AU2012203214A1 - Improved Keys - Google Patents

Abstract

Tuned percussion instruments are musical instruments in which a component of the instrument is struck by a mallet resulting in a complex tone with attributes of musical 5 pitch. Rigid bodies rarely vibrate in such a way as to produce a harmonic frequency sequence. Tuned percussion instruments in which the component primarily responsible for the production of acoustic energy is a rigid body such as a bar, cylinder or disk do not produce harmonic frequency sequences unless the precise geometry of the component primarily responsible for the production of acoustic energy has been adapted in some way 10 so as to tune the modal frequencies. Improved keys and methods of the design of keys are provided for tuned percussion instruments such as xylophones and metalophones. Methods for the design of keys are also provided. Mode 1, transverse (0,2) 291Hz Mode 4, transverse (0,4) 1566Hz Mode 2, transverse (0,3) 800Hz Mode 5, torsional (1,1) 1855Hz Mode 3, lateral (0,2) 1423Hz Mode 6, transverse (0,5) 2582Hz Figure 1.

Description

1 KEYS FOR TUNED PERCUSSION INSTRUMENTS FIELD The invention relates keys for tuned percussion instruments such as xylophones and 5 metalophones. Methods for the design of keys are also provided. BACKGROUND Uniform columns of air and thin strings usually vibrate in such a way as to produce a 10 harmonic frequency sequence. That is to say that the modal frequencies occur at integer multiples of the lowest modal frequency ("The Physics of Musical Instruments", N. Fletcher and T. Rossing, Springer-Verlag, 1991, Chapter 1). Human perception of the pitch of harmonic complex tones has been shown to correlate the frequency of a pitch sensation to the lowest frequency (usually referred to as the "fundamental frequency") of 15 a harmonic frequency sequence ("Music, Cognition and Computerized Sound", P. Cook, MIT Press, 2001, Chapter 5). When vibrating air colunms are contained within tubes that are closed at one end, the air column is forced to vibrate in such a way that only odd numbered harmonic frequencies 20 will resonate. This occurs in clarinets without loss of pitch clarity ("The Physics of Musical Instruments", N. Fletcher and T. Rossing, Springer-Verlag, 1991, Chapter 3). Therefore it can be understood that the human auditory system will interpret a subset of the harmonic series in a similar way to a complete harmonic series of the same order. 25 The state of the art in xylophone keys are those found in modem, orchestral xylophones and marimbas. Orchestral xylophone keys have an arch cut from the underside of the centre of the key to tune the second transverse mode to a ratio of three times the frequency of the first transverse mode. Marimbas usually have the same mode tuned to a ratio of four by the same method of tuning. Tuning of the third transverse mode in 30 marimba keys has been shown to influence the perceived pitch, but this appears to be rarely undertaken ("The Physics of Musical Instruments", N. Fletcher and T. Rossing, Springer-Verlag, 1991, Chapter 19). The state of the art in metalophone keys are also those found in modern vibraphones. 35 Vibraphone keys also have an arch cut from the underside of the centre of the key to tune the second transverse mode to a ratio of four times the frequency of the first transverse mode, and higher frequency modes are rarely tuned ("The Physics of Musical Instruments", N. Fletcher and T. Rossing, Springer-Verlag, 1991, Chapter 19). Aluminium keys used in vibraphones have much longer decay times than wooden keys, 40 and so it is reasonable to expect the un-tuned higher modes to have an increased affect on the pitch of these instruments. The term "xylophone" is known to refer to musical instruments in which keys of wood are struck by a mallet in order to produce a pitched resonant tone. For the sake of 2 convenience "xylophone" shall be used in the following as a general term to include all types of xylophones used in orchestras, music education and elsewhere, including those often referred to as "Marimbas" and even those made from synthetic materials that imitate the sound of wood. "Metalophone" is a more recent term used to describe in a 5 general fashion tuned percussion instruments in which metal keys are struck by a mallet in order to produce a pitched resonant tone. Instruments such as glockenspiels and vibraphones may be included in the general term "metalophones". Metalophones and xylophones generally have keys with a rectangular top surface, with 10 length usually greater than 5 times the width, and width generally greater than twice the depth. The keys are horizontally supported or suspended in a frame from locations equidistant from each end of the key with their largest surface upward. To radiate sound efficiently the keys need to be baffled or resonated. The design of baffles and resonators, where the location on the key where it is struck, and the mallet used to strike the key all 15 affect the perceived tone of the instrument, however the clarity of pitch of the instrument is largely affected by the specific shape of the key. The vibration of keys may be regarded as the linear combination of different motions known as the key's "normal modes of vibration", or simply "modes". Each mode has 20 particular directions and extents of motion in all the regions of the key, which occur with specific frequencies of oscillation. Thus the sound produced by the key can usually be determined to consist of a number of tones, each with a specific frequency and time varying amplitude superimposed upon each other. A resonant sound consisting of multiple tones is commonly referred to as a complex tone. 25 When a key is struck a great many modes will be set in motion. The particular modes set in motion and their time varying amplitude will depend on where the key is struck and the manner in which the key is supported, as displacement is inhibited in some way at the support locations. Many methods and locations of support may be found in examples of 30 xylophones and metalophones. The large ratio of length to width of the keys used in xylophones and metalophones, and the conventional methods of support ensure that the most prominent modes are those in which the key bends along its length in a direction normal to its top surface. In what follows these modes will be referred to as transverse modes. Other modes that generally do not produce audible overtones in xylophones and 35 metalophones include torsional and longitudinal modes, and tranvserse modes where bending occurs normal to the other surfaces of the key. The first 6 modes of an unconstrained rectangular bar are shown in Figure 1. Analytical theories have been developed which accurately predict the frequency of 40 transverse modes in bars ("Sounds, Structures and Their Interaction", Miguel C. Junger and David Feit, MIT press, 1972, Chapter 7). The frequencies of these modes vary approximately as the inverse square of the length of the bar over the geometric region for which these analytical equations hold. 45 From analytical theory it is known that the modes of rigid bodies may be tuned by substantially changing only the effective stiffness of the body with respect to the mode 3 shape being tuned, by substantially changing only the effective mass inertia of the body with respect to the mode shape being tuned, or by both simultaneously. An example of tuning by substantially changing only the mass inertia is the practice of tuning a tubular bell by the addition of an end cap. 5 Tuned percussion instruments are musical instruments in which a component of the instrument is struck by a mallet resulting in a complex tone with attributes of musical pitch. Rigid bodies rarely vibrate in such a way as to produce a harmonic frequency sequence. Tuned percussion instruments in which the component primarily responsible 10 for the production of acoustic energy is a rigid body such as a bar, cylinder or disk do not produce harmonic frequency sequences unless the precise geometry of the component primarily responsible for the production of acoustic energy has been adapted in some way so as to tune the modal frequencies. 15 It is an object of the present invention to provide improved keys and methods of the design of keys in order to overcome the deficiencies of existing keys and key design. SUMMARY 20 Accordingly, in an aspect of the current invention there is provided a key for tuned percussion having a plurality of transverse modal frequencies, wherein the first five transverse modal frequencies are substantially tuned to harmonic frequency ratios of the first transverse modal frequency. A further aspect of the invention provides a key for tuned percussion having a plurality of 25 transverse modal frequencies, wherein the first four transverse modal frequencies are substantially tuned to harmonic frequency ratios of the first transverse modal frequency. A further aspect of the invention provides a key for tuned percussion having a plurality of transverse modal frequencies, wherein the first three transverse modal frequencies are substantially tuned to harmonic frequency ratios of the first transverse modal frequency. 30 Preferably, not more than one consecutive harmonic ratio is omitted. The keys may optionally have an upper top surface and a lower bottom surface wherein and the upper top and lower bottom surfaces of the key are identical, flat and parallel. Further, the key is preferably symmetrical about its centre point or symmetrical along a line through its centre point and parallel with its length. 35 Preferably the keys defined above are constructed from material with physical properties able to support audible transverse modes of vibration, commonly used examples are metal or wood. A further aspect of the invention provides a metalophone key having top and bottom surfaces that are identical, flat, parallel, separated by a distance of 5mm and substantially 40 geometrically similar to a surface defined by lines fitted to the points the rectangular coordinates of which are set out in Table 2.

