In music, 15 equal temperament, called 15-TET, 15-

EDO Edo ( ja, , , "bay-entrance" or "estuary"), also romanized as Jedo, Yedo or Yeddo, is the former name of Tokyo. Edo, formerly a ''jōkamachi'' (castle town) centered on Edo Castle located in Musashi Province, became the ''de facto'' capital of ...

, or 15-ET, is a tempered scale derived by dividing the octave into 15 equal steps (equal frequency ratios). Each step represents a frequency ratio of (=2), 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 Slendro ( jv, ꦱ꧀ꦭꦺꦤ꧀ꦢꦿꦺꦴ, ban, slendro, translit=Sléndro) ( su, salendro, translit=Saléndro) is one of the essential tuning systems used in gamelan instruments that have pentatonic scale. Based on Javanese mythology, ...

scale in Indonesian

gamelan Gamelan () ( jv, ꦒꦩꦼꦭꦤ꧀, su, ᮌᮙᮨᮜᮔ᮪, ban, ᬕᬫᭂᬮᬦ᭄) is the traditional ensemble music of the Javanese, Sundanese, and Balinese peoples of Indonesia, made up predominantly of percussive instruments. T ...

. 15 equal temperament is not a

meantone Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...

system.

History and use

Guitars have been constructed for 15-ET tuning. The American musician

Wendy Carlos Wendy Carlos (born Walter Carlos, November 14, 1939) is an American musician and composer best known for her electronic music and film scores. Born and raised in Rhode Island, Carlos studied physics and music at Brown University before moving ...

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. Blackwood believes that 15 equal temperament, "is likely to bring about a considerable enrichment of both classical and popular repertoire in a variety of styles".Skinner (2007), p.75.

Notation

Easley Blackwood, Jr.'s notation of 15-EDO creates this chromatic scale: B/C, C/D, D, D, E, E, E/F, F/G, G, G, A, A, A, B, B, B/C An alternate form of notation, which is sometimes called "Porcupine Notation," can be used. It yields the following chromatic scale: C, C/D, D, D/E, E, E/F, F, F/G, G, G, A, A, A/B, B, B, C A notation that uses the numerals is also possible, in which each chain of fifths is notated either by the odd numbers, the even numbers, or with accidentals. 1, 1/2, 2, 3, 3/4, 4, 5, 5/6, 6, 7, 7/8, 8, 9, 9/0, 0, 1 In this article, unless specified otherwise, Blackwood's notation will be used.

Interval size

Here are the sizes of some common intervals in 15-ET: 15-ET matches the 7th and 11th harmonics well, but only matches the 3rd and 5th harmonics roughly. The

perfect fifth In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five ...

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 (except for the

septimal tritone A septimal tritone is a tritone (about one half of an octave) that involves the factor seven. There are two that are inverses. The lesser septimal tritone (also Huygens' tritone) is the musical interval with ratio 7:5 (582.51 cents). The greater ...

s). 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 Xenharmonic music is music that uses a tuning system that is unlike the 12-tone equal temperament scale. It was named by Ivor Darreg, from Xenia (Greek ξενία), ''hospitable,'' and Xenos (Greek ξένος) ''foreign.'' He stated that it wa ...

. Unlike 12-ET and 19-ET, 15-ET matches the 11:8 and 16:11 ratios. 15-ET also has a neutral second and

septimal whole tone In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to an 8/7 ratio of frequencies.Andrew Horner, Lydia Ayres (2002). ''Cooking with Csound: Woodwind and Brass R ...

. To construct a major third in 15-ET, one must stack two intervals of different sizes, whereas one can divide both the minor third and perfect fourth into two equal intervals.

References

External links

Ivor Darreg, "15-TONE SCALE SYSTEM" (1991)
''Sonic-Arts.org''.

Noah Jordan: "The Devil" (piano work)

Claudi Meneghin: ''"Tocada"'' (for Two Organs)
{{Musical tuning Equal temperaments Microtonality