# 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 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,[1] magic and miracle[2] 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, ${\displaystyle 2^{24/41}\approx 1.50042}$ is a better approximation to the ratio ${\displaystyle 3/2=1.5}$ than either ${\displaystyle 2^{17/29}\approx 1.50129}$ or ${\displaystyle 2^{7/12}\approx 1.49831}$.

## History and use

Although 41-ET has not seen as wide use as other temperaments such as 19-ET or 31-ET [citation needed], pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague.[3] 41-ET can also be seen as an octave-based approximation of the Bohlen–Pierce scale.

41-ET guitars have been built, notably by Yossi Tamim. The frets on such guitars are very tightly spaced. To make a more playable 41-ET 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. [4] 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 (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-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 Octave 41 1200 2:1 1200 0 Harmonic seventh 33 965.85 7:4 968.83 −2.97 Perfect fifth 24 702.44 3:2 701.96 +0.48 Grave fifth 23 673.17 262144:177147 678.49 −5.32 Septimal tritone 20 585.37 7:5 582.51 +2.85 Eleventh harmonic 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 (super)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 Classic major third 13 380.49 5:4 386.31 −5.83 Tridecimal neutral third, thirteenth subharmonic 12 351.22 16:13 359.47 −8.25 Undecimal neutral third 12 351.22 11:9 347.41 +3.81 Classic 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 (sub)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 Classic 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 Classic 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 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.[5] 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,[6] 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 harmonics
or 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
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

1. ^ "Schismic Temperaments ", Intonation Information.
2. ^ "Lattices with Decimal Notation", Intonation Information.
3. ^ de Klerk, Dirk (1979). "Equal Temperament". Acta Musicologica. 51 (1): 140–150. doi:10.2307/932181. ISSN 0001-6241. JSTOR 932181.
4. ^ "The Kite Guitar ", Xenharmonic Wiki.
5. ^ "Ups and downs notation - Xenharmonic Wiki". en.xen.wiki. Retrieved 2024-08-20.
6. ^ "41edo - Xenharmonic Wiki". en.xen.wiki. Retrieved 2024-08-20.