Last question (for now): How does your system deal with more complex
tonal harmonies? For instance, one of Beethoven's "Diabelli" variations
contains the chord E/G#/C/B-flat. Would it know that the root of the
chord is "C," and adjust accordingly?
--Mark Arnest
This is a wise question. No, this harmony will not be corrected as it is a dissonant structure of an augmented triad, C-E-G# and it is not possible to tune the thirds C-E and E-G# to a pure ratio of 5/4 without bringing G#(Ab)-C to an extreme dissonant distance.But according to the proceding notes this harmonic structure mostly will not be tuned to exact equal temperament but to a slightly unequal tuning.
The background of this tuning behavior is a musical reason and a mathematical reason.
The musical one: Correcting the tuning to just intonation brings only consonant structures e.g. fifths, major thirds, major chords and minor chords to a better sound. Indeed, Hermode Tuning doesn't limit the tuning corrections to these harmonies, it corrects note combinations until 6 different simultaneous ringing notes "if there doesn't occur an internal conflict" as mentioned above. But such complex harmonies, even when being corrected, sound slightly different than with equal temperament, but they sond still dissonant
In case you will ask why we correct also such complex structures: This supports the horizontal stability. You know that in living music the harmonic structures don't follow each other clearly disjoined. The beginning and the ending of the notes overlap each other. Imagine a simple harmonic change from C-E-G to B-D-F#. At the passage from the first to the second structure there could occur for a short moment note combinations of e.g. C-E-G-F# to C-E-G-D-F# to E-B-D-F#. Everyone of this structure has to be and will be corrected in real time.
The mathematic reason: As already mentioned above some note combinations show ambiguos relations. BTW: Hermode Tuning doesn't position the root to equal temperament, correcting only the notes apart from that. All notes of every harmonic structure interact themselves. A major triad in its default position is tuned to (numbers shown as deviation in Cents from equal temperament) root +4, major third -10, fifth +6, a minor triad to root -6, minor third + 10, fifth -4. In both examples the som of deviation is 0. Depending of the effectvely played music this values could be slightly higher or deeper.
You said: Last question... but if any you have questions to this position method, ask please.
Additionally, visiting the following link could be helpful
www.hermode.com/html/tuning-history_en.html
There you will find presented at the end - see right at this page - an oversight over the four fundamental and different ideas in programme controlled tuning. The fourth of them shows Hermode Tuning and comparing its function by note examples and in comparision with the other ideas could be helpful.