The Question

Can a musical system be perfectly in tune in all keys?

Equal temperament sacrifices purity for flexibility. Just intonation sacrifices flexibility for purity.

Why It Resists Closure

You cannot optimize both simultaneously.

The twelfth root of 2 is irrational. Pure intervals are rational. These two worlds cannot be reconciled — only approximated.

Every tuning system is a compromise. The “perfect” piano does not exist.

DomainAcoustics
TypeStructural tradeoff
StatusMathematically constrained

What do you think?