How are Just-Tuned scales beyond those of 12-notes constructed?

Question

I was experimenting with what ratios come up when one continues on the Pythagorean cycling of fifths, and came up with the following chart:

Those ratios beyond the 12-note Pythagorean scale are shown in red. Now, two things come to mind here. I was trying to construct a proper 19-EDO scale. Now, what I noticed is that in the 12-EDO Justly-tuned scaled, there are of course 13 actual ratios, since due to the fundamental theorem of arithmetic, the cycling of fifths can never converge together, thus creating the infamous wolf interval. However, it seems that such a wolf-interval can't really occur in the same way at the ends of the 19-EDO scale. The reason being that 19 is an odd number. With 12-EDO, we get 13 different ratios, 6 in both "directions" plus 1:1, unison. With 19-EDO can't do cycle through an equal number of times in both directions and end up with 20 (i.e. 19+1) different ratios.

I guess my main question is that, is this indeed how the 19-EDO Pythagorean scale should be constructed? Furthermore, what becomes of the wolf-interval caused by the non-equivalence of 588.27¢ and 611.73¢ ? It's not like taking the process further causes these ratios to somehow disappear. Is there a wolf-interval peculiar to 19-EDO? Or is there something special about 12-EDO in that respect?

Answer 1

Your diagram indeed shows a 19-pitch Pythagorean scale.

19 just perfect fifths exceed 11 octaves by an interval of frequency ratio 3¹⁹:2³⁰. This interval is about 137.145 cents. If you want to divide the octave into 19 and have a closed circle of 19 fifths, then your 19 fifths, however you temper them, will add up to exactly 11 octaves. So the question is how to share that 137.145 cents among the 19 fifths. The tuning system you give in your diagram has 18 just fifths, which implies that the other fifth (between the pitches you label "9" and "-9" is narrowed by the full 137.145 cents. This makes it 564.81 cents -- much narrower even than a tritone! There's your wolf.

19-EDO of course shares out the 137.145 cents equally among the 19 fifths, narrowing each one by 7.218 cents.

One thing you might consider is to pick a number of divisions where the amount to temper is less. For example 53 (as suggested by ttw in a comment). 53 just perfect fifths exceed 31 octaves by an interval of frequency ratio 3⁵³:2⁸⁴. This interval is about 3.615 cents -- a much smaller wolf. 53-EDO shares that equally among the 53 fifths, narrowing each one by 0.068 of a cent.