17 equal temperament

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 ).

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

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.

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♭). Play 

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

Below are some intervals in 17-EDO compared to just.

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 12-EDO, B♮ is 11 steps, in 17-EDO B♮ is 16 steps.
interval name size (steps) size (cents) midi just ratio just (cents) midi error
octave 17 1200 2:1 1200 0
minor seventh 14 988.23 16:9 996 −07.77
perfect fifth 10 705.88 Play  3:2 701.96 Play  +03.93
septimal tritone 08 564.71 Play  7:5 582.51 Play  −17.81
tridecimal narrow tritone 08 564.71 Play  18:13 563.38 Play  +01.32
undecimal super-fourth 08 564.71 Play  11:80 551.32 Play  +13.39
perfect fourth 07 494.12 Play  4:3 498.04 Play  −03.93
septimal major third 06 423.53 Play  9:7 435.08 Play  −11.55
undecimal major third 06 423.53 Play  14:11 417.51 Play  +06.02
major third 05 352.94 Play  5:4 386.31 Play  −33.37
tridecimal neutral third 05 352.94 Play  16:13 359.47 Play  −06.53
undecimal neutral third 05 352.94 Play  11:90 347.41 Play  +05.53
minor third 04 282.35 Play  6:5 315.64 Play  −33.29
tridecimal minor third 04 282.35 Play  13:11 289.21 play  −06.86
septimal minor third 04 282.35 Play  7:6 266.87 Play  +15.48
septimal whole tone 03 211.76 Play  8:7 231.17 Play  −19.41
whole tone 03 211.76 Play  9:8 203.91 Play  +07.85
neutral second, lesser undecimal 02 141.18 Play  12:11 150.64 Play  −09.46
greater tridecimal ​2⁄3-tone 02 141.18 Play  13:12 138.57 Play  +02.60
lesser tridecimal ​2⁄3-tone 02 141.18 Play  14:13 128.30 Play  +12.88
septimal diatonic semitone 02 141.18 Play  15:14 119.44 Play  +21.73
diatonic semitone 02 141.18 Play  16:15 111.73 Play  +29.45
septimal chromatic semitone 01 070.59 Play  21:20 084.47 Play  −13.88
chromatic semitone 01 070.59 Play  25:24 070.67 Play  −00.08

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.

References
  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.

the shbobo shtar, based on a Persian tar, has a 17tet fretboard.

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