Similarity & Distance

Two songs are playing on a streaming platform. Song A: jazz piano, tempo 95 BPM, energy 0.31, acousticness 0.88. Song B: acoustic folk, tempo 102 BPM, energy 0.28, acousticness 0.82. Song C: electronic pop, tempo 128 BPM, energy 0.91, acousticness 0.05. Which song is most similar to Song A? Intuitively Song B — both are acoustic, low energy, unhurried. But “intuitively similar” cannot run on a server. It must be converted to a number. That number is a distance metricdistance-metric — a function that takes two feature vectors and returns a single value representing how different they are. The choice of metric encodes an assumption about what similarity means, and different assumptions produce different answers.

Euclidean distance

Imagine the straight-line distance between two points on a map. Euclidean distanceeuclidean-distance extends this idea to any number of dimensions — it is the shortest possible path between two points in feature space.

Square the difference in each feature, sum them, take the square root. Song A and Song B are 0.07 apart — close. Song A and Song C are 1.02 apart — far. Song B is the correct recommendation.

Squaring the differences means that one large gap between songs hurts more than many small ones. A song that differs from the reference by a lot in just one feature — say, tempo — gets pushed far away even if energy, acousticness, and danceability all match closely.

Manhattan distance

Imagine navigating a city grid — you can only walk along streets, never cut diagonally through a block. Manhattan distancemanhattan-distance measures that grid path: the total distance travelled when you move one feature at a time.

Take the absolute difference in each feature and sum them all — no squaring. For Song A and Song B: |0.31 − 0.28| + |0.88 − 0.82| = 0.03 + 0.06 = 0.09.

Because Manhattan does not square differences, a song that is very different in one feature but nearly identical in all others stays closer under Manhattan than under Euclidean. It is more forgiving of a single outlier dimension.

Cosine similarity

Two songs can sound identical but one was recorded quietly and one loudly. Cosine similarityCosine SimilarityThe cosine of the angle between two vectors, computed as their dot product divided by the product of their magnitudes. It ranges from −1 (opposite directions) through 0 (perpendicular) to +1 (identical directions) and is magnitude-invariant — scaling either vector leaves the value unchanged. ignores overall volume — it only asks: do these two songs point in the same direction in feature space? A song with the same ratios of energy to acousticness to tempo is maximally similar by cosine even if its raw values are twice as large.

Compute the dot product of the two feature vectors, divide by the product of their lengths. The result is between −1 and 1: songs pointing in the same direction score near 1; songs pointing in perpendicular directions score near 0.

import numpy as np

def euclidean(a, b):
  return np.sqrt(((a - b) ** 2).sum())

def manhattan(a, b):
  return np.abs(a - b).sum()

def cosine_similarity(a, b):
  return np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))

# Features: [tempo_scaled, energy, acousticness] — z-score normalised
song_a = np.array([0.95, 0.31, 0.88])  # jazz
song_b = np.array([1.02, 0.28, 0.82])  # acoustic folk — similar
song_c = np.array([1.28, 0.91, 0.05])  # electronic pop — different

print(euclidean(song_a, song_b))         # small — songs are close
print(euclidean(song_a, song_c))         # large — songs are far
print(cosine_similarity(song_a, song_b)) # near 1 — same direction
print(cosine_similarity(song_a, song_c)) # lower — different direction

The three metrics are not interchangeable. Choosing Euclidean when cosine is appropriate, or Manhattan when Euclidean matches the data geometry, changes which songs are recommended — and the error is silent. The algorithm returns k results regardless of whether those results are the genuinely most similar songs or just the ones that happen to be numerically close by the wrong definition of close. The playground makes this concrete: switch metrics and watch which songs appear as nearest neighbours.