17 equal temperament

File:Rank-2 temperaments with the generator close to a fifth and period an octave.jpg

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

File:1 step in 17-et on C.mid

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

File:17-tet scale on C.png

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♭).File:17-tet scale on C.mid

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, Chalf sharp/D♭, C♯/Dhalf flat, D, Dhalf sharp/E♭, D♯/Ehalf flat, E, F, Fhalf sharp/G♭, F♯/Ghalf flat, G, Ghalf sharp/A♭, G♯/Ahalf flat, A, Ahalf sharp/B♭, A♯/Bhalf flat, B, C

Interval size

Below are some intervals in 17 Template:Sc compared to just. [[File:Major chord on C.png|thumb|upright=1.3| Major chord on C in 17 Template:Sc : All notes are within 37 cents of just intonation (rather than 14 cents for [[12 equal temperament|12 Template:Sc]]).

17 Template:Sc	File:Major chord on C in 17 equal temperament.mid
just	File:Major chord on C in just intonation.mid
[[12 equal temperament|12 Template:Sc]]	File:Major chord on C.mid

]]

File:Simple I-IV-V-I isomorphic 17-TET.png

I–IV–V–I chord progression in 17 Template:Sc.Template:Sfnp File:Simple I-IV-V-I isomorphic 17-TET.mid Whereas in 12 Template:Sc, B♮ is 11 steps, in 17 Template:Sc, B♮ is 16 steps.

interval name	size (steps)	size (cents)	Template:Sc audio	just ratio	just (cents)	Template:Sc 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	File:10 steps in 17-et on C.mid	3:2	701.96	File:Just perfect fifth on C.mid	+03.93
septimal tritone	08	564.71	File:8 steps in 17-et on C.mid	7:5	582.51	File:Lesser septimal tritone on C.mid	−17.81
tridecimal narrow tritone	08	564.71	File:8 steps in 17-et on C.mid	18:13	563.38	File:Tridecimal narrow tritone on C.mid	+01.32
undecimal super-fourth	08	564.71	File:8 steps in 17-et on C.mid	11:80	551.32	File:Eleventh harmonic on C.mid	+13.39
perfect fourth	07	494.12	File:7 steps in 17-et on C.mid	4:3	498.04	File:Just perfect fourth on C.mid	−03.93
septimal major third	06	423.53	File:6 steps in 17-et on C.mid	9:7	435.08	File:Septimal major third on C.mid	−11.55
undecimal major third	06	423.53	File:6 steps in 17-et on C.mid	14:11	417.51	File:Undecimal major third on C.mid	+06.02
major third	05	352.94	File:5 steps in 17-et on C.mid	5:4	386.31	File:Just major third on C.mid	−33.37
tridecimal neutral third	05	352.94	File:5 steps in 17-et on C.mid	16:13	359.47	File:Tridecimal neutral third on C.mid	−06.53
undecimal neutral third	05	352.94	File:5 steps in 17-et on C.mid	11:90	347.41	File:Undecimal neutral third on C.mid	+05.53
minor third	04	282.35	File:4 steps in 17-et on C.mid	6:5	315.64	File:Just minor third on C.mid	−33.29
tridecimal minor third	04	282.35	File:4 steps in 17-et on C.mid	13:11	289.21	File:Tridecimal minor third on C.mid	−06.86
septimal minor third	04	282.35	File:4 steps in 17-et on C.mid	7:6	266.87	File:Septimal minor third on C.mid	+15.48
septimal whole tone	03	211.76	File:3 steps in 17-et on C.mid	8:7	231.17	File:Septimal major second on C.mid	−19.41
greater whole tone	03	211.76	File:3 steps in 17-et on C.mid	9:8	203.91	File:Major tone on C.mid	+07.85
lesser whole tone	03	211.76	File:3 steps in 17-et on C.mid	10:90	182.40	File:Minor tone on C.mid	+29.36
neutral second, lesser undecimal	02	141.18	File:2 steps in 17-et on C.mid	12:11	150.64	File:Lesser undecimal neutral second on C.mid	−09.46
greater tridecimal Template:Sfrac-tone	02	141.18	File:2 steps in 17-et on C.mid	13:12	138.57	File:Greater tridecimal two-third tone on C.mid	+02.60
lesser tridecimal Template:Sfrac-tone	02	141.18	File:2 steps in 17-et on C.mid	14:13	128.30	File:Lesser tridecimal two-third tone on C.mid	+12.88
septimal diatonic semitone	02	141.18	File:1 step in 17-et on C.mid	15:14	119.44	File:Septimal diatonic semitone on C.mid	+21.73
diatonic semitone	02	141.18	File:2 steps in 17-et on C.mid	16:15	111.73	File:Just diatonic semitone on C.mid	+29.45
septimal chromatic semitone	01	070.59	File:1 step in 17-et on C.mid	21:20	084.47	File:Septimal chromatic semitone on C.mid	−13.88
chromatic semitone	01	070.59	File:1 step in 17-et on C.mid	25:24	070.67	File:Just chromatic semitone on C.mid	−00.08

Relation to 34 EDO

17 Template:Sc is where every other step in the [[34 equal temperament|34 Template:Sc]] scale is included, and the others are not accessible. Conversely 17 Template:Sc is a subset of 34 Template:Sc.

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 projectTemplate:Cbignore
• Template:Replace on YouTube, by Wongi Hwang