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). {{audio|41-tet scale on C.mid|Play}} Each step represents a frequency ratio of 2^1/41, or 29.27 cents ({{Audio|1 step in 41-et on C.mid|Play}}), an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic,[http://x31eq.com/schismic.htm "Schismic Temperaments "], Intonation Information. magic and miracle[http://x31eq.com/decimal_lattice.htm "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, $2^{24/41} \approx 1.50042$ 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 {{Citation needed|date=April 2008}}, pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague.{{Cite journal |last=de Klerk |first=Dirk |date=1979 |title=Equal Temperament |url=https://www.jstor.org/stable/932181 |journal=Acta Musicologica |volume=51 |issue=1 |pages=140–150 |doi=10.2307/932181 |jstor=932181 |issn=0001-6241|url-access=subscription }} 41-ET can also be seen as an octave-based approximation of the Bohlen–Pierce scale.

41-ET guitars have been built, notably by [https://www.yossitamim.com/ Yossi Tamim]. The frets on such guitars are very tightly spaced. To make a more playable 41-ET guitar, an approach called [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf "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. [https://en.xen.wiki/w/The_Kite_Guitar "The Kite Guitar "], Xenharmonic Wiki. 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

[http://hpi.zentral.zone/tonalplexus Tonal Plexus] is designed.

Interval size

Here are the sizes of some common intervals (shaded rows mark relatively poor matches):

class="wikitable"

|align=center bgcolor="#ffffb4"|interval name

|align=center bgcolor="#ffffb4"|size (steps)

|align=center bgcolor="#ffffb4"|size (cents)

|align=center bgcolor="#ffffb4"|midi

|align=center bgcolor="#ffffb4"|just ratio

|align=center bgcolor="#ffffb4"|just (cents)

|align=center bgcolor="#ffffb4"|midi

|align=center bgcolor="#ffffb4"|error

align=center|Octave

|align=center|41

|align=center|1200

|align=center|

|align=center|2:1

|align=center|1200

|align=center|

|align=center|0

align=center|Harmonic seventh

|align=center|33

|align=center|965.85

|align=center|{{Audio|33 steps in 41-et on C.mid|Play}}

|align=center|7:4

|align=center|968.83

|align=center|{{Audio|Harmonic seventh on C.mid|Play}}

|align=center|−2.97

align=center|Perfect fifth

|align=center|24

|align=center|702.44

|align=center|{{Audio|24 steps in 41-et on C.mid|Play}}

|align=center|3:2

|align=center|701.96

|align=center|{{Audio|Just perfect fifth on C.mid|Play}}

|align=center|+0.48

align=center|Grave fifth

|align=center|23

|align=center|673.17

|align=center|

|align=center|262144:177147

|align=center|678.49

|align=center|

|align=center|−5.32

align=center|Septimal tritone

|align=center|20

|align=center|585.37

|align=center|{{Audio|20 steps in 41-et on C.mid|Play}}

|align=center|7:5

|align=center|582.51

|align=center|{{Audio|Lesser septimal tritone on C.mid|Play}}

|align=center|+2.85

align=center|Eleventh harmonic

|align=center|19

|align=center|556.10

|align=center|{{Audio|19 steps in 41-et on C.mid|Play}}

|align=center|11:8

|align=center|551.32

|align=center|{{Audio|Eleventh harmonic on C.mid|Play}}

|align=center|+4.78

align=center bgcolor="#D4D4D4"|15:11 Wide fourth

|align=center bgcolor="#D4D4D4"|18

|align=center bgcolor="#D4D4D4"|526.83

|align=center bgcolor="#D4D4D4"|{{Audio|18 steps in 41-et on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|15:11

|align=center bgcolor="#D4D4D4"|536.95

|align=center bgcolor="#D4D4D4"|{{Audio|Undecimal augmented fourth on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|−10.12

