Pythagorean tuning is somewhat different from what’s described.
In Pythagorean tuning, your E would be 81/64 above C, or equal to four fifths minus two octaves. This is slightly higher than E in the article, and the difference (81/80) is called the syntonic comma.
Different tuning systems were invented in order to resolve this discrepancy, and without these advances, jazz would be radically different. One of the things about jazz is that you see distant movements that only really make sense as enharmonics—like how Coltrane’s “Countdown” uses the familiar ii-V-I, but modulates in major thirds, which only makes sense when you allow the final modulation te be the same as the first—something that only works enharmonically.
I'm the guy who wrote the blog post, and I want to offer some clarity. Usually when people say "Pythagorean" tuning they are talking about three-limit just intonation, not five-limit. In three-limit, the major third is indeed tuned to 81/64, which sounds pretty terrible. That might explain why medieval people thought thirds were a dissonant interval. Five-limit just intonation tunes major thirds to 5/4.
Cool article, by the way! When I sang in choir in undergrad, we were exposed to these subjects. Since I was buried in jazz music, your blog was a cool primer on tuning choices.
For sure, the broad similarity I’m trying to touch on is the focus on mathematical resonance and context of a key. Equal temperament removed a lot of that context, but definitely opened the door for further harmonic experimentation. Giant Steps is also a good example of what you’re talking about too.
In Pythagorean tuning, your E would be 81/64 above C, or equal to four fifths minus two octaves. This is slightly higher than E in the article, and the difference (81/80) is called the syntonic comma.
Different tuning systems were invented in order to resolve this discrepancy, and without these advances, jazz would be radically different. One of the things about jazz is that you see distant movements that only really make sense as enharmonics—like how Coltrane’s “Countdown” uses the familiar ii-V-I, but modulates in major thirds, which only makes sense when you allow the final modulation te be the same as the first—something that only works enharmonically.