41 equal temperament
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In music, 41 equal temperament, often abbreviated 41-tET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 21/41, or 29.27 cents, an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic[1] 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.
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[edit] History & use
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]
[edit] Interval size
Here are the sizes of some common intervals:
| interval name | size (steps) | size (cents) | just ratio | just (cents) | difference |
| perfect fifth | 24 | 702.44 | 3:2 | 701.96 | -0.48 |
| tritone | 20 | 585.37 | 7:5 | 582.51 | -2.85 |
| (11:8) augmented fourth | 19 | 556.10 | 11:8 | 551.32 | -4.78 |
| (15:11) augmented fourth | 18 | 526.83 | 15:11 | 536.95 | 10.12 |
| perfect fourth | 17 | 497.56 | 4:3 | 498.04 | 0.48 |
| tridecimal major third | 16 | 468.29 | 13:10 | 454.21 | -14.08 |
| septimal major third | 15 | 439.02 | 9:7 | 435.08 | -3.94 |
| undecimal major third | 14 | 409.76 | 14:11 | 417.51 | 7.75 |
| major third | 13 | 380.49 | 5:4 | 386.31 | 5.83 |
| undecimal neutral third | 12 | 351.22 | 11:9 | 347.41 | -3.81 |
| minor third | 11 | 321.95 | 6:5 | 315.64 | -6.31 |
| 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 |
| neutral second, lesser undecimal | 5 | 146.34 | 12:11 | 150.64 | 4.30 |
| semitone, septimal diatonic | 4 | 117.07 | 15:14 | 119.44 | 2.37 |
| Septimal chromatic semitone | 3 | 87.80 | 21:20 | 84.47 | -3.34 |
| 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). It thus could be used as an approximation to 11-limit music.
With respect to its matches to intervals of the harmonic series, 41-ET and 31-ET are very similar. The primary difference is that 41-ET distinguishes between the major and minor whole tones, which is connected to the fact that 31-ET is a meantone system whereas 41-ET is not. 41-ET also introduces new intervals, such as the tridecimal minor third (13:11), and it distinguishes between the septimal and undecimal major thirds, which 31-ET does not do. 41-ET also has 6 distinct intervals between a perfect fourth and perfect fifth, whereas 31-ET has only four; the two additional intervals are poor matches to the ratios 15:11 and 22:15.
[edit] Tempering
Intervals not tempered out by 41-ET include the 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).
[edit] References
- ^ Schismatic temperaments
- ^ Lattices with Decimal Notation
- ^ [1] Dirk de Klerk "Equal Temperament", Acta Musicologica, Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150
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