17 equal temperament

Figure 1: 17-ET on the Regular diatonic tuning continuum at P5= 705.88 cents, from (Milne et al. 2007).[1]

In music, 17 tone 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 ( play ). 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.

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

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♭).  Play 
Major chord on C in 17 equal temperament: all notes within 37 cents of just intonation (rather than 14 for 12 equal temperament).  Play 17-et ,  Play just , or  Play 12-et 
I–IV–V–I chord progression in 17 equal temperament.[4]  Play  Whereas in 12TET B♮ is 11 steps, in 17-TET B♮ is 16 steps.

Interval size

interval name size (steps) size (cents) midi just ratio just (cents) midi error
perfect fifth 10 705.88  Play  3:2 701.96  Play  +3.93
septimal tritone 8 564.71  Play  7:5 582.51  Play  −17.81
tridecimal narrow tritone 8 564.71  Play  18:13 563.38  Play  +1.32
undecimal super-fourth 8 564.71  Play  11:8 551.32  Play  +13.39
perfect fourth 7 494.12  Play  4:3 498.04  Play  −3.93
septimal major third 6 423.53  Play  9:7 435.08  Play  −11.55
undecimal major third 6 423.53  Play  14:11 417.51  Play  +6.02
major third 5 352.94  Play  5:4 386.31  Play  −33.37
tridecimal neutral third 5 352.94  Play  16:13 359.47  Play  −6.53
undecimal neutral third 5 352.94  Play  11:9 347.41  Play  +5.53
minor third 4 282.35  Play  6:5 315.64  Play  −33.29
tridecimal minor third 4 282.35  Play  13:11 289.21  play  −6.86
septimal minor third 4 282.35  Play  7:6 266.87  Play  +15.48
septimal whole tone 3 211.76  Play  8:7 231.17  Play  −19.41
whole tone 3 211.76  Play  9:8 203.91  Play  +7.85
neutral second, lesser undecimal 2 141.18  Play  12:11 150.64  Play  −9.46
greater tridecimal ​2⁄3-tone 2 141.18  Play  13:12 138.57  Play  +2.60
lesser tridecimal ​2⁄3-tone 2 141.18  Play  14:13 128.30  Play  +12.88
septimal diatonic semitone 2 141.18  Play  15:14 119.44  Play  +21.73
diatonic semitone 2 141.18  Play  16:15 111.73  Play  +29.45
septimal chromatic semitone 1 70.59  Play  21:20 84.47  Play  −13.88
chromatic semitone 1 70.59  Play  25:24 70.67  Play  −0.08

Relation to 34-ET

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

Sources

  1. ↑ Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 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", p.175, Perspectives of New Music, Vol. 29, No. 2, pp. 166-200.
  4. ↑ Andrew Milne, William Sethares, and James Plamondon (2007). "Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum", p.29. Computer Music Journal, 31:4, pp.15–32, Winter 2007.
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