CA2131691A1 - A musical percussion instrument - Google Patents

Abstract

ABSTRACT

A musical percussion instrument consists of more than three integrally formed, substantially linear, elongate sections, The sections are not colinear, but are substantially coplanar. The sections form a shape, and are made from a material (typically metal or a ceramic material) which has vibrational properties such that, when one section is struck with a mallet, the instrument emit a musically concordant sound. The mathematical analysis leading to alternative designs for such an instrument having five sections (termed a "pentangle"), made by bending mild steel rod, is presented. The sound produced by the instrument may be acoustically or electronically amplified. An array of "pentangles" (60), each tuned to different pitch, may be assembled to provide the equivalent of a keyboard instrument.

Description

2131~91 WO 93rl8503 PCr/AU93tO0101 TITLE: "A MUSICAL PERCUSSION ~NSTRUM~N~
,::
~echn~cal Field Thi~ inventlon relate~ to percussion lnqtruments. More particulsrly, it concerns a muslcal percu8sion lnst~ument containlng a ~erleg of elongate, non-colinear, integrally formed ~ections whlch, when one ~eotlon i5 ctruck with a mallet or beater, emit~ a musically pleasant sound. The quality of the em~tted sound varie~ according to the nature of the mallet (soft or ~ard) and ~he way in whlch the in~trument is struck. When Qtruck wlth a hard beater or mallet, the ln~trument emlt~ a bell-llke s~und containing partial tones in n~arly harmonic relationship.
.~ .
Ba~kground Art The closest prlor art to the present inventlon is the musical instrument commonly ~nown as the "orchestral triangle" or the "perCuQgiOn tr~angle". ~e tradltlonal orche~ral t~iangle produces a characterictlc ~triangle"
sound of indefinlte pitch. The triangle lc not tuned to provide an harmonious ~ound. Although the relation between the mode frequencies of a percusslon triangle could be varied to 80me extent by chang~ng tha ba8e angles and corner curvatures of the trlangle, the extent of such tuning is qulte limited. It i~ po~ble to decign a triangle having a nominal pitch fixed by it~ ovQrall size, so as to bring two other ~ode frequencles into harmonic relat~on with this nomin~l pitch. ~owever, the remaining inharmonic partials in guch a ~tuned~ trlangle are ~ignificantly prominent, and this lg regarded by mu~lcia~s as a llmitation ~o the usefulness of the triangle.
' . ' ~ `~ '' - :
. .

.

:. . .
:. . :: ''. : . . :
: , -: . . - : . :

213~691 ~ :
WO93/1~3 PC~AU93/~0101 - 2 ~
~ . -. .: - ~

P~sclosur~ of the Invent$on It ls an ob~ect of t~e presQnt invent~on to provide a muQical percussion in6trument ha~ing a serieg o elongate, non-collnear, $ntegrally formed ~ectlons that i8 tuned to 5 provide a musically pleas~nt sound when a sectldn 1Q struck ~ ~
by a mallet or beat~r. ~ -: :''~
Thls ob~sctive ~s ac~ieved by providlng a musical percussion lnstrument comprlslng a length or piece of metal or other ~uitable mater~al (for exa~ple, 8 ceramlc materlal) whlch i8 formed lnto a ~ape of more t~an three sectlons wh~ch, when one sectlon i~ struck, wlll emit a ~ ~;
~ound havlng a frequency spectrum whlch i~ mu~cally concordant. The sectlons of metal or other materlal are formed into a shape that is gu~8tantlally plansr, and preferably the frequency spectrum of the em~tt~d sound 1 cUch that at l~ast the first five in-plane modes are substantially harmonlcally related.

Thu~, acco~dln~ to the present lnv~ntion, there is provided a percussion lnstrument comprigln~ a plur~llty of more than three integrally formed elongate sections, said sectlons bein~ substantia~ly co-plan~r, non-col~n~ar, and formed ~rom a mat~rlal whlch, ~t room temper~ture, is rigid and has ~brational properties such that, when one of the ection~ is struck with a mallet, the ln~trument emits a -~
musically concordant ~ound~

Preferably the lnQtrument has flve gectlons, which form a -non-regular ~ymmetrlcal s~ape. A partlcularly ~seful shape ~s ~ne which is mirror symmetrlc about ~e centre po~nt of :,. .: . :

2~3l69~
~093/18503 PCT~AU93J00101 the middl~ section, with $ts end sectlons of equal length, and wlth the lntermediate s~ctions, whlch are between the end ~eot~ons and the centr~l section, a~so of equal length.
The present inventors have termed thi~ structure a ~pentangle" ~ructure. ~n one r~alisation of the "pentangle~ structure, the lQngths of t~e ~ection~ are in the ratios l.00 : l.95 : 0.92 wlth two lncluded angles of approximately 95 and the other two lncluded angles be~ng approximstely 93. In another reall~ation of this format, the lengths of the sectlon~ of th~ pentangle are in the rstlos l.00 : 1 85 : 0.97 and the lncluded angles are approximately 146~ and 10. -The pre~ent invention al~o encompa~-~e~ a percuss~on instrument whlc~ compr~ses an a~embly or array of the lnd~vldual in8truments described above, each ~nstrument in the ~ssembly being tuned to a dlfferent pitch, to provide a desired mu~ical sc8le. F~r example, a two octave scale ~comprislng 25 l~d~vldual instruments) could be provided and played l~ke a xylophone.

~he instrument of the present in~ent~on may be associated w~th a sound radlator to ~ncrea~e the efficiency of the sound production. Such a radlator may be a resonant radiator or a non-re~onant radlator, although lt is preerred that a resonant radlator rather than a wide band, non-resonant radlator ls used with an instrume~t comprising a s~ngIe pentangle. Known ra~lator structures that may be used include tunable pipe or cavlty ( Helmholtz) ~esonators having a flexible diaphra~m coupled to the lnstrument vla ~ - -a thin wire or cord. If the percu~sion instrument :,:: :. : ~ : :

2131691 ~:
PCi`tAU93tW10 - 4 ~

compri~es an array of pentangle~, a broad-b~nd non-re~nant soundbosr~ backed by ~ cavlty 1~ prsfQrrQ~. Electronic amplificatlon of the sound produce~ by lnstrument~
con8tructed in scco~dance w$th the pregent inventlon ls al~o posslble.

