I was about to replay to your post and suddenly John Hasquin posted his reply that said exactly what I meant to say. In fact he has much more credential to assess the validity of you attempts then me as he actually build the horns. In addition (surprise, surprise - a big rarity among the builders) – John actually can hear the things….
I have no reasons to question your results and I actually put your post in site’s “Knowledge Tree”. However, as a practicing person, I recognize 2 aspects that make me skeptical. Let me to name them:
1) Not necessary direct relativity of descriptive science and sonic results. You see, the scientific methods and math approximation are juts DESCRIBE the realty but they do not PRESENT it. Description of realty has one language and in many instances the language of the description takes too much over of Really. We might presume that visual reflections have own rules and evaluate the horn performance for the perspectives of those rules but who said that only those visual reflections rules are playing in the game and who know how they play? We might take an abstract theory of some kind operation that would be perfectly rational as a self-contained theory but how does is relates to a subjective perception of sound? This is a completely another question that is not implicit into the theory of that operation. So, is your profile better then the regular tractrix? I think that only way to do it is to make both of them and look at the actual sonic results. However, the second aspect might defeat it.
2) We in the horn word are in the deep dark and in the deep shit, sort of speaking… Our compression drivers are garbage that made with less precision then merchandises in “Toy’s USA”. Our horns are very poor and very irregular. Move a center of a compression driver for 1mm aside of the center of the horn and you enter the world of such a huge problems that they would be 1000000% more annoying then the issues that you described in your research. Or for instance run your driver against a distortion analyzer while you mis-centering the suspension of a diaphragm in the gap and your will see such a huge magnitude of the problem that you would loose any interest to do anything “precise” in the horn world.
I certainly do not advocate to do the things “without any rational” but in my experiments I use the “live editing techniques” that I have stolen from cinematographers. When a professions editors cuts a film then he uses many rules about duration of the screens: plot, composition, metric, rhythmic, balance, purpose of expressed or implied ideas, continuity, density, dynamics and many-many others. However, there is another methods: a person juts watches a whole film, not edited, and when he or she feel that the screens become too long then he punches a button that marks the location where "is felt” that the screen should be changed. Then, the movie is cut according those marks of the “live editing”.
So, try to do the same: make a horn, give it to another parson and ask him to talk to you from 6 feet into the horn pointed to you. Pay attention to the timbers, harmonics, contrast, accsents, articulation and so on... It is also very nice to stick a horn into your ear and juts “listen the world”.
To me, it is very illustrative test to take a horn to the roof of my downtown home and juts listed through the horn the sound of city. I usually would need 3 seconds to say that something is wrong if something was wrong…
Romy The caT
"I wish I could score everything for horns." - Richard Wagner. "Our writing equipment takes part in the forming of our thoughts." - Friedrich Nietzsche