17-limit

Prime limit

---

← 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 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.

These things are contained by the 17-limit, but not the 13-limit:

• The 17-odd-limit;
• Mode 9 of the harmonic or subharmonic series.

Terminology and notation

Conceptualization systems disagree on whether 17/16 should be a diatonic semitone or a chromatic semitone, and as a result the disagreement propagates to all intervals of HC17.

• In Functional Just System, 17/16 is a diatonic semitone, separated by 4131/4096 from 256/243, the Pythagorean diatonic semitone.
• In Helmholtz–Ellis notation, 17/16 is a chromatic semitone, separated by 2187/2176 from 2187/2048, the Pythagorean chromatic semitone.

The case for mapping it to either category may include:

• Number of steps in the chain of fifths. The diatonic semitone is simpler than the chromatic semitone in the chain of fifths, being -5 steps as opposed to +7 steps.
• Size of the associated formal commas. The formal comma of the chromatic mapping, 2187/2176, is simpler and smaller than that of the diatonic mapping, 4131/4096, though both are generally considered small enough as commas which do not alter the interval category. The chromatic mapping has the advantage of keeping the Pythagorean order of diatonic semitone < chromatic semitone in the intervals of 17.
• Interactions with other primes. On one hand, if 7/4 is known to be a seventh, assigning 17/16 to a second will make intervals 17/14 and 21/17 thirds. This is favorable because 17/14 and 21/17 are important building blocks of tertian harmony. On the other hand, if 5/4 is known to be a third, then 17/16 being a unison will make 17/15 a second and 20/17 a third. This is favorable because 17/15 is the mediant of major seconds of 9/8 and 8/7. The HEJI authors find it generally favorable for harmonics to be positive and subharmonics to be negative in the chain of fifths, possibly in order to make the system integrate better with the 5-limit.

In practice, the interval categories may, arguably, vary by context. One solution for the JI user who uses expanded chain-of-fifths notation is to prepare a Pythagorean comma accidental so that the interval can be notated in either category.

The names tabulated in #Intervals are common names and do not follow this discussion yet.

Edo approximation

Here is a list of edos with progressively better tunings for 17-limit intervals (monotonicity limit ≥ 17 and decreasing TE error): 31, 38df, 41, 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. For a more comprehensive list, see Sequence of equal temperaments by error.

Here is a list of edos which provides relatively good tunings for 17-limit intervals (TE 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.

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.

Music

Bryan Deister

• my tune 216 played on 2/16 at 216hz (2024)

Francium

• "thepresentistheever" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube
• "Bit Of A Sudden Change Of Plan" from You Are A... (2024) – Spotify | Bandcamp | YouTube

Randy Wells

• Green (is not a creative color) (2023)

See also

• Seventeen limit tetrads