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41 Equal Temperament

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Title: 41 Equal Temperament  
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Subject: Equal temperament, Bohlen–Pierce scale, List of pitch intervals, George Secor, 72 equal temperament, Septimal semicomma, Septimal kleisma, Magic temperament, Septimal minor third, Musical temperament
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41 Equal Temperament

In music, 41 equal temperament, abbreviated 41-tET, 41-EDO, or 41-ET, is the 12-ET.

History and use

Although 41-ET has not seen as wide use as other temperaments such as 19-ET or 31-ET , 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

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

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

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