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 . The frets on such guitars are very tightly spaced. 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
Grave fifth23673.17262144:177147678.49−5.32
Septimal tritone20585.377:5582.51+2.85
Eleventh harmonic19556.1011:8551.32+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 (super)major third15439.029:7435.08+3.94
Undecimal major third14409.7614:11417.51−7.75
Pythagorean major third14409.7681:64407.82+1.94
Classic major third13380.495:4386.31−5.83
Tridecimal neutral third, thirteenth subharmonic12351.2216:13359.47−8.25
Undecimal neutral third12351.2211:9347.41+3.81
Classic minor third11321.956:5315.64+6.31
Pythagorean minor third10292.6832:27294.13−1.45
Tridecimal minor third10292.6813:11289.21+3.47
Septimal (sub)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
Classic 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
Classic 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 lesser diesis, septimal diesis, septimal sixth-tone, septimal comma, and the syntonic comma.
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, 1029:1024, and the small diesis.

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.
Chord nameChordNotesAs harmonics
or subharmonics		Homonyms
Sus4C4C-F-G6:8:9F sus2
Sus2C2C-D-G8:9:12 or 9:8:6G sus4
Downmajor or downCvC-vE-G4:5:6
UpminorC^mC-^E-G6:5:4
DownminorCvmC-vE-G6:7:9
Upmajor or upC^C-^E-G9:7:6
UpdimC^dimC-^E-G5:6:7
DowndimCvdimC-vE-G7:6:5
Downmajor7CvM7C-vE-G-vB8:10:12:15
Down7Cv7C-vE-G-vB4:5:6:7
Down add7Cv,7C-vE-G-B36:45:54:64
Up7C^7C-^E-G-^B9:7:6:5
Upminor7C^m7C-^E-G-^B10:12:15:18^E down6
Downminor7Cvm7C-vE-G-vB12:14:18:21
Downmajor6 or down6Cv6C-vE-G-vA12:15:18:20vA upminor7
Upminor6C^m6C-^E-G-^A12:10:8:7^E downdim down7
Downminor6Cvm6C-vE-G-vA6:7:9:10vA updim up7
Updim up7C^dim^7C-^E-G-^B5:6:7:9^E downminor6
Downdim down7Cvdimv7C-vE-G-vB7:6:5:4vE upminor6
Up9C^9C-^E-G-^B-D9:7:6:5:4
Down9Cv9C-vE-G-vB-D4:5:6:7:9