These and other features of the present lnvent~on will be discu~sed ~n more detall in the following descript~on of example~ of the pre~ent inventlon. In the following de~cr~ption, whlch ls providea by wa~ of example only, and which include~ dstails of the deriva~ion of suitable shapes for the integrally formed ~ection~ of an instrument constructed according to the pre3ent invention, reference will be made to the accompanylng dr~wings Brlef Descript~on of the D~a~ln~s Figure 1 illus~ratQs a simplif~ed pentangular ~hape whlch ha~ been used a~ t~e startlng point in thQ m~the~atlcal ~odelling o~ the preferred ghapes of the present lnvention~

~igure 2 i~ a gr~ph ~ow~ng variatlon ~f t~e frequencies of the ~irst few in-plane modes of an instrument constructed 20 ln accordance with the pres~nt lnvent~on, rom a t~n rod ~ent to form i~e ~ect~ons.

Figure 3 sho~g ~olution gurfaces in a 3-dimQnsional configuration space, which ~ referred to ~n the explanatlon of the deriv~tlon of a gu~table shape for the preQent inventlon.
~ ~-.,, " ., .:, ' :' : '"

2131691 ~
WO93~18~3 PCT/AU93/~101 Flgure 4 ~llustrates contours ~n the ~e, ~} su~p~ce, whlch is also referred to in the explanatlon of t~e der~vation of a suitable ~hape for t~e present inventlon.

Figure~ 5 and 6 illugtrate the pentsngul~r ~hapes of two S i~struments constructed in accordance wlth the present ln~ention.

~igure 7 ~s a partly schematic illustratlon of an asfiembly of lnd$vldual pentangular instrument~, con~tructed ln accordance with the present invention.

Deta~led PQscription of Illustrated E~bodl~ent~
Because ~1) the present invention can ~e regardea as a signiflcant improvement of the orchestral trlangle, althou~h used for a dlfferent mus~c~l purpose, (11) the triangle is normally constructed by bendlng ~ ~etal rod, and ~ill) lt ie sxpectsd that t~e present in~entlon will also be con~tructed by bendlng a metal rod, the followlng ~e~criptlon will be malnly dlrected to thlQ ~onstruct~on technlque. However, it should be appreclated that the sections of the integral body wh~ch constitutes the present inventive concept can be formed by castlng a metal or a metal alloy, or by pressure mouldlng and flring a ceramic material. Castlng techniques, and constructlon uslng a mechanically ~trong ceramlc materisl hav~n~ ~ultable vibrational properties (for example, certaln oxide ceramlcs), are expensi~e when compared with the benBing of a metal rod, but (a) they may enable possible problems associatea wlt~ the c~oice of a radius of curvature for a ,: : . .

2131 ~91 ~093/1~503 PC~/AU93J~101 bend in a rod to be avoided, and (b) top ~uallt~ orchestral instruments are never inexpensive.

If the percusslon lnstrument of the present ~nventlon ~s to be made from a ceramic materlal, any suitable ceramic fa~rlcatlon technique may be u~d. Mo~t ceramic bodles, how~ver, are constructed uslng t~e followlng st~ps:
( a ) a finely ground powder of at le~st o~e ceramlc material ~ mixed with a fugitlve binder;
(b) the mixture so formed is moulded to the required shape and pressed (for example, u~ing lso~tatlc pressing ;~
technlques) to form what is known a~ a "gre~n" ~ody:
(c) tho green body is then fired to a tempera~ure at ~ich ~ ;
the ceramic materlal is ~intered (durlng the early ~ -stages of the heatlng to the flring temperature, the fugitive binder is evaporated rom the green body):
and ; ~ -~d) the sintered ceramlc body is allowed to cool to room temperature at a coolin~ rate whlch ~n~ures th~t large ~ ~-cracks in the body are not created.

In the prototypes of the pre~ent ~nvention (all having a pentangular construction), ~he present lnventors have mainly used mild stQel rod hav~ng a diameter of 12.7 mm, -~
wlth the length of rod in indivldual psntan~les ~arying ~rom ~bout 0.5 m to 1.5 m. Mild ~teel rod is not 25 expensive, ls ea~ily wor~ed, and has appropriate :
vibrational propertles as far as lnternal damping and the mechanlcal admlttance of the flnished art~cle 1~ concerned.
It will be appreciated that other metals or ~et~l alloys may be used. lf the instr~ment is to be mdde by metal :, ::
:

-: .: . , ... ;

:
:

-- 2~31~91 W093/1~3 ~cT/A~93loolol casting techniques, bronze ls a particularly useful materlal. -.:
Tradition lly, musical lnstrument deslgn ~ag 2vol~ed wlthout the use of h~gher mathematlcs. PQrcussion instruments constructed in accordance with the p~esent ~nvention may be deslgned by tryin~ different combinatlons of the variable~ a~sociated wlth the lnstru~Qnt. However, mathematical mode}lln~ and analysls has enabled the present - -inventors to construct u~eful implementat~ons o the present lnvention in a relatlvely short time. De~alls o~
the mathematic~l analy~s will now be provided. -;

F~gure 1 ~llustrates, in a simpllf~ed form, a pentangular ~hape formed by bending a thin rod. The independent dimensional parameters are (i) the len~ths of the sectlons which make up the pentangle (al, a2 and a3) and (ii) the lncluded angle~ between ad~acent gections of the pentangle ( e and ~). The dlmensions of the metal rods used in the cons~ruction of the prototype instru~ents suggest that a thin-rod approxlmatlo~ ~s val~d. ~This ls the usual approximation for the beha~iour of beams that ls lmplemented in finite-element package-~.) The next as~umption (slmpliflcation) ~ade for the purpose of t~e mathematlcal modelling is that the $nstrument will be played u~ng a hammer blow ~8vlng a veloclty only in the plane of the sections form~ng the pentangle!. Such an lmpulse should excite only modes lying in the ~lane of the lnstrument. In non-ldeal cases, when other ~ib~atlon modes are excitea, the amplitudes of the 1n-plane moaes w~ll be much greater than tho~e of the non-planar modes. -~

:- . : : : : ~ . ,. . :
, ~ : : -:
: . : : .. . .: . . .

