17 equal temperament

Summary

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

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.

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).

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 equal temperament: all notes within 37 cents of just intonation (rather than 14 for 12 equal temperament)
17-et
just
12-et
I–IV–V–I chord progression in 17 equal temperament.[4]
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
3:2 701.96
+03.93
septimal tritone 08 564.71
7:5 582.51
−17.81
tridecimal narrow tritone 08 564.71
18:13 563.38
+01.32
undecimal super-fourth 08 564.71
11:80 551.32
+13.39
perfect fourth 07 494.12
4:3 498.04
03.93
septimal major third 06 423.53
9:7 435.08
−11.55
undecimal major third 06 423.53
14:11 417.51
+06.02
major third 05 352.94
5:4 386.31
−33.37
tridecimal neutral third 05 352.94
16:13 359.47
06.53
undecimal neutral third 05 352.94
11:90 347.41
+05.53
minor third 04 282.35
6:5 315.64
−33.29
tridecimal minor third 04 282.35
13:11 289.21
06.86
septimal minor third 04 282.35
7:6 266.87
+15.48
septimal whole tone 03 211.76
8:7 231.17
−19.41
whole tone 03 211.76
9:8 203.91
+07.85
neutral second, lesser undecimal 02 141.18
12:11 150.64
09.46
greater tridecimal 23-tone 02 141.18
13:12 138.57
+02.60
lesser tridecimal 23-tone 02 141.18
14:13 128.30
+12.88
septimal diatonic semitone 02 141.18
15:14 119.44
+21.73
diatonic semitone 02 141.18
16:15 111.73
+29.45
septimal chromatic semitone 01 070.59
21:20 084.47
−13.88
chromatic semitone 01 070.59
25:24 070.67
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, 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). JSTOR 833437.
  4. ^ Milne, Sethares & Plamondon 2007, p. 29.

Sources

  • Milne, Andrew; Sethares, William; Plamondon, James (Winter 2007). "Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum". Computer Music Journal. 31 (4): 15–32.

External links

  • "The 17-tone Puzzle — And the Neo-medieval Key that Unlocks It" by George Secor
  • Libro y Programa Tonalismo, heptadecatonic system applications (in Spanish)
  • Georg Hajdu's 1992 ICMC paper on the 17-tone piano project[dead link]
  • "Crocus", 17 equal temperament, 9 tone mode on YouTube, by Wongi Hwang