Some thoughts on resolution

There are 3 types of 7th in 15edo, and as such, with the major 3rd and the lower 2 of these 7ths, we have 2 types of tritone. Both of these 7ths can resolve down 1 or 2 steps to a major or minor 3rd, as in 12 tone music.

Harmonically, when the neutral / middle 7th resolves down one step, the new root is 1 step higher than the previous. This means that the V is going to the augI instead of the I. This is balanced a bit by the fact that the V in this tuning is quite sharp.

I have made a video demo of this accessible from my page: https://www.patreon.com/noahdeanjordan

If all this is agreeable to you , we can then suggest that some of the following may be true:
1. the step size from the 7 of the V to the 3 of the I can be of a variable size, whether it resolves to a i min or a I maj.

2. the V chord could resolve to places other than the I or the vi

3. different tritones may have different tendencies

4. the just harmonic approximation of the tritone or the 7th may determine in part the quality of the resolution

– this last part begs a question:

In 15edo, the lower tritone is a 7/5 or 11/8 perhaps,

and the higher tritone a 16/11 or a 10/7, the undertone versions.

That means that in this tuning, the distinction between these tritone is not 7th harmonic vs 11th harmonic but overtone vs undertone.

On the other hand, the lowest of the 7ths in 15edo is a near 7/4, and the middle 7th a 11/6 or a 9/5 perhaps. And in any event, these are not inverses of each other, but definitely of different harmonic qualities.

So the questions are: does the harmonic limit of an interval affect the quality of the resolution? will the over/undertone nature of an interval affect its quality or functionality? how are these related? particularly if the tritone and the 7th within a resolution are of different harmonic limits. — furthermore, in this 15edo tuning we have a few possibilities for the interpretation of these senses:

1. In the 4:5:6:7 chord (major triad with the lowest 7th (7:4)) – we can interpret it as a 7:5 and a 7:4 or as a 11:8 and a 7:4… will this 7:4 predominance force us to hear the lower tritone as a 7:5? — YES, if this is a 4:5:6:7 chord. But if we don’t analyze this as a full form (because we are not in just intonation), the other approach could be possible.

side note:

to read 4:5:6:7, we are looking at a series of ratios

the ratio between the first two notes, the root and the major 3rd is 4:5 (5:4… 5/4)

the ratio between the 2nd two notes, the M3 and the fifth is 5:6 (6:5, min 3), and the ratio between the root and the fifth is 4:6 or 2:3 (3/2, P5)

the ratio between the final two is 6:7, between the 2nd and the final is 5:7 (7:5, the tritone in question),

and the ratio between the root and the final is 4:7, or 7:4 -> the note we define.
2. In the 12:15:18:22 chord with the neutral 7 (as a 11:6) on the major chord, we get the tritone defined in this sense as 15:22. If we want the large tritone to be an undertone, any of out close approximations of 10:7 or 16:11, we cannot fully analyze the chord this way.  The combination of an overtone and an undertone being analyzed in the same chord has difficulties.

This difficulty is shown in the different of complexity between writing a major and a minor triad.

4:5:6

and

10:12:15..

what happens is you want to write 5:6 (for the minor third), but must make the 2:3 as well, and as 5 is not a multiple of 2, we must multiply 2 to all.

 

More thoughts into this to come..

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