Where Is Register Key Clarinet

JPSJ News Comments 11, 17 (2014) [one Pages]

The Mechanism of the Register Key of the Clarinet

+ Affiliations

Department of Mathematical Sciences, Osaka Prefecture University

Published December 1, 2014

The clarinet has a small but elaborate device, the annals central, which gives the clarinet its large pitch range. The physical mechanism of this small device has remained an open question. A differential and difference equation model provides insight into how the annals key tin can produce specific higher harmonics of tones inside a certain range.

The annals central is a small-scale pigsty with an open/shut mechanism set well-nigh 14 cm from the tip of the mouthpiece on the clarinet (Fig. ane). When pressed to open the key, the pitch of the notation is raised by a 12th (19 semitones). Clarinetists tin can play more than than an octave with this elaborate device. Other woodwinds like the saxophone and oboe accept a similar (only non the same) device chosen an octave fundamental, which extends the range of the tone. The register or octave key not simply expands the potential of the woodwind, it too poses some questions: By what mechanism does a small hole excite the third harmonic for all tones inside a given frequency range? What is the range covered by this device? Why were the particular size and position of the pigsty selected? Are other college harmonics excited? How this small hole works with respect to these questions has remained unanswered.

Fig. 1. The clarinet and the register cardinal (hole) [courtesy of Professor Kin'ya Takahashi (author of Ref. one)].

Takahashi et al. [1] focused on finding some answers to these questions. They used a differential and difference equation model of the clarinet. The oscillation in wind instruments is well modeled by this blazon of equation [two]. However, from a mathematical viewpoint their assay is difficult considering the model equation is intrinsically a multi attractor system, i.e., even with the same parameter sets, the behavior of the arrangement depends on the initial condition, which should exist a function of all (or role) of the past states of the organisation.

The Takahashi et al. model consists of ii filibuster terms and a dumping term. The equation was suggested by Ikeda and Mizuno for the optical resonator [3] to clarify the complex behavior of a nonlinear optical arrangement. The ii systems resemble each other in that the competition betwixt the two delay terms causes a kind of frustration in selecting an oscillation way. Analysis of the model shows that even harmonics suffer from negative delayed feedback and odd college harmonics, including the tertiary harmonic, are relatively emphasized. The register central creates conditions in which only the 3rd higher harmonic becomes dominant. Afterwards the mathematical analysis, the correspondence betwixt the mathematical model and the actual musical instrument was discussed, where they considered the geometry of the musical instrument and the acoustic characteristics were adamant by the reflection function. Finally, they discussed why but the 3rd higher harmonic is excited in the clarinet.

The results of this commodity can be applied to other woodwinds and provide insight into the machinery of mode option in multiple delayed systems.

References

[1] K. Takahashi, Thou. Goya, and S. Goya, J. Phys. Soc. Jpn. 83, 124003 (2014). x.7566/JPSJ.83.124003 Link, Google Scholar
[2] N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer, New York, 1998) 2d ed. Crossref, Google Scholar
[3] K. Ikeda and M. Mizuno, Phys. Rev. Lett. 53, 1340 (1984) x.1103/PhysRevLett.53.1340; Crossref;, Google ScholarM. Mizuno and K. Ikeda, Physica D 36, 327 (1989). ten.1016/0167-2789(89)90088-2 Crossref, Google Scholar

Writer Biographies

Virtually the Author: Tsuyoshi Mizuguchi

Tsuyoshi Mizuguchi received his Ph.D. in physics from Kyoto University in 1993 and after spent 2 years as a postdoctoral young man in Kyoto and Sendai. In 1995, he became an Banana Professor at Kyoto University. In 2001, he moved to Osaka Prefecture University as a lecturer and is now an associate professor. His research interest is nonlinear science, including pattern germination and commonage behavior of biological individuals.