The Feed-DrumGRAME Rencontre Musicales Pluridisciplinaries 2006 Retrieved on August 18, 2009. is an imperial bass drum with a system of electronic conditioning of the skin conceived by composer Michelangelo Lupone and coproduced between CRM - Centro Ricerche Musicali, Rome and Istituto Gramma in L'Aquila.
Through the principle of feed-back, the signal produced by the excitation of the skin returns to the skin in the form of acoustic pressure. The result is an infinitely sustained sound. The system controls the damping of the movement of the skin, and therefore the decay rate of the sound, and permits isolation of high frequency modes by the combined action of nodes present on the skin and the amount of feed-back input energy.Universidad Complutense de Madrid The design on the skin surface is a simplified map of the oscillatory modes based on the Bessel functions. The map was limited to 13 diameters and 8 nodal circles, the latter divided into even semicircles (to the left) and odd semicircles (to the right).Max Planck Institut - Informatik See also Skin-Act.
Experimental work Although the skin of an imperial bass drum allows the excitation of a considerable number of high-frequency modes, their duration in time is not normally appreciable by the listener, apart from the timbral contribution to the attack phase of the sound. The possible variations of the mode of emission, adequate for a sufficient acoustic response of the resonator (shell), are limited and with scarce modulability. The basic frequency, obtained by the tension of the skins, upper and lower, each bound to the edges with 16 mechanical tie rods, is influenced by the non-homogeneous distribution of the tensioning forces which contributes to render complex the spectrum of the real modes.
Experiments were made at the Centro Ricerche Musicali - CRM in RomeGran Cassa and the Adaptive Instrument Feed-Drum - Computer Music Modeling and Retrieval - Seno, Lorenzo - Lupone, Michelangelo - Springer Verlag 2006 - CRM - Centro Ricerche Musicali and at the Istituto Gramma in L'Aquila with the aim of achieving these objectives:
Variation of basic frequency through application of nodal constraints to the skin Identification of timbres on the basis of type, mode, and point of excitation Sound modulation through glissandos, vibratos, portamento, and rhythmic micro-articulation Continuous or step variations of the dynamics, on the basis of the type of damping applied to the skin
Traditional bass drums obviously do not provide these characteristics. To explore the timbre richness of the attack phase and isolate the vibrational modes, the makers created a system to electronically manipulate the skin. Through the principle of feedback, the system returns the signal produced by the excitation of the skin back to the skin as acoustic pressure. This produces infinite prolongation of the sound. The system controls the damping the skin, and therefore the decay rate of the sound, and permits isolating high frequency modes by the combined action of the nodes present on the skin and of the amount of feed-back input energy.
The stability of the signal obtained with this conditioning system makes it possible to experiment and design on the skin surface a preliminary simplified map of the oscillatory modes based on the Bessel’s functions. The map is limited to 13 diameters and 8 nodal circles, the latter divided into even semicircles (to the left) and odd semicircles (to the right). Electronic conditioning of the instrument leaves the topology and primary acoustic features unaltered, but increases the scope of the vibrational criteria and control. This made it possible to distinguish the different pitches of various modes, to obtain the emission of long notes, which could be modulated as those emitted by a stretched string, and to adapt the acoustic energy independently of the emitted frequencies.INIST/CNRS
To maintain agile execution and adequate reproducibility of the phenomena, the first classification of sounds and performance techniques was limited to the use of fingers, hands and arms (Fig. 4). During the composition of Gran Cassa by Michelangelo Lupone, experiments were also made with objects of different shapes and dimensions occupying wider or multiple nodal sections. This increased the sound possibilities. However, the complexity of the vibrational phenomena required analysis of the instrument's mechanical parts to identify and reduce dispersions and non-linear contributions from vibrations of the structural materials and their combinations.
Given these complications, it was decided to plan and realize a new instrument, the Feed-Drum (Fig. 5), for the purpose of not only extending the acoustic possibilities, but also of permitting the ergonomic use of new executive techniques. In particular, the vibrational attitude was transformed by eliminating the lower skin, a decision which simplified the tuning of the instrument’s basic frequency (30 Hz) and reduced the excitation rise time in the upper modes. A synthetic membrane was applied with isotropic characteristics and high flexibility on which the previously described map was drawn, with colors that made the areas of performance more visible. The shell and the tensioning hoop were realized in steel and aluminium; in particular, the tensioning hoop was made stiffer while the height was reduced and the adhesion surface increased. The suspension system was realized in such a way as to separate the Feed-Drum completely from the supporting structure on the ground; all the mechanical parts, which were in contact with one another, were separated by an intermediate layer of antivibrational material.Conservatory of Music Alfredo Casella L'Aquila Retrieved on August 21, 2009
Theory of operation The behaviour of the Feed-Drum is extremely complex and many of its aspects remain to be clarified. Below are the known elements, those conjectural, and those still undefined.
