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

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.
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.
Blackwood, Easley (Summer, 1991). "Modes and Chord Progressions in Equal Tunings", p.175, Perspectives of New Music, Vol. 29, No. 2, pp. 166-200.
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

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

Secor, George. "The 17-tone Puzzle — And the. Neo-medieval Key That Unlocks It".
Microtonalismo Heptadecatonic System Applications
Georg Hajdu's 1992 ICMC paper on the 17-tone piano project
ProyectoXVII Heptadecatonic System Applications project XVII - Peruvian
"Crocus" on YouTube, by Wongi Hwang

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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.