Re: Issue...I think...
Could be that the portions of the strings between the bridge and the tailpiece, or between the nut and the tuning pegs, are vibrating when you play your E strings. There are li'l dampening devices, from rubber grommets, to leather strips woven through the strings, to felt installed under the tailpiece cover, to combat these unwanted tones.
Try just putting some sound-absorbent material around the strings between the bridge and tailpiece, and see if the "brief and subtle tone" persists.
Allen Hopkins
Gibsn: '54 F5 3pt F2 A-N Custm K1 m'cello
Natl Triolian Dobro mando
Victoria b-back Merrill alumnm b-back
H-O mandolinetto
Stradolin Vega banjolin
Sobell'dola Washburn b-back'dola
Eastmn: 615'dola 805 m'cello
Flatiron 3K OM
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