W093/1~03 PCT/AUg3tO~tOI

One ~urther simplificat~on ~hat i~ msde for the ini~lal mathematlcal analyQ~ i~ the a~sumptlon th~t the ~orners of the pentangle ~tructure are ~arp corners. Thia assumption ellminate~ one parameter, t~e corner curvature, and allows an analyt$c solutlon for the stra$ght-rod 8ect!ions, whl~h can be Joined by a~propriate boundary cond$tlons at the corners.

Now the propagatlon of tranQverse elastic waveQ along a thin rod is descrlbed by the equation ~ __ 4P ~ (1) a~ Ea2 ~2 ..
, ' - '. '~, .- ' whe~e y $s the dlsplacement normal to the rod, ~ i6 ~he co-ordlnate measur~ng leng~h along the rod, p ls the dens~ty and E the Young'Q modulus of the rod materlal, and a ls the rad$us of the rod. ThlQ equatlon, appropriately supplemented ~or long$tudinal motion a snown below, descrlbes the behaviour of ea~h stralght seCtion of rod in the pentangle, and s~mply leaves appropriate c~ndltions to be imposed (i) at the benas where two rods meetland (ii) at the free end~ of the pentangle. In this way ~n analyt~c ~olutlon to thl-q approxlmate repr~sentatlon of the real problem can be achieved in ~uch a way as to allow s~mple and rapld calculatlon of the normal mode fre~uencies.

Inspection of Equation tl~ shows t~at its general solution can be written in the form :" ~; ~, ',. ..

, ~'''';

:-. : ~ . - . -WO 93/18~03 PCI/AU93/00101 y~(x) '~ I~nCO~ nc4~;h y"
whe~e k is the wave number, given, from Equati~ns (1) and ~2), ~n terms o the ~n~ular frequency ~ by t~e relati~nship~
4 4~2p E~
In Equatlon (2), which refers to a sect~on of ~Oa labelled by ~he subscript n, the quantitie8 on, ~n~ Yn and ~ are constants, the values of w~ch ars determihed by the ~oundary conditions at the two ends of thl~ ~ection of rod.

E~uation (2), however, describes only displacements normal to the axis of the rod. To complete the description of ~he ~ibrations, the possib$11ty of dl placement parallel to t~e axis of t~e rod must be allowed. ~or the section n of the rod, the BymbOl ~n ls u~ed to denote a parallel dlsplacement. Ea~h ~ is taken to be consta~t along the length of the relevant rod, wh~ch means that the possibillty of longitudlnal waves in the rod ~aterial is ignored. This ls physically justifled, s~nce the frequencies of the normal moaes assoo~ated with ~-longitudinal waves are much higher than thosl o ~ending modes, and they are therefore ou~slde the frequency range in which lie the mode~ to be tuned.
: ':
Since there are five strai~t section~ of rod in t~e pentangle structure, there are Z5 unknown coefficients.

W093/~3 2 1 3 1 6 9 1 ~CT/~U93/0010]

- 1 0 ~

~ote, however, that the pentangle -Qtructure ha~ a plane of mlrror symmetry, ghown by t~e axls OC ln F~gureil, ~nd this implies that the modeR must be either s*mmetric or antiQymmetric in relation to thi8 plane. Applying thls S condltion reauces the number of unknown coef1cient~ to 15.

These 15 coefflcients, however, are not su~ficlent to describe the dynamics of the problem. In pa*ticular, an appropriate balancing of force~ ~nd moments at the corners o the pentangle structure is re~uired. ~he ~ending moments are properly de~cribed in term-~ of (l) the elastic moduli, (~i) the rod radius, and (i~i);the second derivative of the normal displacement y. The shear forces s~milarly involve the elastlc ~oduli and the third derivatives of the normal displacement. A de criptlon of the tension forces in the rod, ~owever, ~equlres the lntroductlon of tension force~ T, which vary along the lengt~ o each rod. Being concerned only wlth ma~chlng condltions at t~e corners, therefore, introduc~ a further S lndependent quantitles. ~hese can be deslgnated, wlth reference to Figure 1, ln terms of the symbol used for ths corner (or the centre 0~ and the rod number, as TO, T " TA, TB, ~B The ten~on at the free end C clearly vanishes.
Symmetry con~iderations allow tensions for one half of the pentangle o~ly to ~e specified.

Addlng the S tension quantlties to the 15 lndependent dlsplacement parameters glves the total of 20 independent par~meters necessary to speclfy the dyn~m~cs of the simpltfied thin-rod model. ~hu~ 20 linea~ly lndependent equations relat~ng these quantities are now requlred. Once ' ' '~; ' ~ :"

- -;

these have been written down and solved, there is presented a nonlinear equation in the wave number k, or equivalently in the frequency ~, the ~olutlon.~ of whlch ara t~e mode frequencies for the pent~ngle.

Now for each of the two bends A and B (gee Fi~ure 1), it is requlred that the rods or sections ~oin together in a contlnuou.~ manner S2 equations) and that their siopes ~y/~x match (1 equ~tion), ~o that the bend angle i~ not distorted (distortion of the be~d angle would take an ininlt~ moment about the ~oln point). ~urthermore, cons~deratlon of a t~ny element of rod at the bend shows that, if it~ motion is to remain flnlte, the bend~ng moments ln the ~wo rods a~
the ~oin must be equal, lmply~ng contlnuity of a2y~x2 (1 equation) Flnally, the forces exerted on the~elemen~ by the two rods must balance in two orthogonal directions ln the plane (~1vlng 2 equations involving a3y/dx3,.T, and the bend angle). This gives 6 equations at A and ~ further 6 equatlon~ at ~, maklng 12 equatlons in all.

