15 equal temperament
Encyclopedia
In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the octave into 15 equal steps (equal frequency ratios). Each step represents a frequency ratio of 21/15, or 80 cents
. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of 5 equal divisions of the octave, each of which resembles the Slendro
scale in Indonesian
gamelan
.
used 15-ET as one of two scales in the track Afterlife from the album Tales of Heaven and Hell. Easley Blackwood Jr.
has written and recorded a suite for 15-ET guitar.
15-ET matches the 7th and 11th harmonics well, but only matches the 3rd and 5th harmonics roughly. The perfect fifth
is more out of tune than in 12-ET, 19-ET
, or 22-ET
, and the major third in 15-ET is the same as the major third in 12-ET, but the other intervals matched are more in tune. 15-ET is the smallest tuning that matches the 11th harmonic at all and still has a usable perfect fifth, but its match to intervals utilizing the 11th harmonic is poorer than 22-ET
, which also has more in-tune fifths and major thirds.
Although it contains a perfect fifth as well as major and minor thirds, the remainder of the harmonic and melodic language of 15-ET is quite different from 12-ET, and thus 15-ET could be described as xenharmonic. Unlike 12-ET and 19-ET, 15-ET matches the 11:8 and 16:11 ratios, and does not match the 7:5 or 10:7 tritone
. 15-ET also has a neutral second
and septimal whole tone
. In order to construct a major third, one must stack two intervals of different sizes, whereas one can divide both the minor third and perfect fourth into two equal intervals.
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...
. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of 5 equal divisions of the octave, each of which resembles the Slendro
Slendro
Slendro is a pentatonic scale, one of the two most common scales used in Indonesian gamelan music, the other being pélog.-Tuning:...
scale in Indonesian
Culture of Indonesia
Indonesian culture has been shaped by long interaction between original indigenous customs and multiple foreign influences. Indonesia is central along ancient trading routes between the Far East and the Middle East, resulting in many cultural practices being strongly influenced by a multitude of...
gamelan
Gamelan
A gamelan is a musical ensemble from Indonesia, typically from the islands of Bali or Java, featuring a variety of instruments such as metallophones, xylophones, drums and gongs; bamboo flutes, bowed and plucked strings. Vocalists may also be included....
.
History and use
Guitars have been constructed which use 15-ET tuning. The American musician Wendy CarlosWendy Carlos
Wendy Carlos is an American composer and electronic musician. Carlos first came to notice in the late 1960s with recordings made on the Moog synthesizer, then a relatively new and unknown instrument; most notable were LPs of synthesized Bach and the soundtrack for Stanley Kubrick's film A...
used 15-ET as one of two scales in the track Afterlife from the album Tales of Heaven and Hell. Easley Blackwood Jr.
Easley Blackwood Jr.
Easley Blackwood, , is a professor of music, a concert pianist, a composer of music, some using unusual tunings, and the author of books on music theory, including his research into the properties of microtonal tunings and traditional harmony.Blackwood was born in Indianapolis, Indiana...
has written and recorded a suite for 15-ET guitar.
Interval size
Here are the sizes of some common intervals in 15-ET:| interval name | size (steps) | size (cents) | just ratio | just (cents) | error | audio |
| 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... |
9 | 720 | 3:2 | 701.96 | +18.04 | |
| 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... |
7 | 560 | 7:5 | 582.51 | −22.51 | |
| 11:8 wide fourth | 7 | 560 | 11:8 | 551.32 | +8.68 | |
| 15:11 wide fourth | 7 | 560 | 15:11 | 536.95 | +23.05 | |
| 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... |
6 | 480 | 4:3 | 498.04 | −18.04 | |
| septimal major third | 5 | 400 | 9:7 | 435.08 | −35.08 | |
| 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... |
5 | 400 | 5:4 | 386.31 | +13.69 | |
| 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... |
4 | 320 | 6:5 | 315.64 | +4.36 | |
| 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... |
3 | 240 | 7:6 | 266.87 | −26.87 | |
| 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... |
3 | 240 | 8:7 | 231.17 | +8.83 | |
| 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:... |
2 | 160 | 11:10 | 165.00 | −5.00 | |
| lesser undecimal neutral second | 2 | 160 | 12:11 | 150.63 | +9.36 | |
| septimal diatonic semitone | 1 | 80 | 15:14 | 119.44 | −39.44 | |
| just diatonic semitone | 1 | 80 | 16:15 | 111.73 | −31.73 | |
| 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.... |
1 | 80 | 21:20 | 84.46 | −4.47 | |
| just chromatic semitone | 1 | 80 | 25:24 | 70.67 | +9.33 | |
| septimal third-tone Septimal third-tone A septimal 1/3-tone is an interval with the ratio of 28:27, which is the difference between the perfect fourth and the supermajor third. It is about 62.96 cents wide. The septimal 1/3-tone can be viewed either as a musical interval in its own right, or as a comma; if it is tempered out in a given... |
1 | 80 | 25:24 | 62.96 | +17.04 |
15-ET matches the 7th and 11th harmonics well, but only matches the 3rd and 5th harmonics roughly. The 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 more out of tune than in 12-ET, 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 22-ET
22 equal temperament
In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the tempered scale derived by dividing the octave into 22 equal steps . Each step represents a frequency ratio of 21/22, or 54.55 cents ....
, and the major third in 15-ET is the same as the major third in 12-ET, but the other intervals matched are more in tune. 15-ET is the smallest tuning that matches the 11th harmonic at all and still has a usable perfect fifth, but its match to intervals utilizing the 11th harmonic is poorer than 22-ET
22 equal temperament
In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the tempered scale derived by dividing the octave into 22 equal steps . Each step represents a frequency ratio of 21/22, or 54.55 cents ....
, which also has more in-tune fifths and major thirds.
Although it contains a perfect fifth as well as major and minor thirds, the remainder of the harmonic and melodic language of 15-ET is quite different from 12-ET, and thus 15-ET could be described as xenharmonic. Unlike 12-ET and 19-ET, 15-ET matches the 11:8 and 16:11 ratios, and does not match the 7:5 or 10:7 tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
. 15-ET also has a 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:...
and 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...
. In order to construct a major third, one must stack two intervals of different sizes, whereas one can divide both the minor third and perfect fourth into two equal intervals.