17 equal temperament


In music, 17 tone equal temperament is the tempered scale derived by dividing the octave into 17 equal steps. Each step represents a frequency ratio of, 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.

History and use

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

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.
in 17 equal temperament. Whereas in 12-EDO, B is 11 steps, in 17-EDO B is 16 steps.
interval namesize size midijust ratiojust midierror
octave1712002:112000
minor seventh14988.2316:9996−7.77
perfect fifth10705.883:2701.96+3.93
septimal tritone8564.717:5582.51−17.81
tridecimal narrow tritone8564.7118:13563.38+1.32
undecimal super-fourth8564.7111:8551.32+13.39
perfect fourth7494.124:3498.04−3.93
septimal major third6423.539:7435.08−11.55
undecimal major third6423.5314:11417.51+6.02
major third5352.945:4386.31−33.37
tridecimal neutral third5352.9416:13359.47−6.53
undecimal neutral third5352.9411:9347.41+5.53
minor third4282.356:5315.64−33.29
tridecimal minor third4282.3513:11289.21−6.86
septimal minor third4282.357:6266.87+15.48
septimal whole tone3211.768:7231.17−19.41
whole tone3211.769:8203.91+7.85
neutral second, lesser undecimal2141.1812:11150.64−9.46
greater tridecimal -tone2141.1813:12138.57+2.60
lesser tridecimal -tone2141.1814:13128.30+12.88
septimal diatonic semitone2141.1815:14119.44+21.73
diatonic semitone2141.1816:15111.73+29.45
septimal chromatic semitone170.5921:2084.47−13.88
chromatic semitone170.5925:2470.67−0.08

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.