It may be quite old.
The medieval Welsh Book of Taliesin is from the 14th century but the poems it contains may date from the 10th century.
It contains (in translation) a reference to "the golden pipes of Lleu" which looks to have the meaning "Lleu's singing".
The Four Ancient Books of Wales, ed. William F. Skene, p.509, Abela Publishing 2011 (reprint)
OED also has 1851, which directly uses the metaphor of human singing voices as organ pipes:
† people-organ n. Obs. rare a body of people seen figuratively as constituting a church organ.
- 1851 E. B. Browning Casa Guidi Windows i. xx. 56 This..teacher, will.. build the golden pipes and synthesize This people-organ for a holy strain.
"people, n." OED Online. Oxford University Press, December 2014. Web. 9 January 2015.
Following up on the possible Welsh connection, I've found a translation of Welsh lyrics in "Musical and poetical relicks of the Welsh Bards", Edward Jones, 1808, p.100:
How sweet, Isgywer*, is thy charming sound,
Which makes the youthful heart with transport bound
Thy various notes, mellifluous and strong,
Flow tuneful as the golden pipes of song!
[*] Iſgywer = small harp
Jones is obviously quoted from an existing song here, but the footnote just says "D. Samwell", so I'm not sure where he's getting it from or how old it is - or whether the translation is made by Jones himself, or if it's pre-existing. It would be nice to know how literal (or not) this translation is - do we have anyone here who can read Welsh?