17 equal temperament

Figure 1: 17-ET on the syntonic temperament’s tuning continuum at P5= 705.88 cents, from (Milne et al. 2007).

In music, 17 tone 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 2^1/17, or 70.6 cents (<phonos file="1 step in 17-et on C.mid">play</phonos>). 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.

17-ET is the tuning of the syntonic temperament in which the tempered perfect fifth is equal to 705.88 cents, as shown in Figure 1 (look for the label "17-TET"). On an isomorphic keyboard, the fingering of music composed in 17-ET is precisely the same as it is in any other syntonic tuning (such as 12-ET), so long as the notes are spelled properly -- that is, with no assumption of enharmonicity.

History

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♭). <phonos file="17-tet scale on C.mid">Play</phonos>

Major chord on C in 17 equal temperament: all notes within 37 cents of just intonation (rather than 14 for 12 equal temperament). <phonos file="Major chord on C in 17 equal temperament.mid">Play 17-et</phonos>, <phonos file="Major chord on C in just intonation.mid">Play just</phonos>, or <phonos file="Major chord on C.mid">Play 12-et</phonos>

I-IV-V-I chord progression in 17 equal temperament. <phonos file="Simple_I-IV-V-I_isomorphic_17-TET.mid">Play</phonos> Whereas in 12TET B♮ is 11 steps, in 17-TET B♮ is 16 steps.

Interval size

interval name	size (steps)	size (cents)	midi	just ratio	just (cents)	midi	error
perfect fifth	10	705.88	<phonos file="10 steps in 17-et on C.mid">Play</phonos>	3:2	701.96	<phonos file="Just perfect fifth on C.mid">Play</phonos>	+3.93
septimal tritone	8	564.71	<phonos file="8 steps in 17-et on C.mid">Play</phonos>	7:5	582.51	<phonos file="Lesser septimal tritone on C.mid">Play</phonos>	−17.81
tridecimal narrow tritone	8	564.71	<phonos file="8 steps in 17-et on C.mid">Play</phonos>	18:13	563.38		+1.32
undecimal super-fourth	8	564.71	<phonos file="8 steps in 17-et on C.mid">Play</phonos>	11:8	551.32	<phonos file="Eleventh harmonic on C.mid">Play</phonos>	+13.39
perfect fourth	7	494.12	<phonos file="7 steps in 17-et on C.mid">Play</phonos>	4:3	498.04	<phonos file="Just perfect fourth on C.mid">Play</phonos>	−3.93
septimal major third	6	423.53	<phonos file="6 steps in 17-et on C.mid">Play</phonos>	9:7	435.08	<phonos file="Septimal major third on C.mid">Play</phonos>	−11.55
undecimal major third	6	423.53	<phonos file="6 steps in 17-et on C.mid">Play</phonos>	14:11	417.51	<phonos file="Undecimal major third on C.mid">Play</phonos>	+6.02
major third	5	352.94	<phonos file="5 steps in 17-et on C.mid">Play</phonos>	5:4	386.31	<phonos file="Just major third on C.mid">Play</phonos>	−33.37
tridecimal neutral third	5	352.94	<phonos file="5 steps in 17-et on C.mid">Play</phonos>	16:13	359.47	<phonos file="Tridecimal neutral third on C.mid">Play</phonos>	−6.53
undecimal neutral third	5	352.94	<phonos file="5 steps in 17-et on C.mid">Play</phonos>	11:9	347.41	<phonos file="Undecimal neutral third on C.mid">Play</phonos>	+5.53
minor third	4	282.35	<phonos file="4 steps in 17-et on C.mid">Play</phonos>	6:5	315.64	<phonos file="Just minor third on C.mid">Play</phonos>	−33.29
tridecimal minor third	4	282.35	<phonos file="4 steps in 17-et on C.mid">Play</phonos>	13:11	289.21	<phonos file="Tridecimal minor third on C.mid">play</phonos>	−6.86
septimal minor third	4	282.35	<phonos file="4 steps in 17-et on C.mid">Play</phonos>	7:6	266.87	<phonos file="Septimal minor third on C.mid">Play</phonos>	+15.48
septimal whole tone	3	211.76	<phonos file="3 steps in 17-et on C.mid">Play</phonos>	8:7	231.17	<phonos file="Septimal major second on C.mid">Play</phonos>	−19.41
whole tone	3	211.76	<phonos file="3 steps in 17-et on C.mid">Play</phonos>	9:8	203.91	<phonos file="Major tone on C.mid">Play</phonos>	+7.85
neutral second, lesser undecimal	2	141.18	<phonos file="2 steps in 17-et on C.mid">Play</phonos>	12:11	150.64	<phonos file="Lesser undecimal neutral second on C.mid">Play</phonos>	−9.46
greater tridecimal 2/3-tone	2	141.18	<phonos file="2 steps in 17-et on C.mid">Play</phonos>	13:12	138.57		+2.60
lesser tridecimal 2/3-tone	2	141.18	<phonos file="2 steps in 17-et on C.mid">Play</phonos>	14:13	128.30		+12.88
septimal diatonic semitone	2	141.18	<phonos file="1_step_in_17-et_on_C.mid">Play</phonos>	15:14	119.44	<phonos file="Just chromatic semitone on C.mid">Play</phonos>	+21.73
diatonic semitone	2	141.18	<phonos file="2 steps in 17-et on C.mid">Play</phonos>	16:15	111.73	<phonos file="Just diatonic semitone on C.mid">Play</phonos>	+29.45
septimal chromatic semitone	1	70.59	<phonos file="1_step_in_17-et_on_C.mid">Play</phonos>	21:20	84.47	<phonos file="Septimal chromatic semitone on C.mid">Play</phonos>	−13.88
chromatic semitone	1	70.59	<phonos file="1_step_in_17-et_on_C.mid">Play</phonos>	25:24	70.67	<phonos file="Just chromatic semitone on C.mid">Play</phonos>	−0.08

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.

External links

• Secor, George. "The 17-tone Puzzle — And the. Neo-medieval Key That Unlocks It".
• Microtonalismo Heptadecatonic System Applications
• Georg Hajdu's 1992 ICMC paper on the 17-tone piano project
• ProyectoXVII Heptadecatonic System Applications project XVII - Peruvian