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

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?

• One thing you can do is to take the 19-tone scale and select the 12 closest notes to just tuning. Or use a 31 or 53 tone scale. – ttw Nov 5 '16 at 1:51
• (a) I think that there are some typeoes in your question, e.g. I don't understand what "bother directions" are. Even so, I don't really understand what you specific answerable question is. – Dave Nov 6 '16 at 18:29
• @Dave meant to say "in both directions", just a typo, fixed – junius Nov 7 '16 at 3:13