41 equal temperament


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. 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, magic and miracle 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

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. 41-ET can also be seen as an octave-based approximation of the Bohlen–Pierce scale.
41-ET guitars have been built, notably by . To make a more playable 41-ET guitar, an approach called omits every-other fret while tuning adjacent strings to an odd number of steps of 41. Thus, any two adjacent strings together contain all the pitch classes of the full 41-ET system. The Kite Guitar's main tuning uses 13 steps of 41-ET between strings. With that tuning, all simple ratios of odd limit 9 or less are available at spans at most only 4 frets.
41-ET is also a subset of 205-ET, for which the keyboard layout of the
is designed.

Interval size

Here are the sizes of some common intervals :
interval namesize size midijust ratiojust midierror
octave4112002:112000
harmonic seventh33965.857:4968.83−2.97
perfect fifth24702.443:2701.96+0.48
septimal tritone20585.377:5582.51+2.85
11:8 wide fourth19556.1011:8551.32+4.78
15:11 wide fourth18526.8315:11536.95−10.12
27:20 wide fourth18526.8327:20519.55+7.28
perfect fourth17497.564:3498.04−0.48
septimal narrow fourth16468.2921:16470.78−2.48
septimal major third15439.029:7435.08+3.94
undecimal major third14409.7614:11417.51−7.75
Pythagorean major third14409.7681:64407.82+1.94
major third13380.495:4386.31−5.83
tridecimal neutral third, inverted 13th harmonic12351.2216:13359.47−8.25
undecimal neutral third12351.2211:9347.41+3.81
minor third11321.956:5315.64+6.31
Pythagorean minor third10292.6832:27294.13−1.45
tridecimal minor third10292.6813:11289.21+3.47
septimal minor third9263.417:6266.87−3.46
septimal whole tone8234.158:7231.17+2.97
diminished third8234.15256:225223.46+10.68
whole tone, major tone7204.889:8203.91+0.97
whole tone, minor tone6175.6110:9182.40−6.79
lesser undecimal neutral second5146.3412:11150.64−4.30
septimal diatonic semitone4117.0715:14119.44−2.37
Pythagorean chromatic semitone4117.072187:2048113.69+3.39
diatonic semitone4117.0716:15111.73+5.34
Pythagorean diatonic semitone387.80256:24390.22−2.42
20:19 wide semitone387.8020:1988.80−1.00
septimal chromatic semitone387.8021:2084.47+3.34
chromatic semitone258.5425:2470.67−12.14
28:27 wide semitone258.5428:2762.96−4.42
septimal comma129.2764:6327.26+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. 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. Although not as accurate, it can be considered a full 15-limit tuning as well.

Tempering

Intervals not tempered out by 41-ET include the diesis, septimal diesis, septimal sixth-tone, septimal comma, and the syntonic comma.
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, 1029:1024, and the small diesis.