How to make a 5th

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

 

 

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