align=center|27:20 Wide fourth

|align=center|18

|align=center|526.83

|align=center|{{Audio|18 steps in 41-et on C.mid|Play}}

|align=center|27:20

|align=center|519.55

|align=center|{{audio|Wolf fourth on C.mid|Play}}

|align=center|+7.28

align=center|Perfect fourth

|align=center|17

|align=center|497.56

|align=center|{{Audio|17 steps in 41-et on C.mid|Play}}

|align=center|4:3

|align=center|498.04

|align=center|{{Audio|Just perfect fourth on C.mid|Play}}

|align=center|−0.48

align=center |Septimal narrow fourth

|align=center |16

|align=center |468.29

|align=center |{{Audio|16 steps in 41-et on C.mid|Play}}

|align=center |21:16

|align=center |470.78

|align=center |{{Audio|Twenty-first harmonic on C.mid|Play}}

|align=center |−2.48

align=center|Septimal (super)major third

|align=center|15

|align=center|439.02

|align=center|{{Audio|15 steps in 41-et on C.mid|Play}}

|align=center|9:7

|align=center|435.08

|align=center|{{Audio|Septimal major third on C.mid|Play}}

|align=center|+3.94

align=center bgcolor="#D4D4D4"|Undecimal major third

|align=center bgcolor="#D4D4D4"|14

|align=center bgcolor="#D4D4D4"|409.76

|align=center bgcolor="#D4D4D4"|{{Audio|14 steps in 41-et on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|14:11

|align=center bgcolor="#D4D4D4"|417.51

|align=center bgcolor="#D4D4D4"|{{Audio|Undecimal major third on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|−7.75

align=center|Pythagorean major third

|align=center|14

|align=center|409.76

|align=center|{{Audio|14 steps in 41-et on C.mid|Play}}

|align=center|81:64

|align=center|407.82

|align=center|{{Audio|Pythagorean_major_third_on_C.mid|Play}}

|align=center|+1.94

align=center|Classic major third

|align=center|13

|align=center|380.49

|align=center|{{Audio|13 steps in 41-et on C.mid|Play}}

|align=center|5:4

|align=center|386.31

|align=center|{{Audio|Just major third on C.mid|Play}}

|align=center|−5.83

align=center bgcolor="#D4D4D4"|Tridecimal neutral third, thirteenth subharmonic

|align=center bgcolor="#D4D4D4"|12

|align=center bgcolor="#D4D4D4"|351.22

|align=center bgcolor="#D4D4D4"|{{Audio|12 steps in 41-et on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|16:13

|align=center bgcolor="#D4D4D4"|359.47

|align=center bgcolor="#D4D4D4"|{{Audio|Tridecimal neutral third on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|−8.25

align=center|Undecimal neutral third

|align=center|12

|align=center|351.22

|align=center|{{Audio|12 steps in 41-et on C.mid|Play}}

|align=center|11:9

|align=center|347.41

|align=center|{{Audio|Undecimal neutral third on C.mid|Play}}

|align=center|+3.81

align=center|Classic minor third

|align=center|11

|align=center|321.95

|align=center|{{Audio|11 steps in 41-et on C.mid|Play}}

|align=center|6:5

|align=center|315.64

|align=center|{{Audio|Just minor third on C.mid|Play}}

|align=center|+6.31

align=center|Pythagorean minor third

|align=center|10

|align=center|292.68

|align=center|{{Audio|10 steps in 41-et on C.mid|Play}}

|align=center|32:27

|align=center|294.13

|align=center|{{Audio|Pythagorean_minor_third_in_scale.mid|Play}}

|align=center|−1.45

align=center|Tridecimal minor third

|align=center|10

|align=center|292.68

|align=center|{{Audio|10 steps in 41-et on C.mid|Play}}

|align=center|13:11

|align=center|289.21

|align=center|{{Audio|Tridecimal minor third on C.mid|Play}}

|align=center|+3.47

align=center|Septimal (sub)minor third

|align=center|9

|align=center|263.41

|align=center|{{Audio|9 steps in 41-et on C.mid|Play}}

|align=center|7:6

|align=center|266.87

|align=center|{{Audio|Septimal minor third on C.mid|Play}}

|align=center|−3.46

align=center|septimal whole tone

|align=center|8

|align=center|234.15

|align=center|{{Audio|8 steps in 41-et on C.mid|Play}}

|align=center|8:7

|align=center|231.17

|align=center|{{Audio|Septimal major second on C.mid|Play}}

|align=center|+2.97

align=center bgcolor="#D4D4D4"|Diminished third

|align=center bgcolor="#D4D4D4"|8

|align=center bgcolor="#D4D4D4"|234.15

|align=center bgcolor="#D4D4D4"|{{Audio|8 steps in 41-et on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|256:225