4 A further aspect of the invention provides a metalophone key having top and bottom surfaces that are identical, flat, parallel, separated by a distance of 4mm and substantially geometrically similar to a surface defined by lines fitted to the points the rectangular coordinates of which are set out in Table 4. 5 A further aspect of the invention provides a xylophone key having top and bottom surfaces that are identical, flat, parallel, separated by a distance of 6mm and substantially geometrically similar to a surface defined by lines fitted to the points the rectangular coordinates of which are set out in Table 6. The invention also encompasses keys having a shape that is substantially resealed from 10 the geometry described herein. In yet a further aspect of the invention, there is provided a method for designing a key for tuned percussion having a plurality of transverse modal frequencies, wherein the first four or five transverse modal frequencies are substantially tuned to a sub-set of a harmonic sequence wherein not more than one consecutive harmonic ratio is omitted, the method 15 comprising the steps of selecting an initial shape and using the initial shape in an optimisation procedure for modifying the key shape. According to the method, the optimization procedure preferably comprises finite element modeling options and shape optimisation options for modifying the key shape such that it becomes an harmonic key. 20 The method of the invention provides for designing a key for tuned percussion, the method comprising the steps of: a. defining a current key shape as an initial shape; b. selecting a number of modes to be tuned in the current set of objectives; c. selecting a desired value and range for the frequency of each of the selected 25 modes to fall within; d. modifying the current key shape in accordance with the optimisation method which will cause the values of the selected modes to move toward the desired value; e. repeating step "d" as many times as necessary for the values of the current 30 objective set to all lie within the desired range for each value; f. if all the modal frequencies to be tuned in the key are not suitably tuned, selecting a new set of modes to be tuned in the current set of objectives or manually adapting the initial shape, otherwise saving the optimised geometry; g. repeating steps "c" to "f" in relation to the current set of objectives.

