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Fascination of Blues Harp Physics
Alfred Förtsch

The “Mississippi Saxophone”, as the blues harp (harp, harmonica) was sometimes called in former times, is a tiny instrument with great expressiveness. It has a rich overtone spectrum which can be modified by suitable configurations of the player’s vocal tract and by playing amplified, cupping the microphone and the harp with both hands.

There are three kinds of tones that can be played on the instrument, namely normal notes, bends and overbends. Playing with relaxed embouchure results in normal blow respectively normal draw notes. Playing frequencies of such normal notes are somewhat below the eigen frequency of the blow respectively draw reed. Geometries of the player’s vocal tract similar to those when pronouncing the vowels [i:] or [u:] are connected with bend notes (draw bends on the lower channels and blow bends on the higher channels of the harp) and overbends (overblows on the lower holes and overdraws on the higher holes). Normal notes on the lower channels can be continuously bent down, whereas overbends seem to “pop up”.

Physically the player and the instrument form a complex interactive system with the generated notes as collective modes. As is customary in physics, it is assumed that this system can be broken down into separable, interacting subsystems.

The comparatively “easy part” of blues harp physics are the two free reeds in the channel, each of these modelled by a clamped beam, oscillating nearly sinusoidally in its first transversal mode. On the other hand, there is the air flow from the lungs of the player through his vocal tract and the instrument into the surroundings (for blow notes) or vice versa (for draw notes) with the player’s and the instrument’s bodies and the surroundings defining boundary conditions.

For the air flow inside the vocal tract there is a hierarchy of models developed  for speech analysis and artificial speech synthesis, from formant analysis using standing waves to numerical computational methods in fluid dynamics. Sound emission, however, is very different from speech production. Blues harp sound is generated by fluctuating air jets through the gaps between the oscillating reeds and their support plates. And, in contrast to speech generation, the role of the vocal folds can by no means be compared with the part played by the two oscillating reeds in the channel.

Is there hope for some, at least partially, intuitive understanding of blues harp physics, or is there only blind faith in numerical solutions of the Navier-Stokes equations?

One intuitive argument could be that there is some kind of feedback process between the air flow through the reed gaps, the pressure differences between the respective reed surfaces and the reed movements.
Playing normal blow notes on channels #1 to #6, only the blown-closed reed seems to be involved in the feedback process. Neglecting resonance-like properties of the oral cavity (normal notes are played with relaxed embouchure), this single free reed might be compared to an accordion reed. Accordions sound because of a feedback process between reed motion and air flow, caused intuitively by the inertia of accelerated air in the upstream region (Ricot, Caussé, Misdariis).
Playing overblows on channels #1 to #6 , only the blown-open reeds seems to be involved in the feedback process. Assuming resonance-like properties of the oral cavity, this single reed might be compared to a single free reed in an Asian harp (sheng, khaenn, bawu), coupled to a resonance tube. Those instruments can be intuitively understood in a simple generator-resonator model similar to woodwind instruments like the clarinet or the saxophone (Cottingham, Dieckman).
Playing bend notes, both reeds are actively involved in the feedback process. It is easy to bend normal draw notes continuously down on channels #1 to #6. So there should be a common physical model for normal draw and draw bends. Trying to play a blow bend on channels #1 to #6 will either result in no well-defined sound at all or in a overblow, which seems to “pop up”. Therefore a satisfying model for the harmonica should actually explain all kinds of playable notes, including an explanation why overblows prevail over blow bends. A simple resonator model (Johnston) won’t do these jobs (there is no resonator for normal notes, and blow bends and overblows appear in Johnston’s model as two competing options on an equal footing).

Laurent Millot  offers a comprehensive model for playing the blues harp, assuming a non-linear approach for the acoustic properties of vocal tract and instrument from the outset (see his project “Physics of free reeds instruments: from modeling to numerical simulation” on ResearchGate).

A 3D printed vocal tracts model based on MRT measurements of an expert musician (Barrett) performing note bends were shown to precisely reproduce the same pitch changes (Ganzow, Rossing, Egbert). Such measurements and models should allow for further experiments (e. g. flow visualizations) or testing various theoretical models (resonator model, various approximations in fluid dynamics).

In summary:
Compared to the saxophone, the exploration of the “Mississippi Saxophone” includes a number of challenges. There are two reeds (instead of one), coupled to air flowing through a highly complex resonator, resulting in sound generated by air jets which leave the channel through both reed gaps, entering into the volume between comb and reed plates. The two reeds work differently (one opening and one closing reed) with one reed dominating or with both reeds cooperating. The vocal tract is involved in sound production in a way more or less similar to speech production. Its geometry and its physical properties (e. g. viscosity) influence playing frequency and tone. Properties of the instrument like the size of the channels, the reeds and (very important) the gaps between reeds and reed plates as well as material properties of the reeds will appear in quantitative calculations. If the instrument is played acoustically, the listener will appreciate the far field of the emitted sound. If the blues harp is played amplified, the microphone in the player’s hands will record the near field inside the cupped volume.

Despite all this complexity there should still be hope for intuitive interpretations of at least some parts of physical models for the blues harp. One aim of this website will be to present such intuitive explanations.

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