17-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. The case for it being a diatonic semitone includes:
- The diatonic semitone is simpler than the chromatic semitone in the chain of fifths, being -5 steps as opposed to +7 steps, and the associated comma 4131/4096 is small enough to be considered a comma which does not alter the interval category.
- 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.
In Helmholtz-Ellis notation, 17/16 is a chromatic semitone, separated by 2187/2176 from 2187/2048, the Pythagorean chromatic semitone. The case for it being a chromatic semitone includes:
- 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 (decreasing TE error): 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.
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
- "thepresentistheever" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube
- "Bit Of A Sudden Change Of Plan" from You Are A... (2024) – Spotify | Bandcamp | YouTube