5 The invention described herein further incorporates keys produced in accordance with the methods described above. The invention also further encompasses tuned percussion such as metallophones and xylophones comprising keys described herein. 5 Through the design of keys with more modal frequencies substantially at harmonic frequency ratios, the strength, clarity and stability over time of the pitch is increased. Tuning more modes to lower order harmonics also produces a "fuller" tone from keys; that is to say a tone in which more of the acoustical energy is in the low to middle range of the frequencies of concern in music (around 30 - 4,000Hz). 10 An additional advantage of the present designs is that they can be manufactured at a cost that the users of tuned percussion instruments can afford. The key designs of the present invention are keys in which the top and bottom surfaces are identical, flat, parallel shapes. This allows for the keys to be cut from sheets or boards that are commonly available in 15 industrial economies at relatively low cost. A harmonic key is a single such key which may be incorporated into a frame with other harmonic keys to become a musical instrument of varying pitch. 20 In this specification, a reference to the modal frequencies being substantially in an harmonic sequence means that a frequency series substantially conforms to the ratios 1, 2, 3 etc with respect to and including the fundamental of the harmonic sequence. Since it is established practice to use pitched musical instruments with overtones that are tuned to only the odd harmonics, keys wherein at least 3 modes are substantially tuned to a sub-set 25 of an harmonic sequence where not more than one consecutive harmonic is omitted, are referred to hereafter as a harmonic keys. DESCRIPTION OF THE FIGURES Fig 1: A representation of the first 6 transverse modes of an unconstrained rectangular 30 bar. Shading shows contours of displacement. The first number in the name of each mode refers to the number of nodal lines in the Z direction and the second number refers to the number of nodes in the X direction. Fig 2: A representation of the top view of an embodiment of the current invention showing the shape of the top and bottom surfaces of key 1 in Example 1. 35 Fig 3: A representation of the top view of the initial key shape manually created for the embodiment of the invention described in Example 2 Fig 4: A representation of the top view of the embodiment of the invention showing the shape of the top and bottom surfaces of key 3 of Example 2. Fig 5: A representation of the top view of the embodiment of the invention showing the 40 shape of the top and bottom surfaces of the key of Example 3. Fig 6: A spectrum from a 4mm thick stainless steel key 3 of Example 2. The key has a fundamental frequency of 328 Hz and was recorded at different times after onset with mode allocations based on FEA analysis.

