Dickinson doesn't get too wild with the sound games in this poem, but she does slip some in on us for kicks. The first thing we notice is a couple of prime uses of alliteration. The first line is the most glaring example. We're sure you didn't miss these double Ms:
Much Madness is divinest Sense— (1)
Here, Dickenson proves the power of alliteration to pack the perfect punch. (Did you see what we did there? Yeah, we're not that clever.) By kicking the poem off with repeated consonant sounds at the beginning of the first two words, she puts a major emphasis on those words. It's a double punch that establishes exactly what we're supposed to be paying attention to. It aggressively sets the tone for this kind of in-your-face poem.
We find some more alliteration later in the poem:
Demur—you're straightway dangerous— (7)
Here, the two Ds help bookend the line and highlight the contrast the line contains. How can somebody who only demurs (mildly disagrees) be called dangerous? By choosing words that begin with the same letter to get across these ideas, the poet links them sonically in our minds (yeah, she's sneaky like that), which helps to hammer home the ridiculousness of majority's unfair labeling process.
We also hear a lot of consonance in the poem, mainly with the repeated S sounds within words. Let's look at the first three lines together this time. Read it out loud, and you can really hear it:
Much Madness is divinest Sense—
To a discerning Eye—
Much Sense—the starkest Madness— (1-3)
It's almost like the poem is hissing at us, which seems to feed into poem's biting, sarcastic tone. Dickinson uses the sounds in this poem to unsettle us and to directly attack. The poem strikes fast with its keen observations and leaves us running for the anti-venom before we know what hits us.