17/16: Difference between revisions
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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. 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, it is treated as the next basic consonance after 13 and 15. | 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. 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, it 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]], 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 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 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¢. | ||
== Terminology and notation == | == Terminology and notation == | ||
There exists a disagreement in different conceptualization systems on whether 17/16 should be a [[diatonic semitone]] or a [[chromatic semitone]]. In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from [[256/243]], the Pythagorean diatonic semitone. In [[Helmholtz-Ellis notation]], it is a [[chromatic semitone]], separated by [[2187/2176]] from [[2187/2048]], the Pythagorean chromatic semitone. The term "large septendecimal semitone" omits the diatonic/chromatic part and only describes its melodic property i.e. the size. | There exists a disagreement in different conceptualization systems on whether 17/16 should be a [[diatonic semitone]] or a [[chromatic semitone]]. In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from [[256/243]], the Pythagorean diatonic semitone. It is also called the '''minor diatonic semitone''' (→ [[Wikipedia: 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 [[2187/2048]], the Pythagorean chromatic semitone. The term "large septendecimal semitone" omits the diatonic/chromatic part and only describes its melodic property i.e. the size. | ||
In practice, the interval category may, arguably, vary by context. One solution for the JI user who uses expanded [[circle-of-fifths notation]] is to prepare a [[Pythagorean comma]] accidental so that the interval can be notated in either category. | In practice, the interval category may, arguably, vary by context. One solution for the JI user who uses expanded [[circle-of-fifths notation]] is to prepare a [[Pythagorean comma]] accidental so that the interval can be notated in either category. | ||