15 equal temperament

Guitars have been constructed for 15-ET tuning. The American musician Wendy Carlos used 15-ET as one of two scales in the track Afterlife from the album Tales of Heaven and Hell.David J. Benson, Music: A Mathematical Offering, Cambridge University Press, (2006), p. 385. {{ISBN|9780521853873}}. Easley Blackwood, Jr. has written and recorded a suite for 15-ET guitar.Easley Blackwood, Jeffrey Kust, Easley Blackwood: Microtonal, Cedille (1996) ASIN: B0000018Z8. 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.

Easley Blackwood, Jr.'s notation of 15-EDO creates this chromatic scale:

B{{music|sharp}}/C, C{{music|sharp}}/D{{music|flat}}, D, D{{music|sharp}}, E{{music|flat}}, E, E{{music|sharp}}/F, F{{music|sharp}}/G{{music|flat}}, G, G{{music|sharp}}, A{{music|flat}}, A, A{{music|sharp}}, B{{music|flat}}, B, B{{music|sharp}}/C

Ups and Downs Notation,{{Xenharmonic wiki|Ups_and_downs_notation}} Accessed 2023-8-12. uses up and down arrows, written as a caret and a lower-case "v", usually in a sans-serif font. One arrow equals one edostep. In note names, the arrows come first, to facilitate chord naming. This yields this chromatic scale:

B/C, ^C/^D{{music|b}}, vC{{music|#}}/vD,

D, ^D/^E{{music|b}}, vD{{music|#}}/vE,

E/F, ^F/^G{{music|b}}, vF{{music|#}}/vG,

G, ^G/^A{{music|b}}, vG{{music|#}}/vA,

A, ^A/^B{{music|b}}, vA{{music|#}}/vB, B/C

Chords are spelled differently. C–E{{music|b}}–G is technically a C minor chord, but in fact it sounds like a sus2 chord C–D–G. The usual minor chord with 6/5 is the upminor chord. It's spelled as C–^E{{music|b}}–G and named as C^m. Compare with ^Cm (^C–^E{{music|b}}–^G).

Likewise the usual major chord with 5/4 is actually a downmajor chord. It's spelled as C–vE–G and named as Cv.

Porcupine Notation significantly changes chord spellings (e.g. the major triad is now C–E♯–G♯). In addition, enharmonic equivalences from 12-EDO are no longer valid. It yields the following chromatic scale:

C, C{{music|sharp}}/D{{music|flat}}, D, D{{music|sharp}}/E{{music|flat}}, E, E{{music|sharp}}/F{{music|flat}}, F, F{{music|sharp}}/G{{music|flat}}, G, G{{music|sharp}}, A{{music|flat}}, A, A{{music|sharp}}/B{{music|flat}}, B, B{{music|sharp}}, C

One possible decatonic notation uses the digits 0-9. Each of the 3 circles of 5 fifths is notated either by the odd numbers, the even numbers, or with accidentals.

1, 1{{music|sharp}}/2{{music|flat}}, 2, 3, 3{{music|sharp}}/4{{music|flat}}, 4, 5, 5{{music|sharp}}/6{{music|flat}}, 6, 7, 7{{music|sharp}}/8{{music|flat}}, 8, 9, 9{{music|sharp}}/0{{music|flat}}, 0, 1

In this article, unless specified otherwise, Blackwood's notation will be used.

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 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 tritones). 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. 15-ET also has a neutral second and septimal whole tone. 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.

45 equal temperament, which has a step size of 26.67 cents, is a threefold subdivision of 15 equal temperament. It has a perfect fifth of 693.33 cents, which is quite flat, but is still more accurate than the 720-cent fifth of 15-ET. Its best major third is 373.33 cents, which is slightly more accurate than the 400-cent one.

45-ET is an important example of a flattone temperament. Thus, it tempers out the syntonic comma like meantone temperaments do, but has a different mapping for intervals involving the seventh harmonic. Ordinarily, in meantone temperaments the 7:4 ratio is equated with the augmented sixth, whereas in 45-ET, this interval is instead equated with the diminished seventh, due to the smaller size of the chroma. This makes pieces involving chromatic alterations sound quite different.

45-ET is practically equivalent to an extended 2/5-comma meantone. Although it is harmonically less accurate than meantone temperaments like 31-ET or 19-ET, having a flatter fifth than both, it is still more accurate than 12-ET. {{cite web|url=https://en.xen.wiki/w/45edo|title=45edo}}

105 equal temperament (a sevenfold division of 15-ET) gives a meantone temperament with a fifth of 697.14 cents. It has a chromatic semitone equal to 80 cents, one step of 15-ET. {{cite web|url=https://en.xen.wiki/w/105edo|title=105edo}}

270 equal temperament does not see as much practical use as it is an extremely fine equal division with a step size of 4.44 cents, eighteen of which make up one step of 15-ET. Nonetheless, it is of academic interest due to being exceptionally accurate for an equal temperament of its size, particularly in the 15 odd limit.{{cite web|url=https://en.xen.wiki/w/270edo|title=270edo}}

{{reflist}}

• [https://web.archive.org/web/20080704060740/http://sonic-arts.org/darreg/dar35.htm Ivor Darreg, "15-TONE SCALE SYSTEM" (1991)], Sonic-Arts.org.
• [https://web.archive.org/web/20121025054304/http://home.comcast.net/~brentishere/15noteequaltempermenttutorial.html Brewt: "Fifteen note equal temperament tutorial"].
• [https://noahjordan.ca/portfolio/the-devil/ Noah Jordan: "The Devil" (piano work)].
• [https://www.youtube.com/watch?v=7_lJ6oVs6Yo Claudi Meneghin: "Tocada" (for Two Organs)].

{{Microtonal music}}

{{Musical tuning}}

Category:Equal temperaments

Category:Microtonality