18/17
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In 17-limit just intonation, 18/17 is the small septendecimal semitone of about 99 ¢. It is very close to 12edo's "half step" of 100¢, and fairly close to the "large septendecimal semitone" of 17/16 (~105¢).
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 18/17 specifically:
- In the Functional Just System, it is a chromatic semitone, separated by 4131/4096 from the Pythagorean augmented unison (2187/2048).
- In Helmholtz-Ellis notation, it is a diatonic semitone, separated by 2187/2176 from the Pythagorean minor second (256/243).
The term small septendecimal semitone omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the large septendecimal semitone of 17/16.
Temperaments
Vincenzo Galilei (1520-1591) proposed a tuning based on eleven 18/17 semitones and one larger semitone of about 111.5 ¢ (the octave complement)[1]. This well temperament provides seven wide perfect fifths of about 705.2 ¢ and five narrow perfect fifths of about 692.7 ¢, whose distribution is maximally even instead of grouping together the wide and the narrow fifths along the circle of fifths, as is often the case in other well temperaments.
The following linear temperaments are generated by a ~18/17 in the 2.3.5.17 and 2.3.5.17.19 subgroups:
Some 12th-octave temperaments treat ~18/17 as the period, including compton's 17-limit extension.
See also
- 17/9 – its octave complement
- 17/12 – its fifth complement
- Gallery of just intervals
- List of superparticular intervals
References
- ↑ Barbour, J. Murray. Tuning and temperament: a historical survey, p. 57.