Triads

One chord you do not have to argue for

Foundations established that a single note is a stack of partials at whole-number multiples of its fundamental. Take a low C at 65.41 Hz and count upward. The fourth partial is 261.62 Hz, the fifth is 327.03, the sixth is 392.44.

Those three frequencies are a major triad. Not approximately, and not by convention. They are three consecutive members of a series that every vibrating string produces on its own, and they are the same numbers Foundations lesson 5 used for the just major triad when it compared tunings.

Which makes major the easy case

A major triad is the ratio 4 : 5 : 6. Its lower interval is 386.3 cents and its upper is 315.6, and both of those are simple ratios: 5:4 and 6:5. The chord is low in the harmonic series, made of small whole numbers, and physically present in every note you have ever heard.

That is about as strong an argument as this subject offers, and it is worth being clear that it is the last one of its kind. Everything from lesson 2 onwards is structure rather than acoustics.

And minor the awkward one

The minor triad is 10 : 12 : 15. Those are not small numbers, and they do not sit next to each other anywhere in the series. Reverse the two intervals of a major triad, so the minor third is underneath instead of on top, and that is what you get: 315.6 cents then 386.3, exactly the major triad's intervals in the opposite order.

So minor is a rearrangement. Calling it an inversion is wrong, and calling it the mirror image of major is worse, though you will meet people who say so.

The mirror story is worth killing properly, because it is elegant and it is wrong. It says that if major comes from the overtone series, minor comes from an undertone series running downward from a note. Riemann built a whole theory on it in the nineteenth century. The problem is that no undertone series exists. A string vibrates in halves, thirds and quarters of its length, which produces frequencies going up; nothing about it produces frequencies going down. The symmetry is real as arithmetic and has no physical referent, which is a good description of quite a lot of music theory and a reason to check.

Scoring the family

Foundations lesson 3 built a model of sensory roughness and used it to show why some intervals sound smooth: partials that nearly coincide beat against each other, and the model counts that. It was never told anything about music.

It works on three notes as easily as two. The figure below runs it over the five common triads, and the first thing to notice is that it gets the obvious part right without being asked.

Where it stops working, and why that matters

Diminished scores well above major and minor, which is what anyone would report. Major scores slightly below minor, which is also about right. Switch the tuning to just intonation and every chord that has one gets smoother, which is Foundations lesson 5's claim about what tempering costs, showing up again as a number.

Now look at the augmented triad. In equal temperament the model puts it at the bottom of the list, smoother than major, and it does the same with a sine spectrum and with a square one.

Part of that is an artefact, and switching the tuning exposes it. Move to just intonation and the major triad drops to 0.287 while the augmented one stays at 0.295, because it has no just version to improve into, and the ordering flips. So some of its apparent smoothness was never its own: it was equal temperament roughening everything it was being compared against.

But only some. Even after that correction it sits level with minor and below sus4, and anybody who has heard an augmented triad knows it is the least restful chord on the list by a wide margin. So either the model is broken, or it is answering a different question from the one we are asking. It is the second.

Roughness measures beating between nearby partials. An augmented triad is three stacked major thirds, and a major third is a reasonably smooth interval, so there is not much beating to find. What makes the chord unsettling is not friction at all. It is that three equal intervals divide the octave symmetrically, so no note in it is more fundamental than any other: it has no root. Play it and your ear cannot decide what key it is in, because it is equally at home in several and settled in none.

That is a structural fact about intervals and symmetry, not a psychoacoustic one, and no amount of partial-counting will produce it.

One arithmetic curiosity, for the same reason

The augmented triad has no just-intonation version, and the reason is a small piece of arithmetic in the same family as the Pythagorean comma from Foundations lesson 4.

Stack three pure major thirds: (5/4)³ = 1.9531. An octave is 2. The three thirds fall short by 41.1 cents, which is nearly half a semitone and enormously audible. So an augmented triad that closes the octave cannot be built from pure thirds. It exists because equal temperament made every major third 13.7 cents sharp, and three of those errors add up to precisely the gap.

It is a chord that only exists because of a compromise, which is a fair warning about what the rest of this course is going to be like.