17 equal temperament

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Figure 1: 17-ET on the regular diatonic tuning continuum at P5=705.88 cents.[1]

In music, 17 equal temperament is the tempered scale derived by dividing the octave into 17 equal steps (equal frequency ratios). Each step represents a frequency ratio of 17√2, or 70.6 cents. 17-ET is the tuning of the regular diatonic tuning in which the tempered perfect fifth is equal to 705.88 cents, as shown in Figure 1 (look for the label "17-TET").

History and use

Alexander J. Ellis refers to a tuning of seventeen tones based on perfect fourths and fifths as the Arabic scale.[2] In the thirteenth century, Middle-Eastern musician Safi al-Din Urmawi developed a theoretical system of seventeen tones to describe Arabic and Persian music, although the tones were not equally spaced. This 17-tone system remained the primary theoretical system until the development of the quarter tone scale.[citation needed]

Notation

Notation of Easley Blackwood[3] for 17 equal temperament: intervals are notated similarly to those they approximate and enharmonic equivalents are distinct from those of 12 equal temperament (e.g., A♯/C♭).File:17-tet scale on C.mid

Easley Blackwood Jr. created a notation system where sharps and flats raised/lowered 2 steps. This yields the chromatic scale:

C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C

Quarter tone sharps and flats can also be used, yielding the following chromatic scale:

C, Chalf sharp/D♭, C♯/Dhalf flat, D, Dhalf sharp/E♭, D♯/Ehalf flat, E, F, Fhalf sharp/G♭, F♯/Ghalf flat, G, Ghalf sharp/A♭, G♯/Ahalf flat, A, Ahalf sharp/B♭, A♯/Bhalf flat, B, C

Interval size

Below are some intervals in 17 EDO compared to just.

Major chord on C in 17 EDO : All notes are within 37 cents of just intonation (rather than 14 cents for 12 EDO).
17 EDO File:Major chord on C in 17 equal temperament.mid
just File:Major chord on C in just intonation.mid
12 EDO File:Major chord on C.mid
I–IV–V–I chord progression in 17 EDO.[4] File:Simple I-IV-V-I isomorphic 17-TET.mid Whereas in 12 EDO, B♮ is 11 steps, in 17 EDO, B♮ is 16 steps.
interval name size
(steps) 		size
(cents) 		MIDI
audio 		just
ratio 		just
(cents) 		MIDI
audio 		error
octave 17 1200 00 2:1 1200 00 0
minor seventh 14 988.23 16:9 996.09 −07.77
harmonic seventh 14 988.23 7:4 968.83 +19.41
perfect fifth 10 705.88 File:10 steps in 17-et on C.mid 3:2 701.96 File:Just perfect fifth on C.mid +03.93
septimal tritone 08 564.71 File:8 steps in 17-et on C.mid 7:5 582.51 File:Lesser septimal tritone on C.mid −17.81
tridecimal narrow tritone 08 564.71 File:8 steps in 17-et on C.mid 18:13 563.38 File:Tridecimal narrow tritone on C.mid +01.32
undecimal super-fourth 08 564.71 File:8 steps in 17-et on C.mid 11:80 551.32 File:Eleventh harmonic on C.mid +13.39
perfect fourth 07 494.12 File:7 steps in 17-et on C.mid 4:3 498.04 File:Just perfect fourth on C.mid −03.93
septimal major third 06 423.53 File:6 steps in 17-et on C.mid 9:7 435.08 File:Septimal major third on C.mid −11.55
undecimal major third 06 423.53 File:6 steps in 17-et on C.mid 14:11 417.51 File:Undecimal major third on C.mid +06.02
major third 05 352.94 File:5 steps in 17-et on C.mid 5:4 386.31 File:Just major third on C.mid −33.37
tridecimal neutral third 05 352.94 File:5 steps in 17-et on C.mid 16:13 359.47 File:Tridecimal neutral third on C.mid −06.53
undecimal neutral third 05 352.94 File:5 steps in 17-et on C.mid 11:90 347.41 File:Undecimal neutral third on C.mid +05.53
minor third 04 282.35 File:4 steps in 17-et on C.mid 6:5 315.64 File:Just minor third on C.mid −33.29
tridecimal minor third 04 282.35 File:4 steps in 17-et on C.mid 13:11 289.21 File:Tridecimal minor third on C.mid −06.86
septimal minor third 04 282.35 File:4 steps in 17-et on C.mid 7:6 266.87 File:Septimal minor third on C.mid +15.48
septimal whole tone 03 211.76 File:3 steps in 17-et on C.mid 8:7 231.17 File:Septimal major second on C.mid −19.41
greater whole tone 03 211.76 File:3 steps in 17-et on C.mid 9:8 203.91 File:Major tone on C.mid +07.85
lesser whole tone 03 211.76 File:3 steps in 17-et on C.mid 10:90 182.40 File:Minor tone on C.mid +29.36
neutral second, lesser undecimal 02 141.18 File:2 steps in 17-et on C.mid 12:11 150.64 File:Lesser undecimal neutral second on C.mid −09.46
greater tridecimal ⁠ 2 / 3 ⁠-tone 02 141.18 File:2 steps in 17-et on C.mid 13:12 138.57 File:Greater tridecimal two-third tone on C.mid +02.60
lesser tridecimal ⁠ 2 / 3 ⁠-tone 02 141.18 File:2 steps in 17-et on C.mid 14:13 128.30 File:Lesser tridecimal two-third tone on C.mid +12.88
septimal diatonic semitone 02 141.18 File:1 step in 17-et on C.mid 15:14 119.44 File:Septimal diatonic semitone on C.mid +21.73
diatonic semitone 02 141.18 File:2 steps in 17-et on C.mid 16:15 111.73 File:Just diatonic semitone on C.mid +29.45
septimal chromatic semitone 01 070.59 File:1 step in 17-et on C.mid 21:20 084.47 File:Septimal chromatic semitone on C.mid −13.88
chromatic semitone 01 070.59 File:1 step in 17-et on C.mid 25:24 070.67 File:Just chromatic semitone on C.mid −00.08

Relation to 34 EDO

17 EDO is where every other step in the 34 EDO scale is included, and the others are not accessible. Conversely 34 EDO is a subset of 17 EDO.

References

  1. ↑ Milne, Sethares & Plamondon 2007, pp. 15–32.
  2. ↑ Ellis, Alexander J. (1863). "On the Temperament of Musical Instruments with Fixed Tones", Proceedings of the Royal Society of London, vol. 13. (1863–1864), pp. 404–422.
  3. ↑ Blackwood, Easley (Summer 1991). "Modes and Chord Progressions in Equal Tunings". Perspectives of New Music. 29 (2): 166–200 (175). doi:10.2307/833437. JSTOR 833437.
  4. ↑ Milne, Sethares & Plamondon (2007), p. 29.

Sources

External links

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