I should disclose that my pseudonym is a misnomer on both counts, as I am neither a poet nor a musician. However with that having been said, I would suppose that your teacher has devised a silly trick question. Synesthesia is most appropriately a medical condition, which originally referred to "the production of a sensation located in one place when another place is stimulated" according to the Century Dictionary and Cyclopedia's definition of the word1, which may be one of the earliest definitions of the word as it was printed in 1891*.
Today, it most commonly refers sensory input as if it was another, such as tasting an aroma, or as the Oxford English Dictionary puts it*:
c. Production, from a sense-impression of one kind, of an associated mental image of a sense impression of another kind: see quot 1903. …
1903 F. W. H. Myers Human Personality 1. p. xl, Vestiages of the primitive undifferentiated sensitivity persist in the form of synæsthesiæ, e.g. when the hearing of an external sound carries with it, by some arbitrary association of ideas, the seeing of some form or colour.
This makes sense when you consider that most of the senses are associated with certain body parts, which is to say, you smell with your nose, you see with your eyes, you taste with your tongue and you hear with your ears.
Regarding the poetic application of the word, there does seem to be a tropic use mentioned specifically in the American Heritage Dictionary Fifth Edition where this might seem to be applicable:
- The description of one kind of sense impression by using words that normally describe another.
The trope makes sense; however, there are a couple of problems with this definition:
The first is that it is the very last use listed, meaning it is the least used definition the American Heritage Dictionary recognizes, since like most other modern dictionaries these days, they sort by popularity. Moreover it is not currently recognized by Oxford Living Dictionaries, Merriam-Webster, Random House Webster or Collins, so it strikes me as a highly dubious fringe usage.
However let us posit for a moment that this definition is indeed valid, at least for the purposes of poetry. After-all, that is the class's subject, the American Heritage Dictionary is a very reputable source and more recently updated than other printed dictionaries (although I would have expected Oxford's website to have been the trendsetter here).
Does it really apply in this case? I doubt it does. First, we would need to consider silence a type of sound, which is rather questionable. Then we would like to liken that so-called sound to another sense. Yes, an argument could be made that "golden" can be used refer to color, but that is not the intended meaning of “silence is golden”, and I also very much doubt that it is the normal meaning too. Both of those would probably refer to to the word pertaining to gold as a highly valued metallic substance, with a special emphasis on highly valued. The Phrase Finder link lambie provided leaves little room for doubt on the matter of intention, especially since it indicates the original proverb was “speech is silver; silence is golden.”
Granted, this objection is somewhat questionable on the grounds of what the most normal definition is, as the British Dictionaries, Collins and Oxford both list the color first. Nevertheless the remaining three American dictionaries list the metal first, and if it comes down to an American English versus the Queen's English, the U.S.A. is more than twice as populous as Canada, the U.K., Australia and New Zealand combined. It was approximately 318 million vs 128 million the last time I added the totals up, so overall it is more probable that the American preference is the more popular one.
Moreover, even if we really go out on a limb to be accommodating, and accept that it is synesthesia too, does this matter? Are these mutually exclusive categories? I doubt it. As a matter of fact, this form of synesthesia would probably have to be a sort of metaphor, since it likens one thing to another through transference of characteristics. Consider the definition of metaphor from
The Century Dictionary and Cyclopedia:
A figure of speech by which, from some supposed resemblance or analogy, a name, an attribute, or an action belonging to or characteristic of one object is assigned to another to which it is not literally applicable; the figurative transfer of a descriptive or affirmative word or phrase from one thing to another; implied comparison by transference of terms; as, the ship spread its wings to the breeze; "Judah is a lion's whelp,". Gen. xlix. 9 If Jacob had said "is like or resembles a lion's whelp," the expression would have been a simile instead of a metaphor. …
Also, Sumelic points out that Lectures on the Study of Language by Hanns Oertel in 1901 even refers to it as a type of metaphor directly:
The second class of metaphors which ought to receive an exhaustive treatment is the transfer of terms from one sense sphere to another. These rest upon what James calls "the principle of analogous feeling stimuli" and are illustrated by phrases like "a sharp tone", "loud colors", "a high note." The phenomenon of synesthesia has received rather full treatment at the hand of psychologists, but its reflection in language has not yet received adequate treatment by lexicographers.
However, it is also worth noting that a metaphor is usually a sort of direct comparison. If this example is what the American Heritage Dictionary used as a basis for its definition, then it is probably worth noting that the sort of description meant is the application of an adjective that more appropriately relates to one sense to a noun that more appropriately relates to another. These take entirely different forms "This is that" as in "Silence is Golden", which is an entirely different structure than the adjective noun pairings Hans Oertel exemplified.
As for the other options, Oxymoron is ruled straight out as a possibility, since there is not even an apparent contradiction, let alone a false one. I am also reluctant to call it symbolism or metonymy, since we are not substituting one word for another.
So out of the four words, I think metaphor is the best option.
* This information is sourced from A New English Dictionary on Historical Principles, volume 9, part 2, which was published in 1919 by Clarendon Press at Oxford, which is part of the first edition of the Oxford English Dictionary for those of you who did not already know.