Seventeen limit tetrads

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The 17th harmonic, octave reduced to the frequency ratio 17/16, is about 105¢. It is likely to sound dissonant against the fundamental, which is perhaps one reason why Just Intonation composers usually stop at the 13-limit or lower. Another interval of 17 that can sound just as dissonant is 18/17, about 99¢. Thus, 17/16 also clashes with 9/8. Therefore, one approach for 17-limit harmony is to filter the total list of chords to exclude these small steps, resulting in non-rooted harmonies.

Every otonal tetrad within the 17-limit is listed in the table below as the simplest possible subset of harmonics 2-17; there are 56 in total. The columns to the right of the chords provide different cutoffs for filtering out small steps. For example, chords with a 16/15 cutoff do not contain any interval smaller than or equal to 16/15 (which includes 17/16 and 18/17) among the dyads. Since this eliminates harmonics 2 and 9 right away, it also eliminates chords containing 10/9 and 9/8. (This is why there is a jump from 12/11 to 9/8 in the table below.)

no cutoff

56 tetrads

16/15 cutoff

20 tetrads

15/14 cutoff

16 tetrads

14/13 cutoff

13 tetrads

13/12 cutoff

10 tetrads

12/11 cutoff

7 tetrads

9/8 cutoff

4 tetrads

17/15 cutoff

1 tetrad

2:3:5:17
2:3:7:17
2:3:9:17
2:3:11:17
2:3:13:17
2:3:15:17
2:5:7:17
2:5:9:17
2:5:11:17
2:5:13:17
2:5:15:17
2:7:9:17
2:7:11:17
2:7:13:17
2:7:15:17
2:9:11:17
2:9:13:17
2:9:15:17
2:11:13:17
2:11:15:17
2:13:15:17
3:5:7:17 x x x x x x x
3:5:9:17
3:5:11:17 x x x x
3:5:13:17 x x x
3:5:15:17 x x x x x x
3:7:9:17
3:7:11:17 x x x x
3:7:13:17 x x
3:7:15:17 x
3:9:11:17
3:9:13:17
3:9:15:17
3:11:13:17 x x x
3:11:15:17 x x x x
3:13:15:17 x x x
5:7:9:17
5:7:11:17 x x x x x
5:7:13:17 x x
5:7:15:17 x
5:9:11:17
5:9:13:17
5:9:15:17
5:11:13:17 x x x x x
5:11:15:17 x x x x x
5:13:15:17 x x x x x x
7:9:11:17
7:9:13:17
7:9:15:17
7:11:13:17 x x
7:11:15:17 x
7:13:15:17 x
9:11:13:17
9:11:15:17
9:13:15:17
11:13:15:17 x x x x x x