17-limit

Prime limit

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← 13-limit  17-limit  19-limit →

The 17-limit consists of just intonation intervals whose ratios contain no prime factors higher than 17. It is the 7th prime limit and is thus a superset of the 13-limit and a subset of the 19-limit. It adds to the 13-limit a semitone of about 105¢ – 17/16 – and several other intervals between the 17th harmonic and the lower ones.

The 17-limit is a rank-7 system, and 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 needed.

Edo approximations

A list of edos with progressively better tunings for 17-limit intervals: 46, 58, 72, 103, 111, 121, 140, 171, 183, 217, 224, 270, 311, 354, 400, 422, 460, 494, 581, 742, 764, 814, 935, 954 and so on.

Another list of edos which provides relatively good tunings for 17-limit intervals (relative error < 5.4%): 46, 58, 72, 87, 94, 103, 111, 121, 130, 140, 171, 183, 190g, 212g, 217, 224, 243e, 270, 282, 301, 311, 320, 328, 342f, 354, 364, 373g, 383, 388, 400, 414, 422, 441, 460, 494, 525, 535, 540, 552g, 566g, 571, 581, 597, 624, 639, 643, 653, 684, 692, 711, 718, 742, 764, 814, 822, 836(f), 863efg, 867, 882, 908, 925, 935, 954 and so on.

Note: wart notation is used to specify the val chosen for the edo. In the above list, "190g" means taking the second closest approximation of harmonic 17.

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 also

• 17-odd-limit
• Seventeen limit tetrads