Why is bass pronounced like base? If bass sounds like base, why isn't ass pronounced as (nearly) ace instead?
Checking etymonline.com, we find:
late 14c., of things, "low, not high," from L.L. bassus "short, low" (see base (adj.)). Meaning "low in social scale or rank" is recorded from late 14c. Of voices and music notes, from mid-15c. (technically, ranging from the E flat below the bass stave to the F above it), infuenced by It. basso. Meaning "lowest part of a harmonized musical composition" is from mid-15c. Meaning "bass-viol" is from 1702; that of "double-bass" is from 1927.
And looking at base, the adjective:
late 14c., "low, of little height," from O.Fr. bas "low, lowly, mean," from L.L. bassus "thick, stumpy, low" (used only as a cognomen in classical Latin, humilis being there the usual word for "low in stature or position"), possibly from Oscan, or Celtic, or related to Gk. basson, comparative of bathys "deep."
So bass in the musical sense has its origins in base, and the current spelling is influenced by the Italian basso. In other words, the spelling changed out from under the pronunciation.
Bass as in a bass guitar is indeed pronounced the way you say. However, the fish named bass is pronounced just the way it is spelled (with a short a, rhymes with gas).
Asking why in English generally isn't all that productive. The issue with English is that it has been a written language for a very long time. Pronunciations of words often change over time, but they remain spelled the same way for historical reasons. Then sometimes we borrow words verbatim from other languages where the rules are completely different. Bass could be either, as a lot of our technical music words came unchanged from French and Italian back in the late Renaisance.
Bascially, words are just spelled the way they are spelled and you have to memorize them all. If you find one pronounced in some way analogous to how it is spelled, consider yourself lucky.
Ted and Robusto answer the direct question.
Let me just add: There are a lot of general pronunciation rules in English. But there are also a lot of exceptions that have to be memorized.
I've seen statistics on what percentage of English words are spelled phonetically that vary all over the place, from 25% to 90%+. Perhaps that discrepancy comes from your definition of what constitutes "spelled phonetically". i.e. how many rules do you consider? If you insist on one sound per letter, the number of words that that helps you pronounce correctly will be small. Add in all the two-letter combinations, "ph", "ng", "oo", etc, and it goes up considerably. Add in a few more rules, like "e at the end makes the vowel long", and it will go up yet more.
I recall musing once that if you add enough pronunciation rules, then all words are spelled phonetically ... it's just that many rules would apply to only 1 or 2 words. Well, I guess you'd still got stuck on words that have one spelling but two pronunciations. Like, say, "bass".
In practice, it makes sense to learn the rules that apply to many many words, and then memorize the words that are not covered by those rules. There's problem a formula you could work out here for the optimimum number of rules, based on how many words each new rule covers.
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Studying English pronunciation,
I will teach you in my verse
Sounds like corpse, corps, horse and worse.
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