24/17: Difference between revisions

| Name = smaller septendecimal tritone

In [[17-limit]] [[just intonation]], '''24/17''' is the '''smaller septendecimal tritone''', measuring very nearly 597¢. It is the [[mediant]] between [[7/5]] and [[17/12]], the "larger septendecimal tritone". The two septendecimal tritones are each 3¢ away from the 600¢ half-octave, and so they are well-represented in all even-numbered [[EDO]] systems, including [[12edo]]. Indeed, the latter system, containing good approximations of the 3rd and 17th harmonics, can use the half-octave as 24/17 and 17/12 in close approximations to chords such as 8:12:17 and 16:17:24. [[22edo]] is another good EDO system for using the half-octave in this way.

There exists a disagreement in different conceptualization systems on whether 24/17 should be an augmented fourth or a diminished fifth. In [[Functional Just System]], it is an augmented fourth, separated by [[4131/4096]] from [[729/512]], the Pythagorean augmented fourth. In [[Helmholtz-Ellis notation]], it is a diminished fifth, separated by [[2187/2176]] from [[1024/729]], the Pythagorean diminished fifth. The term "small septendecimal tritone" omits the distinction 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.