In music, 41 equal temperament, often abbreviated 41-tET, 41-EDO, or 41-ET, is the tempered

Musical temperament

In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

 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

Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

 , an interval close in size to the septimal comma

Septimal comma

A septimal comma is a small musical interval in just intonation that contains the number seven in its prime factorization. There is more than one such interval, so the term septimal comma is ambiguous, but it most commonly refers to the interval 64/63....

. 41-ET can be seen as a tuning of the schismatic

Schismatic temperament

The schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament or Helmholtz temperament.-Comparison with other tunings:...

 and miracle

Miracle temperament

In music, miracle temperament is a regular temperament discovered by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones. Because 15:14 and 16:15 are equated, their ratio 225:224 \left is tempered out, and two secors give an 8:7...

 temperaments. It is the smallest equal temperament whose perfect fifth

Perfect fifth

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

 is closer to just intonation

Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

 than that of 12-ET.

History and use

Although 41-ET has not seen as wide use as other temperaments such as 19-ET

19 equal temperament

In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...

 or 31-ET

31 equal temperament

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...

 , pianist and engineer Paul von Janko

Paul von Janko

Paul von Jankó was a Hungarian pianist and engineer.He first studied mathematics and music in Vienna, where he was a pupil of H. Schmitt, J. Krenn and Anton Bruckner. He then moved to Berlin where he during the years 1881 and 1882 studied mathematics at the city's University, and piano with H....

 built a piano using this tuning, which is on display at the Gemeentemuseum

Gemeentemuseum Den Haag

The Municipal Museum is an art museum, located in The Hague, Netherlands.The museum was built by the Dutch architect H.P. Berlage. It is renowned for its large Mondrian collection, the largest in the world...

 in The Hague

The Hague

The Hague is the capital city of the province of South Holland in the Netherlands. With a population of 500,000 inhabitants , it is the third largest city of the Netherlands, after Amsterdam and Rotterdam...

. Also, the keyboard layout of the
Tonal Plexus is designed with 41-ET in mind.
41-ET can also be seen as an octave-based approximation of the Bohlen-Pierce scale

Bohlen-Pierce scale

The Bohlen–Pierce scale is a musical scale that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the diatonic scale. Compared with octave-repeating scales, its intervals are more consonant with certain types of acoustic spectra. It was...

.

Interval size

Here are the sizes of some common intervals:

interval name	size (steps)	size (cents)	midi	just ratio	just (cents)	midi	error
perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...	24	702.44		3:2	701.96		+0.48
septimal tritone Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7...	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 Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...	17	497.56		4:3	498.04		−0.48
tridecimal major third	16	468.29		13:10	454.21		+14.08
septimal major third Septimal major third In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal...	15	439.02		9:7	435.08		+3.94
undecimal major third	14	409.76		14:11	417.51		−7.75
major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...	13	380.49		5:4	386.31		−5.83
undecimal neutral third Neutral third A neutral third is a musical interval wider than a minor third but narrower than a major third . Three distinct intervals may be termed neutral thirds:...	12	351.22		11:9	347.41		+3.81
minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...	11	321.95		6:5	315.64		+6.31
tridecimal minor third	10	292.68		13:11	289.21		+3.47
septimal minor third Septimal minor third In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...	9	263.41		7:6	266.87		−3.46
septimal whole tone Septimal whole tone In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies. It is about 231 cents wide in just intonation. Although 24 equal temperament does not match this interval particularly well, its...	8	234.15		8:7	231.17		+2.97
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 Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:...	5	146.34		12:11	150.64		−4.30
septimal diatonic semitone	4	117.07		15:14	119.44		−2.37
diatonic semitone	4	117.07		16:15	111.73		+5.34
septimal chromatic semitone Septimal chromatic semitone In music, a septimal chromatic semitone or minor semitone is the interval 21:20 . It is about 84.47 cents. The septimal chromatic semitone may be derived from the harmonic series as the interval between the twentieth and twenty-first harmonics....	3	87.80		21:20	84.47		+3.34
chromatic semitone	2	58.54		25:24	70.67		−12.14
28:27 semitone	2	58.54		28:27	62.96		−4.42
septimal comma Septimal comma A septimal comma is a small musical interval in just intonation that contains the number seven in its prime factorization. There is more than one such interval, so the term septimal comma is ambiguous, but it most commonly refers to the interval 64/63....	1	29.27		64:63	27.26		+2.00

shaded rows mark poor matches

As the table above shows, the 41-ET both distinguishes between and closely matches all intervals involving the ratios in the harmonic series

Harmonic series (music)

Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

 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

Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

 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

Neutral second

A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:...

 (11:10).

Tempering

Intervals not tempered out by 41-ET include the septimal diesis

Septimal diesis

In music, septimal diesis is an interval with the ratio of 49:48 , which is the difference between the septimal whole tone and the septimal minor third. It is about 35.7 cents wide, which is narrower than a quarter-tone but wider than the septimal comma...

 (49:48), septimal sixth-tone

Septimal sixth-tone

In music, septimal sixth-tone is a septimal sixth-tone, an interval with the ratio of 50:49 , about 34.98 cents, which in just intonation is the difference between the lesser septimal tritone, and its inversion, the greater septimal tritone ....

 (50:49), septimal comma

Septimal comma

A septimal comma is a small musical interval in just intonation that contains the number seven in its prime factorization. There is more than one such interval, so the term septimal comma is ambiguous, but it most commonly refers to the interval 64/63....

 (64:63), and the syntonic comma

Syntonic comma

In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...

 (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

Septimal kleisma

In music, the ratio 225/224 is called the septimal kleisma .It is a minute comma type interval of approximately 7.7 cents. Factoring it into primes gives 2−5 32 52 7−1, which can be rewritten 2−1 2 . That says that it is the amount that two major thirds of 5/4 and a septimal...

(225:224).