soundPlay  4:3 498.04 About this soundPlay  −0.48
septimal narrow fourth 16 468.29 About this soundPlay  21:16 470.78 About this soundPlay  −2.48
septimal major third 15 439.02 About this soundPlay  9:7 435.08 About this soundPlay  +3.94
undecimal major third 14 409.76 About this soundPlay  14:11 417.51 About this soundPlay  −7.75
Pythagorean major third 14 409.76 About this soundPlay  81:64 407.82 About this soundPlay  +1.94
major third 13 380.49 About this soundPlay  5:4 386.31 About this soundPlay  −5.83
tridecimal neutral third, inverted 13th harmonic 12 351.22 About this soundPlay  16:13 359.47 About this soundPlay  −8.25
undecimal neutral third 12 351.22 About this soundPlay  11:9 347.41 About this soundPlay  +3.81
minor third 11 321.95 About this soundPlay  6:5 315.64 About this soundPlay  +6.31
Pythagorean minor third 10 292.68 About this soundPlay  32:27 294.13 About this soundPlay  −1.45
tridecimal minor third 10 292.68 About this soundPlay  13:11 289.21 About this soundPlay  +3.47
septimal minor third 9 263.41 About this soundPlay  7:6 266.87 About this soundPlay  −3.46
septimal whole tone 8 234.15 About this soundPlay  8:7 231.17 About this soundPlay  +2.97
diminished third 8 234.15 About this soundPlay  256:225 223.46 About this soundPlay  +10.68
whole tone, major tone 7 204.88 About this soundPlay  9:8 203.91 About this soundPlay  +0.97
whole tone, minor tone 6 175.61 About this soundPlay  10:9 182.40 About this soundPlay  −6.79
lesser undecimal neutral second 5 146.34 About this soundPlay  12:11 150.64 About this soundPlay  −4.30
septimal diatonic semitone 4 117.07 About this soundPlay  15:14 119.44 About this soundPlay  −2.37
Pythagorean chromatic semitone 4 117.07 About this soundPlay  2187:2048 113.69 About this soundPlay  +3.39
diatonic semitone 4 117.07 About this soundPlay  16:15 111.73 About this soundPlay  +5.34
Pythagorean diatonic semitone 3 87.80 About this soundPlay  256:243 90.22 About this soundPlay  −2.42
20:19 wide semitone 3 87.80 About this soundPlay  20:19 88.80 About this soundPlay  −1.00
septimal chromatic semitone 3 87.80 About this soundPlay  21:20 84.47 About this soundPlay  +3.34
chromatic semitone 2 58.54 About this soundPlay  25:24 70.67 About this soundPlay  −12.14
28:27 wide semitone 2 58.54 About this soundPlay  28:27 62.96 About this soundPlay  −4.42
septimal comma 1 29.27 About this soundPlay  64:63 27.26 About this soundPlay  +2.00

As the table above shows, the 41-ET both distinguishes between and closely matches all intervals involving the ratios in the harmonic series up to and including the 10th overtone. This includes the distinction between the major tone and minor tone (thus 41-ET is not a meantone tuning). These close fits make 41-ET a good approximation for 5-, 7- and 9-limit music.

41-ET also closely matches a number of other intervals involving higher harmonics. It distinguishes between and closely matches all intervals involving up through the 12th overtones, with the exception of the greater undecimal neutral second (11:10). Although not as accurate, it can be considered a full 15-limit tuning as well.

Tempering[edit]

Intervals not tempered out by 41-ET include the diesis (128:125), septimal diesis (49:48), septimal sixth-tone (50:49), septimal comma (64:63), and the syntonic comma (81:80).

41-ET tempers out the 100:99 ratio, which is the difference between the greater undecimal neutral second and the minor tone, as well as the septimal kleisma (225:224), 1029:1024 (the difference between three intervals of 8:7 the interval 3:2), and the small diesis (3125:3072).

References[edit]

  1. ^ "Schismic Temperaments ", Intonation Information.
  2. ^ "Lattices with Decimal Notation", Intonation Information.
  3. ^ [1] Dirk de Klerk "Equal Temperament", Acta Musicologica, Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150