They are related to the mathematical properties of the harmonic series, but they are definitely not rooted in it.
If it was true that western musical ideas of consonance is rooted in simple harmonic ratios, and that our musical scales were based on the harmonic series, then we should be seeing subminor thirds (6:7) and supermajor seconds (7:8) everywhere. They are way more "consonant" than major seconds (9:10), or minor seconds (15:16). But in reality, they are seen as exotic and wrong.
Unfortunately, nobody said "western musical ideas of consonance is rooted in simple harmonic ratios". The whole point of Plomb & Levelt's work (recommended if you have no read it) is that provides an explanation for consonance that has simple harmonic ratios as an existential but insufficient component. From their work, what matters is not the ratios of the two fundamentals, because there are essentially no natural tones that consist of only the fundamental (1). Instead, consonance/dissonance is a result of the sum of the consonsance relationships between each pair of partials in the two tones.
Sethares extended their conclusions a bit by noting that since the partial spectrum is the definition of timbre, the most consonant/dissonant intervals would vary by timbre, which they claimed is observed in the real world.
(1) Indeed, the first step of their work, defining the "human dissonance response curve" relies on being able to use a signal generator to create pure sine tones.
If it was true that western musical ideas of consonance is rooted in simple harmonic ratios, and that our musical scales were based on the harmonic series, then we should be seeing subminor thirds (6:7) and supermajor seconds (7:8) everywhere. They are way more "consonant" than major seconds (9:10), or minor seconds (15:16). But in reality, they are seen as exotic and wrong.