At the free end C, the bendlng moment and the ~hear force must both vanlsh, giving d2y/~x2 = O and a3y/ ax3 = O (a further 2 equations). $be tension force has already been set equal to zero by not lncluding lt among the unknowns.

The longltudinal mot~on of each rod or section under the influence of the d~fference between the tenslons at its two ends is now considered, ~ls difference in tensions ls equal to the product of the mas~ and acceleration of the rod, and is thus proportional to t~e displacement ~uantity 213169~ `

~), the rod length, cross-~ectton and denslty, ~nd the square of the frequency (~ equation ).

Th~ gives a total of 17 llne~rly indepondent oquat~ons.
The remalnlng 3 equatlons needed to dQtermlne the ~- -parameters nece~sar~ to spaclfy the dynam$c~ of the thln-rod model are derlved for the point 0 on the symmetry plane, and are dlffer~nt for symmetric and antisy~metric modes. ~or the ~ymmetric modes case, clearly ~y~x ~ O and a3y/ax3 e 0~ w~ile the necess~ty for a gtatlonary centre of mass requireis that ~ - 0. For the antis~mmetrlc;modex case, s~mmetry dictates that y . 0, ~2y/aX2 ~ O, ~nd ~'y/~x~ ~ 0 In either css~, 3 additional e~uatlons ar~ provldsd.

The 20 equation~ are homogeneous, since no external orce~
are ~nvolved, and the necessary and sufflcl~nt condltlon that they have a real solutlon 18 that the dQtermlnant of the matrix of thelr coef~icients ~hould vanlsh. T~l~
determinant 18 complicated, ~or lt wlll be seen from Equation ~2) thst the coe~ficlents ~nvolve quantltle-~ such a~ co8 ~a and cosh ka. The present in~entor~ u~ed one of 20 the computer programs publlshed in the book by W H Press, P Flannery, S ~ Teukolsky ~nd W T Vetterling, e~tltled "Numerical Recipe~" (Cambr~dge University Preg~, New ~ork 1986, page 39), for e~aluating a deter~in~nt once its -~
element~ are given nu~rical form, by cho~oslng~a vaLue of k. The selected program was used to search for those values of k for whlch the determinant vanigh~s. Equation -(3) was then be used, with values of the ela~tic constant~
lnserted, to ~on~ert the~ k values to frequencles. A
~eparate ~omputer program to perform th~s operatlon wa~
.
.: ..~ ,.

: :: , : .: - -W093/18503 PCT/AU93/~101 - 13 - .

written. It gave t~e flrst 6 or 7 mode fre~ue~cie~ ~o good precision in only a few minutes on ~n A~-compatible microcomputer u~lng Microsoft QulckBasic. The ~peed of thi~ part of the analy~l~ cou~d have been further $mproved by first using algebraic manipulation to reduce.the rank of the determinant. There w~ no problem a~o~t requir~ng extra constralnt~ and ellmlnatlng r~gld-body modes a~ there is in 80me lmplementatlon8 of the cbrresponding finite-element calculation.

The results of a calculatlon uslng thi~ an~lytlc approach are given in Figure 2, whlch ghows th~ ~nrlation of the first 6 mode frequencles as a 1 m length of 12.7 mm diameter steel rod is progregglvely bent lnto a.rectangular 8hape and then unbent in the oppo~ite order. ~n Figure 2, t~e angles e and ~ are a~ defined ln F~gure l ~nd the sectlon lengths are cho-~en ~o that R2~=aJ4l=2, 3/a Clearly thQre are large chan~es in the relative,frequencies of the modes, sugge~ting that t~ere is a strong llkelihood that a shape mlght be achieved w~ich gives a nearly harmonic relation~hip between gome appreciabl~ number of the frequencies of the modes. H~ving determine~ a suitable s~ape to first order, u~lng thi~ analytlc appro*imatlon ~nd the tuning philosophy outllne~ below, ~t ig ~ relatively straightforward exercise to use a finlte-element package to -: .:: , .:

: : - ~ ~-.: :

.:

W093tl13~03 21 31 691 Pcr/Aug3/oolol ~,, ~ ,.
- 14 ~

ref~ne the shape by lncluding the finite curvature of the corners.

As noted above, ~y restrictin~ the ln~tru~ent of the present invention to gymmetr1~al ghap~ and leavlng aslde S the possibll~ty of changing the curvature~ ~t the bends, lt ls al~ar from Flgure l th~t flve parameter~ (a~, a2 ~ a3, e and ~) are available for tuning the lnstrument. ~is suggests that it is possible to tune f~ve mo~es or, mo~e usefully, a bas~c pitch and four mode-frequency ratios relative to lt. ~his is essentially the numbe~ of modes explicltly tuned ln a church bell or a carillon bell.
:
In the pentangle ~hape illustrated in Flgure l, the basic pitch is determined by the overall length of the eomb~ned sections ta, 1 2a2 ~ 2a3). It iq convenlent to take the parame~er set, for tuning thls lnstrument, to be (R2" R3"
e, ~ ), where R2l = a~/a, and R31 = a~a~

In percus~lon lnstrumentq of the bell or ~ong family, the sound can be li~tened to in two wa~s, known:as ~olistic li~tening and an~lyt~cal llsten~ng. In holisti,c li~tening, the perception ls of a well-def~ned mu~cal pitch and a characterlstic musical timbre or tone-quallty~ In `analytical llstening, the perception ls of ~he set of lnd~vidual parti~ls making up the ~ound, ~or a successful mu~ical instrument, t~e relations~ip between the partial~
ha~ to be ~uch as to encourage holistic listening, and this ls most readily ~chieved if t~e most prominQnt partials ha~e frequencies ~n integral (har~onic), or nearly integral frequency relationsh~p.
;,, :.. ~.::. ..:
:~,:,. : , . :.