The oscillation modes of a circular non-rigid membrane—pegged down and stretched along its rim—are known from literature. In a conservative model (that is, without dissipations and acoustic irradiations and therefore “in vacuum” ), the oscillation modes of a membrane of radius a have the form in cylindrical coordinates
(1)
where: and
and where are Bessel functions of the first kind and of an order is an arbitrary phase dependent on the initial conditions (there cannot be any privileged directions, since the problem applies to a circular symmetry).
Owing to the constraint on the rim, , where is the radius of the membrane; this allows calculating k (wave number) which is discrete and dependent on two indices (m,n): , where is the nth root of the Bessel function of order .
Hence (1) becomes .
Determination of the wave number is therefore possible by determining the roots of the Bessel function of the first kind. Once the roots and wave numbers are determined, the angular frequencies peculiar to the modes are given by: where c is the velocity propagation of transverse waves in the membrane, where is the stretching force of the rim and is the surface density of the membrane. However, can easily be estimated on the basis of the frequency of mode , the lowest of all (basic frequency), taking into account that :
For the Feed-Drum, and , and therefore .
Irrespective of the order of the Bessel functions, the root base tends to for ; in addition, Bessel functions of different order do not have coincident roots (an important consideration for the purpose of the Feed-Drum). The exact calculation of the roots can only be realized numerically, a task which is not particularly difficult given the oscillatory character (even if not periodical) of Bessel functions. In fact, the roots of these functions are each comprised between a maximum and a minimum or vice versa.
Calculations of the frequencies for the modes up to 5 octaves above the “basic frequency” (960 Hz for the Feed-Drum) gives the following distributions of frequencies and modal density:
The index is responsible for the creation of nodal diameters, the index for that of nodal circles. In general, the pattern of the modes is simply correlated to the indices, as can be seen from the diagrams given below.
Conditioning system and implementation Excitation of the membrane is via a loudspeaker (Ø = 45 cm.) and an 11 cm-long wave guide (designed to convey maximum acoustic pressure between the center and 1/3 of the radius); that is, fairly short as far as the form factor is concerned. It proved fairly easy to obtain, in addition to the 30 Hz basic frequency, the 68.9 Hz frequency corresponding to the mode (0,2). It was on the contrary impossible to obtain the frequency of 47.8 Hz corresponding to the mode (1,1). At these frequencies, the behavior of the air excited by the loudspeaker can presumably be schematized with a piston motion, which exercises an almost uniform pressure on the membrane. A uniform excitation is poorly compatible with the modal form (1,1). The loudspeaker was driven by an electric power signal, generated by a feed-back system that sampled the signal issued by a piezoceramic sensor placed on the rim and detecting deflection of the membrane. In this way a “multimodal” oscillator was obtained generating a feed-back on a resonant element, the membrane. The loop gain was controllable by a pedal.Corriere della Sera
Intonation of the upper modes Intonation is through the combined action of the negative feed-back gain and of the pressure on one or two points of a nodal line. The effect of the pressure can be schematized in a first approximation as dual: on one hand the introduction of a constraint on the pressure points, on the other a shift of the “work point” of the membrane around a slightly higher tension, and therefore an increase of the transverse wave velocity . Consequently, all the frequencies move upwards. It is a matter of a shift mechanism, a “pitch-shift”, in the sense that the frequencies of the modes are all multiplied by a common factor, therefore leaving unchanged their relationships. In point of fact this effect has been encountered in practice and is utilized for obtaining the vibrato. The term “pitch-shift” is however improper in this case, since the spectrum of the partial tones of the membrane is not harmonic and as a result a pitch is not definable.
The apposition of constraint points (z = 0) has the effect, as a rule, of inhibiting every mode which has no set of nodal lines passing through all the aforesaid points, not even with an opportune choice of .