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^ⓘ). 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]}

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

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^ⓘ	+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.

External links

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

Sources

^ 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.

[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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