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 norm (log2 nd)	8.08746
Weil norm (log2 max(n, d))	8.17493
Wilson norm (sopfr(nd))	25
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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).

It could also be reasonable to treat 17/16 as the formal comma for prime 17 in its own right, as it is roughly the same size as the 3-limit accidental 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.

Approximation

Edo approximations for 17/16 (104.96 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
11	1\11	109.09	+4.14	+3.79
12	1\12	100.00	-4.96	-4.96
22	2\22	109.09	+4.14	+7.58
23	2\23	104.35	-0.61	-1.16
24	2\24	100.00	-4.96	-9.91
34	3\34	105.88	+0.93	+2.63
35	3\35	102.86	-2.10	-6.12
45	4\45	106.67	+1.71	+6.42
46	4\46	104.35	-0.61	-2.33
57	5\57	105.26	+0.31	+1.46
58	5\58	103.45	-1.51	-7.28
68	6\68	105.88	+0.93	+5.25
69	6\69	104.35	-0.61	-3.49
79	7\79	106.33	+1.37	+9.04
80	7\80	105.00	+0.04	+0.30

Temperaments

The 17/16 interval generates srutal archagall with a half-octave period, where it also represents 16/15 and 18/17. With a full-octave period it generates septendesemi, the 80 & 103 temperament. It is also a good generator for lithium with a 1/3-octave period, as well as the less accurate august.

It is very close to 7 steps of 80edo, and is tempered as such in many 80th-octave temperaments.

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