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 sound Play  Each step represents a frequency ratio of 21/41, or 29.27 cents (About this sound Play ), 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.

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
perfect fifth 24 702.44 About this sound Play  3:2 701.96 About this sound Play  +0.48
septimal tritone 20 585.37 About this sound Play  7:5 582.51 About this sound Play  +2.85
11:8 wide fourth 19 556.10 About this sound Play  11:8 551.32 About this sound Play  +4.78
15:11 wide fourth 18 526.83 About this sound Play  15:11 536.95 −10.12
27:20 wide fourth 18 526.83 About this sound Play  27:20 519.55 +7.28
perfect fourth 17 497.56 About this sound Play  4:3 498.04 About this sound Play  −0.48
septimal narrow fourth 16 468.29 About this sound Play  21:16 470.78 −2.48
septimal major third 15 439.02 About this sound Play  9:7 435.08 About this sound Play  +3.94
undecimal major third 14 409.76 About this sound Play  14:11 417.51 About this sound Play  −7.75
pythagorean major third 14 409.76 About this sound Play  81:64 407.82 About this sound Play  +1.94
major third 13 380.49 About this sound Play  5:4 386.31 About this sound Play  −5.83
inverted 13th harmonic 12 351.22 About this sound Play  16:13 359.47 About this sound Play  −8.25
undecimal neutral third 12 351.22 About this sound Play  11:9 347.41 About this sound Play  +3.81
minor third 11 321.95 About this sound Play  6:5 315.64 About this sound Play  +6.31
pythagorean minor third 10 292.68 About this sound Play  32:27 294.13 About this sound Play  −1.45
tridecimal minor third 10 292.68 About this sound Play  13:11 289.21 About this sound Play  +3.47
septimal minor third 9 263.41 About this sound Play  7:6 266.87 About this sound Play  −3.46
septimal whole tone 8 234.15 About this sound Play  8:7 231.17 About this sound Play  +2.97
whole tone, major tone 7 204.88 About this sound Play  9:8 203.91 About this sound Play  +0.97
whole tone, minor tone 6 175.61 About this sound Play  10:9 182.40 About this sound Play  −6.79
lesser undecimal neutral second 5 146.34 About this sound Play  12:11 150.64 About this sound Play  −4.30
septimal diatonic semitone 4 117.07 About this sound Play  15:14 119.44 About this sound Play  −2.37
diatonic semitone 4 117.07 About this sound Play  16:15 111.73 +5.34
pythagorean diatonic semitone 3 87.80 About this sound Play  256:243 90.22 About this sound Play  −2.42
septimal chromatic semitone 3 87.80 About this sound Play  21:20 84.47 About this sound Play  +3.34
chromatic semitone 2 58.54 25:24 70.67 −12.14
28:27 semitone 2 58.54 28:27 62.96 −4.42
septimal comma 1 29.27 About this sound Play  64:63 27.26 About this sound Play  +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).

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