We're not talking about folks getting down to some uplifting gospel music. These tunes "oppress" rather than uplift. And with them comes a heaviness that the speaker can't help but feel burdened by. They also inflict a kind of "Heavenly Hurt" that reveals the speaker's "internal difference" or conflict even more.
Lines 3-4: The light is compared to the "Heft" of "Cathedral Tunes," which suggests the sort of heaviness that comes with these tunes as well as the light. So we get the sense of there being a kind of relationship between the light and the tunes for the speaker.
Lines 5-6: The light and cathedral tunes inflict a "Heavenly Hurt" that leaves no scar. So the effect of the light or tunes' heaviness and oppression is something that is internal and unable to be seen.