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I think of purple as encompassing both indigo (blueish purple) and violet (pinkish purple).
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I agreed with you, and then went to the Wikipedia pages for both. I might have changed my mind now.

https://en.wikipedia.org/wiki/Purple

https://en.wikipedia.org/wiki/Violet_(color)

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I think using violet as a name for the entire color-range around (~128, 0, 255) is also common. So in a sense purple is an element of the violet color-range. But as points they are distinct. I think purple is more specific - as a color-range it'd cover less area.
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Purple has a lot more red. (157, 0, 255) vs. (128, 0, 255). Good to have learned something today...
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