6 Fig 7: A photograph of a metallophone constructed using the key of Example 3 shown in Figure 5. Fig 8: A representation of the top view of the embodiment of the invention showing the shape of the top and bottom surfaces of the key of example 4. 5 Fig 9: A spectrum of a key of example 4 as shown in figure 8. DETAILED DESCRIPTION 10 The acoustically important modes are those in which the direction of motion is normal to the top and bottom surfaces of the key, as these modes will set the nearby air in motion to the extent that audible sound is radiated away from the key surface. Thus in what follows, it is to be understood that reference to "modes" will be only to those modes in which the direction of motion is normal to the key's top and bottom surfaces. Similarly, a reference 15 to "frequencies" is a reference only to frequencies due to modes in which vibration occurs in a direction normal to the key's top and bottom surfaces. In this specification a reference to the first, say, three modes is a reference to the lowest frequency mode, the second lowest frequency mode and the third lowest frequency mode. Other references to the number of modes are to be construed similarly. 20 In this specification, references to "mode sequence" and "frequency sequence" for a key are references to lists of modes and frequencies respectively in order of ascending modal frequency. In this specification, references to frequencies "being tuned", and similar expressions, are references to modal frequencies that are desired to be modified to 25 substantially adopt particular values. For example in a tuned bell a number of modal frequencies have substantially "been tuned" to particular frequencies or ratios of frequencies with desirable musical attributes. Finite element methods for numerically estimating the normal modes of a solid body, and 30 their associated frequencies, are known. These methods can be used when the geometry of the solid body is too complex to be solved by analytically derived equations. In finite element methods the body is notionally divided into many elements. The geometry of this division is known as a "mesh". Individual elements are geometrically defined by "nodes" which are points on the boundaries of elements at the intersection points of the mesh. 35 Elements may have a wide range of properties by virtue of the equations defining their mechanical properties and the values of variables used to model material properties. It is common to use so-called "solid elements" with eight nodes to define a volume and so called "shell elements" with four nodes to define a surface. A shell thickness must also be provided in the mathematical definition of shell elements. 40 Numerical methods like finite element analysis, or analytical methods are only capable of estimating the mode shapes and frequencies of a particular, given geometry. Tuning the modal frequencies of a rigid body requires that modal analysis be undertaken for a range of geometries of that body. It is known to use so-called "shape optimisation methods" in 45 conjunction with finite element analysis to tune rigid bodies. In this instance, shape optimisation in conjunction with finite element analysis uses certain optimisation rules to 7 successively modify the geometry of a finite element model of a rigid body until the analysis indicates that the frequency of various modes have reached the desired values known as the "objectives". Each successive modification and analysis is called an "iteration" of the optimisation process. Certain geometrical properties of the model or 5 frequency values of modes may be defined as "constraints" during the shape optimisation process. The optimisation process will not allow geometries in which constraints are violated. It is known to use shape optimisation in conjunction with finite element analysis to tune 10 the modes of bells ("The Design of Bells with Harmonic Overtones" by McLachlan, Nigjeh and Hasell and "Improvements in or related to bells, US Patent 6,915,756 B1). Shape optimisation has also been applied to numerical models of xylophone bars to alter the undercut arch and tune the first three modes to ratios of 1:4:10 ("Designing Musical Structures Using a Constrained Optimisation Approach" and "Optimal Undercuts for the 15 Tuning of Percussive Beams" by J. Petrolito and K. A. Legge) and the ratios 1:3:6, 1:4:8 9, 1:5:10-13 ("Nonuniform beams with harmonically related overtones for use in percussion instruments" by F.Orduia-Bustamente). Five transverse modes were tuned to the frequency ratios 1:2:4:8:16 for vibraphone bars with complex undercut surface shapes in "Optimal Design and Physical Modelling of Mallet Percussion Instruments" by L.L. 20 Henrique and J.Antunes. The optimisation procedures according to the present invention will be described below. Preferably the optimisation procedure according to the present invention comprises the steps of: 25 a. defining a current key shape as an initial shape; b. selecting a number of modes to tuned in the current set of objectives; c. selecting a desired value and range for the frequency of each of the selected modes to fall within; 30 d. modifying the current key shape in accordance with the optimisation method which will cause the values of the selected modes to move toward the desired value; e. repeating step "d" as many times as necessary for the values of the current objective set to all lie within the desired range for each value. 35 f. If all the modal frequencies to be tuned in the key are not suitably tuned, selecting a new set of modes to be tuned in the current set of objectives or manually adapting the initial shape, otherwise saving the optimised geometry; g. Repeating steps "c" to "f' in relation to the current set of objectives. 40 The optimisation procedure must determine the so-called "step direction" at each iteration. The step direction is the modification to be made to the key shape during the given iteration. Preferably the optimisation method uses gradient methods to determine the step direction. Preferably the method of conjugate gradient is used, although the method of steepest descent may also be used. 45 8 The optimisation procedure undertaken in this preferred embodiment used computer software called ReShape by Advea Engineering Pty Ltd (a company incorporated in Victoria, Australia). The invention is expressly not limited to the use of this package to affect the optimisation method according to the present invention. 5 In the preferred embodiment, raw sensitivities are calculated analytically at each node of the finite element mesh. Raw sensitivity at a particular node is a measure of the rate of change of the objective (ie the frequency of a chosen mode) with respect to a change in the node position. Raw sensitivity is also very sensitive to mesh shape, and so sensitivity 10 is then recalculated with respect to a generalised coordinate system via mesh-geometry associativities, which in the preferred embodiment were so-called "influence functions". These functions are also used to apply the geometric changes required by the optimisation procedure to a range of finite element nodes about a node selected as a control point. The magnitude of displacement across the range of the influence function is calculated by a 15 relationship to proximity to the selected node defined by a spline curve. Therefore the operator can choose a few nodes as control points and a coordinate system by which to define the influence functions, and the geometric changes in the model will be smoothed and averaged accordingly. Multiple influence functions with overlapping domains are blended and boundary nodes are locked (with an automatically defined influence 20 function) so that an appropriate overall relationship between the user defined control points and the model mesh is created (ReSHAPE User's Manual, Version 98a by Josef Tomas). In the preferred embodiment the step direction is calculated from the sensitivity vectors 25 calculated with respect to the influence function previously defined by the user by the selection of the coordinate system, design domain and control points. The step direction must include the sensitivity of all the modes currently being tuned. In the optimisation procedure, when a mode that has been chosen to be tuned is within the desired frequency range it ceases to be referred to as an objective and becomes a constraint. A shape change 30 normal to the sensitivity vector for a mode will produce the minimum change to the frequency of that mode. The calculated step direction for the objective is projected onto hypersurfaces normal to the sensitivity vectors of all constrained modes to determine the step direction in each iteration of the optimisation process. 