Wow...no one seems to care about our 'microtonal' demands. I second that, some improvement must be done regarding Pianoteq's 'cents or ratio' view, at least, in order to do some justice to modern western music theorists.
Indeed, for some strange reason, Pianoteq's Werckmeister III scale preset includes 10 rationals, whereas (unless other sources report differently) there should be a total of 8 pythagoreans and no more 'just intonation' ratios on top of those 8, since 4 intervals are supposed to be 1/4comma-flat 'fifths', distributed across the dodecatonic octave-period (4 irrational intervals out of 12, with those 4 requiring logarithmic values, necessarily).
I disagree with the emphatic statements claiming that the 29th harmonic has nothing to do with anything in this case. It has at least something to do in some degree, otherwise no algorithm would have ever found any useful 29-limit approximation close to anything of interest here.
However, you made the point. Assuming the comma 'fraction' be pythagorean's rather than syntonic's (due to the 696.1c 'fifth' being displayed in the same preset), then the resulting 'major sixth' irrational interval should be measuring ≈ 888.27c, rather than being described by 1044/625 (≈ 888.23c).
The latter (Pianoteq's rational approximation, as we see) is quite spot-on in terms of 'difference' (≈ 0.04c), despite making no sense, because 1044/625 misinforms the user right away about the intended historical temperament. As a consequence too, the .1 rounding makes that interval 888.2c instead of 888.3c. Same goes for last step interval (2349/1250), which is displayed as 1092.1c when clicking on 'cents', while certainly it can't be an integer, but an irrational interval to be more fairly rounded at 1092.2c.
Funny enough, that 1044/625 value I was unable to find with tools other than Pianoteq. Checked Yacavone xen-calc, ScaleWorkshop (versions 1 & 2, they use different approximation algorithms) and Untwelve javascript interval calculator. For example, a more 'docile' 29-limit ratio, namely, the semi-convergent 87/52, appears in there, though it exceeds the expected ≈ 888.27c irrational 'major sixth' a little bit (by ≈ 2.73c).
All in all, there is quicker common result, that is, the convergent 11-limit ratio of 147/88 (≈ 888.29c) which is a better approximation (≈ 0.02c 'difference'). As a finer result after that, Untwelve outputs 1034/619, a 619-limit ratio (whose numbers kind catch 1044/625). But that's some heinous, out-of-the-question high prime.
I have no idea where does that 1044/625 (29-limit) come from, but I read that the 29th harmonic is used as a maximum prime-limit in the Scala / ScaleWorkshop v2 'approximate by ratios' function, for computational reasons. Maybe Modartt can tell what's happening on their own front, whether this be related or not?
What else? I think the request of improvement is fine, but, considering also that there might already be better software for scale design and analysis, we need to posit it better, understanding where the best trade-off lies for the sake of both ends (you know, maybe it's not a very easy feature to be implemented as it might look, or maybe it is too time-consuming, down the schedules, etc).
Definitely what has been said about Pianoteq's Werckmeister III preset inaccuracy is worth to note, and leaving as it is is not the best idea. Anyway I would like to add the following: setting largest prime alone without any minimum prime-limit as well (something which ScaleWorkshop v1 allows instead) makes no sense in my opinion. A further improvement of your request, then, would be to allow both min & max prime-limit to be set.
Even more, one could enlist a specific prime-subgroup (for example, 2.3.11), also limiting the results to cut more or less monstrosities eventually arising from exponential ratios. Only in this case the 'temperament view' section would become something more of a scale design nature, rather than a circle graph with more simple ratio display. And monzo notation also might come in handy depending on cases but then again we would get back to square one...too much progress is of no use.
It would be nice if a bit more users were joining discussions like this. Other than that, thanks for your suggestion. Hopefully gets heard. The approximation behavior never bothered me really, but now that I've seen what it does, actually, it does a little bit.
