17 equal temperament

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 2^1/17, 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.

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

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 

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 12TET B♮ is 11 steps, in 17-TET B♮ is 16 steps.

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	+3.93
septimal tritone	8	564.71	Play	7:5	582.51	Play	−17.81
tridecimal narrow tritone	8	564.71	Play	18:13	563.38		+1.32
undecimal super-fourth	8	564.71	Play	11:8	551.32	Play	+13.39
perfect fourth	7	494.12	Play	4:3	498.04	Play	−3.93
septimal major third	6	423.53	Play	9:7	435.08	Play	−11.55
undecimal major third	6	423.53	Play	14:11	417.51	Play	+6.02
major third	5	352.94	Play	5:4	386.31	Play	−33.37
tridecimal neutral third	5	352.94	Play	16:13	359.47	Play	−6.53
undecimal neutral third	5	352.94	Play	11:9	347.41	Play	+5.53
minor third	4	282.35	Play	6:5	315.64	Play	−33.29
tridecimal minor third	4	282.35	Play	13:11	289.21	play	−6.86
septimal minor third	4	282.35	Play	7:6	266.87	Play	+15.48
septimal whole tone	3	211.76	Play	8:7	231.17	Play	−19.41
whole tone	3	211.76	Play	9:8	203.91	Play	+7.85
neutral second, lesser undecimal	2	141.18	Play	12:11	150.64	Play	−9.46
greater tridecimal 2/3-tone	2	141.18	Play	13:12	138.57		+2.60
lesser tridecimal 2/3-tone	2	141.18	Play	14:13	128.30		+12.88
septimal diatonic semitone	2	141.18	Play	15:14	119.44	Play	+21.73
diatonic semitone	2	141.18	Play	16:15	111.73	Play	+29.45
septimal chromatic semitone	1	70.59	Play	21:20	84.47	Play	−13.88
chromatic semitone	1	70.59	Play	25:24	70.67	Play	−0.08

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

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.