35 In designing a harmonic key using an optimisation procedure it is necessary to have an initial bar shape in which the transverse modal frequencies are suitably close to the desired values. It was realised that to arrive at such an arrangement that the desired values of transverse modal frequencies should be chosen that are a reasonably close fit with the modal frequencies of a bar with suitable dimensions for a musical instrument. That is to 40 say that the objective of the optimisation should be physically achievable within the domain of possible geometries that will be practical for a range of acoustical, ergonomic and manufacturing reasons. Accordingly the desired values were chosen after a number of trials such that the fundamental of the harmonic sequence was the first transverse mode and the next 3 transverse modes were tuned to the ratios 2:4:6. 45 9 In carrying out the optimisation procedure according to the present invention it is sometimes necessary to observe the behaviour of various modes during the optimisation. Since the modes being optimised are similarly sensitive to some geometric changes (such as overall length) they are likely to behave in a correlated fashion during the optimisation. 5 This behaviour may influence the success of an optimisation procedure for a particular set of modes. Where an optimisation procedure has failed to meet a set of objectives it may be possible to undertake the optimisation of a different set of modes first before returning to the modes for which the optimisation failed. The optimisation procedure for these modes may then succeed as it is starting from a different initial shape. 10 It was realised that geometrical changes tended to compress or expand the frequency range across which the first 4 transverse modal frequencies occur. With this in mind the initial shape of the bar could be chosen in which the modal frequencies were more widely spread than the desired values, with the knowledge that substantially lowering the 15 frequency ratio of say the third mode to the fundamental would also lower the frequency ratio of other transverse modes. Small changes in the frequency ratios of the modes could then be made independently of each other to fine-tune the key. Having compressed the frequency range of the first four transverse modes in the first 20 optimisation run, it was also realised that the transverse modes of the key could be independently fine-tuned by mass inertial effects when the complex top surface geometry included protrusions from the side of the bar. These protrusions could be located near a maximum of vibration of each mode and so alter the mass inertia of that mode more than other modes by moving their centre of mass slightly toward or away from the anti-node. 25 That is to say a shape was chosen in which the independence of the sensitivity functions of the modes being tuned was increased. Where the shape optimisation procedure was unable to attain the desired frequencies it was further realised that the tuning could be improved if mass was removed by manually deleting elements within the main body of the key near maxima of modes the frequency of which needed to be increased. 30 Three examples of the principles relating to the optimisations of harmonic keys according to the present invention will be described in the following. In the first two of these examples the material model chosen for the finite element analysis is steel (elastic modulus of 206 GPa, Poisson coefficient of 0.3 and density of 7.85e-9 tonne/mm 3 ). When 35 materials such as wood or composites are used the modulus of elasticity may vary in the directions of the length and width of the key. In the third example a key is designed using material properties of Brazilian rosewood. The harmonic key may be symmetrical about its centre point (Figure 2a) or symmetrical 40 only along a line through its centre point and parallel with its length (Figure 2b). The harmonic key may be made in any material with physical properties able to support audible transverse modes of vibration. Should the frequency of the fundamental frequency need to be changed without changing 45 the depth of the key, it can be scaled in the direction of its length according to the relationship that the frequencies of transverse mode are proportional to the inverse square 10 of the key length. Small tuning errors introduced by this process can then be corrected using an optimisation procedure if necessary. Otherwise the frequency of the key can be changed without affecting the frequency ratios of its modes by scaling in three dimensions. 5 The present invention will now be further described with reference to the following examples, which are intended for the purpose of illustration only and are not intended to limit the generality of the subject invention as hereinbefore described. 10 Example 1 As an example of the principles relating to the optimisations of harmonic keys according to the present invention, the optimisation procedure, starting geometry, final geometry, initial modal frequencies and final modal frequencies are described in the following. 15 A finite element model of an initial bar was created using solid elements. The length, width and depth of the bar were 300mm, 25mm and 5mm respectively. A technique known as a "symmetry condition" was used in the modelling to ensure the lengthwise symmetry of the bar was maintained during the optimisation procedure. The nodes along one edge of the bar parallel to its length (hereafter referred to as the "inside edge") were 20 constrained in the finite element model so that they were unable to move in the direction normal to this edge. The bar then effectively behaves as if it were a bar of 50mm width unable to move in the lateral direction (see Figure 1). This limitation of motion has no effect on the behaviour of the transverse modes being tuned in the optimisation procedure. 25 The optimisation procedure was undertaken with respect to a Cartesian coordinate system in which the X direction was parallel to the length of the key and located along its inside edge and the Y direction was normal to the top surface of the key. The domain for shape changes was the X and Z direction only. Twelve nodes equally spaced along the other 30 edge of the bar parallel to its length (the "outside edge") on the top surface were selected for the control points of the influence function. The influence function applied shape changes equally to nodes located on the top and bottom surfaces. In this way shape changes to the outside edge of the model were substantially similar to changes applied symmetrically to a 50mm width bar. The finite element nodes comprising the top and 35 bottom surface of the bar were constrained from moving in the Y direction during the optimisation so that these surfaces remained flat and parallel. After a number of optimisation runs elements were manually deleted from the model to create holes near the maxima of transverse modes the frequencies ratios of which needed 40 to be increased with respect to the first mode. Three further control points were added on the top surface along the edge of these holes so that their shape may vary during subsequent optimisation runs. Table 1 shows the modes tuned and their initial and final frequencies, frequency ratios and the error from perfect harmonic ratios. Figure 2 shows the shape of the top and bottom surfaces the harmonic key design and Table 2 lists the 45 coordinates of the nodes that define the outside edge of the key and the shape of the 11 holes. Only /4 of the nodes are listed in Table 2 as the key is symmetrical about its centre point. The key is symmetrical about the point X = 142.7mm, Y = 2.5mm, Z=0mm. X values 5 greater than 142.7 are calculated by applying the formula X 1 = 142.7 + (142.7 - Xo) where X 0 is taken from above. The Z values corresponding to X 1 are unchanged from those corresponding to X 0 . Another set of Z values for each X that defines the other symmetrical edge are the negative of the values shown above. A second surface identical to the one defined above is created by duplicating all of the points shown and calculated 10 above with a Y value of 5mm.

