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If I know the color temperature of a given source of light is it possible to predict the spectral composition of the light (in other words the specific wavelengths / frequencies that make up the light emitted)?

Although I'm not completely sure about this but my intuition says that it is not predictable because a bulb rated as 6,500 kelvin could have a spectral composition of a number of different wavelengths and still have the same color temperature. For example, 50% of 400nm + 50% of 600nm may give the same color temp as 50% 450nm + 50% 550nm.

Is my reasoning correct?

What about the opposite scenario if I knew all the wavelengths that make up a given light source can I predict the color temperature?

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