Partridge (e.g.) shows “not much chop”, supposedly from “no chop” = “inferior” (Australian), presumably originally “not having a chop”, where “chop” (as in “first-chop” = “first-rate” etc.) originally meant “stamp” or “seal [of quality/approval]” in Anglo-Indian and Anglo-Chinese, apparently from Hindi “chhap” = “stamp"/"brand".

The relationship of the [near-]synonymous “no[t much] cop” is not clear (at least not to me), but maybe this was originally a variant, or maybe unrelated.

I don’t know that either “no[t much] c[h]op” has ever been usual in the US.

The other “chop[s]” mentioned above may be unrelated.

Incidentally, on brief search of old Australian newspapers, I see seemingly relevant examples of “no chop” from 1840 and of “not much chop” from 1857.

Silver coins, such as U.S. Trade Dollars, used during the later 19th century in trade with the far East (with China in particular), , are frequently die-stamped with various marks - these are called “chop marks”, and may (or may not) have indicated that the coins had been verified, or evaluated, by somebody. I’m fairly sure that this has little, if any at all, relation to the subject under discussion - but of course I could be wrong. Certainly these chop marks are a long way from U.S. musician’s slang. So is the word “chopper”, a U.K. slang word (once common, may now be in desuetude) for the male genital organ, which, if one wanted to wallow really deep in utterly speculative etymology, might be connected with the use of the word “axe” for a saxophone (in Brave New World, erotic music was produced by sexophones - etymology alert!). Fortunately, I don’t have to write a thesis (on this or anything else), and will simply remind my fellow word-worriers (or perhaps “word-harriers” might be a better appellation) that “cop” can mean other things as well - for instance, in the Yiddish vocabulary of non-Yiddish speakers, “cop” can also mean “head” (as in “use your cop, you idiot") - derived from the Yiddish kopf, derived from the German kopf, “head”. Now “head” can mean a whole lot of things, too, including the cloaca (to coin a euphemism) of a submarine. Perhaps there is some connection here with “not much chop/cop” (how active are one’s bowels, full fathom five or more below, with depth-charges going off all around one, I wonder?)

Er, right .....

DARE doesn’t mention “no c(h)op / not much c(h)op” but has the following entry under chop n3 (from Hindi chap, stamp, brand, transferred from trade usage), as has been suggested earlier:

1.Quality; esp in phr first chop highest quality.
1810(1912) Bell Journey to OH 49, He thinks himself a gentleman of the first chop, & takes the liberty of coining words for himself.  1844 Stephens High Life in NY 1.48(1958 Taylor-Whiting Dict. Amer. Proverbs), My boys aint men of the common chop.

"First chop” may well be a distinctive Americanism - The Times online has an instance of it in 1838, but it comes in a book extract where the writer, Thomas Haliburton, is using idioms already apparently associated with American speech. 

“I know a judge of the State Court of New York, a first chop man, too, give it up, and take the office of clerk in the same identical court.” (Because public office was so ill-rewarded in America.  Allegedly.)

I should have said that DARE is the Dictionary of American Regional English. Apologies for causing confusion.

No, I understood that - I was just surprised that a word apparently originating in trade with the Far East should have come into American English first.  I’d have thought Britain would a) have had a bigger trade in imported silks and b) been in a better position to adopt a word from Hindi.

I was surprised to learn axe first referred to a sax, not that I’m doubting it. I thought it was from ‘70s macho electric-guitar rock with the neck resembling the handle and the body the blade! As here. Embarrassing now, but Bill Nelson once had guitar chops.