In 15 tone equal temperament (15edo / 15-ed2). The “perfect” 5th is tuned to 720¢. 720¢ is almost exactly 47:31, or decreasingly, 44:29, 41:27, 38:25, 35:23, 32:21, 29:19, 26:17, 23:15, 20:13, 17:11, 14:9, 11:7, 8:5, 5:3, 2:1.
In between any two intervals described in Just Intonation, an interval between them can be found by adding the numerators and the denominators. For example: between 3:2 and 4:3, we have (3+4):(2+3) or 7:5. In the first pattern we are finding intervals closer and closer to the 3:2 starting from the octave.
The first are familiar: the 2/1 octave, the 5/3 major sixth, the 8/5 minor sixth. Then they become more abstract: we could suggest 11:7, 14:9 and 17:11 to be subminor sixths, but this is challenging territory; what is 11:7? It describes an interval generated by the 11th harmonic up and the 7th harmonic down. Is some amount of the characteristic of the subminor 6th to depend on a relation between the 7th and the 11th harmonic? Or maybe 14:9 is more characteristic of the sound?
I pose this as an open question? Which is more consonant and in which contexts (for context may lead to many answers), 14:9 or 11:7? And further, is it solely larger number that makes more dissonant or complex intervals? Or is the composition (and prime composition) a large factor as well? And do these have functional relations?
In addition, 32:21 may be described as a diminished 6th, a minor sixth lowered by a chromatic semitone. In this case, 8:5 lowered by 21:20, creating 32:21.
Back to the idea of the 5th.
It is evident that by analysis of Just Intonation from this standpoint that a 5th must be tuned quite sharp for it to appear to be something other than a fifth. Where is this bound and how is it defined?
**this is coming from the many debates on xen alliance II fb group of whether or not the 15-ed2 fifth is good/usable/etc or not
Noah Dean Jordan 