Alliteration is the repetition of a consonant sound within a line of poetry. Similarly, assonance is the same repetition within a line of poetry, but the sound is that of a vowel. Consonance, therefore, is the combination of both assonance and alliteration given that consonance is the recurrence of similar sounds when in close proximity.
The use of alliteration and assonance was first highlighted in Anglo-Saxon poetry and epics. The scops (the singers of the poems and epics) found that (assumptive based upon critical research of the period) the repetition of similar sounds translated well into song.
Therefore, the use of both alliteration and assonance (or consonance when joined) creates a sing-song aspect when used in poetry. The words tend to roll off of the tongue easier and create a very specific mood (one of elegiac (sad remembrance) or bold (when used when speaking of the heroic battles of Epic heroes)).
In Louis Gluck's poem "Epithalamium," alliteration and assonance add to the mournful mood of the poem. The poem speaks to memories, pain and death (typical of the elegiac and lyrical poems seen in the Anglo period). The opposites proposed by Gluck offer a hard contrast to the joy of marriage and the pain of abuse.
Therefore, the speaker's voice carries a mournful and almost hymn-like tone. The alliteration and assonance support the tone of the poem.
There really is not a lot of alliteration, or assonance and consonance, in this poem. The two main instances of alliteration here occur in the lines "green hill in gold light" and "here is my hand that will not harm you." As mentioned before, this alliteration adds a sing-song quality to the poem, which, to delve into this a bit more, could mirror the traditional vows of a wedding ceremony, especially the line "here is my hand that will not harm you." The way that this line mimics wedding vows is particularly bitter, since the speaker hints that her husband actually is harming her with either physical or emotional abuse. Thus, Gluck could be mocking the ways in which we often blindly take part in cultural traditions and rituals, reciting our vows without actually meaning them or understanding what they mean.
While the assonance and consonance in this poem definitely add to its sing-song quality, they also create a harsh, biting sound. For example, there is both assonance and consonance in the line "the terrible charity of marriage" with the repeated r's and the short e sound across terrible, charity, and marriage. As a result of this assonance and consonance, this line takes on, again, a bitter tone. We can almost picture Gluck's speaker spitting out her words here.