12 Table 1. The Tuning of the Harmonic Metalophone Key 1 in Example 1. Mode No. Type Initial bar Harmonic Key Freq. (Hz) Freq Ratio Freq.(Hz) Freq Ratio Error (%) 1 0,2 291 1 444 1 2 0,3 800 2.75 869 1.96 2 3 0,4 1566 5.38 1793 4.04 1 4 0,5 2582 8.87 2652 5.97 0.5 5 Table 2. Node locations for Harmonic Metalophone Key 1 in Example 1. outline end hole centre hole Xmm Zmm Xmm Zmm Xmm Zmm Xmm Zmm 1.68 0.00 82.37 25.85 61.61 0.00 137.37 0.00 0.00 3.37 86.30 29.08 61.43 2.07 137.35 4.83 2.36 4.81 87.36 32.21 1 61.25 4.06 137.33 8.78 4.26 3.45 88.43 35.33 61.08 6.11 142.69 8.92 6.42 2.56 89.49 38.46 58.67 7.82 8.67 1.97 90.64 41.53 61.80 8.84 11.03 1.90 92.85 43.99 64.82 9.84 13.39 1.86 93.85 46.95 67.84 10.82 15.74 1.96 96.38 48.83 70.87 11.80 18.11 2.07 99.52 48.96 73.83 12.95 20.46 2.28 102.15 47.27 76.79 14.12 22.82 2.51 103.31 44.38 79.74 15.30 25.17 2.76 102.75 41.22 82.70 16.50 27.52 2.84 103.16 38.15 88.61 18.93 29.88 2.78 105.43 34.93 88.78 15.16 32.24 2.71 106.26 33.65 88.94 11.38 33.23 5.67 108.14 31.12 89.10 7.62 34.23 8.64 110.11 28.60 89.25 3.86 34.94 11.66 111.24 26.73 89.42 0.00 35.08 14.77 113.45 23.86 37.05 17.12 115.67 21.00 40.12 17.14 121.04 20.35 42.08 14.82 126.46 20.69 44.71 13.32 131.87 21.05 50.05 14.46 137.28 21.44 54.96 15.51 142.70 21.82 59.95 16.13 64.94 16.70 69.97 17.24 74.24 19.81 78.39 22.70 13 Example 2 Another example of the principles relating to the optimisations according to the present 5 invention, the optimisation procedure, starting geometry, final geometry, initial modal frequencies and final modal frequencies are described in the following. With the knowledge gained from the designs typified by Example 1, a second design was undertaken in which an initial key shape was manually derived from Example 1. It was 10 learnt from physical prototypes of Example 1 that the best location to strike the key in order to excite all the modes with approximately equal energy was the end of the key. Therefore in Example 2 the initial shape of the key had an enlarged area at one end. Generally the remainder of the initial shape was calculated to remove the holes from the design and reduce the maximum width of the key whilst minimizing the changes in the 15 average mass of the key over any sections of its length over both directions from its centre. The optimisation procedure was carried out in a similar fashion to that described for Example 1. The data in the columns for the initial key and harmonic key 2 in Table 3 shows the 20 modes tuned and their initial and final frequencies, frequency ratios and the error from perfect harmonic ratios. Figure 3 shows the shape of the top and bottom surfaces of the initial key design. The shape created for harmonic key 2 was then resealed for manufacture from 4mm thick steel and holes for suspension points added before a final optimisation process was carried out to arrive at the harmonic key 3 design shown in 25 Figure 4. The data in the columns for harmonic key 3 in Table 3 shows the modes tuned, their final frequencies, frequency ratios and the error from perfect harmonic ratios. Note that a further transverse mode (mode 5) was tuned to substantially a harmonic ratio of the first mode in Example 2. This modal frequency is not claimed as an essential aspect of a harmonic key as there are two missing harmonics between it and the next highest 30 transverse modal frequency. However it is believed to be an advantage to tune as many transverse modal frequencies to harmonic ratios as possible. Table 4 lists the coordinates of the nodes that define the outside edge of the key and the shape of the holes for harmonic key 3. 35 The key is symmetrical about a line along X where Y = 2mm, Z=Omm. Another set of Z values for each X that defines the other symmetrical edge are the negative of the values shown above. A second surface identical to the one defined above is created by duplicating all of the points shown and calculated above with a Y value of 4mm.

