Adding valve pipe lengths does not add linearly to frequency drops, forcing compromise tuning

Evidence

The decrease in frequency required to lower a note by one semitone is therefore 5.95%, and this is also the factor by which a pipe should be lengthened by the second valve on a brass instrument. Depressing the first valve only should lower the f0 and hence lengthen the pipe by 12.25%, since the frequency ratio for two semitones is the square of that for one semitone (1.05952 = 1.1225). Depressing the first and second valves together will lengthen the pipe by 18.2% (12.25% + 5.95%), which is not sufficient for three semitones, since this requires the pipe to be lengthened by 18.9% (1.05953 = 1.1893). The player must lip notes using this valve combination down in pitch. The third valve is also set nominally to lower the f0 by three semitones, but, because of the requirement to add a larger length the farther down that is progressed, it is set to operate with the first valve to produce an accurate lowering of five semitones. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1258

Categorization

Category: Causal Relationships

• XREF: Connects to the physics of the harmonic series and equal temperament tuning — the 1.05946 ratio per semitone appears throughout acoustics
• SHARE: A trumpeter friend would appreciate why the 1-2 combination is always slightly flat and needs lipping

Related

• Brass instruments compensate for fixed valve lengths with player-adjustable tuning (similarity 0.84)
• Trumpets need a movable tuning slide because fixed third-valve length is inaccurate (similarity 0.81)
• Trombone slides overcome tuning compromises that valves require (similarity 0.81)

Broader Connections

• Local bore changes tune instrument modes by their position relative to pressure nodes #expands (0.80) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1225