Phase lineage represents the subset of wavelengths required to produce 'usable' constructive interference along various points of a scale. We can use a guitar to describe it's application.Can you explain what you are talking about?
What is "phase lineage" as you refer to it here, and what does it matter?
Where did that startlingly precise surface area come from?
Why is the significance of a "chime potential", what does it add to the ideas of resonance and standing waves?
Without a stated hypothesis and definitions of terms, how can you have a "theory" of anything?
The frame of reference will be as follows: the guitar is a right handed version and is held in the conventional manner. The nut is the left limit and the bridge is the right. Strike one string just sufficiently to set it in audible motion and take note of the amount of time it takes for the sound to diminish beyond your hearing ability. A useful approximation is all that is needed. This is called an open strike and produces the set of fewest and longest waves required to sustain the smoothest and most efficient path to equilibrium.
Find the halfway point between the nut and the bridge. This is usually right above the 12th fret. Touch the string but don't push it down to the fingerboard. With a different finger, strike the string sufficiently to set it in motion, taking note of the duration. If you are close enough to the sweet spot, you will clearly hear the harmonic chime and it's duration will be reasonably close to the sustain of the open string.
The energy from the strike is more or less simultaneously transmitted toward the nut and the bridge. Some of the energy that does not get absorbed into the hardware will make it's way back towards it's source of emission where it's peaks and valleys sum to constructive interference with the waves coming back from the other end.
The more rigorous the craftsmanship of the luthier (maker of stringed instruments), the better your chances of finding chimes.
If you repeat this process at the 7th fret, you will find another sweet spot (chime potential).
You will hear the same pitch raised by one octave. On the guitar that I am using the hardware contact points for the G string are about 12 3/4 inches equidistant from the 12th fret so the interval ratio is 1:1. This is probably the most commonly played chime on guitar.
The distance from the 7th fret to the bridge looks like about 17 3/16". 7th fret to nut is about 8 1/2"
There seems to be a discrepancy and there is. If you can hear the chime, you would expect a 2:1 ratio but this is clearly not precisely the case. In this case, the sweet spot is actually a little to the right of the 7th fret. The frets have very little to do with the wave action unless the string is depressed. If they are metal, they will contribute at least a little to the reverberation.
Fretted stringed instruments do not generally produce perfect pitch at every location on the fingerboard. For one thing, depressing the string may produce a slightly 'wrong' intonation.
Discrepancies in intonation are brought about by a number of things such as: uneven wear on the frets, neck bow and string diameter. To compensate, many commercially produced guitars are bridged by independent saddles so that you can adjust the length of each string.
Even after all that you will likely still get some dissonance which, in the right measure can actually compliment a chord. The more the two waves are said to be in phase, the more you get what sounds like a hum. When they are just the right amount out of phase, you can get more of a pleasing 'growl'. This growl sells records and is sought after by those who understand it's value.
In this case, the saddles provide an adjustable constant that can help to integrate unwanted dissonance into a fuzzy state of pleasing imperfection.
Phase lineage describes a family of cycles that are related by proportionality. The same reasoning that is used to describe how these proportions give us chimes is used to describe why electrons must gain a specific amount of energy to go to a higher state.
0.84 fm ^2 may seem precise but do not be startled. It is actually just a guess based on what I gathered the current consensus is if you could measure the diameter of a proton.