, ,.:~ ~ - - ~:

, - . ...
~': ': . :-" ' ', W093/18~03 PCT/AU93/00l0l Such a relatlonship can be wrltten as a produ~t of pr~mes 2n3~5~ where n, m, s ... are ~mall po~ltive or negative lntegers. The de~ree of accuracy of the requlred tuning is hl~he~t lf only the factor 2 is involved (that ~s, when the tunln~ produce~ promlnent partials ln oct~ve~ ) . The degr~e of accur~cy ls atrly critlcal if both 2 and 3 occur (that 1~, the promlnent partials arè ~n fifth4 and fourths), and lt is much 1~8s critical if 2, 3 and 5 occur, to tnclude ma~or and minor th~rd~. Tuning o~ inter~als lnvolvlng 7, 10 or higher prime~, ls very uncritical as far a~ consonance ~ -~
i~ con~erned. The exact sequence of p~rtial tones in the sound, and their relatlve strengths, has a gr~at be~iring on the sound quallty, ~ does also the strength and general freguency distribution of the untunea higher partlal~
whlCh, generally, are not heard analyt~cally.

~able 1 sets out the harmonic (or "~ust") frequency r~t~os fo~ the musical pltches of concern in tunlng the pentangle lnstrument illu~trated in Figure l -Table 1 20Harmonic or "Just" Pitch Ratlo~
(~irst llne relative to Cl, ~econd line rels~lve to C3 ) .

C, 1 F, G, ~ ~ ~ C, , G. C~ ~, , ~ -~
1 Y4 4~ ~/2 4 s 6 8 10 12 16 ~5 20 24 25 ~25 3~ ~75 1 125 15 2 25 3 4 ~8 5 6 ': ' ' ~ ' .' ~ '''' ' , .. , : ~ . ~ - , . ~, : ::

2131~91 WO 93/1 8~iO3 PCI /AU93/00101 ,: ;:'' In Table 1, the subscripts to the pitch ~ymb,ol~ re~er to the octave ln whlch they occur, C~ belng "mlddle C" and C
~eing the lowest ~ on the plano keyboard. ~he notes of a keyboard instrument, 8UC~ as the plano, are tu~ed to "equal S temperament~, ln which all the fifths are flattened (tempered) b~ about 0.1 per cen~ so th~t all twel~e notes of the ~cale have the same frequency ratlo 21~2 to their ~ ~ :
nelghbours. This results in ma~or and mlnor thlrds which `~
differ from the harmonic rat~os 5/4 and 6/5 b~ about 1 per cent ~o produce a sat~ 8fylng bell-llke ~ound with a pentangle instrument of the present invsntlon, the aim i~ to tune at least four prom~nent partials $nto small-i~teger ratios with the particulsr partial tone ~ generally the strongest low partial - that is taken ~s the no~inal p~tch of the bell. If the aim i8 to produce 8 sound l~ke a church bell, then it iQ also highly desi~able to ~nclud~ a minor-third lnterval (6:5 or one of ~ts octaves) relative to this ~ ~-nominal , ~ ~ " - :,.:
For a straight rod of radius a and lengt~ L wlth free ends, the frequency of the nth mode i~ g~ven by -f A ~ (n+ ~ 5) . ....... ~.
".;~',''': .~' '' '''',~
where A ls a constant dependlng upon the den~y of the rod material and its ~oung's modulus. The frequencies of the modes thus have ratlos close to a sequence ~hich can ~e most helpfully written 0.36 : 1.00 : 1.96 : 2.56 : 3.24 ~
" :: ~, -~ ~ .:::

:
.
: ~ -:: . :::

.:.. ...
:~. ~:, , ~ '-W093/18~03 PCT~AU93/~101 , ~ .

. The lowest frequency 18 ~ell removed ~ro~ the others and is not radlated very efflclently, so that it ~ 8 logical to take the frequency of the second mode as defin~ng the nomlnal pltch. From Figure 2 ~t can ~e seen that the lowest mode of a bent rod l~ simllarly lsolated ln relat~ve frequency from the upper modes, so that the ~ame nomlnal pitch a3slgnment may be adopted for the ln~trument of the present invent~on. It -~hould al~o ~e noted that the second mode l~ also generally taken as defining the,p~tch of a church bell, the first mode beln~ called the "hum" or undertone.

The next step in the selection of a tuned configuration of a pentangle, ~herefore, i~ to explore the 4,dlmensional parameter space {R21, R", e, ~} gnd f$nd configurations for wh~ch the frequency ratios, relative to the ~econd mode a~
nominal, have the required slmple form. ~hls task i~
potentially very extensive nume~ically, but it can be slmplified greatly by proceedlng one mode at a time and 4y add~ng parameters one ~t a time a~ ~ollows.

~he suh-nomlnal fir~t mode freguency is of llttle importance, .~ince a sound re80n8tor (radiator) coupled with the instrument can be tuned to the second mode and will then radiate little at thl~ lower frequency. This ~ub-nom~nal frequency, therefore, can be neglected and only 2S the higher mode frequencieg relative to mode l considered~
Assuming reasonable values for two of t~e paramete~s, say the side-length ratios R2~ and R31, allows numerical exploration o the 2-parameter {B, ~ configuration space by computin~ mode frequencies o~er a grid of about 5 x 5 , ~, 2131 ~91 W093/18~3 PCTJAU93tO0101 polnts and dr~wing ( e, ~ ~ contours along whlch ths requlred harmonic frequency relat~onghip~ are met. The bage plane of the 3-d~men~ional configuratlon space shown in ~lgure 3 is ~n example of such contours for ~odes 3 and ~ at acceptable frequency ratlo~ such as 2:l or 3:l relati~e to mode 2. ~f a ~olution to the tuning problem exists, then these two contourg must cross at a polnt A w~thin the acce~sible {e, ~} space.

