What is the basis of the 'Sweetened' tunings?
Does it move the pythagorean comma dependent on the octave you are playing in?
Or something like that?
A lot of the schemes I've seen around the internet a claimed to be sent from the heavens yet make neither harmonic or mathematical sense. Anything that makes harmonic sense simply doesn't work on a fretted instrument as it needs to be even tempered. (the known inaccuracy averaged through the scale)
Sure you can sweeten some frequency relations but at the expense of making the majority of chords worse than they already are from a harmonic perspective.
The expectation bias introduced from being exposed several centuries worth of chromatically tuned instruments also makes it very difficult to asess the harmonic value of a truly sweetened harmonic series. The 7th harmonic for example has all but been eradicated from most forms of western music because it don't sound pretty enough, even to the point where the piano's hammer hits is design to eliminated as much 7th harmonic resonance as possible. If we ever could hear a truly 'sweetened' tuning it would likely sound horrible to most of us.
I limit my experimentation on this stuff to microtonal keyboards and listening to exponents of classical microtonal instruments such as the voice and viol based strings mostly. Those guys are really using their ears!
There have been many experts in this field since and including Pythagoras and nobody seems to have come up with anything conclusive yet. I'm figuring from that it is a pretty tricky subject defining a truly sweetened tuning and an impossibility on a previously fretted instrument.
The only thing that makes any sense at all within our current system is to change A=440 to.....think of a number.