17 equal temperament

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

1 step in 17-ET

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.

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 equal temperament: all notes within 37 cents of just intonation (rather than 14 for 12 equal temperament)

17-et
just
12-et

I–IV–V–I chord progression in 17 equal temperament. 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		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
whole tone	03	211.76		9:8	203.91		+07.85
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-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

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.

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