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 (equal frequency ratios). Each step represents a frequency ratio of 2^1/41, 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, $2^{24/41}\approx 1.50042$ is a better approximation to the ratio $3/2=1.5$ than either $2^{17/29}\approx 1.50129$ or $2^{7/12}\approx 1.49831$.

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 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
harmonic seventh	33	965.85		7:4	968.83		−2.97
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
tridecimal neutral third, 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
diminished third	8	234.15		256:225	223.46		+10.68
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
Pythagorean chromatic semitone	4	117.07		2187:2048	113.69		+3.39
diatonic semitone	4	117.07		16:15	111.73		+5.34
Pythagorean diatonic semitone	3	87.80		256:243	90.22		−2.42
20:19 wide semitone	3	87.80		20:19	88.80		−1.00
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 wide 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). 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 (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).