41 equal temperament

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 (equal frequency ratios). Each step represents a frequency ratio of 2^1/41, or 29.27 cents ( Play ), an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic and miracle temperaments. It is the smallest equal temperament whose perfect fifth is closer to just intonation than that of 12-ET.

History and 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. Also, the keyboard layout of the Tonal Plexus is designed with 41-ET in mind. 41-ET can also be seen as an octave-based approximation of the Bohlen-Pierce scale.

Interval size

Here are the sizes of some common intervals:

interval name	size (steps)	size (cents)	midi	just ratio	just (cents)	midi	error
perfect fifth	24	702.44	Play	3:2	701.96	Play	+0.48
septimal tritone	20	585.37	Play	7:5	582.51	Play	+2.85
11:8 wide fourth	19	556.10	Play	11:8	551.32	Play	+4.78
15:11 wide fourth	18	526.83	Play	15:11	536.95		−10.12
27:20 wide fourth	18	526.83	Play	27:20	519.55		+7.28
perfect fourth	17	497.56	Play	4:3	498.04	Play	−0.48
tridecimal major third	16	468.29	Play	13:10	454.21	Play	+14.08
septimal major third	15	439.02	Play	9:7	435.08	Play	+3.94
undecimal major third	14	409.76	Play	14:11	417.51	Play	−7.75
major third	13	380.49	Play	5:4	386.31	Play	−5.83
undecimal neutral third	12	351.22	Play	11:9	347.41	Play	+3.81
minor third	11	321.95	Play	6:5	315.64	Play	+6.31
tridecimal minor third	10	292.68	Play	13:11	289.21	Play	+3.47
septimal minor third	9	263.41	Play	7:6	266.87	Play	−3.46
septimal whole tone	8	234.15	Play	8:7	231.17	Play	+2.97
whole tone, major tone	7	204.88	Play	9:8	203.91	Play	+0.97
whole tone, minor tone	6	175.61	Play	10:9	182.40	Play	−6.79
lesser undecimal neutral second	5	146.34	Play	12:11	150.64	Play	−4.30
septimal diatonic semitone	4	117.07	Play	15:14	119.44	Play	−2.37
diatonic semitone	4	117.07	Play	16:15	111.73		+5.34
septimal chromatic semitone	3	87.80	Play	21:20	84.47	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	Play	64:63	27.26	Play	+2.00

shaded rows mark poor matches

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 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).