But it's impossible to explain without getting into like, what is even the problem solved by tuning systems. Without the intuition that comes from making music, programmers and engineers see the fractions & obvious series and get too fixated on finding the "perfect" system. When these are much more physical tools, created over time to make certain processes easier. Tuning systems are more like a woodworker's knives than like the unit circle: being perfect does not make them better tools for creation if they are already fit enough.
^^ from the article. this kind of thing bothers me. what is the devil? for whom is it the devil? music doesnt havent to be "christian" to "sound good".
It was nearly or actually impossible to harmonize or progress tritones while following the medieval "rules" for church music composition. So it is someone complaining about their work, not a statement about musical morality or whatever.
Xenharmlib has deep support for intervals, chords, scales, non-standard notations, and advanced topics like non-Western harmonics, diatonic set theory, and non-octave-repeating systems and also allows for the mathematical manipulation of ratios and structures (harmonic exploration).
PyTuning allows for generating scales from ratios/cents, EDO, just intonation, and custom temperaments and facilitates calculations such as frequency ratios, comma approximations, and temperament comparisons, aligning with the article's derivations and trade-off discussions at a similar depth.
I hope to see more math of music articles in the future .. its a fascinating subject indeed!