17/16

Interval information

Ratio	17/16
Subgroup monzo	2.17 [-4 1⟩
Size in cents	104.9554¢
Name	large septendecimal semitone minor diatonic semitone
Color name	17o2, iso 2nd
FJS name	[math]\displaystyle{ \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
[sound info]
Open this interval in xen-calc

English Wikipedia has an article on:

Minor diatonic semitone

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. It is objectively correct, as in a scale-agnostic context such as pure JI, and especially for an octave-reduced prime that isn't defined by separating any other intervals, the diatonic/chromatic distinction is meaningless.

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