No tuning system can make all diatonic scale intervals perfectly consonant

Evidence

Just intonation is based purely on triads, where the frequencies of the triads C-E-G, G-B-D, and F-A-C have the ratio 4:5:6 and octaves are always tuned to 2:1. Unfortunately, with this method, many intervals are not in tune. For example, the interval D-A should have the ratio 1.5, but in this tuning it is 1.48, which is audibly mistuned. The Pythagorean tuning system has the greatest number of pure fifths and octaves. Unfortunately, the thirds are then mistuned. … 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…, 1306

Categorization

• XREF: The irreducible tuning tradeoff parallels the mathematical impossibility results in other domains—no perfect compromise satisfying all constraints simultaneously, like Arrow-style impossibility theorems. — connects to: No evidence supports preferring 432 Hz piano tuning over 440 Hz — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 2054; No pure integer-ratio system can tune musical intervals without compromise.; No musical tuning system can be proven superior to another (+6 more)
• corroborated: 2 independent passages support this claim — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1306

Broader Connections

• No keyboard tuning can make every key sound justly intonated #qualifies (0.95)
• Equal temperament allows all keys but in-tunes no interval #corroborates (0.95)
• Only the octave stays perfectly in-tune in equal temperament #exemplifies (0.75)
• Twelve stacked perfect fifths fail to close the octave exactly #exemplifies (0.75)