Glad that was of help, thank you for replying as well. However I must confess my own perplexities about the request of improvement here. I'm still interested in understanding this with the best of my abilities. So I ran a quick test confronting all MTS-ESP 'Transpose' modes vs Pianoteq 'Change root key of current temperament'.
This topic is controversial, please allow me to present a more comprehensive post than before (which will require some time at reading). I will not go any more into dynamic micro-tuning specifics like it happened with my previous post as it also seems that you are simply not looking for 'automatic adaptive tuning' (such as that which BitKlavier Spring Tuning provides) but rather you want to manually set a specific new 'root' for the 'modulation' (re-tuning or re-mapping) to be more or less immediately applied each time, depending on the moment the next note-on takes place. This would translate in that you might want to choose 'New notes only (no pitch-bending)' over Continuous retuning (pitch-bending)' if Pianoteq was running as a MTS-ESP Client, contrary to what I've said before...whatever.
With that being said, based on my recent results (if I didn't miss anything) it looks like you are wrong with your first assumptions on MTS-ESP. Did you give it a try? Pianoteq behaves not like the "Modulate" mode but exactly like the "Mode Rotate" mode with a reversed +/- value. Also "Mode Rotate" mode in MTS-ESP does not allow any 'drift' nor any 'adaptive tuning' option whatsoever, so we would better not mix things up.
The acoustic result is identical (except for that insignificant detail I told you in the last part of my answer before) but the way these two pieces of software achieve it is not. MTS-ESP "Mode Rotate" mode needs a negative 'macro value' to match the positive increment of Pianoteq's 'root' (MIDI notes have nothing to do with 'roots' in music, just saying). That is, when you set a different 'root' by means of the Pianoteq function alone, and you go for a MIDI note higher than, say, the initial 60 (C4) - hence, say, 61 (Db4) - then, the 'macro value' within MTS-ESP "Mode Rotate" must go to the reversed direction (-1 in this case), or things won't match. And so on (needless to say, Pianoteq must deal with a total common n-tone scale and be loading the corresponding .scl and .kbm files).
It's hard to figure out what you personally mean by 'modulation' without more specific examples. The "Interval matrix (modes)" in Scale Workshop v2 browser app is what I mean by specific examples now, because it gives you a good vision of what happens at least in this specific case. Also, both the 'diapason' value in Pianoteq and the MTS-ESP 'reference note frequency' are completely left untouched by my computer when running this test. So technically I'm unable to register any 'diapason' change. What you are saying about diapason and modulation in relation to Pianoteq 'Change root key of current temperament' facts may be needing some clarification?
If I understood this correctly, Pianoteq does what it does by changing what is referred to as the 'reference note' in MTS-ESP or the 'MIDI note for base frequency' in ScaleWorkshop 2 (these are all .kbm file parameters) while leaving all .scl values untouched. First, the user queries the program for this 'root change' action to be taken (according to the MIDI event he/she must set in the Options -> MIDI, under 'Global MIDI mapping'), then selects the 'root' (MIDI note) by pressing some MIDI note, and, after that, the change immediately occurs in the digital realm, whereas the re-tuning ('modulation' or whatever) can be acoustically appreciated only following the next striked note-on, and provided that the specific MIDI note pressed for the query doesn't lead to a mapped frequency resulting in the start of the period (or modulo-resetting, because this 'wraps' the scale around, whereby no modulation can actually be heard, as if the initial step 0 was meant).
So the 'shifting of the root' (for a lack of better terms) can be indicatively assumed by looking at the note labels 'rotating' towards one direction or the other under the 'Keyboard mapping' section found in the 'advanced tuning window' in Pianoteq (all this without MTS-ESP being engaged, of course). We all know those note labels mean potentially nothing (just values from 0 to 127).
Conversely, MTS-ESP brings the same, identical result as Pianoteq own 'Change root key of current temperament' at the conditions explained above, but gives you a whole bunch of information about the 'modulation', since the 'tuning table' values (.scl file parameters, basically) are dynamically and instantly changed. The change is reflected by the 'Macro Adjustment' column values as well as by the 'step frequency' column values (the latter, in Hz).
The 'Macro Adjustments' can be displayed either in cents interval or in frequency ratio (decimals) format, also either relatively to the 'tonic' (1/1, or step 0 of your scale preset, eventually modified) if the "Display Macro Adjustment as" option is set to 'Absolute'; alternatively, relatively to each step (showing each adjusted inter-step interval size) if set to 'Difference'. Again, this situation with all the possible changes thereof can be grasped as a whole through Scale Workshop v2 "Interval matrix (modes)" available in the "Analysis" window.
As a side note, ODDSound YouTube video about the 'Transpose' mode informs you (along with its own sequel dedicated to 'adaptive tuning') about the existence of a preset such as that of "Adaptive JI". It seems that the maneuver required for you to instruct the software on the precise 'root change' can be customized into more usable and quick ways than those available by means of Pianoteq action & MIDI event global managing (see, for example, the 'Use Lowest Played Note' option demonstration), for all that it might turn out to be that their software is not strictly needed for your specific purpose here (you know).
All in all, some change is reflected in the MTS-ESP line graph / circle graph as well (although questionably, from a practical point-of-view), whereas Pianoteq, in its current implementation of the 'Change root key of current temperament' function, will not display any change at all, for the reasons implicated above (while its own circle graph is definitely more appealing to the eye).
So here's where my perplexity lies. If you are asking Modartt for an improvement in the mere visual sense, it's one thing. If you are asking them for an improvement which instead involves achieving a totally different acoustic effect than what Pianoteq and MTS-ESP (in "Mode Rotate" mode) both achieve, then I would expect you to report here, in return, which actually is the MTS-ESP transpose / modulation mode that does what you request (if there is one).
From what I know, 'extended just intonation' is a tuning system like that which Ben Johnston used for his String Quartets works, and, the "Transpose + Modulate" mode within MTS-ESP - let us forget about drifts and adaptive stuff for the moment - sounds like the way to go, if the composer's interest is to reproduce the same just intonation 'lattice' only potentially 'extended' to any absolute frequency contained in the initial set of absolute frequencies that must come with the initial mode/scale.
Please, let us know. Looking forward to be hearing best results from you. And good luck with your experiments!
