41 equal temperament
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In music, 41 equal temperament, often abbreviated 41-tET, 41-EDO, 41-ET, is the tempered scale derived by dividing the octave into 41 equal-sized steps. Each step represents a frequency ratio of 21/41, or 29.27 cents. This tuning is the smallest equal temperament with a closer match to the perfect fifth than the standard 12 equal temperament.[1][2] It can be viewed as a schismatic temperament[3], and a miracle temperament[4][5].
[edit] History & 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.[6]
[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 |
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. These close fits make 41-ET a good approximation for 5- and 7-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] References
- ^ http://thinkzone.wlonk.com/Music/12Tone.htm
- ^ [1] Noah Hearle, "The Relationship Between Mathematics and Musical Temperament", Nahoo.
- ^ http://x31eq.com/schismic.htm
- ^ http://x31eq.com/decimal_lattice.htm
- ^ [2] Joe Monzo, "Miracle (temperament family)" Encyclopedia of Microtonal Music Theory.
- ^ [3] Dirk de Klerk "Equal Temperament", Acta Musicologica, Vol. 51, Fasc. 1 (Jan. - Jun., 1979), pp. 140-150
| Tunings | edit | ||||
| Pythagorean · Just intonation · Harry Partch's 43-tone scale | |||||
| Regular temperaments | |||||
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| Irregular temperaments | |||||
| Well temperament | |||||
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