How to understand a minor chord using the harmonic series?

Question

I play bass guitar and I have often wondered why a bass note would synchronise well with both major and minor chords of say a piano when we consider the harmonic series.

For instance, the C Major is C, E, G and this can be understood from the harmonic series as having ratios of 4:5 (a major third) and 2:3 (a fifth). And the C minor is C, E♭, G with ratios of 5:6 (a minor third) and 2:3 (a fifth).

My questions are 1) How does C Minor fit in the harmonic series? 2) The C bass note works well with both major and minor. How is it possible when we consider the overtones, frequencies, wavelengths, etc.?

Answer 1

Overview

One approach that people sometimes take is to peer into the harmonic series and search for triads. When we constrain our search to root position triads, here's what we find (in order from the triad we find first to the triad we find last):

1. major triad
2. minor triad
3. augmented triad
4. diminished triad

This is how I'm going to do to answer your question: by looking at the harmonic series and determining how high up the series we must look to finally see each root position triad. There's not a black and white answer that a triad either does or doesn't fit in with the harmonic series. Instead, there are degrees to which a triad fits into the harmonic series. The degree of "synchronization" depends on how high up the triad appears within the harmonic series.

How to Find a Triad in the Harmonic Series (Using Frequency Ratios)

As you know, here are the first few frequency ratios:

• the frequency ratio 1:2 makes an octave (e.g., A4:A5)
• the frequency ratio 2:3 makes a fifth (e.g., A4:E4)
• the frequency ratio 3:4 makes a fourth (e.g., A4:D4)
• the frequency ratio 4:5 makes a major third (e.g., A4:C♯4)
• the frequency ratio 5:6 makes a minor third (e.g., A4:C4)

This interval list isn't in a random order--if we put all of these ratios together into a single string, we get the harmonic series: 1:2:3:4:5:6:7:8, etc. Let's pick C1 as the starting point, and write out the harmonic series. I'll stop the series once we have seen a root position triad. The notes in the C1 harmonic series are C1 C2 G2 C3 E3 G3. We can stop there, because you'll recognize that the last three notes (C3 E3 G3) spell out a root position C major triad.

We could keep going higher and higher up the harmonic series in this same fashion--writing out the next notes and looking for more triads. But the problem is that the notes in the harmonic series don't match up perfectly with the notes on a piano. (This is referred to as inharmonicity, and that mismatch produces "fake" triads that appear to have the correct notes of a triad but don't have the correct frequency ratios.) So instead of simply writing out all of the notes, let's use the frequency ratios to find where the triads are within the harmonic series.

A major chord like C3 E3 G3 consist of two intervals: a major third (from C3 to E3) and then a minor third (from E3 to G3). The interval ratios for a major triad are 4:5:6. Writing the ratios as a single string allows us to see all of the intervals:

• 4:5:6 describes C3:E3:G3
• 4:5 (from C3 to E3) is a major third
• 5:6 (from E3 to G3) is a minor third
• 4:6 (from C3 to G3) simplifies/reduces to 2:3 and thus is a perfect fifth

Now let's find the interval ratios for a root position minor triad. Every minor triad (like E G B) contains a minor third (from E to G) and then a major third (from G to B). This gives an interval ratio of 5:6 (a minor third) followed by 4:5 (a major third). Combining these two ratios into a single string gives 10:12:15.¹

Finally we can see where these triads appear in the harmonic series, because now we know the interval ratios for both a root position major triad and a root position minor triad. Let's look at the harmonic series again:

At this point, we can see that the major triad occurs lower in the harmonic series than the minor triad.

Understanding What a Triad's Position Means

There is a general rule: the higher up the harmonic series a triad appears, the less synchronized the overtones/harmonics will be within that triad. I won't prove this rule, but hopefully it will make some intuitive sense. (Perhaps some insight would be gained by reflecting on the notes in a C harmonic series and comparing those notes to the 4:5:6 → C:E:G major chord and the 10:12:15 → E:G:B minor chord.) Diminished and augmented chords appear even higher up the harmonic series and thus have even less synchronizing of overtones/harmonics than a minor chord. This is partly why diminished and augmented chords sound more "wrong" than a minor chord.

Summary

To summarize: root position major triads and root position minor triads both are found in the harmonic series. However, major triads occur lower in the series and minor triads occur higher in the series. The position of the triad impacts how "synchronized" the triad's overtones, harmonics, frequency ratios, etc. will be. In particular, the higher up a triad occurs within a harmonic series, the less "synchronized" the triad's overtones, harmonics, frequency ratios, etc. will be.

Footnote

¹Here are the intervals for a minor chord like E G B:

• 5:6, a minor third (E:G)
• 4:5, a major third (G:B)

But we can't string these two intervals together yet because the note G has been assigned two different numbers: we're calling it a 6 in the first ratio and a 4 in the second ratio. We need to take the second ratio and change the values so that the first number is a 6 instead of a 4. But we can't change just the first number from a 4 to a 6--we have to multiply the second number by the same scaling factor so that the original ratio is preserved. Here's how we do that:

So a minor chord has the ratios 5 : 6 and 6 : 7.5. Putting these together, we get 5 : 6 : 7.5. But the harmonic series uses only whole numbers, so let's multiply everything by two to remove the decimal: