17/16: Difference between revisions
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{{Wikipedia|Minor diatonic semitone}} | {{Wikipedia|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 | In [[17-limit]] [[just intonation]], '''17/16''' is the 17th [[harmonic]], [[octave reduced]], and may be called the '''large septendecimal semitone'''. Measuring about 105{{cent}}, it is close to the [[12edo]] semitone of 100{{cent}}, 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 [[13/1|13]] and [[15/1|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 | 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{{c}}. The difference between them is [[289/288]], about 6{{c}}. 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. | 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{{c}}. 17/16 is also almost exactly 1/8 of [[13/8]], with the difference between 13/8 and (17/16)<sup>8</sup> being approximately 0.9{{c}}. The difference between ten 17/16's and [[11/6]] is approximately 0.2{{c}}, while the difference between thirteen 17/16's and [[11/5]] is approximately 0.6{{c}}. | ||
== Terminology and notation == | == Terminology and notation == | ||
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For 17/16 specifically: | For 17/16 specifically: | ||
* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243|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. | * In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243|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{{c}}. | ||
* In [[ | * In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048|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. | 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 == | |||
{{Interval edo approximation|17/16}} | |||
== 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 == | == See also == | ||
Latest revision as of 21:14, 29 November 2025
| Interval information |
minor diatonic semitone
reduced,
reduced harmonic
[sound info]
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 | 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