17-limit

From Xenharmonic Wiki
Jump to: navigation, search

In 17-limit Just Intonation, all ratios in the system will contain no primes higher than 17. The 17-limit adds to the 13-limit a "minor ninth" of about 105¢ -- 17/16 -- and several other intervals between the 17th overtone and the lower ones.

The 17-prime-limit can be modeled in a 6-dimensional lattice, with the primes 3, 5, 7, 11, 13, and 17 represented by each dimension. The prime 2 does not appear in the typical 17-limit lattice because octave equivalence is presumed. If octave equivalence is not presumed, a seventh dimension is need.

EDOs which provides an excellent tuning for 17-limit intervals are: 46, 58, 72, 80, 94, 111, 121, 149, 159, 183, 217, 253, 270, 282, 301, 311, 320, 354, 364, 383, 388, 400, 414, 422, 436, 441, 460, 494, 525, 535, 581, 597, 615, 624, 639, 643, 653, 684, 692, 718, 742, 764, 771, 776, 814, 860, 867, 882, 894, 908, 925, 935, 954, 981, 995, and 997 among others.

17-limit Intervals

Here are all the 21-odd-limit intervals of 17:

Ratio Cents Value Color Name Name
18/17 98.955 17u1 su unison small septendecimal semitone
17/16 104.955 17o2 so 2nd large septendecimal semitone
17/15 216.687 17og3 sogu 3rd septendecimal whole tone
20/17 281.358 17uy2 suyo 2nd septendecimal minor third
17/14 336.130 17or3 soru 3rd septendecimal supraminor third
21/17 365.825 17uz3 suzo 3rd septendecimal submajor third
22/17 446.363 17u1o3 sulo 3rd septendecimal supermajor third
17/13 464.428 17o3u4 sothu 4th septendecimal sub-fourth
24/17 597.000 17u4 su 4th lesser septendecimal tritone
17/12 603.000 17o5 so 5th greater septendecimal tritone
26/17 735.572 17u3o5 sutho 5th septendecimal super-fifth
17/11 753.637 17o1u6 solu 6th septendecimal subminor sixth
34/21 834.175 17uz6 suzo 6th septendecimal superminor sixth
28/17 863.870 17uz6 suzo 6th septendecimal submajor sixth
17/10 918.642 17og7 sogu 7th septendecimal major sixth
30/17 983.313 17uy6 suyo 6th septendecimal minor seventh
32/17 1095.045 17u7 su 7th small septendecimal major seventh
17/9 1101.045 17o8 so octave large septendecimal major seventh

To avoid confusion with the solfege syllable So, the so 2nd, 5th and 8ve are sometimes called the iso 2nd, 5th and 8ve.

see Harmonic Limit, seventeen limit tetrads