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]

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.^[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, identical to ups and downs notation for 17-EDO. ((10*7) mod 17 = 2.) 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.^[4] 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 a subset of 34 EDO, equivalent to every other step in the 34 EDO scale.