17 Equal Temperament
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17 Equal Temperament
Figure 1: 17-ET on the Regular diatonic tuning continuum at P5= 705.88 cents, from (Milne et al. 2007).[1]

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 , or 70.6 cents ( ). 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.[]

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

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?).
Major chord on C in 17 equal temperament: all notes within 37 cents of just intonation (rather than 14 for 12 equal temperament). , , or
I-IV-V-I chord progression in 17 equal temperament.[4]  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   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 -tone 02 141.18   13:12 138.57   +02.60
lesser tridecimal -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.