No pure integer-ratio system can tune musical intervals without compromise.

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 (#expands, via linked claim)

Categorization

• XREF: connects to: Uneven intervals make Just intonation scales impossible to transpose freely.; No tuning system can make all diatonic scale intervals perfectly consonant — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1306; Musical scales universally favor intervals with small integer frequency ratios

Related

• No tuning system can make all diatonic scale intervals perfectly consonant (similarity 0.95)
• No tuning system can make all diatonic intervals have simple ratios (similarity 0.93)
• No musical tuning system can be proven superior to another (similarity 0.91)
• Just intonation leaves many intervals audibly mistuned like the D-A fifth (similarity 0.88)
• Uneven intervals make Just intonation scales impossible to transpose freely. (similarity 0.86)
• No perfect tuning system exists; all musical temperaments are compromised (similarity 0.85)
• Just intonation rested on ideological mysticism about small integer ratios (similarity 0.84)
• Simple integer frequency ratios between notes produce the greatest measured consonance (similarity 0.83)
• Simple integer frequency ratios produce local consonance maxima between tones (similarity 0.82)
• Pythagorean tuning generates scales purely from whole-number ratios of the fifth (similarity 0.82)
• Pure ratio intervals on three chords generate the entire just diatonic scale (similarity 0.82)
• Musical scales universally favor intervals with small integer frequency ratios (similarity 0.82)
• Simple integer frequency ratios minimize partial interference and thus dissonance (similarity 0.81)
• Small-integer ratios cannot govern all musical intervals consistently (similarity 0.81)
• Pythagoras’ simple-ratio tuning system proved ultimately unsatisfactory. (similarity 0.80)

Broader Connections

• No tuning system can make all diatonic scale intervals perfectly consonant #expands (0.95) — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1306
• Musical scales universally favor intervals with small integer frequency ratios #complicates (0.85)