No tuning system can make all diatonic intervals have simple ratios

Evidence

We can thus conclude that no single method exists that tunes the seven notes of the diatonic scale so that all the intervals between the notes have intervals with simple integer ratios. Some methods have been developed to overcome this problem. For instance, the mean-tone temperament alters the Pythagorean tuning so that the fifths are tuned a bit narrower, which overcomes some problems of the system. — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1307

Categorization

• QUESTION: Worth chasing: how does equal temperament’s slightly mistuned major third actually affect perceived consonance versus just intonation in practice?
• CANDY: The fact that no tuning system can make every interval perfectly in tune is a nice pub-fact about music.
• XREF: connects to: Fixed additive valve lengths cannot correctly tune all brass notes — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1258