josepp - I think you are looking for something that doesn't exist. There is nothing unusual in the melodies or harmonies in sacred harp or shape note singing. It's generally straightforward 3- or 4-part harmony of fairly standard melodies. The only feature that is slightly unusual is that the melody is carried by one of the middle parts, but this also happens in other settings.

The distinctive sound, which you said earlier that you could always recognise, comes from the voices, and particularly the mix of high and low voices singing the same part. The performance at a fairly constant mezzoforte adds to the effect.

It would be possible to imitate the sound with instruments, but I see little point. A string quartet couldn't do the sound justice, unless they found a way of double stopping each part in octaves. An orchestra could certainly do it, and it might have some interest as a short quote in a longer piece, but for anything over 5 minutes it would be a waste of an orchestra's time.

Phil