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During bending, self-excited oscillations of the air flow and the two reeds in the channel occur within a very short time with both reeds oscillating with equal frequency nearly from the very beginning. The essay **Transient response of a half-tone bend on channel #3 of a blues harmonica** can be downloaded here.

While searching for essays on the physics of the blues harp, the first paper I came across was by *Robert B. Johnston*:

Johnston, Robert B. "Pitch control in harmonica playing." Acoustics Australia 15.3 (1987): 69-75.

Johnston makes substantial use of formulas from a paper by* Neville H. Fletcher* from 1979:

Fletcher, Neville H. "Excitation mechanisms in woodwind and brass instruments." Acta Acustica united with Acustica 43.1 (1979): 63-72.

Consequently, Johnston's work can only be "understood" if one has understood Fletcher's essay. Result of my occupation with the work of **Fletcher** is this essay Fletcher 1979, in which first the mathematical derivation of the formulas used by Johnston is calculated in detail (a nice application of elementary analysis). This presentation was inspired by a book by *Robert D. Klauber: Student friendly quantum field theory*, which actually fulfills its promise given in the title.

- Linear acoustics and AC physics are closely related. Fletcher explains self-excited feedback in wind instruments by analogy with electrical feedback circuits using components with negative differential resistance (such as the Gunn diode).
- From today's point of view, Fletcher derives the formulas used by Johnston as a linear approximation for harmonic perturbations. Neglecting one of the terms introduced by Fletcher (which he himself omits in later works), his formulas for the admittance of the mouthpiece are identical to the formulas used today in linear stability analysis, as found in the textbooks by
*Hirschfeld*or*Chaigne/Kergomard*. This is presented in a "student-friendly" way. - Work by
*Dieckman*and*Cottingham*on Asian mouth organs successfully uses an approach going back to*v. Helmholtz*, who considered the admittance of the generator only to be the dynamic response of the oscillating reed. This approach is compared with Fletcher's approximation formulas. - Fletcher's formulas with and without mass term as well as the admittance as dynamic response are applied exemplarily to
*channel #4 of a blues harp in C*in analogy to Johnston's work.

- My paper Johnston 1987 prepares the theoretical part of Johnston's paper in a "student-friendly" way.
- Furthermore, Johnston's formulas are applied to
*channel #4 of a C-Harp*. In particular, the curves for the real part of the admittance at first glance appear different from Johnston's example. Actually the curves are related by the damping factor as a parameter. - For a resonator with two modes and for a cylinder as resonator, we qualitatively investigate to what extent Johnston's phase condition yields playing frequencies that deviate noticeably from the resonance frequency of the resonator.
- To actually verify Johnston's approach, one would need measured admittances or admittances calculated from MRI pictures of the vocal tract. To my knowledge, such data are not currently available.

Physical studies on playing the blues harmonica model the reeds as harmonic 1-point oscillators. The damping constant is needed for the equation of motion of a reed. In this paper two methods are presented to determine this damping constant without expensive laboratory equipment. An exemplary investigation is made of the draw reed in the fourth channel of a blues harmonica in C. In fact, after a transition phase, the decaying reed oscillation can be described as a monofrequent falling exponential curve. It can be seen that both methods, which are fundamentally different in their approach, provide the same results for frequency and relaxation time within this range, which indicates that they are indeed suitable for measuring the damping constant.

Blues harp reeds are free reeds. They can move freely due to the slots in the reed plate and oscillate almost sinusoidally in the first transverse mode of a clamped beam. The motion of a reed tip (and thus reed motion) can be modeled by an abstract 1-point oscillator equation, where effective values have to be used for some parameters. This linear equation can be used to compare the influence of different forces on reed motion.

The oscillation is generated and kept going by interaction with the fluctuating air pressure in the channel. Here, the amplitude of the oscillatory component of the pressure force on the reed in the oscillator equation is an order of magnitude smaller than the amplitude of the elastic restoring force. The influence of the pressure force is explained in frequency domain by a slight phase shift of the fundamental oscillations of reed and pressure, which provides for the necessary energy supply and for a playing frequency deviating from the resonant frequency of the reed. Apart from that, a blues harp reed oscillates "almost by itself" due to the interplay of elastic restoring force and mass inertia. This is ultimately due to the fact that the reed oscillates unhindered and weakly damped.

In a supplementary section, it is argued why it is better not to call the pressure force acting on the reed a "Bernoulli force", even if it is related to a Bernoulli equation.

This article collects experimental findings on playing the blues harp, which a physical model should thus be able to explain. The leitmotif will be the question whether and how suitably defined resonance properties of the vocal tract influence playing frequencies.

When playing blow or draw notes on a channel of a blues harp, the airflow through the player's respiratory tract and through the instrument together with the closing and the opening reed in the channel perform self-excited oscillations with a common playing frequency. This is not obvious, instead two closing reeds in the channel generate self-excited oscillations close to their respective eigenfrequencies.

Advanced players are able to change the pitch of normal notes on the blues harmonica by appropriate variations of the vocal tract geometry. MRI images show a narrow constriction between tongue and palate together with a well-defined front cavity volume. A measured correlation between playing frequency of various bend notes and front cavity volume suggest that constriction and front cavity together form a Helmholtz resonator, whose resonance frequency has a decisive influence on the periodicity of the self-excited oscillations of airstream and reeds. On the contrary, measurements with a blues harmonica excited by a tube resonator suggest a less intuitive interaction between resonator and reeds. For the player's vocal tract as a resonator, the phase angle of its admittance is postulated to define playing frequency. Normal notes sound with any arbitrary relaxed embouchure, so they should be explainable without reference to a resonator. On the other hand, it is possible to bend a normal draw note continuously down, so physical models for normal and for bend notes should also be connected by continuously changing some common parameter.

It is possible to bend draw notes on two or three neighbouring channels or even octaves (using tongue split) simultaneously with one common embouchure. Changing rapidly between inhaling and exhaling, one can play a draw bend and an overblow on the same channel with one embouchure, whereas playing frequencies lie one third apart. On the lower channels, an overblow and a blowbend can be played simultaneously, although the blowbend sounds significantly louder and always prevails over the blowbend during the transition to a single note.

The latter listed experiments contradict the intuitive notion that the playing frequencies of bends and overbends are simply determined by a single resonant frequency of the vocal tract near the playing frequency. On the other hand, comparable geometries of the oral tract occur when speaking vowels, whistling, singing overtones, or bending notes on the saxophone, where they have been shown to act via resonant properties of the enclosed air volume. It would therefore be desirable to conduct experiments that can simultaneously record the geometry of the vocal tract and the resonance behavior of the air flowing through it, the oscillations of the two reeds and the sound emitted when playing the blues harp.