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 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,"Schismic Temperaments ", Intonation Information. magic and miracle"Lattices with Decimal Notation", Intonation Information. 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-TET has not seen as wide use as other temperaments such as 19-TET or 31-TET , pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague. 41-TET can also be seen as an octave-based approximation of the Bohlen–Pierce scale.

41-TET guitars have been built, notably by Yossi Tamim. The frets on such guitars are very tightly spaced. To make a more playable 41-TET guitar, an approach called "The Kite Tuning" omits every-other fret (in other words, 41 frets per two octaves or 20.5 frets per octave) while tuning adjacent strings to an odd number of steps of 41. "The Kite Guitar ", Xenharmonic Wiki. Thus, any two adjacent strings together contain all the pitch classes of the full 41-TET system. The Kite Guitar's main tuning uses 13 steps of 41-TET (which approximates a 5/4 ratio) between strings. With that tuning, all simple ratios of odd limit 9 or less are available at spans at most only 4 frets.

41-TET is also a subset of 205-TET, 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 namesize (steps)size (cents)midijust ratiojust (cents)midierrorOctave4112002:112000Harmonic seventh33965.857:4968.83−2.97Perfect fifth24702.443:2701.96+0.48Grave fifth23673.17262144:177147678.49−5.32Septimal tritone20585.377:5582.51+2.85Eleventh harmonic19556.1011:8551.32+4.7815:11 Wide fourth18526.8315:11536.95−10.1227:20 Wide fourth18526.8327:20519.55+7.28Perfect fourth17497.564:3498.04−0.48Septimal narrow fourth16468.2921:16470.78−2.48Septimal (super)major third15439.029:7435.08+3.94Undecimal major third14409.7614:11417.51−7.75Pythagorean major third14409.7681:64407.82+1.94Classic major third13380.495:4386.31−5.83Tridecimal neutral third, thirteenth subharmonic12351.2216:13359.47−8.25Undecimal neutral third12351.2211:9347.41+3.81Classic minor third11321.956:5315.64+6.31Pythagorean minor third10292.6832:27294.13−1.45Tridecimal minor third10292.6813:11289.21+3.47Septimal (sub)minor third9263.417:6266.87−3.46septimal whole tone8234.158:7231.17+2.97Diminished third8234.15256:225223.46+10.68Whole tone, major tone7204.889:8203.91+0.97Whole tone, minor tone6175.6110:9182.40−6.79Lesser undecimal neutral second5146.3412:11150.64−4.30Septimal diatonic semitone4117.0715:14119.44−2.37Pythagorean chromatic semitone4117.072187:2048113.69+3.39Classic diatonic semitone4117.0716:15111.73+5.34Pythagorean diatonic semitone387.80256:24390.22−2.4220:19 Wide semitone387.8020:1988.80−1.00Septimal chromatic semitone387.8021:2084.47+3.34Classic chromatic semitone258.5425:2470.67−12.1428:27 Wide semitone258.5428:2762.96−4.42Septimal 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 (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 lesser 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 100:99, 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).

Notation Using extended pythagorean notation results in double and even triple sharps and flats. Furthermore, the notes run out of order. The chromatic scale is C, B, A/E, D, C, B, E, D... These issues can be avoided by using ups and downs notation. The up and down arrows are written as a caret or a lower-case "v", usually in a sans-serif font. One arrow equals one step of 41-TET. In note names, the arrows come first, to facilitate chord naming. The many enharmonic equivalences allow great freedom of spelling.

C, ^C, ^^C/vvC/vD, vC/D, C/^D, ^C/^^D/vvD, vD, D, ^D, ^^D/vvD/vE, vD/E, D/^E, ^D/^^E/vvE, vE, E, ^E/vvF, ^^E/vF, F, ^F, ^^F/vvF/vG, vF/G, F/^G, ^F/^^G/vvG, vG, G, ^G, ^^G/vvG/vA, vG/A, G/^A, ^G/^^A/vvA, vA, A, ^A, ^^A/vvA/vB, vA/B, A/^B, ^A/^^B/vvB, vB, B, ^B/vvC, ^^B/vC, C

Chords of 41 equal temperament Because ups and downs notation names the intervals of 41-TET, it can provide precise chord names. The pythagorean minor chord with 32/27 on C is still named Cm and still spelled C–E–G. But the 5-limit upminor chord uses the upminor 3rd 6/5 and is spelled C–^E–G. This chord is named C^m. Compare with ^Cm (^C–^E–^G).

+ Various 7-limit chords of 41-TET Chord name Chord Notes As harmonicsor subharmonics Homonyms Sus4 C4 C-F-G 6:8:9 F sus2 Sus2 C2 C-D-G 8:9:12 or 9:8:6 G sus4 Downmajor or down Cv C-vE-G 4:5:6 Upminor C^m C-^E-G 6:5:4 Downminor Cvm C-vE-G 6:7:9 Upmajor or up C^ C-^E-G 9:7:6 Updim C^dim C-^E-G 5:6:7 Downdim Cvdim C-vE-G 7:6:5 Downmajor7 CvM7 C-vE-G-vB 8:10:12:15 Down7 Cv7 C-vE-G-vB 4:5:6:7 Down add7 Cv,7 C-vE-G-B 36:45:54:64 Up7 C^7 C-^E-G-^B 9:7:6:5 Upminor7 C^m7 C-^E-G-^B 10:12:15:18 ^E down6 Downminor7 Cvm7 C-vE-G-vB 12:14:18:21 Downmajor6 or down6 Cv6 C-vE-G-vA 12:15:18:20 vA upminor7 Upminor6 C^m6 C-^E-G-^A 12:10:8:7 ^E downdim down7 Downminor6 Cvm6 C-vE-G-vA 6:7:9:10 vA updim up7 Updim up7 C^dim^7 C-^E-G-^B 5:6:7:9 ^E downminor6 Downdim down7 Cvdimv7 C-vE-G-vB 7:6:5:4 vE upminor6 Up9 C^9 C-^E-G-^B-D 9:7:6:5:4 Down9 Cv9 C-vE-G-vB-D 4:5:6:7:9

References

External links 41edo on Xenharmonic Wiki