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17 equal temperament

(Redirected from 17 tone equal temperament)

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.

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

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

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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

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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♭).

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

Interval size

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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
just
12 EDO
 
I–IV–V–I chord progression in 17 EDO.[4] 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 2:1 1200 0
minor seventh 14 988.23 16:9 996.09 −7.77
harmonic seventh 14 988.23 7:4 968.83 19.41
perfect fifth 10 705.88 3:2 701.96 3.93
septimal tritone 8 564.71 7:5 582.51 −17.81
tridecimal narrow tritone 8 564.71 18:13 563.38 1.32
undecimal super-fourth 8 564.71 11:8 551.32 13.39
perfect fourth 7 494.12 4:3 498.04 −3.93
septimal major third 6 423.53 9:7 435.08 −11.55
undecimal major third 6 423.53 14:11 417.51 6.02
major third 5 352.94 5:4 386.31 −33.37
tridecimal neutral third 5 352.94 16:13 359.47 −6.53
undecimal neutral third 5 352.94 11:9 347.41 5.53
minor third 4 282.35 6:5 315.64 −33.29
tridecimal minor third 4 282.35 13:11 289.21 −6.86
septimal minor third 4 282.35 7:6 266.87 15.48
septimal whole tone 3 211.76 8:7 231.17 −19.41
greater whole tone 3 211.76 9:8 203.91 7.85
lesser whole tone 3 211.76 10:9 182.40 29.36
neutral second, lesser undecimal 2 141.18 12:11 150.64 −9.46
greater tridecimal ⁠ 2 / 3 ⁠-tone 2 141.18 13:12 138.57 2.60
lesser tridecimal ⁠ 2 / 3 ⁠-tone 2 141.18 14:13 128.30 12.88
septimal diatonic semitone 2 141.18 15:14 119.44 21.73
diatonic semitone 2 141.18 16:15 111.73 29.45
septimal chromatic semitone 1 70.59 21:20 84.47 −13.88
chromatic semitone 1 70.59 25:24 70.67 −0.08

Relation to 34 EDO

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

References

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  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

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