Optimization of valve tube lengths for brass instruments

Abstract

The intonation deviations from the well-tempered scale are presented for valve brass instruments. Compensating and full double instruments are considered.The lengths of the valve tubes are optimized by minimizing the overall root mean squared (rms) intonation error using the method established by Robert Young^{[4]} in 1967. Open note detuning is considered in each case and improves the intonation. The worst three valve instrument without open note detuning has an rms deviation of 24 cents. The best of the three and four valve instruments is a three valve compensating euphonium tuned 3 cents sharp on the open notes and flat on valves 2, 1 and 3 by 3, 3 and 13 cents respectively. Although it lacks the extended range of the four valve instrument, it exhibits only 2 cents rms deviation on single valves and less than 3 cents on valve combinations. The comparisons also include several three and four valve systems including one having a switch valve that descends by 6 rather than 5 semitones. An intonation improvement for the triple waldhorn is presented also.