|align=center bgcolor="#D4D4D4"|223.46

|align=center bgcolor="#D4D4D4"|{{Audio|Just diminished third on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|+10.68

align=center|Whole tone, major tone

|align=center|7

|align=center|204.88

|align=center|{{Audio|7 steps in 41-et on C.mid|Play}}

|align=center|9:8

|align=center|203.91

|align=center|{{Audio|Major tone on C.mid|Play}}

|align=center|+0.97

align=center|Whole tone, minor tone

|align=center|6

|align=center|175.61

|align=center|{{Audio|6 steps in 41-et on C.mid|Play}}

|align=center|10:9

|align=center|182.40

|align=center|{{Audio|Minor tone on C.mid|Play}}

|align=center|−6.79

align=center|Lesser undecimal neutral second

|align=center|5

|align=center|146.34

|align=center|{{Audio|5 steps in 41-et on C.mid|Play}}

|align=center|12:11

|align=center|150.64

|align=center|{{Audio|Lesser undecimal neutral second on C.mid|Play}}

|align=center|−4.30

align=center|Septimal diatonic semitone

|align=center|4

|align=center|117.07

|align=center|{{Audio|4 steps in 41-et on C.mid|Play}}

|align=center|15:14

|align=center|119.44

|align=center|{{Audio|Septimal diatonic semitone on C.mid|Play}}

|align=center|−2.37

align=center|Pythagorean chromatic semitone

|align=center|4

|align=center|117.07

|align=center|{{Audio|4 steps in 41-et on C.mid|Play}}

|align=center|2187:2048

|align=center|113.69

|align=center|{{Audio|Pythagorean apotome on C.mid|Play}}

|align=center|+3.39

align=center|Classic diatonic semitone

|align=center|4

|align=center|117.07

|align=center|{{Audio|4 steps in 41-et on C.mid|Play}}

|align=center|16:15

|align=center|111.73

|align=center|{{Audio|Just diatonic semitone on C.mid|Play}}

|align=center|+5.34

align=center|Pythagorean diatonic semitone

|align=center|3

|align=center|87.80

|align=center|{{Audio|3 steps in 41-et on C.mid|Play}}

|align=center|256:243

|align=center|90.22

|align=center|{{Audio|Pythagorean_minor_semitone_on_C.mid|Play}}

|align=center|−2.42

align=center|20:19 Wide semitone

|align=center|3

|align=center|87.80

|align=center|{{Audio|3 steps in 41-et on C.mid|Play}}

|align=center|20:19

|align=center|88.80

|align=center|{{Audio|Novendecimal augmented unison on C.mid|Play}}

|align=center|−1.00

align=center|Septimal chromatic semitone

|align=center|3

|align=center|87.80

|align=center|{{Audio|3 steps in 41-et on C.mid|Play}}

|align=center|21:20

|align=center|84.47

|align=center|{{Audio|Septimal chromatic semitone on C.mid|Play}}

|align=center|+3.34

align=center bgcolor="#D4D4D4"|Classic chromatic semitone

|align=center bgcolor="#D4D4D4"|2

|align=center bgcolor="#D4D4D4"|58.54

|align=center bgcolor="#D4D4D4"|{{Audio|2 steps in 41-et on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|25:24

