Talk:39edo

39dfgijk

If you don't want the second row of odd harmonics (or prime harmonics if switched to that), you should also get rid of "39edo can be usefully mapped onto the val 39dfgijk", since the argument about higher harmonics being too inaccurate would make this val not so useful. Lucius Chiaraviglio (talk) 03:04, 2 April 2026 (UTC)

Octave compression certainly makes the higher harmonics more accurate, though one needs to be careful about intervals with many powers of 2 (and also 11, since it loses accuracy at that level of compression). A second row won't do too much harm, so I guess adding it back is fine. Overthink (talk) 03:54, 2 April 2026 (UTC)

Also, when you added the second table you added an extra line between the templates, which makes them more spaced apart than they should be. --Overthink (talk) 03:57, 2 April 2026 (UTC)

39 isn't a dual-7 edo

39d is clearly the best val up to the 11-limit and 39 patent should not be put in the interval table as a competing column (39df might be considered as the 13-limit mapping tho that's besides the point), for mostly the same reason 44d should not as I showed in Talk: 44edo.

To be clear, the question of a dual-prime edo concerns whether two mappings are nearly equally valid. If one mapping is considerably more accurate, it is hard for one to hear the other mapping as a valid approximation to the same set of intervals, since their presence in the same tuning system means the difference in quality is highlighted thru contrast. As such, for many edo articles we present a main mapping most useful for composition. This mapping is discussed at length in the theory section and put in the interval table. The distinction of a main mapping and various ancillary mappings is a consistent feature of edo articles on this wiki.

The ancillary mappings can also be used, and may be interesting for various reasons. I think they deserve to be discussed briefly in the theory section. However, we can't afford to put whatever we think is potentially or marginally useful in the interval table, cuz human readers have limited attention resource and wish to spend it on the best things. A less valid mapping in the interval table means divided attention and less efficiency of presenting information.

For example, in 145edo, there is this short sentence discussing the utility of a less accurate mapping: "The 145c val provides a tuning for magic which is nearly identical to the POTE tuning." But the main mapping is discussed in the rest of the article.

The reasons that 39d commends itself as the main mapping are mostly the same as that for 44. Specifically:

• The sharp 3, 5, 11 justifies the sharp 7. The interactions of 7 with 3, 5, 9, 11, and 15 all favor the sharp mapping. Iow 7 itself is the only inconsistently mapped interval in the 11-limit 15-odd-limit. While this is also true for 34edo, which is treated as dual-7, 39edo differs from 34edo in that the other primes and especially the 5 are very sharp, which brings us to …
• With the flat 7, the 7/5 will have 93% error and the 15/14 will have 112% error, whereas with the sharp 7, the maximum error comes from 7 itself, only 51%.
• TE error for 39d: 2.43 cents; 39dee: 3.13 cents; 39: 3.79 cents. Note that 39dee has a lower error than 39, so if 39dee isn't reasonable to consider, neither is 39 logically.

The only difference here is that the flat-7 mapping is a patent val. On that account one might argue that the mapping is of some special importance. I think the value of patentness has been overstated in the community at large. What we mean by a patent val is really using the closest approximation for the basis elements, but basis elements can change. For example, many ppl consider 5/3 and/or 7/6 to be as important in composition as 5/4 and 7/4, and one can generate the 7-limit with 2, 3, 5/3, and 7/6. In this basis, the patent val for 39edo isn't the same as the one found for 2, 3, 5, and 7. In fact it's the sharp-7 mapping. That reveals the lack of unique significance of patent vals in practice (and in math, as every GPV is demonstrably patent in some way); as such the importance of a mapping solely from being a patent val in this specific case is baseless from a broader perspective.

—FloraC (talk) 21:51, 29 May 2026 (UTC)

Overall, I agree that 39d feels more natural to use. However, the wiki is supposed to present info from a neutral point of view rather than pushing a perspective. Not everyone agrees that more accurate necessarily means "better", and that patent vals are completely arbitrary. People often think of the octave as the equivalence interval, so they want to keep it pure. The pure-octave patent val with prime harmonics as basis entries feels like the most natural mapping to use for many people, even if it is less accurate overall. The patent val isn't completely uninteresting, supporting structures like immunity and triforce. 39edo is a medium-sized edo, and someone who uses it very much may not be focused on accuracy.

Overall, I think 39 and 39d should have about equal coverage, with structures in both presented. The page definitely should explain how 39d improves accuracy of many intervals, and it should be up to the reader to decide which perspective they agree with, and which mapping to use. --Overthink (talk) 17:08, 30 May 2026 (UTC)

That sounds very sensible — put my vote in for that as well. Lucius Chiaraviglio (talk) 18:36, 30 May 2026 (UTC)

I have no opinion on 39et in particular, but I will agree that "patentness" is a completely arbitrary quality and the procedure of rounding is merely heuristic to get something that usually works. – Sintel🎏 (talk) 19:47, 30 May 2026 (UTC)