Rafael, that is what I prefer Tooner to do. I will not challenge him on whatever he comes up with. I'll leave that to anyone else who wants to do that or who wants to agree with him. I am firmly resolved and satisfied in my approach to tuning ET as an idea that is practical and which works for the people I am trying to help to learn to tune ET.

I have mucked around a bit in my spare time here experimenting with Bill's suggested sequence (on a challenging piano no doubt) and would have to say it was near impossible to not let my biases interfere. Still, I could see it being useful as an alternative or even complimentary method of getting a decent temperament for those who are new to it or struggling with other methods. I say this not because it worked any better for me than what I have learned and used for many years, but rather, it did not reveal any large hidden surprises or errors that left me in a lurch.
Tuning temperaments are like finding your way out of a desert. Having a good sequence is much like having a map. Fixed theoretical beat rates on the other hand are analogous to a mirage in a desert. Knowing that its just a mirage does not change the fact that it still looks like real water; it will always be the logical place to go to a new traveler if everything else still looks like desert. The problem lies with the assumption that the water/mirage will lead you in a straight line out of the desert...which in most cases does not.
I'm sure that the best people to assess the value of using this method are the very people in need of it, neophytes. Those who have learned to do it with other methods like myself have the added problem of divorcing ourselves of our biases and in a sense "unlearn" and subsequently devalue what we had ultimately conquered. With human pride being what it is, its not an easy path to take. It would be a worthwhile experiment for a tuning school to separate its students into 2 groups with one using this method exclusively, just to see if does better than the other group.

Thank you Emmery for your statements. Your last suggestion is quite good. I have always said that if people are successful with another method which is habitual, then that is a better approach for that individual. So, I am quite pleased that you gave it a try. I am sure that the equal beating approach was unfamiliar to you but I think you can imagine that for an aural tuning novice, it is no more unfamiliar than any other method.

I do get the comment quite often, "These 4ths are equal beating now but they beat faster than I am used to hearing." I cover that in my written material but I am sure that comment can be read and forgotten easily. The fact is that at the point when they are heard that way, those 4ths are still 2 cents wider from where they will end up at the end of the process, so they naturally will sound too fast at that point. It is important therefore to have strictly pure intervals where they are directed and strictly equal beating where that is directed. The more closely those directions are adhered to, the better the end result will be.

This process is specifically designed to tune a tuning exam detuned piano and NOT for fine tuning. It follows that it can be useful for a pitch change. Once all 13 notes of the temperament have been initially tuned and error is noted, it would not be productive to start the ET via Marpurg sequence all over again. That would actually involve detuning notes which are already where they need to be or very nearly so.

It is better at that point to follow some other known sequence step by step and whenever a note is found to be slightly incorrect, to correct it at that point and use whatever checks are known and available at that time. If the result is very nearly perfect and only slightly flawed, diagnostic checks that prove an error can be used to make small corrections.

I've tried both (being a hobby mathematican, theory teacher, pianoplayer and former chess player that was the only solution that could get me to sleep at night :-D

I started with the counting of beats in F3-A3-C#4-F4-A4, just to find out they very seldom lined up as they should, except if I got into "ultrastrech mode" and adjusted the F4 very wide. On the contrary, when in similar situations the F3 is rised to get a beat interval of 6bps, even 5bps, things fell into place. I learned to like bills idea of not counting, just comparing the contigious 3rd sentence! The feel for the slight acceleration is musical in some way, and hey - I come from the land of Räikkönen and Häkkinen, so i know what gears ratio is :))

Regards,
Pati