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above which reflective behavior changes to non-reflective behavior is called the “cutoff frequency.” Benade soon pub- lished some mouthpiece and output trumpet spectra and a
13 roughhigh-pass-filter-shapedtransmissionresponse .Backat
the University of Illinois, I made more swept sine wave trans-
mission response measurements, this time again on a trom-
bone. The resulting transmission curve minima lined up well
with values taken from the filter response previously calculat-
ed from the trombone tone spectral analysis. But while agree-
ment was good for frequencies below cutoff, it was not for fre-
quencies above cutoff. Again, the nonlinear effect seemed
obvious. However, I had doubts about my performance-condi-
tion measurements because (a) computing the ratio with weak
mouthpiece harmonics above cutoff is tricky, and (b) some
prominent musical acoustics researchers (e.g., Backus and
Hundley and Elliott et al.) had predicted that significant non- 14-15
linearity was unlikely . Nevertheless, in 1995 Hirschberg
and colleagues finally verified the nonlinearity effect—due to
16
shock waves . A thorough predictive analysis was later done
17
by Thompson and Strong . It turns out that high-amplitude
traveling waves “steepen” as they move down the pipe, thus increasing the upper harmonic content. Synthesis methods based on this idea have been devised by Vergez and Rodet,
18-19
Msallam et al., and others at IRCAM in Paris . In the mean-
time, Andrew Horner and I devised a filter-like multiple-
wavetable brass synthesis model based on a variable spectral
20
envelope . The increased speed and memory of computers
that transpired between the 1960s and the 1990s had made a strictly linear source-filter model, which proved to be incor- rect, unnecessary.AT
References for further reading:
1. Daniel W. Martin, “Lip vibrations in a cornet mouthpiece,” J. Acoust. Soc. Am. 13, 305-308 (1942).
2. John Backus and T.C. Hundley, “Harmonic generation in the trumpet,” J. Acoust. Soc. Am. 49, 509-519 (1971).
3. Shigeru Yoshikawa, “Acoustical behavior of brass player’s lips,” J. Acoust. Soc. Am. 97, 1929-1939 (1995).
4. Seiji Adachi and Masa-aki Sato, “Trumpet sound simulation using a two-dimensional lip vibration model,” J. Acoust. Soc. Am. 99, 1200-1209 (1996).
5. David C. Copley and William J. Strong, “A stroboscopic study of lip vibrations in a trombone,” J. Acoust. Soc. Am. 99, 1219- 1226 (1996).
6. Mark T. McLaughlin, Lowell J. Eliason, and R. Dean Ayers, “Inexpensive apparatus for studies of the lip reed,” J. Acoust. Soc. Am. 116 (4, Pt. 2), 2593 (2004).
7. David Luce and Melville Clark, Jr., “Physical correlates of brass-instrument tones,” J. Acoust. Soc. Am. 42, 1232-1243 (1967).
8. Daniel W. Martin, “A physical investigation of the perform- ance of brass wind instruments,” Ph.D. dissertation, Dept. of Physics, Univ. of Illinois, Urbana, IL (1941).
9. James W. Beauchamp, “Nonlinear characteristics of brass tones,” J. Acoust. Soc. Am. 46, 98 (1969).
10. James W. Beauchamp, “Analysis of simultaneous mouthpiece and output waveforms of wind instruments,” Audio Engineering Society Preprint No. 1626 (1980).
11. James W. Beauchamp, “Wind instrument transfer responses,” J. Acoust. Soc. Am. 83, S120 (1988).
12. James W. Beauchamp, “Inference of nonlinear effects from spectral measurements of wind instrument sounds,” J. Acoust. Soc. Am. 99, 2455 (1996).
13. A.H. Benade, “Physics of brasses,” Scientific American, July, 24-35 (1973).
14. John Backus and T.C. Hundley, “Trumpet air-column over- loading,” J. Acoust. Soc. Am. 47, 131 (1970).
15. Stephen Elliott, John Bowsher, and Peter Watkinson, “Input and transfer response of brass wind instruments,” J. Acoust. Soc. Am. 72, 1747-1760 (1982)
16. A. Hirschberg, J. Gilbert, R. Msallam, and A.P.J. Wijnands, “Shock waves in trombones,” J. Acoust. Soc. Am. 99, 1754- 1758 (1996).
17. Michael W. Thompson and William J. Strong, “Inclusion of wave steepening in a frequency-domain model of trombone sound production,” J. Acoust. Soc. Am 110, 556-562 (2001).
18. Christophe Vergez and Xavier Rodet, “New algorithm for nonlinear propagation of a sound wave, application to a phys- ical model of a trumpet,” J. Signal Processing 4, 79-87 (2000).
19. R. Msallam, S. Dequidt, R. Caussé, and S. Tassart, “Physical model of the trombone including nonlinear effects. Application to the sound synthesis of loud tones,” Acustica 86, 725–736 (2000).
20. James W. Beauchamp and Andrew Horner, “Wavetable inter- polation synthesis based on time-variant spectral analysis of musical sounds,” Audio Engineering Society Preprint 3960 (1995).
James Beauchamp received bache-
lors and masters degrees from the
University of Michigan in 1960
and 1961 and a Ph.D. from the
University of Illinois at Urbana-
Champaign (UIUC) in 1965, all in
electrical engineering. He joined
the faculty of the UIUC
Department of Electrical and
Computer Engineering in 1965,
and in 1969 took a joint appoint-
ment with UIUC's Dept. of ECE
and its School of Music. From 1965 until he retired in 1997 he taught courses in electronics, acoustics, audio, electronic music, and computer music in both departments. In the 1960s and 1970s he worked on the design of analog and hybrid synthesizers, but in the mid-1960s he began research on computer analysis and synthesis of musical sounds which led to the design of digital synthesis models. In his most recent efforts, as an emeritus professor at the UIUC, he has focused on perception of musical timbre and on automatic transcription and voice separation of polyphonic music. He is a fellow of the Audio Engineering Society (1981) and the Acoustical Society of America (1999) and is currently chair of the ASA's Technical Committee on Musical Acoustics. Further information can be found at http:// ems.music.uiuc.edu/beaucham/.
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