Why does the G# work so well when the rest of the song is in C major? Why, if the song is in A minor (the relative minor to C) does the G# become the note that turns it into a Harmonic minor? The answer would be based on the Harmonic series.

Here is the Harmonic series as explained in Wikipedia’s page: https://en.wikipedia.org/wiki/Harmonic_series_(music)

The G# we are discussing is based on the 13th element of the series. In a non-tempered instrument, this would be the +41 Cents indicated, meaning that it would be a fairly sharp note.

Let’s Ignore Octave for a moment and just write out the notes of the Series.

C C G C E G Bb C D E Gb G Ab Bb B C C# D D# E

You can think about how much color a given note has by how far down the series you need to go to get to that note. The 1, 3, 5 and Dom 7 all are in the Early part of the series. That Dom 7 is fairly flat (-31 cents) from the tempered tuning. I think it is fascinating that the Minor Third we hear so often in a minor chord is so far down the Harmonic series. A C minor uses the Eb, which is the 19th note of the series. However, if we think of the A minor, then the Third is the base note of the series, while the root of the chord is based on the 27th note of the series. There must be a simpler explanation for that.

The E minor chord is much earlier in the series…the E is the 5th note of the series, and the G is the 3rd. But the B (the V of the E minor) is the 27th note of the series.I suspect that what we actually hear is not based on a single harmonic series, but on multiples. If we were to translate the C scale shown above to the E scale, then the V of the chord is, again, the 3rd note of the series. Same with the A minor. So You would have to transpose around the harmonic series to see where the G# fits relative to the other notes of the chords.

I suspect that what you are going to find is that the Chords are based on multiple Harmonic series. For example, the B that makes up the Major 7th in the C maj chord is very closely related to the G in the Harmonic series based on the G root. It does not even show up in the first 20 notes of the Harmonic series based on C.

Looking again at our series of notes, we can see the earliest Minor chord is the G minor. The G is the 3rd and 6th entry of the Series, so very strong root. The Bb is the 7th entry and the D is 9th entry.

The sequence also shows why a Dom7 with a Ninth added to it sounds so strong. The Chord Ninth is the D, coincidentally it is also the 9th element in the series.

I also find it quite interesting that the E to Bb tritone shows up so early in the series: 5th to 7th entries. This sound is so strong and jarring, but it is right up front.