: Why the difference between interval ratios and the harmonic series? I've observed that the harmonic series which are the most consonant pitches of the fundamental such as the note A for this example, with A's first harmonic

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It's because the fourth is the inversion of the fifth. E is the third harmonic of A, and C# is the fifth harmonic of A, but A is the third harmonic of D.

So the ratio of the ascending fifth (the interval, that is, in this example A to E) is 3:2. The ascending major third (for example A to C#) is a ratio of 5:4, and the ratio of the ascending fourth (A to D) is 4:3. Notice that the odd factor is in the denominator here, because A is an overtone of D rather than D being an overtone of A.

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The harmonic series is not in strict order of pitch consonance with the fundamental. This can be seen from the fact that there are harmonics that are two, three, four.... octaves above the fundamental, all of which would themselves be very consonant with the fundamental, and yet have harmonics either side of them representing pitches that are less consonant.

Consonance and dissonance are subjective sensations which aren't necessarily even particularly well-defined. Plomp and Levelt's 1965 paper discusses some of the difficulties in pinning down what consonance really is, and discusses how their curve of interval consonance was calculated. I thought it was obtained directly from experiments on people, but on reading the paper, that isn't the case - it's derived from experiments using simple tones (sine waves).