17/16
| Ratio | 17/16 |
| Subgroup monzo | 2.17 [-4 1⟩ |
| Size in cents | 104.95541¢ |
| Name | large septendecimal semitone minor diatonic semitone |
| Color name | 17o2, iso 2nd |
| FJS name | [math]\text{m2}^{17}[/math] |
| Special properties | superparticular, reduced, reduced harmonic |
| Tenney height (log2 nd) | 8.08746 |
| Weil height (log2 max(n, d)) | 8.17493 |
| Wilson height (sopfr(nd)) | 25 |
| Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.28419 bits |
[sound info] | |
| open this interval in xen-calc | |
In 17-limit just intonation, 17/16 is the 17th harmonic, octave reduced, and may be called the large septendecimal semitone. Measuring about 105¢, it is close to the 12edo semitone of 100¢, and thus 12edo can be said to approximate it closely, although an even better approximation is available in 23edo. In a chord, it can function similarly to a jazz "minor ninth" – for instance, 8:10:12:14:17 (although here the interval is 17/8, which is a little less harsh sounding than 17/16). In 17-limit JI, 17/1 is treated as the next basic consonance after 13 and 15.
17/16 is one of two superparticular semitones in the 17-limit; the other is 18/17, the small septendecimal semitone, which measures about 99¢. The difference between them is 289/288, about 6¢. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.
17/16 is almost exactly 1/3 of the 6/5 minor third. The difference between 6/5 and three 17/16 semitones is 24576/24565, an interval of approximately 0.8¢. 17/16 is also almost exactly 1/8 of 13/8, with the difference between 13/8 and (17/16)8 being approximately 0.9¢. The difference between ten 17/16's and 11/6 is approximately 0.2¢, while the difference between thirteen 17/16's and 11/5 is approximately 0.6¢.
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. See 17-limit for a detailed discussion.
For 17/16 specifically:
- In Functional Just System, it is a diatonic semitone, separated by 4131/4096 from the Pythagorean minor second (256/243). It is also called the minor diatonic semitone, which contrasts the 5-limit major diatonic semitone of 16/15 by 256/255, about 6.8¢.
- In Helmholtz-Ellis notation, it is a chromatic semitone, separated by 2187/2176 from the Pythagorean augmented unison (2187/2048).
The term large septendecimal semitone omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.
See also
- 32/17 – its octave complement
- 24/17 – its fifth complement
- 17/8 – same interval, one octave higher
- Gallery of just intervals
- List of superparticular intervals