Defining harmonic relatedness

Linear music notation is good for reading, but it doesn’t tell you everything you want to know about underlying musical structure. Notes that are close to each vertically are not necessarily the most closely related. The concept of harmonic relatedness is a complex one, but there’s an excellent tool for beginning to get a handle on it: the circle of fifths.

The left circle above shows the chromatic circle, the pitch sequence you find on the piano. The right circle shows the circle of fifths. Each note is a fifth higher or a fourth lower than its counterclockwise neighbor, and each note is also a fourth higher or a fifth lower than its clockwise neighbor.

Unlike a lot of music theory you learn in school, the circle of fifths is not some arbitrary Western European cultural convention. There’s actual science behind it. If two notes are adjacent on the circle of fifths, it means they have a lot of overtones in common. If you know what overtones are, you can skip the next few paragraphs. Otherwise, read on.

The scientific term for a musical pitch is a frequency. The frequency is the rate at which the guitar string or speaker cone or whatever is vibrating in order to produce sound. But musical instruments don’t vibrate at just one frequency at a time. Usually, the vibration consists of many overlapping frequencies, all at different amplitudes. The loudest and lowest frequency, the fundamental, is the note you think you’re playing. The other notes are the overtones, and they’re all quieter and higher-pitched than the fundamental.

The overtones of a musical sound have a specific mathematical relationship: they’re all simple integer multiples of the fundamental. If a guitar string is playing concert A, a frequency of 440 Hz, then its overtones are at 880 Hz (440 * 2), 1,320 Hz (440 * 3), 1,760 Hz (440 * 4), and so on. If you can isolate the overtones, you discover that they are all musical pitches in their own right. The overtone series of A consists of A an octave higher, E above that, another A above that, C-sharp above that, yet another A, and so on up to inaudibility. You can have overtones that aren’t integer multiples of the fundamental too. Those are called inharmonics. Pitched instruments are designed to minimize inharmonics, but you hear them in bells, cymbals, and noisy synth sounds.

No one knows exactly why we like the sounds we do. But just about every world culture seems to like octaves and fifths, and that is probably due to the overtone series. Two notes an octave apart share so many overtones that we hear them as being “the same” note. Take a look at the spectrogram of two notes played on the violin: C on the left, and G on the right. The dotted lines show their shared harmonics.

In general, the closer two notes are to each other on the circle of fifths, the more their respective overtone series will overlap. Shared overtone series make our brains light up with recognition. Here in the Western world, we experience that overlap as consonance. Note that consonance is not quite the same thing as “sounding good,” since dissonance sounds good in many situations. Think of consonance as being like sweetness. Everybody likes sweetness, and little kids always prefer it, but adults sometimes prefer bitterness or sourness.

So is that it? Can we just say that notes are more consonant if they’re closer to each other on the circle of fifths, and more dissonant if they’re further away? I’d be inclined to say yes, but that raises some interesting questions. By the circle of fifths measure, the intervals go from most consonant to most dissonant in this order:

• fourths and fifths
• major seconds and minor sevenths
• minor thirds and major sixths
• major thirds and minor sixths
• minor seconds and major sevenths
• tritones

If you know your Western classical theory, you might be scratching your head right now. Major seconds and minor sevenths are more consonant than major thirds? Maybe so! Maybe the nineteenth century Europeans are just wrong on this one. Or maybe the circle of fifths standard is too simplistic. The major third is special because it is itself low in the overtone series. If you play a C and an E together, the E is going to light up your pattern-recognition sense by reinforcing the fifth harmonic of C.

The issue of the major third aside, the circle of fifths closeness standard explains a lot. The major scale and all of its modes are contiguous on the circle of fifths. So are all of the pentatonic scales. If you space notes as far from each other on the circle as you can, you get such weird-sounding things as augmented and diminished chords. And certainly, when you’re talking about key centers, the circle of fifths standard maps very neatly onto our intuitive feeling of relatedness.

While the circle of fifths thing is a useful heuristic, not every culture defines consonance and dissonance the way ours does. Not every culture even uses our tuning system, much less our harmony rules. And even within our culture, we have the blues, which sounds good in spite of breaking all of the Western harmony rules. Cultural context always trumps whatever formal system of music theory we can devise. But at least the circle of fifths points us in the right direction.

Here are the scales we’re using in the aQWERTYon, grouped by how contiguous they are on the circle of fifths. Organizing them this way agrees with my intuitive sense of how I’d group them from “least weird” to “most weird” for the most part.

No gaps

• Major
• Dorian
• Phrygian
• Lydian
• Mixolydian
• Natural minor
• Locrian
• Major pentatonic
• Minor pentatonic
• Supermode
• Chromatic

Two gaps

• Blues
• Melodic minor
• Lydian dominant
• Altered

Three gaps

• Harmonic minor
• Phrygian dominant
• Harmonic major

Four gaps

• Diminished
• Octatonic

Six gaps

• Whole tone

What do you think? Is harmonic minor really that much “weirder” than Locrian? Is grouping scales this way a worthwhile exercise? Let me know in the comments.