It's also a really strong example of "missing the forest for the trees".
Deriving Western music traditions from first principles does not help you to understand why, what, and how any of it actually works.
Deriving Western music traditions from first principles does not help you to understand why, what, and how any of it actually works.
However, first principles explain “why” a major chord sounds the way it does, and “why” a minor chord has more tension.
Let’s look at the five-limit just intervals: 2/1, 3/2, 4/3, 5/4, and 6/5. 2/1 is a perfect interval; two notes an octave apart sound nearly the same. 3/2 and 4/3 are also perfect intervals, but there’s a little more tension and beating than with 2/1. 5/4 has a little more tension than 4/3, and 6/5 has even more tension. I just explained, using first principles, the octave, perfect fifth, perfect fourth, major third, and minor third. That’s most of the scale right there.
Now, let’s make a chord. Experience has shown that three notes is more interesting to the human ear than just two notes. That in mind, we want three distinct notes where the intervals between notes have a minimum of tension. Take the root, take 5/4 of that, then take 3/2 of the root. That’s a major chord. The reason we use 5/4 instead of 4/3 is because the interval between 5/4 and 3/2 is 6/5; the interval between 4/3 and 3/2 is 9/8. Since 9/8 has more tension than 6/5, that gives us a suspended chord, and it’s better to use a major chord with less tension to make a simple chord progression.
Now that we have our first chord, we can hear that having a 1-chord song becomes repetitive very quickly to the human ear. We can make the song more dynamic and interesting by giving it three chords: I, IV, and V. Take that chord: 1, 5/4, and 3/2 (a major chord). Move all three notes up 4/3, because 4/3 is a perfect interval with a minimum of tension. That’s our IV chord (F chord in the key of C major) Next, from the root position, instead of moving up the chord 4/3, we instead move it up 3/2. That’s a V (or dominant) chord—a G chord when played in the key of C major.
Now, from first principles, namely that the intervals 3/2, 4/3, and 5/4 sound pleasant to the human ear, that 6/5 has more tension, and 9/8 has even more tension (so we avoid suspended chords for now), we have a I IV V chord progression, which is the basis for countless rock and popular songs.
If we take all of the individual notes from the I IV V chord progression, that gives us the seven notes of the major scale, after explaining that, once we cross the octave threshold, we can transpose an octave down without affecting the musicality of the notes. We might even explain chord inversions here.
So instead of just parroting “I IV V makes a nice simple chord progression”, we now know the underlying physics which make I IV V sound so nice to the human ear, and how to get the major scale from those chords. I prefer understanding both the why and the how.
(I would have used two slightly detuned sawtooth waves instead of a sine wave for most of the examples if I were to write an article like the one this discussion links to, explaining this has a more pleasant, piano or violin like sound to it. Maybe even a low pass filter to tame the harmonics. Another option is to run a single sawtooth wave through a Solina style chorus, but explaining two detuned sawtooth waves is easier than trying to explain the Solina chorus effect)
Music theory is useful because it establishes a vocabulary.
Other ancient tuning systems are based on three limit tuning:
https://en.wikipedia.org/wiki/Shi%27er_l%C3%BC
https://en.wikipedia.org/wiki/Pythagorean_tuning
Whether the human ear naturally prefers 5/4 (5-limit) over 81/64 (Pythagorean) for the major third, and 6/5 over 32/27 for the minor third, is still something I don’t have solid research on.
As it turns out, the modern western scale is more closely aligned with Pythagorean tuning than with five-limit tuning. In particular, the Pythagorean thirds are closer to modern 12-tone equal tempered thirds than they are to five limit thirds.
https://www.medieval.org/emfaq/harmony/pyth5.html
>>>Around 1300, the English theorist Walter Odington proposes that major and minor thirds (81:64, 32:27) have ratios close to 5:4 and 6:5 respectively, and that singers lean toward these simpler ratios. His observation may reflect the style of much English polyphony of the period, where thirds often have a pervasive role in the texture and may even serve as closing sonorities.<<<
https://casfaculty.case.edu/ross-duffin/just-intonation-in-r...
>>>Beginning with Ramos de Pareja in 1482,Fn11 the simple ratios of Just intonation began to be cited by theorists and used in their divisions of the monochord. [...] an octave is 2:1, a fifth is 3:2, a fourth is 4:3, a pure major third is 5:4, a pure minor third is 6:5. These “six-limit” ratios—the senarius, or senario, as Zarlino calls them—are the basic harmonic intervals. Zarlino attributed almost mystical significance to the senario<<<
https://en.wikipedia.org/wiki/Gioseffo_Zarlino
https://journals.sagepub.com/doi/10.1177/000313131706700215?...
This article is paywalled, but to summarize, it makes the case that the 5:4 and 6:5 ratios are ratios musicians naturally drift towards in polyphonic contexts.
It reminds me a bit of regular polygons or simple geometry like circles- and it makes sense the Greeks would be interested in both- things that are pleasant and can combine and tile well because of their symmetric and regular properties.
I think my "pushback" was more just at the phrasing of "why" something sounds "good"- I do think when we listen to music, we bring our past and our culture into it, and whether a chord sounds "sad" or "happy" depends on a lot more than just the harmonic structure- I wonder about this concept of "tension" you bring up, I can't tell if you mean something emotional or more mechanical. Is that like something you understand just by crunching the numbers on the ratios?