The Hymn Tune Index lists 4 tunes called "Martyrs" that are Common Meter & published before 1820. If I'm understanding this, 3 are variants of a tune, so there are really just 2. Either could be the tune mentioned, but 330a is also called "Martys Scotch." Following a tune from incipits is kind of mind blowing but I think that the one labeled 330a is the 'Old Martyrs' from Cyberhymnal.

Martyrdom seems to have missed the 1820 cut off date to be in the index. I could not find it.

Peter Reynolds, I was looking for information on 'Sing Psalms.' From what I could find it's all metrical Psalms & would not have Amazing Grace. Does it use the tune with one of the metrical psalms? Is there a recording of it?