14 Table 3. The Tuning of the Harmonic Metalophone Keys in Example 2. Mode Type Initial Key Harmonic Key 2 Harmonic Key 3 5 No. Freq. Freq. Freq. Freq Error Freq. Freq Error (Hz) Ratio (Hz) Ratio (%) (Hz) Ratio (%) 1 0,2 437 1 440 1 - 261 1 2 0,3 884 2.02 885 2.01 0.5 525 2.01 0.5 3 0,4 1781 4.08 1763 4.01 0.25 1054 4.04 10 4 0,5 2676 6.12 2644 6.01 0.1 1596 6.11 1.8 5 0,6 3905 8.94 3932 8.94 0.6 2373 9.09 1 15 Table 4. Node locations for Harmonic Metalophone Key 3 in Example 2. outline endholel Xmm Zmm Xmm Zmm Xmm Zmm Xmm Zmm -0.15 0.00 115.45 31.01 244.95 32.41 318.97 10.18 0.62 5.18 115.27 36.02 243.97 27.26 318.87 5.17 6.94 3.94 122.10 37.64 242.84 22.28 321.51 15.46 18.25 2.61 129.20 37.84 241.35 16.96 324.47 15.12 26.15 2.68 135.83 38.27 247.75 15.63 323.48 10.10 35.46 3.13 141.72 39.99 256.43 11.14 324.97 5.06 42.88 3.21 143.59 34.20 266.94 9.03 endhole2 48.78 5.09 145.44 28.53 277.86 7.96 306.26 9.65 46.86 10.27 156.56 27.38 288.80 8.87 301.78 4.71 44.90 16.02 166.52 28.22 296.73 10.70 313.75 10.02 51.43 15.44 177.17 29.31 304.10 12.60 310.73 5.00 60.73 12.32 188.56 28.81 311.67 14.57 suspension point 1 70.59 9.71 198.79 27.59 317.14 16.26 237.41 36.87 80.73 8.25 208.82 26.25 322.15 20.21 229.18 34.88 88.84 10.86 220.47 27.18 325.62 20.16 242.47 35.04 99.64 9.41 220.99 32.49 327.48 18.96 237.08 32.60 108.23 11.45 221.35 37.71 327.22 15.24 suspension point 2 117.54 15.55 229.16 37.40 327.72 10.08 94.63 2.73 116.54 20.73 238.29 42.03 329.03 5.08 105.74 0.00 115.90 25.90 243.21 39.20 330.44 0.00 87.38 0.00 245.67 37.63 15 Example 3 Another example of the principles relating to the optimisations according to the present invention, the optimisation procedure, starting geometry, final geometry, initial modal 5 frequencies and final modal frequencies for a xylophone key are described in the following. The harmonic metalophone key 3 design in Example 2 was adopted as the starting geometry for tuning a harmonic xylophone key by shape optimisation where the top and 10 bottom surfaces of the key are identical, flat and parallel. The bottom surface was offset by 2mm to make the key 6mm thick and a material model for Brazilian rosewood was used. The material properties were obtained from the paper "Comparison between modal analysis and finite element modeling of a marimba bar" by I. Bork, A. Chaigne, L.C Trebuchet, M. Kosfelder and D. Pillot. The optimisation was undertaken using an 15 isotropic material property according to this paper, and the resulting design was then analysed with the orthotropic material properties also described in this paper. Insignificant differences were obtained for analyses using the two material properties for the transverse modes being tuned in the optimisation. 20 Table 5 shows the modes tuned and their initial and final frequencies, frequency ratios and the error from perfect harmonic ratios. Figure 5 shows the shape of the top and bottom surfaces the harmonic xylophone key design and Table 6 lists the coordinates of the nodes that define the outside edge of the key and the shape of the holes. 25 The key is symmetrical about a line along X where Y = 3mm, Z=Omm. Another set of Z values for each X that defines the other symmetrical edge are the negative of the values shown above. A second surface identical to the one defined above is created by duplicating all of the points shown and calculated above with a Y value of 6mm. 30 Table 5. The Tuning of the Harmonic Xylophone Key in Example 3. Mode Type Initial bar Harmonic Key No. Freq. (Hz) Freq Ratio Error (%) Freq.(Hz) Freq Ratio Error (%) 1 0,2 261 1 - 330 1 2 0,3 525 2.01 0.5 660 2 2 3 0,4 1054 4.04 1 1320 4.0 1 4 0,5 1596 6.11 1.8 1980 5.97 0.5 16 Table 6. Node locations for Harmonic Xylophone Key in Example 3. Outline Endhole 1 Xmm Zmm Xmm Zmm Xmm Zmm Xmm Zmm 330.44 0.00 245.40 37.51 115.37 30.05 325.07 5.18 329.18 5.20 242.92 39.24 116.19 24.91 323.65 10.44 327.96 10.35 237.96 42.18 117.20 19.74 324.71 15.59 327.54 15.67 229.07 37.46 118.44 14.64 321.68 15.98 327.80 19.48 221.53 37.59 109.05 9.99 319.08 10.57 325.86 20.68 221.05 32.39 100.08 7.57 318.97 5.30 322.27 20.73 220.54 27.04 88.80 8.84 Endhole 2 317.18 16.83 209.02 26.04 80.64 7.04 310.79 5.14 311.54 15.13 198.81 27.53 70.39 8.80 313.78 10.44 303.62 13.34 188.50 28.75 60.38 12.02 306.04 10.11 296.26 11.22 177.23 29.35 51.74 16.78 301.75 4.86 288.53 9.04 166.53 28.05 44.73 17.42 Suspension point 1 277.71 7.80 156.36 26.83 46.48 9.35 237.03 32.50 266.80 8.00 145.32 28.42 48.70 4.59 242.33 34.89 256.20 10.23 143.59 34.29 42.78 2.90 237.37 36.82 246.96 14.76 141.75 40.17 35.43 2.73 229.11 34.94 240.60 16.61 135.84 38.79 26.21 2.34 Suspension point 2 242.19 21.93 129.15 38.36 18.22 2.42 105.74 0.00 243.54 26.91 121.85 37.73 6.81 4.37 94.81 1.96 244.69 32.11 114.90 35.17 -0.90 9.94 87.38 0.00 -0.15 0.00 5 Example 4 It is desirable to have the keys as close together as possible to reduce the effort required to play keyed instruments. In a further development of the harmonic key design, the optimization process was started from a shape in which the protrusions began from a narrow spine along the length of the key in order to reduce the width of each key. The 10 resultant harmonic key is shown in Figure 8. It is preferable to use aluminium because it is lighter and the lower partials couple better with air and so are louder. This produces a less bright sound and focusses perception on the tuned lower order modes (Rossing (2000) Science ofPercussion Instruments, World 15 Scientific: Singapore) Various minima of the transverse modes occur at different locations close to /4 and /4 of the length of the key. Suspending the key at the exact location of a vibratory minimum for a particular mode will minimize the damping of that mode. So it is possible to 20 suspend the key over a range of locations that enhance the decay time of different modes of vibration. In the example shown in figure 8, slots were provided to allow the point of suspension to be adjusted by the user to alter the decay times of different modes.

