A Computer Program for Brass Instruments Optimization Part II ...

A Computer Program for Brass Instruments Optimization Part II. Applications, Practical Examples Paul Anglmayer, Wilfried Kausel University of Music and Performing Arts Vienna, Institut für Wiener Klangstil, Singerstrasse 26, A-1010 Wien, Austria Summary: This part of the paper deals with practical experiences with the brass optimization program presented in Part I. Examples have been picked out to illustrate its application to real world problems. Although the user interface of the software is simple and easy to understand optimization results are strongly dependent on well selected starting conditions, a good parameter selection and optimization goals which are not impossible to meet.

GENERAL CONSIDERATION The first step of an optimization is to enter the instrument’s initial geometry. To keep the computing time as short as possible it makes sense to reduce the number of breakpoints as much as possible. In areas where the bore diameter does not change too much distances between outline points do not need to be small. Areas with a considerable opening angle like the bell need a much narrower grid. Because of the difficulties to obtain accurately measured bore data of a real instrument (without destroying it) the idea of using the optimizer to reconstruct the true geometry of an instrument presents itself. Attempts have been made to improve the accuracy of a measurement by optimization with promising results. When coordinates are selected to be released for the optimizer it is essential to first consider carefully if a modification in this place later on can actually be applied to the instrument by the instrument maker and makes sense at all from his point of view. The better the starting position of an optimization the faster one will get the desired result. Available information about a good initial position and about areas where modifications are most promising should be taken into consideration. This can be the treasure of experience of a good instrument maker or the knowledge about sound pressure amplitudes of standing waves in the instrument which can be calculated from the bore list as well (1). Pressure amplitudes of standing waves can be used to determine areas where a modification will mainly effect only those resonance frequencies which should be moved (2). The target function can be obtained in several ways. If there is an intonation problem which should be cured then the required intonation alteration can be determined by measuring the true input impedance of the instrument (3) derive its intonation and compare it with a reference instrument or a certain tuning scale. Alternatively a subjective assessment by the player can be used as the starting point of any modification. Normally it will not be possible to combine all contradicting targets, like perfect intonation, prescribed impedance envelope curve and most restrictive modifications of the instrument’s geometry. Therefore it is necessary to establish priorities.

TABLE 1: Intonation error in Cent of trumpet in Bb before and after the optimization Note (sounding) F4 Bb4 D5 F5 Bb3 before optimization: -2 -3 14 -2 8 after optimization: -1 3 7 -4 2

Bb5 -1 0

D6 -13 -8

Weights can be assigned to indicate the significance of a certain improvement. Usually intonation weights of notes played with rarely used valve combinations or of which the pitch can easily be modulated by the player using his mouth or a special tuning slide (trigger) can be decreased. OPTIMIZATION OF A TRUMPET IN B-FLAT The first experiment has been made using a custom made trumpet in B-flat. This instrument is straight without any bend and has no valve. The goal was the improvement of the measured intonation of the instrument. For practical reasons modifications have been restricted to the insertion of a metal sleeve (length 4 cm, wall thickness 0.5 mm) in the cylindrical part of the trumpet. The simple task was to find the optimum position of the sleeve. Table 1 shows that even this little intervention caused a significant improvement of intonation. OPTIMIZATION OF A NATURAL TRUMPET IN D-FLAT A more difficult application of the optimizer was the improvement of the intonation of a natural trumpet in D-flat built by the company Egger (CH). The goal was to shift the 11th and the 13th resonance peak. Each of those peaks is positioned between two notes, both of which can be played modulating the pitch using the lips. It is not equally easy to modulate the pitch of a tone upwards or downwards which is the reason that the optimum position of the peak is not centered between the notes F#5 and G5 for the 11th and between the notes A5 and Bb5 for the 13th resonance peak (all notes are given as they sound). The proposed position for the 11th resonance peak was 60 Cents above F#5 and for the 13th peak 80 Cents above the A5. The intonation of all other tones should also become better, if possible – at least not significantly worse. The first attempt was a pure intonation optimization. The target positions of all resonance peaks have been entered and no other constraints have been specified. Modifications have been allowed in all cylindrical parts of the instrument. After a while the optimizer showed perfect intonation meaning that all intonation targets have been met. Looking at the input impedance of the modified instrument which is shown in figure 1 it can be recognized that the proposal is not very useful. The shape of the curve is an indication for playing properties which are at least very unusual. In a second attempt the shape of the impedance curve was included in the optimization. It turned out that obtaining an excellent intonation, keeping the shape of the impedance curve the same and restricting the modifications to certain areas and a certain shape was not an easy task to do for the program. It took several attempts and some trial and error until finally a reasonable compromise could be found which was partially implemented in the instrument by the manufacturer.

TABLE 2: Comparison of target, initial and final intonation 11. Peak: Target intonation F#5 +60 Cent Initial intonation F#5 +67 Cent Measured intonation after modif. F#5 +69 Cent

13. Peak: A5 +80 Cent A5 +57 Cent A5 +70 Cent

Although not all proposed modifications have been made – only those which were considered to be significant – a visible improvement of the crucial tones was achieved as can be seen in the measurement results presented in table 2. The complete impedance curve is shown in figure 2. [kOhm]

[kOhm]

[Hz]

[Hz] FIGURE 1: Calculated impedance after the first optimization

FIGURE 2: Measured impedance of the modified trumpet

Especially the 13th resonance peak (A5 resp. Bb5) could be improved by as much as 13 Cent while all other tones stayed basically the same. The modified instrument was also played by experienced musicians who generally attested good playing and sound properties. COMPLETE OPTIMIZATION OF A TRUMPET IN B-FLAT In this example all valve combinations of a standard trumpet have been optimized in parallel. The goal was a perfect intonation of all playable notes. No restrictions have been imposed on the impedance envelope and the complete geometry was released for modification. [Cent]

[Cent]

[Hz] FIGURE 3: Initial intonation of a 3 valve trumpet in B-flat (all playable notes)

[Hz] FIGURE 4: Optimized intonation of a 3 valve trumpet in B-flat (all playable notes)

The initial intonation is shown in Figure 3 while intonation and impedance results are presented in Figures 4 and 5. The shown results have been obtained by a mainly unattended run of the optimizer taking about 24 hours on a 200 MHz Pentium II machine.

Figure 6 shows the geometry of the optimized instrument. Although no restrictions have been imposed on the coordinate parameters the cross-section after optimization seems reasonable. It shows certain peculiarities especially in the bell which are not usually found in modern trumpets. The interesting fact is that similar shapes do exist in some historic instruments. [Ohm]

[Hz]

FIGURE 5: Input impedance of optimized trumpet in B-flat (all valves depressed) [m]

[m]

FIGURE 6: Cross-section of optimized trumpet in B-flat (all valves depressed)

REFERENCES (1) Mapes-Riordan, D., J. Audio Eng. Soc. 41(6), pp 471 – 482,(1993). (2) Smith, R. A., G. J., Nature, 262, pp 761 – 765, (1976). (3)Widholm, G., “Brass Wind Instrument Quality Measured and Evaluated by a new Computer System,“ in Proceedings of the 15th International Congress on Acoustics, Trondheim, Vol. III, pp. 517-520, (1995).