The state or quality of having one's head uncovered; the condition of not wearing a hat or head covering.
From 'bareheaded' plus the suffix '-ness,' which creates nouns describing states or qualities; this construction became common in English around the 16th century.
The fact that we have a word specifically for 'the condition of not wearing a hat' tells you how important hat-wearing used to be—you only create elaborate vocabulary for things that truly matter to your culture.
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