17 equal temperament

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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 172, or 70.6 cents (About this soundplay ).

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[edit]

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[edit]

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). About this soundPlay 

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[edit]

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). About this soundPlay 17-et , About this soundPlay just , or About this soundPlay 12-et 
I–IV–V–I chord progression in 17 equal temperament.[4] About this soundPlay  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 About this soundPlay  3:2 701.96 About this soundPlay  +03.93
septimal tritone 08 564.71 About this soundPlay  7:5 582.51 About this soundPlay  −17.81
tridecimal narrow tritone 08 564.71 About this soundPlay  18:13 563.38 About this soundPlay  +01.32
undecimal super-fourth 08 564.71 About this soundPlay  11:80 551.32 About this soundPlay  +13.39
perfect fourth 07 494.12 About this soundPlay  4:3 498.04 About this soundPlay  03.93
septimal major third 06 423.53 About this soundPlay  9:7 435.08 About this soundPlay  −11.55
undecimal major third 06 423.53 About this soundPlay  14:11 417.51 About this soundPlay  +06.02
major third 05 352.94 About this soundPlay  5:4 386.31 About this soundPlay  −33.37
tridecimal neutral third 05 352.94 About this soundPlay  16:13 359.47 About this soundPlay  06.53
undecimal neutral third 05 352.94 About this soundPlay  11:90 347.41 About this soundPlay  +05.53
minor third 04 282.35 About this soundPlay  6:5 315.64 About this soundPlay  −33.29
tridecimal minor third 04 282.35 About this soundPlay  13:11 289.21 About this soundplay  06.86
septimal minor third 04 282.35 About this soundPlay  7:6 266.87 About this soundPlay  +15.48
septimal whole tone 03 211.76 About this soundPlay  8:7 231.17 About this soundPlay  −19.41
whole tone 03 211.76 About this soundPlay  9:8 203.91 About this soundPlay  +07.85
neutral second, lesser undecimal 02 141.18 About this soundPlay  12:11 150.64 About this soundPlay  09.46
greater tridecimal ​23-tone 02 141.18 About this soundPlay  13:12 138.57 About this soundPlay  +02.60
lesser tridecimal ​23-tone 02 141.18 About this soundPlay  14:13 128.30 About this soundPlay  +12.88
septimal diatonic semitone 02 141.18 About this soundPlay  15:14 119.44 About this soundPlay  +21.73
diatonic semitone 02 141.18 About this soundPlay  16:15 111.73 About this soundPlay  +29.45
septimal chromatic semitone 01 070.59 About this soundPlay  21:20 084.47 About this soundPlay  −13.88
chromatic semitone 01 070.59 About this soundPlay  25:24 070.67 About this soundPlay  00.08

Relation to 34-ET[edit]

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[edit]

  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.

External links[edit]