17 equal temperament

In music, 17 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.

Figure 1: 17-ET on the regular diatonic tuning continuum at P5=705.88 cents.

1 step in 17-ET

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. 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 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, 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 EDO : All notes are within 37 cents of just intonation (rather than 14 cents for 12 EDO).

17 EDO
just
12 EDO

I–IV–V–I chord progression in 17 EDO. Whereas in 12 EDO, B♮ is 11 steps, in 17 EDO, B♮ is 16 steps.

interval name	size (steps)	size (cents)	MIDI audio	just ratio	just (cents)	MIDI audio	error
octave	17	1200 00		2:1	1200 00		0
minor seventh	14	988.23		16:9	996.09		−07.77
harmonic seventh	14	988.23		7:4	968.83		19.41
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
greater whole tone	03	211.76		9:8	203.91		07.85
lesser whole tone	03	211.76		10:90	182.40		29.36
neutral second, lesser undecimal	02	141.18		12:11	150.64		−09.46
greater tridecimal ⁠ 2 / 3 ⁠-tone	02	141.18		13:12	138.57		02.60
lesser tridecimal ⁠ 2 / 3 ⁠-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 EDO

17 EDO is where every other step in the 34 EDO scale is included, and the others are not accessible. Conversely 34 EDO is a subset of 17 EDO.

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. doi:10.1162/comj.2007.31.4.15. S2CID 27906745 – via mitpressjournals.org.

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
• "Crocus", 17 equal temperament, 9 tone mode on YouTube, by Wongi Hwang