17 Turning now to Figure 9, the spectrum of a narrow harmonic key shown in figure 8 that was cut from 10 mm thick aluminium showing peaks at frequencies within 2% of frequency ratios of 1:2, 1:3, 1:4, and 1:6 to the lowest frequency partial. The peak at a 5 ratio of 1:3 was due to a tortional mode and all other peaks are due to transverse modes. The torsional mode was tuned in final optimization step after the optimization of the transverse modes. Any discussion of documents, devices, acts or knowledge in this specification is included 10 to explain the context of the invention. It should not be taken as an admission that any of the material formed part of the prior art base or the common general knowledge in the relevant art on or before the priority date of the claims herein. "Comprises" or "comprising" and grammatical variations thereof when used in this 15 specification are to be taken to specify the presence of stated features, integers, steps or components or groups thereof, but do not preclude the presence or addition of one or more other features, integers, steps, components or groups thereof. Those skilled in the art will appreciate that the invention described herein is susceptible 20 to variations and modifications other than those specifically described. It is to be understood that the invention includes all such variations and modifications. The invention also includes all of the steps, feature, compositions and compounds referred to or indicated in this specification, individually or collectively and any and all combinations of any two or more of said steps or features. It is also to be understood that 25 the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting.

Claims (20)

1. A key for tuned percussion having a plurality of transverse modal frequencies, wherein the first five transverse modal frequencies are substantially tuned to 5 harmonic frequency ratios of the first transverse modal frequency.

2. A key for tuned percussion having a plurality of transverse modal frequencies, wherein the first four transverse modal frequencies are substantially tuned to harmonic frequency ratios of the first transverse modal frequency.

3. A key for tuned percussion having a plurality of transverse modal frequencies, 10 wherein the first three transverse modal frequencies are substantially tuned to harmonic frequency ratios of the first transverse modal frequency.

4. The key according to any one of claims 1, 2 or 3 wherein not more than one consecutive harmonic ratio is omitted.

5. The key according to any one of the preceding claims having an upper top surface 15 and a lower bottom surface wherein and the upper top and lower bottom surfaces of the key are identical, flat and parallel.

6. The key according to any one of the preceding claims wherein the key is symmetrical about its centre point.

7. The key according to any one of the preceding claims wherein the key is 20 symmetrical along a line through its centre point and parallel with its length.

8. The key according to any one of the preceding claims wherein the key comprises material with physical properties able to support audible transverse modes of vibration.

9. The key according to claim 8 wherein the material is metal or wood. 25

10. A metalophone key having top and bottom surfaces that are identical, flat, parallel, separated by a distance of 5mm and substantially geometrically similar to a surface defined by lines fitted to the points the rectangular coordinates of which are set out in Table 2.

11. A metalophone key having top and bottom surfaces that are identical, flat, 30 parallel, separated by a distance of 4mm and substantially geometrically similar to a surface defined by lines fitted to the points the rectangular coordinates of which are set out in Table 4.

12. A xylophone key having top and bottom surfaces that are identical, flat, parallel, separated by a distance of 6mm and substantially geometrically similar to a 35 surface defined by lines fitted to the points the rectangular coordinates of which are set out in Table 6. 2

13. A key having a shape that is substantially resealed from the geometry described in any one of claims 10, 11 or 12.

14. A method for designing a key for tuned percussion having a plurality of transverse modal frequencies, wherein the first four or five transverse modal frequencies are 5 substantially tuned to a sub-set of a harmonic sequence wherein not more than one consecutive harmonic ratio is omitted, the method comprising the steps of selecting an initial shape and using the initial shape in an optimisation procedure for modifying the key shape.

15. The method according to claim 14 wherein the optimization procedure comprises 10 finite element modeling options and shape optimisation options for modifying the key shape such that it becomes an harmonic key.

16. A method for designing a key for tuned percussion, the method comprising the steps of: a. defining a current key shape as an initial shape; 15 b. selecting a number of modes to be tuned in the current set of objectives; c. selecting a desired value and range for the frequency of each of the selected modes to fall within; d. modifying the current key shape in accordance with the optimisation method which will cause the values of the selected modes to move toward 20 the desired value; e. repeating step "d" as many times as necessary for the values of the current objective set to all lie within the desired range for each value; f. if all the modal frequencies to be tuned in the key are not suitably tuned, selecting a new set of modes to be tuned in the current set of objectives or 25 manually adapting the initial shape, otherwise saving the optimised geometry; g. repeating steps "c" to "f" in relation to the current set of objectives.

17. A key produced in accordance with the method of any one of claims 14 to 16.

18. A key copied from a key according to any one of claims I to 13 or 17, or keys 30 designed in accordance with the method of any one of claims 14 to 16.

19. A metallophone comprising keys according to any one of claims I to 13, 17 or 18.

20. A key for tuned percussion substantially as hereinbefore described according to figures 2 to 5 and 8.