Fixed additive valve lengths cannot correctly tune all brass notes
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. Five semitones is equivalent to 33.51% (1.05955 = 1.3351), and subtracting the lowering produced by the first valve gives the extra pipe length required from the third valve as 21.26% (33.51 to 12.25%), which is rather more than both the 18.2% available from the combination of the first and second valves and the 18.9% required for an accurate three-semitone lowering. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1258
Categorization
- XREF: connects to: Trumpets need a movable tuning slide because fixed third-valve length is inaccurate — David Howard, Jamie Angus, coustics and Psychoacoustics, 1283; Brass instruments compensate for fixed valve lengths with player-adjustable tuning; Adding valve pipe lengths does not add linearly to frequency drops, forcing compromise tuning (+3 more)
Broader Connections
- Trumpets need a movable tuning slide because fixed third-valve length is inaccurate
#exemplifies(0.92) — David Howard, Jamie Angus, coustics and Psychoacoustics, 1283 - Adding valve pipe lengths does not add linearly to frequency drops, forcing compromise tuning
#expands(0.90) - Trombone slides overcome tuning compromises that valves require
#expands(0.88) - Brass instruments compensate for fixed valve lengths with player-adjustable tuning
#qualifies(0.80) - No tuning system can make all diatonic intervals have simple ratios
#corroborates(0.78) — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1307 - No tuning system can make all diatonic scale intervals perfectly consonant
#corroborates(0.78) — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1306