The phase space may now be extended to three d~men~lon~ by calculating a ~imilar Qet of acceptable ( e, ~ ) contours for additional valueQ of one o f the remalnlng parameters - f or example, R2~. Thls allows gurfaces in the .~di~enslonal { e ~ ~ ~ ~2~ } cpace to be drawn, corre3ponding to acceptable values of the f~equency ratios for modes 3 andl4, as shown in ~igure 3. These two surfaces will intersect in a curve AB, if a ~olution indeed exl~ts.

third ~urface can be drawn in the space o~ ~igure 3 corresponding to an approprlate ratio for the frequency of mode 5. If ~hls surface cuts the solution curve AB for modes 3 and 4, for example at the polnt S, then thi~ point repre~ent~ a.satisfactory 801ution for modes 3, 4 and 5 relat~ve to mode 2. ~he process can then be continued by including the remalnlng parameter, extending polnt S ~nto a curve, and seeking an lntersect~on with the sur~ace for mode 6 at an a~ceptabls frequency ratio.

The advantage of this procedure ~8 th~t ~t limlts the amount of conflgurat~on 8pace that mu~t be explored at each Rtep to that near ~ previously establlshed curve or .. . . - . , .-WO 93~18503 PCI~AU93/00101 eurf ace, Rnd thu~ greatly reduces the calcul~tlon time invol~ed. In fact, it ls an effectlve proc~dure, when the approx~mate locatlon of the-~olutlon point ~n configuratlon ~ -space has been ldent~f~ed, to search ~or the solutlon lteratlvely in the orthogon~l ~e, ~} and {R2l. R3l}
~ub-~paces ln turn. Thu~ Flgur~ 4 shows such a sectlon ln {e, ~} epace for ~21 S 2, R3~ ~ l, whlch is close to, but not coincident wlth, the exact solutlon ratio. It is clear that there are three reg$ons in thls sub-cpace~ marked S~
Sll and S~2 on Figure 4, whlch are close to multiple lntersection~ of the lnd~vidual solutlon surfsces for particular modes. Exploratlon for a solutlon can then be limited to the immediat~ vicinlty of these regions.

Using the method~ outlined above, these soljutlon~ were refined for the idealiged sharp-corner tunlhg problem.
Other solutlons may exlet for ~re~tly dlfferent side-length -~
ratios, slnce the exploratlon was not completely exhau~tive. Only t~e ~olutlon ~ssoclated wlth region Sl was e~sentially ex~ct. In the case of regions SI~ ~nd S1l~, the solution surfaces ~o not all p~8S through the solutlon point, but si~ply approach closely ~o lt. ~eta~ls of the lnltlal ~olutlons are g~ven ln Tables 2 and 3. .

~able 2 Inltial Conflguratlons R2l ~31 e Solution I 1.95 0.92 96 ; 93 Solution II 2.20 1.07 135 46 -~
Solution III 1.85 0.97 146 lO~
. . ~
' ~ ' :':
' ':
.- .: . .:, 2~69~ : ~
WO93J1~3 PCT/AU93/00101 Table 3 Inlt~al Mode Frequency Ratlo~

Solution I(0.35) 1.00 2.00 3.00 4.80 6.00 5 Pitches (F~) C3 C. G~ E~s ~5 ,; ;~

Sslution II (0- 2)1.00 1.51 1.98 2.99 4.80 Pltches (E1-Fl)Cg G3 C, G, E6s 10 Solutlon III (0.31)1.00 1.25 1.46 2.49 4.04 Pitches (E,) C3 E3 G3 ~ C~

With Solutlon I (see Table 2), the top ~ngles are clo-~ed so much that the two free ends of the pentangl~e 3tructure 15 o~erlap. Thls means t~at the pentangle -~tructure must be ;~
bent sllghtl~ out of plane so th~t there ~s adequate separatlon of the free ends. Modes 2 to 6 cnn be tuned exactly, ln the sharp-corner approximation, and~gl~e a well spread 8et of modes, includlng a minor third~at mode 5.
The nominal pitcheg o~ the~e notes are indicated, taklng the nominal pitch of the pentangle structure ~s a whole to be tenor-C (~). Unfortunately, the sub-nominal mode 1 has a dlssonant pitch clo~e to F~ 2 ~ but thls can be lgnored, for the reasons given above.

Solution 1~ gl~es an lnstrument of "coat hahger" shape ~hich, from Table 3, has a well distributed set of mode frequencies. The sub-nom~nal, ln thls case, is well located near a harmonic frequency. Once a~ain there ~s '' ' ~.

;
' ' .

: . . . :
:. : . : : :

WO93/18~03 2131 6 91 pCT/Aus3~lOl - 21 ~
': , -~lnor thlrd (this time at mode 6) and, s1nce mode 3 has the frequency xatio ~.5, there may be an implied un~amental at frequency rstio 0 5, an octave below the nomlnB~ pltch for psychophyslcal rea~ons.

5 In Solutlon III, whlch ~as a very fla~tened shape because of the small value of ~, the mode frequencles 8re den~ely clustered $n the range 1.0 to 2.5 and contain n~ less than three maJor t~ird-~ relative to the nomlnal p~tch. The subjective pitch may agaln be below t~e nomlnal pltch because of the close spac~ng of these mode freq~ency ratio~ The sh~pe of t~is pentangle, however, i8 not s~tisfactory for practical reasons, partlcularly when rounded corners come to be con~idered. For thls rea~on, Solu~ion III was not pursue~ by the present lnventors.

lS The final ~tep ln the ma~hemat~cal modelllng, exerclse, which would not be required if the percusslon ~n~trument 1s ca~t or moulded wlth ~harp corners, 1~ to modify the Solutions I and II to lnclude the effectQ of corner rounding. For thls part of the dQsign exexclse, the flnite-element pac~age "StrandS" (produced by G & D
Computing, Sulte 307, 3 Small Street, Ultlmo, NSW 2007, Australia) was used, again on an AT compatible microcomput~r. This package ls particularly sultable for this calculatlon because lts structure allows access to all the flles and executable modules, 80 that it is a relatlvely easy task to wrlte a batch program to perform the necessary exploratlon of configuratlon space in the immediate vlcinity o~ the inltial Solution~ I and II. It does not requlre the ~nclusion of external con~tralnts on ' , ` ' ~'"~

2131691 ~ ~
WO93~18~03 PCT/AU93t00101 ' - 22 ^

the pent~ngle. Optimi~atlon of the pent~ngl~ deslgn f or corn~r rounding effects, using thls approach, typically takes only a few hour~e.

