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> Pick any three musical notes, even on a microtonal scale, and they form a pleasing chord.

I cannot speak to the color part, but the music part doesn't ring true to me (pun intended). For example, if I pick the first, the second and the seventh of the Phyrigian scale, it sounds pretty jarring to me (like, playing E, F, and D at the same time). And, this is staying pretty much within the scales in western classical music (although I doubt anybody would play it, I bet somebody made it context). Also, extended chords with > 3 notes sound pretty fun and has been the staple of western music for a while (blues, jazz for easy-to-reach examples).

D-E-F and its relatives are used everywhere. Examples

Pink Floyd Comfortably Numb First chord of verse

Eye of the Tiger (Survivor) First chord of verse (“Rising up” lyric)

You can say it’s a different bass note, and so it is. There is something about the chocolate-dill-cayenne example as well that if the harmony was 90% cayenne it probably wouldn’t work. Dill, it would work a bit better. Chocolate, it would work the best.

Same with the foyer. Dominant brass might be tougher than dominant evergreen or mint.

If you play those notes in a close voicing they can be jarring (although I went through a phase of composing with close clusters of tones just like that in my 20s), but if you instead play D, F up a minor third and then put the E a major seventh above the F, you end up with a chord that functions like Dmin9 (or more accurately DminAdd9). The missing fifth gets added psychoacoustically (jazz pianists have taken advantage of this for ages by playing the 3-7 of a chord in their left hand and the melody in the right hand, letting the bass cover the root and the brain of the listener add the 5).

Consistently applying a design language is probably more important than the design language (e.g. your rule of threes) itself.

Except for exceptions to your design rule which will form the finishing touch. :)

Another nice design palette rule that works well is black or dark gray, white or light gray, one nature color (wood, rock, ...), and one vivid color. (That's a set of 4.)

I agree globally. I'll keep my remarks to myself they're not crucial, but here's another example I've put to test many times and you can try every day:

Never wear outfits with more than 3 colors.

I have simplified my garderobe tremendously by choosing garments mostly of neutral colors and some of brighter colors.

By limiting to simple fabrics, I can match almost anything with anything else from my drawers and still have nice results.

It works incredibly well.

The rule of three is not about juxtaposing any three random elements, it's about taking three things which are related to each other. It's not any three things producing harmony, it's a human selecting three things which produce some effect. This is true for rhetoric because the rhetorician has a purpose he wants to achieve.

And another nitpick: It seems perfectly possible to pick three colors which don't work well together. If two of the colors clash with each other, it won't matter what the third one is. With a chord, I suppose (without confidence) that it works because you're hearing all three notes together, but that's not how colors work: if you see the two clashing colors side by side, it's not visually pleasing, even if the third color works with each of those clashing colors individually.

> With any three things, you have harmony to humans.

In math, specifically dynamical systems, a lot of interesting stuff starts appearing after some variable goes up from 2 to 3, such as Period 3 Implies Chaos[1] for discrete maps, and the chaos in the 3 bodies problem. In continuous time systems, having 3 variable is the necessary condition for chaos to occur. My personal guess and take away is that having 3 of something is the bare minimum for nontrivial interactions to happen.

[1]https://sci-hub.se/https://doi.org/10.2307/2318254

> Any three simple flavors (chocolate, dill, and cayenne?) may be weird but interesting but not overwhelming.

I take it you don't like Indian food then. :)