17 equal temperament

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

Template:Stack 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 $\sqrt[17]{2}$, 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.Template:Citation needed

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., ATemplate:Music/CTemplate:Music).File:17-tet scale on C.mid

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, DTemplate:Music, CTemplate:Music, D, ETemplate:Music, DTemplate:Music, E, F, GTemplate:Music, FTemplate:Music, G, ATemplate:Music, GTemplate:Music, A, BTemplate:Music, ATemplate:Music, B, C

Quarter tone sharps and flats can also be used, yielding the following chromatic scale:

C, CTemplate:Music/DTemplate:Music, CTemplate:Music/DTemplate:Music, D, DTemplate:Music/ETemplate:Music, DTemplate:Music/ETemplate:Music, E, F, FTemplate:Music/GTemplate:Music, FTemplate:Music/GTemplate:Music, G, GTemplate:Music/ATemplate:Music, GTemplate:Music/ATemplate:Music, A, ATemplate:Music/BTemplate:Music, ATemplate:Music/BTemplate:Music, B, C

Interval size

Below are some intervals in Template:Nobr compared to just.

Major chord on C in Template:Nobr: All notes are within 37 cents of just intonation (rather than 14 cents for Template:Nobr).

Template:Nobr	File:Major chord on C in 17 equal temperament.mid
just	File:Major chord on C in just intonation.mid
Template:Nobr	File:Major chord on C.mid

I–IV–V–I chord progression in Template:Nobr.Template:Sfnp File:Simple I-IV-V-I isomorphic 17-TET.mid Whereas in Template:Nobr BTemplate:Music is 11 steps, in Template:Nobr BTemplate:Music is 16 steps.

interval name	size (steps)	size (cents)	Template:Sc audio	just ratio	just (cents)	Template:Sc audio	error
octave	17	1200 Template:0Template:0		2:1	1200 Template:0Template:0		0
Major seventh	16	1129.41		27:14	1137.04		-7.62
Major seventh	16	1129.41		15:8	1088.27		+41.14
Neutral seventh	15	1058.82		11:6	1049.36		+9.46
minor seventh	14	988.23		16:9	996.09		−Template:07.77
harmonic seventh	14	988.23		7:4	968.83		+19.41
major sixth	13	917.65		27:16	905.87		+11.78
neutral sixth	12	847.06		13:8	840.53		+6.53
minor sixth	11	776.47		14:9	764.92		+11.55
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	+Template:03.93
augmented fourth (notational)	9	635.29		16:11	648.68		-13.39
septimal tritone	Template: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	Template:08	564.71	File:8 steps in 17-et on C.mid	18:13	563.38	File:Tridecimal narrow tritone on C.mid	+Template:01.32
undecimal super-fourth	Template:08	564.71	File:8 steps in 17-et on C.mid	11:8Template:0	551.32	File:Eleventh harmonic on C.mid	+13.39
perfect fourth	Template:07	494.12	File:7 steps in 17-et on C.mid	4:3	498.04	File:Just perfect fourth on C.mid	−Template:03.93
septimal major third	Template: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	Template:06	423.53	File:6 steps in 17-et on C.mid	14:11	417.51	File:Undecimal major third on C.mid	+Template:06.02
major third	Template: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	Template:05	352.94	File:5 steps in 17-et on C.mid	16:13	359.47	File:Tridecimal neutral third on C.mid	−Template:06.53
undecimal neutral third	Template:05	352.94	File:5 steps in 17-et on C.mid	11:9Template:0	347.41	File:Undecimal neutral third on C.mid	+Template:05.53
minor third	Template: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	Template:04	282.35	File:4 steps in 17-et on C.mid	13:11	289.21	File:Tridecimal minor third on C.mid	−Template:06.86
septimal minor third	Template: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	Template: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	Template:03	211.76	File:3 steps in 17-et on C.mid	9:8	203.91	File:Major tone on C.mid	+Template:07.85
lesser whole tone	Template:03	211.76	File:3 steps in 17-et on C.mid	10:9Template:0	182.40	File:Minor tone on C.mid	+29.36
neutral second, lesser undecimal	Template:02	141.18	File:2 steps in 17-et on C.mid	12:11	150.64	File:Lesser undecimal neutral second on C.mid	−Template:09.46
greater tridecimal Template:Nobr	Template: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	+Template:02.60
lesser tridecimal Template:Nobr	Template: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	Template: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	Template: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	Template:01	Template:070.59	File:1 step in 17-et on C.mid	21:20	Template:084.47	File:Septimal chromatic semitone on C.mid	−13.88
chromatic semitone	Template:01	Template:070.59	File:1 step in 17-et on C.mid	25:24	Template:070.67	File:Just chromatic semitone on C.mid	−Template:00.08

Relation to 34 EDO

Template:Nobr is a subset of Template:Nobr

Sources

• Template:Cite journal

External links

• 17edo on Xenharmonic Wiki
• "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:YouTube, by Wongi Hwang