When corner curv~ture effects are to be lncluded in the S mathemat~cal moaelling, it ~s necessary to define how the ;~
curvatures ~re to ~e me~sur~d. The choice i~ (l) between centre~ of curvature, (ii) between the intersections of the -~
axeQ of the rod segments, and (iii) in ~OmQ o~her way. ~he choice wlll have a signlficant lnfluence upon the results when the corner radil are large or the bend angle~ are large. ~;

For steel rod, the m~nimum re~onab1y achievable bend radiu~ corresponds to bending the rod around i~elf, glv~ng a neutral-section bend radiu~ about equal to the rod lS diamet~r, so that corrections to gtraight-sidQ lengt~s oP
at least thi-Q magnltude ~re lnvolved. S~nce the bend radius introduces an abQolute scale into the p~oblem, it ls neces~ary to define the total length of the rod, w~ich was ta~en to be lO00 mm. If the bend radius iq taken a~ lS mm for 12.7 mm d~ameter rod, then l~ttle ch~nge in the shape of the pentangle ~s required.

However thi~ leaves no flexibility in bend radius ~or simply-Qcaled emaller pentangleq. Accordingly, the neutral-section bend ~a~ius r was taken to be 26 mm for a l000 mm rod (corresponaing to an ~nternal bend radiu~ of 2~ mm), thu~ ~llowing f or tighter bends in smaller pentangles made from the same ro~ stock. Wlth these ~ ;
assumption~, Sol~tions 1 ~nd II were reflned as indicated ., ;:

, WO 93/18503 PC~/~U93tO0101 ~ ~ ~
. : .

: . :: ,.:

above. It wa~ found that the 26 mm radius adopted required signlficant change~ in both segment l~ngths ~nd bend angles. The flnal Sol~tions are shown in Tab~e 4, whl~h gives the stra~ght-llne sectlon length~, o~ d~tances S between centre~ of curvature fo~ the corner~, together w~th the bend angles.

~abl~ 4 Flnal Practical Designs -~

10Dimenglonsa~(mm) a2(mm) a3( mm) e ~ r~mm) .. . .. _ _ . :. ,.. :.. :, . .
De~lgn Solut$on I 103 258 109 90 90 26 Design Solutlon II 113 25~ 100 136 19 26 In the c3se of Solutlon ~, rounding o the corners ~equires only 8mall ad~ustments to the gtralght-~lde lengths and a reduction in the ~nclde angleg to approach again the deslgn mode frequencles to an accuracy of ~ette~ than 2 psr cent.
The resultant s~ape ls shown ln ~igure 5~ It i~
essantially rectangula~, wlth a large side overlap. The ad~usted form of Solution II is illustrated in Figure 6.
In thls case, the rounding o~ the lower corners produoes a considerable chang~ ln mass distribution. This neces~tates a considerable reduct~on of th!e angle relatlve to the sharp-corner conflgurat~on. Ne~erthele~s, the original calculated mode frequencies ~re regained to better than 1 per cent. The extent of this angular change, however, further supports the view that a pentangle corresponding to Solution III probably could not be made by simple bending of ~ rod.

'. ' W093J1~3 PCTtAU93/00101 - 24 ~

~y relaxing the planar con~tralnt~ appl~ed to the finite-elemQnt ~olutlon, lt ls posslblQ, ~lthou~h of limited practical l~portance, to ~alculate the `frequencies of the out-of-pl~ne vlbration mode~ in 8ddi~ion to the planar mode~. The existence of thssQ lnharmonlc out-o~-plane modes allows the p~rformer a degrQQ of control of the timbr~ of the tn~trument, slnce lt can be struck to mlnlmlse or to maxlmlse the amplltude of these mode~
relat~ve to the ha~monic in-plane mode~

10 The pentangleg as deglgned above requlre no hand-tunlng, their mode frequencles ~eing defined by their baslc shape.
The same is true to some extent of traditio~al church bells, but it is almo~t universal practice to flne-tune t~e mode frequencles o~ church b~lls by turning ~mall amounts of metal off t~e lnterlor surface of the bell on a lat~e, followlng recipe~ whlch have been establlshea by long experlence. Cle~rly the same sort of procedure could be u~ed witn pentangles, both to reduce the re~idual tunlng dlscrepancles of t~e first s~x mode~ and perhap~ also to tu~e some of the higher modes.

A practioal approach to this tunin~ problem for the case of bell~ has been developed R G J Mllls, and i~ describea ln his paper entltled "Tunlng of Bell~ by a Llnear Progra~mlng Method" which wa~ publ~shed in the Jourhal of the Acoust~c~l Socie~v of Amerlca, Volume 85, pa~Q 2630-2633 (1989). Thl~ approach lnvolves evaluatlng ths effect on the tuning of all modes of th~ removal of a ~mall amount of metal from each of a large number of bands along the interior surf~ce o the bell. A ltne~r programm~ng :: :: : ~ : : :
, .

WO 93/18503 P(~/~U93tO0101 procedure i~ then used to define a metal-removal schedule that w~ 11 produce opt~mal tuning. Such a procedure may be readily implementQd $or a bent rod pentangle by uslng the finlte-element package to determine the effect of flling metal off the rod ~n varlous locations. W~th cagt metal or moulded ceram~c pentangles, ~lne tunlng c~n ~lso be efected by v~rying the cross-sect~onal s~ze ~nd shape of the ~ectlon~ of the pentangle, or by includlng a wlde-~ect~on "weight" at an appropriate location on a ;
section. It ~ also theoretically possible to f~ne tune a metal pentangle by weld$ng a "weight~' to lt. However, ~t ~--is t~e belief of the pre~ent ~nventors that, in practice, fine-tuning will be generally unnecessa~y. -Once a suitable design ha~ been achieved for a pentangle of 15 some as~umed sl~e, lt is pos~ible to scale this design to the s~ze~ necessary to produce a required set of nom~nal p~tches for a musical scale. There are three possible approaches.