|align=center bgcolor="#D4D4D4"|70.67

|align=center bgcolor="#D4D4D4"|{{Audio|Just chromatic semitone on C.mid|Play}}

|align=center bgcolor="#D4D4D4"|−12.14

align=center|28:27 Wide semitone

|align=center|2

|align=center|58.54

|align=center|{{Audio|2 steps in 41-et on C.mid|Play}}

|align=center|28:27

|align=center|62.96

|align=center|{{audio|Septimal minor second on C.mid|Play}}

|align=center|−4.42

align=center|Septimal comma

|align=center|1

|align=center|29.27

|align=center|{{Audio|1 step in 41-et on C.mid|Play}}

|align=center|64:63

|align=center|27.26

|align=center|{{Audio|Septimal comma on C.mid|Play}}

|align=center|+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{{music|sharp}}, A{{music|triplesharp}}/E{{music|tripleflat}}, D{{music|flat}}, C{{music|sharp}}, B{{music|doublesharp}}, E{{music|doubleflat}}, D... These issues can be avoided by using ups and downs notation.{{Cite web |title=Ups and downs notation - Xenharmonic Wiki |url=https://en.xen.wiki/w/Ups_and_downs_notation |access-date=2024-08-20 |website=en.xen.wiki}} 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{{music|sharp}}/vD{{music|flat}}, vC{{music|sharp}}/D{{music|flat}}, C{{music|sharp}}/^D{{music|flat}}, ^C{{music|sharp}}/^^D{{music|flat}}/vvD, vD,
D, ^D, ^^D/vvD{{music|sharp}}/vE{{music|flat}}, vD{{music|sharp}}/E{{music|flat}}, D{{music|sharp}}/^E{{music|flat}}, ^D{{music|sharp}}/^^E{{music|flat}}/vvE, vE,
E, ^E/vvF, ^^E/vF,
F, ^F, ^^F/vvF{{music|sharp}}/vG{{music|flat}}, vF{{music|sharp}}/G{{music|flat}}, F{{music|sharp}}/^G{{music|flat}}, ^F{{music|sharp}}/^^G{{music|flat}}/vvG, vG,
G, ^G, ^^G/vvG{{music|sharp}}/vA{{music|flat}}, vG{{music|sharp}}/A{{music|flat}}, G{{music|sharp}}/^A{{music|flat}}, ^G{{music|sharp}}/^^A{{music|flat}}/vvA, vA,
A, ^A, ^^A/vvA{{music|sharp}}/vB{{music|flat}}, vA{{music|sharp}}/B{{music|flat}}, A{{music|sharp}}/^B{{music|flat}}, ^A{{music|sharp}}/^^B{{music|flat}}/vvB, vB,
B, ^B/vvC, ^^B/vC, C

=Chords of 41 equal temperament=

Because ups and downs notation names the intervals of 41-TET,{{Cite web |title=41edo - Xenharmonic Wiki |url=https://en.xen.wiki/w/41edo#Intervals |access-date=2024-08-20 |website=en.xen.wiki}} it can provide precise chord names. The pythagorean minor chord with 32/27 on C is still named Cm and still spelled C–E{{music|flat}}–G. But the 5-limit upminor chord uses the upminor 3rd 6/5 and is spelled C–^E{{music|flat}}–G. This chord is named C^m. Compare with ^Cm (^C–^E{{music|flat}}–^G).

Chord name	Chord	Notes	As harmonics or subharmonics	Homonyms
class="wikitable" \|+ Various 7-limit chords of 41-TET
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{{music\|flat}}-G	6:5:4
Downminor	Cvm	C-vE{{music\|flat}}-G	6:7:9
Upmajor or up	C^	C-^E-G	9:7:6
Updim	C^dim	C-^E{{music\|flat}}-G{{music\|flat}}	5:6:7
Downdim	Cvdim	C-vE{{music\|flat}}-G{{music\|flat}}	7:6:5
Downmajor7	CvM7	C-vE-G-vB	8:10:12:15
Down7	Cv7	C-vE-G-vB{{music\|flat}}	4:5:6:7
Down add7	Cv,7	C-vE-G-B{{music\|flat}}	36:45:54:64
Up7	C^7	C-^E-G-^B{{music\|flat}}	9:7:6:5
Upminor7	C^m7	C-^E{{music\|flat}}-G-^B{{music\|flat}}	10:12:15:18	^E down6
Downminor7	Cvm7	C-vE{{music\|flat}}-G-vB{{music\|flat}}	12:14:18:21
Downmajor6 or down6	Cv6	C-vE-G-vA	12:15:18:20	vA upminor7
Upminor6	C^m6	C-^E{{music\|flat}}-G-^A	12:10:8:7	^E{{music\|flat}} downdim down7
Downminor6	Cvm6	C-vE{{music\|flat}}-G-vA	6:7:9:10	vA updim up7
Updim up7	C^dim^7	C-^E{{music\|flat}}-G{{music\|flat}}-^B{{music\|flat}}	5:6:7:9	^E{{music\|flat}} downminor6
Downdim down7	Cvdimv7	C-vE{{music\|flat}}-G{{music\|flat}}-vB{{music\|flat}}	7:6:5:4	vE{{music\|flat}} upminor6
Up9	C^9	C-^E-G-^B{{music\|flat}}-D	9:7:6:5:4
Down9	Cv9	C-vE-G-vB{{music\|flat}}-D	4:5:6:7:9

References

Category:Equal temperaments

Category:Microtonality