The Harmonic Seventh Chord

Traditional western music theory embraces chords with frequency ratios that are products of powers of 2, 3, and 5… but not higher primes. Barbershop quartets go further!

These quartets love four-part harmony using dominant seventh chords. A ‘dominant seventh chord’ contains a note, its major third, its perfect fifth, and its minor seventh. In traditional harmony these have frequencies proportional to 4, 5, 6, and … not 7, because that would be against the rules, but instead some number close to it. Barbershop quartets break the rules and use 7.

Gage Averill writes:

Society arrangers believe that a song should contain anywhere from 35 to 60 percent dominant seventh chords to sound ‘barbershop’ — and when they do, barbershoppers speak of being in ‘seventh heaven.’

Art Merill writes:

There’s a chord in a barbershop that makes the nerve ends tingle….We might call our chord a Super-Seventh! … The notes of our chord have the exact frequency ratios 4–5–6–7. With these ratios, overtones reinforce overtones. There’s a minimum of dissonance and a distinctive ringing sound. How can you detect this chord? It’s easy. You can’t mistake it, for the signs are clear; the overtones will ring in your ears; you’ll experience a spinal shiver; bumps will stand out on your arms; you’ll rise a trifle in your seat.

The frequency ratio

is called the harmonic seventh. In Pythagorean tuning, where we only get to use the primes 2 and 3, the closest simple approximation is the Pythagorean minor seventh:

which is sharp by a factor of

This number is called the septimal diesis.

Of course we can also use the Pythagorean minor seventh in 5-limit tuning (often called just intonation), where we get to use the primes 2, 3 and 5. Then it’s called the small just minor seventh. But in 5-limit tuning we also have a better simple approximation to 7/4 = 1.75, called the large just minor seventh:

This is sharp by a factor of

This number is called the septimal comma or Archytas’ comma.

By the way, it’s essentially just a coincidence that the minor seventh sounds good with a frequency ratio involving the number 7. It’s called a minor seventh because it’s the seventh note in the minor scale! In just intonation, the interval called a major third involves the number 5 (it’s 5/4), while the interval called a perfect fifth involves the number 3 (it’s 3/2). So, be careful!

For more, watch this:

For much more about 7-limit tuning, which only uses the primes 2, 3, 5, and 7, read this:

• Kyle Gann, The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician, University of Illinois Press, 2019.

For a bit less, try this:

• Wikipedia, 7-limit tuning.