Talk:44edo

Dual-7 edo?

Not sure this edo needs to be treated as dual-7. The flat mapping makes little sense to use. —FloraC (talk) 09:00, 8 May 2026 (UTC)

Sharp 7 works with the sharp 3 and sharp 13 (assuming opposites sides of the fraction bar). Flat 7 works with the flat 5 and flat 11 (again assuming opposite sides of the fraction bar). Lucius Chiaraviglio (talk) 11:36, 8 May 2026 (UTC)

I know. But the sharp 7 is still better overall. —FloraC (talk) 10:28, 9 May 2026 (UTC)

But you really need the flat 7 in combination with the commonly used 5, which is flat, but not flat enough for a wart. And you need it in combination with the commonly used 3 which is sharp, when both are on the same side of the fraction bar. Lucius Chiaraviglio (talk) 15:18, 9 May 2026 (UTC)

Well I don't think any inconsistency implies there's a point to treat the edo as dual-prime. The question of a dual-prime edo is rather if the two interpretations are about as valid, and in this case I don't think so. For 2.3.7, consider that 39edo, which has an even sharper 7, isn't treated as dual-7 on this wiki, which I think is very fair and should be taken as a sign that the archy mapping prevails over semaphore in this range. The flat 5 here isn't enough to shift the sharp tendency to justify a flat mapping of 7. This is confirmed in Graham Breed's temperament finder, where 44d doesn't come up in the 7-limit at all and 44e comes before 44d in the 11- to 19-limit. —FloraC (talk) 15:50, 9 May 2026 (UTC)

39edo has the advantage that all of the flat harmonics (at least up to 23) are flat enough that you can just wart them all and call it a day. But 44edo has some harmonics that are just moderately sharp or moderately flat (not really working with a wart), so since 7 is far enough off to be wartable, might as well make it dual. Lucius Chiaraviglio (talk) 16:04, 9 May 2026 (UTC)

I'd say even for the full 7-limit, and up to the 19-limit, the 7 isn't ambiguous. Note that only 7/5 and 11/7 are made closer by switching to the flat mapping, at the cost of all the rest. Another thing that clearly favors the sharp mapping is that it achieves 19-odd-limit monotonicity whereas the flat mapping fails at 14/13 and 15/14. Finally, as I said, 44e has a lower error than 44d, so if 44e isn't reasonable to consider, neither is 44d. —FloraC (talk) 18:26, 9 May 2026 (UTC)

The 11th harmonic has only -21.5¢ relative error, so flipping it is going to introduce much more relative error than flipping the 7th harmonic, which has +47.6¢ relative error. If you were going to apply a wart to just 1 harmonic and use only the warted val, it might make sense to flip the 11th harmonic (as done with 27e), but you wouldn't want to use 44d alone, but in strategic alternation with 44 patent (thus, dual-7 rather than solely alternate-7). Lucius Chiaraviglio (talk) 12:00, 10 May 2026 (UTC)

I gave you three reasons why this edo should not be considered dual-7. You're only addressing one of them, so my point still stands. Now in my defense on this particular one, I'd like to point you to the TE errors, which Graham's temperament finder uses as criteria. These for 19-limit 44e and 44d are 2.58 cents and 2.64 cents, respectively, so while the 11th harmonic itself has a high relative error in the sharp mapping, it is not so with the intervals in the 11- to 19-limit on average. —FloraC (talk) 18:34, 10 May 2026 (UTC)

To add to your comment, for example, flipping the 11th harmonic gives it +78.5% error. However, flipping the 7th makes 7/6 have -78.5% error (almost exactly equal and opposite to the warted 11), and gives 9/7 104.7% error. Overall the sharpness of the 3rd harmonic alone means 44edo should be treated as having a sharp tendency.

And personally, 7/5 and 11/7 being consistent isn't alone enough to justify 44d. The 44e mapping does seem somewhat reasonable, mapping 11/9 to the neutral third. Also, I feel like 22edo being a subset of 44 is critical, and the dual mapping just doesn't seem to fit.

And also I was originally planning to respond directly but got an edit conflict. --Overthink (talk) 19:06, 10 May 2026 (UTC)

With respect to the temperament finder, does it consider the use of alternate vals in combination, or only in isolation? If the latter, then it would not show the benefits of switching between flat 7 and sharp 7. Also, 7/5 and 11/7 (and their octave complements) are not the only ratios that would benefit from having a flat 7 option. For ratios with 5 and 7 on opposite sides of the fraction bar, 28/25 (and 25/14) comes to mind. Add in ratios of 21 (which I didn't give enough coverage to above), and the case for the flat 7 option (not constant flat 7) becomes even better, as in 21/20 and 22/21 (and their octave complements). Lucius Chiaraviglio (talk) 04:41, 11 May 2026 (UTC)

You're making circular reasoning by saying one should consider the system as dual-prime to justify this consideration itself. Of course, some intervals get closer with the flat mapping, but more intervals get worse. The interactions of harmonic 7 with most lower harmonics favor the sharp mapping: 1, 3, 9, 13, 15, 17, 19, and by the time you get to 21, 23 or 25 you may as well take a look at 27.

So, with this thorough analysis, it should be clear that 44 is the "canonical" val. If someone has a use for 44d, 44e, or any other low-accuracy alternatives, like some notable temperaments they support, I advise discussing them secondarily in the theory section.

—FloraC (talk) 10:16, 11 May 2026 (UTC)

Yeah I noticed that too but didn't bother to note down the exact percentages. —FloraC (talk) 10:16, 11 May 2026 (UTC)

I don't get why you're calling what I say "circular reasoning". 44edo has 2 quite flat and 3 quite sharp prime harmonics in the range up to 13, of which the 7th harmonic is the one that sticks out the most, so it makes sense to toggle that one between flat and sharp as needed. Lucius Chiaraviglio (talk) 15:41, 11 May 2026 (UTC)

> I don't get why you're calling what I say "circular reasoning".

I've made it very clear. You said: "does it consider the use of alternate vals in combination, or only in isolation? If the latter, then it would not show the benefits of switching between flat 7 and sharp 7." You essentially need the temperament finder to consider the system dual-prime to justify doing so. That is circular reasoning.

> 44edo has 2 quite flat and 3 quite sharp prime harmonics in the range up to 13, of which the 7th harmonic is the one that sticks out the most, …

Let's not go in circles cuz Overthink and I have done a more comprehensive analysis. To summarize:

1. The interactions of harmonic 7 with most low harmonics favor the sharp mapping. These are 1, 3, 9, 13, 15, 17, and 19, vs 5 and 11. The sharpness of 3 in particular implies a sharp tendency, which the harmonic 7 follows.
2. In terms of monotonicity limits, 44 has 19 whereas 44d only 13, tuning 14/13 < 15/14.
3. In terms of TE errors, 44 < 44e < 44d in the 11- to 19-limit.
4. In terms of maximum errors, 44d's 9/7 is more than 100% off.

I hope that helps.

—FloraC (talk) 17:23, 11 May 2026 (UTC)