JavaScript (ES6), 153 bytes

n=>n?(a="AugmentedDiminishedMinorMajorPerfectSecondThirdFourthFifthSixthSeventh".match(/.[a-z]+/g))["51324"[n-9]^n%13%4]+' '+a[n+3*(n>11)+19>>2]:'Unison'

Try it online!

How?

The case \$n=0\$ is processed separately.

We use the repeating sequence Diminished / Minor / Major / Augmented for \$1\le n\le 8\$ and for \$14\le n\le 21\$. The corresponding indices in the lookup array are \$[1,2,3,0]\$. This is done by reducing \$n\$ modulo \$13\$ and then modulo \$4\$.

We actually use the same calculation for \$9\le n\le 13\$, but we apply a bitwise XOR to get the correct indices \$[4,3,0,2,4]\$, corresponding to Perfect (Fourth) / Major (Fourth) / Augmented (Fourth) / Minor (Fifth) / Perfect (Fifth).

This gives:

"51324"[n - 9] ^ n % 13 % 4

We then need to append Second to Seventh with indices \$5\$ to \$10\$ in the lookup array. The expected sequence is:

     n | 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21
-------+---------------------------------------------------------------
 index | 5  5  5  5  6  6  6  6  7  7  7  8  8  9  9  9  9 10 10 10 10

Basically, each index is repeated 4 times but 3 indices are omitted between \$n=11\$ and \$n=12\$.

This can be obtained with:

n + 3 * (n > 11) + 19 >> 2

Charcoal, 95 94 bytes

⪫Ｅ§⪪”$＆⊞ςq×Ｐ5μ·⸿§▷"↑Ｗ4⪪9Ｑλς”²⊖Ｎ§⪪”↶⌈∨Ｓ⎚χ]¹↶Ｅ´üêＢ⭆↓ＨηW◧⁻◨~,⦄Zi≦r⊗Þπ↖Z):⊕À´Ａeδ∨8:≧#$/ＶMχ)Ａ” ⌕αι 

Try it online! Link is to verbose version of code. Explanation:

    ”...”                   Compressed lookup table
   ⪪     ²                  Split into substrings of length `2`
  §                         Index by
           Ｎ                Input integer
          ⊖                 Decremented
 Ｅ                          Map over letters
             ⪪”...”         List of words
            §               Indexed by
                    ⌕αι     Uppercase letter index
⪫                           Join with spaces
                            Implicitly print

Note that the lookup table is 43 letters long so that the last entry only has one letter; this is the one that is then selected for an input of 0.

Previous 95-byte solution:

Ｎθ∧θ§⪪”↶↧*y³↨Cⅈ～➙…⟧⊖⁺¡κＹ⁸⁴Σ1|R6；” §”)⊞⮌i›‽”θＭ→§⪪”↶⌈∨l⟲Ｅχ⁼⁰θ℅ＢNμ↨%),ρ℅«◨ TＨＹ¬O⬤ε” §”)➙:⌊B/Ｍ⁷ê\`”θ

Try it online! Link is to verbose version of code. Explanation:

Ｎθ

Input the distance.

∧θ§⪪”...” §”...”θ

If it's not zero, cyclically index into the compressed table 4012301234231 to find the index into the compressed list of words diminished minor major augmented perfect.

Ｍ→

Leave a space between the two words.

§⪪”...” §”...”θ

Index into the compressed table 0111122223334455556666 to find the index into the compressed list of words unison second third fourth fifth sixth seventh.

05AB1E, 71 70 bytes

.•10°ÝΣËðw¶\ƵUʒ•…‡®¢ÑíÀ«#•”Öζт'µ���5вè“…Þ‰ª¦ƒ³äÇïÖµ“#•γ¤æ•ÅΓø.•I¢#€•šÅв»

Outputs in lowercase.

Try it online or verify all test cases.

Explanation:

.•10°ÝΣËðw¶\ƵUʒ• # Push compressed string "augmented diminished "
 …‡®¢ÑíÀ         # Push dictionary string "major minor perfect"
        «        # Append them together
         #       # Split this string on spaces:
                 #  ["augmented","diminished","major","minor","perfect"]
•”Öζт'µ•        '# Push compressed integer 160475945053335
        5в       # Convert it to base-5 as list:
                 #  [1,3,2,0,1,3,2,1,4,2,1,3,4,1,3,2,0,1,3,2,0]
          è      # 0-based index each into the previous list of strings
“…Þ‰ª¦ƒ³äÇïÖµ“   # Push dictionary string "second third fourth fifth sixth seventh"
              #  # Split it by spaces as well
•γ¤æ•            # Push compressed integer 4333444
     ÅΓ          # Run-length decode the list of strings and digits of this integer
ø                # Create pairs of the two list of strings
 .•I¢#€•         # Push compressed string "unison"
        š        # Prepend it to the list of pairs
Åв               # Convert the (implicit) input-integer to this custom base as list
  »              # Join each inner pair by spaces, and then the strings by newlines,
                 # which simply pushes the only string in the list in this case
                 # (after which it is output implicitly as result)

See this 05AB1E tip of mine (all four sections) to understand why .•10°ÝΣËðw¶\ƵUʒ• is "augmented diminished "; …‡®¢ÑíÀ is "major minor perfect"; •”Öζт'µ• is 160475945053335; •”Öζт'µ•5в is [1,3,2,0,1,3,2,1,4,2,1,3,4,1,3,2,0,1,3,2,0]; “…Þ‰ª¦ƒ³äÇïÖµ“ is "second third fourth fifth sixth seventh"; •γ¤æ• is 4333444; and .•I¢#€• is "unison".