How many degrees of spread does this tool's analogous palette cover?
Sixty degrees total, from -30 to +30 around the base hue, with five fixed points spaced 15 degrees apart. This window is the classic color-theory consensus for "analogous": wide enough to vary, narrow enough to still read as one family.
Should I use an analogous or complementary color for an error alert?
Neither: an error calls for a semantically fixed color (usually red), outside the brand's harmony system, so it is never confused with a calm, low-urgency state. The analogous palette is for the dozens of neutral badges that should not compete for attention.
Why do saturation and lightness stay the same across the five tones?
To keep visual weight equal across every tone: only hue varies (-30 to +30 degrees), so no badge ends up looking more saturated or lighter than its neighbor by accident. If you need hierarchy between states, change to a different hue on the wheel, not the saturation inside the same analogous family.
Does the analogous palette guarantee enough contrast for text on top of these colors?
No, and that is the most common trap in badge design: five hues close together tend to share similar lightness, so the same white or black text can pass the WCAG test (4.5:1 for normal text) on one color and fail on its 15-degree neighbor. Check each text/background pair individually.