Conceptually the most straightforward appro~h 18 ~imply to Qcale all the dimensions of the pentangle (rod length, diameter and bend radius) uniformly. From Equ~tion (4), the mode frequencies will then all var~ in~ersely with the scale factor. The pract~cal d$fficulty with this apProach is t~at lt requires a different rod diameter and bend radius for each pentangle o~ the ~et.

At the other extreme, ~oth the rod diameter a~d the bend raaiu~ mlght be ~ept constant and only the lengths of the rod section~ scaled. This would require performlng a .:, , .
~.' ' ~ " '' ...... ~ .. : : ., ~, : ~ , .. . .

::
2~ 31691 W093/18503 PCT/AU93~0010]

: . . ~ .:

~eparate finlte-element optlmlsatlon to lncorporate corner rounding for each different memher of the ~et, and the limitation~ imposed by a fixed bend radlus mig~t mean that the overtOnQ structure of the rQsulting pentangles ml~ht ~ve to chan~e at certain no~ nal pltches. This is not very ~at~sfactory.
The most practlcally appeallng scallng approach, ~herefore, i~ to use the same diameter rod for all pentangles, scaling rod length and bend radius, and to use once more the scaling law Equation (4), which shows that the mode frequencieQ in th$s case vary inversely aq the ~quare of the 5cale factor. This ha~ econom~c advantage~, even though a different bending dle has to be made for each size of pentangle~

Table S ~hows the measured mode frequency ratlos of the fir~t two pentangle lnstruments constructed b~ the pre~ent inventor~ accordin~ to the calculated curved corner deslgns. The measured deviatlons from the calculated frequenc~es can be ascribed in large mcasure to small deviatlons fro~ the deslred geometr~ of the pentangle, slnce the rod was bent b~ hand ln a ~imple Jig. Agreement is certainly adequate to valld~te the deslgn pr~nc~ples.

:: ~ .

.

. ' . ~, .. : , , :-,: ~ : : . : :

:

2 1 3 1 6 9 1 ~

W0~3/18503 ~CT/AU93/0010 - 27 ~
`' -'''~" ~,' ~ ', ', " , ''' Table 5 ~ .
Calculate~ and Me~sured F~equency R~tio~

~, . ~ , .
Mode Nu*~ers 1 2 3 4 5 6 S - .. -,~
Design Solution ~ 0.35 1.00 1.96 3.05 4.82 6.13 Pent~ngle I 0.35 1.00 2.01 3.05 4.79 5.93 (measured) Out-of-plane 0.39 1.43 2.40 ? 4.~7 9.57 ;~
(measured) '.
Design Solutlon I} 0.32 1.001.50 1.99 3.04 4.76 Pentangle ~I 0.33 1.00 1.491.96 3.05 4.75 (measured) -15 Out-of-plane 0 83 1.06 1.422.57 4.03 5.17 ..
(measured) .

Also ~hown ~n ~able 5 ure the fre~uencies of the out-of-plane modes, which form an inharmon~c se~ies . ;
~nterlacing those o~ the planar modes. In practlce, and despite the lack of planar~ty of the pentangle of ~i~ure 5 :
mad~ necessary ~y overlap of the ends in Solutlon I, lt is eas~ly posslble to execute the stri~e so as to exolte almost exclusively the in-plane or the out-o-plane modes. .
The large inh~rmonicity of the out-of~plane modes creates a very dlerent sound, and thus places some .lnterestln~
effects in the hanas of the performer.

WO 93rl8~3 PCT/AU93~00101 ~-"

' In ~udglng mu~lcal effectlveness, the view can be taken ths~ ~imul~tlon of the g~und of a tra~lt~onal we~tern European church bell l~ being attempted, or it can slmply be requi~ed that the ~ound be pleasant, a matter whlch can be ~udged only su~ectively.

For a tradltional church bell, the first f~ve modes are (i) the ~um or undertone, wlth frequency l:2 relatlve to ~-- the second mode, (ii) the fundamental or pr~me, which i5 .
the reference frequency, (lii) the t$erce or minor thtrd (6:5), (lv) the qulnt or perfect flfth, and (v) the nominal or octave (2:11- Clearly neither of the f~rst t~o pentangles constructed by the present inventors comes close to this set of frequencles. Solution I contalns the ~;
correct frequency ratlos, includlng the mlnor thlrd, ~o within power~ of 2, except in the case of the ~lrst mode, but they are spread over ~everal oc~aves rather than belng concentrated. Solution I~ has more closely clustered mode frequencies ~nd again has a minor third. Solution Il~, lt will be noted, has mode frequencle6 clustered ~oxe like 20 those of a church bell but, a8 explai~ed above, this design ~ :
has not been implemented for practlcal reasons. ~he ~;
mu~ical effec~lvenes~ of the present inventlon, therefore, cannot be a m~tter of exact slmulatlon but must rely upon the production of an approprlate ~ub~ectlvely bell-llke ~ound.

To ~um~arlse, a method for optlmising the ~hape of a pentangular framework $n order that lts flrst few modes sho~ld form a 3erles with harmonically related frequencies has been de~crlbed above. The method is relatively . . : , . . . .

, .: ~

213~691 W093~1~3 PC~AU93/00101 ~tr~lghtforward and csn be generalised to tune elther a different set of modes for the pentangle, or to tune a ~et of modes ln a framework of dlfferent geometry, although the number of a~ail~ble parameters lncreaseg rgpi~ly lf many mose sectlonQ are used.

Slnce making the flr~t two "pentangles~, the present inventors have constructea a percusslon lnstrument wlth thlrteen "pentangles", as shown schematically in Figure 7.
Each pentangle 60 i~ constructed in the form shown in ~lgure 6 and is suspended from a frame 61 us~ng a nylon thread. The pentangles are also connected to a ~road-bana, non-resonant soundboard 62, backea ~y a cavlty or sound box 63, to enhance the sound tr~n~mis~on of the ~nstrument.
Both the soundboard and the sound box are tapered from the 15 ~a~ to the treble end, snd are coupled to the pent~n~le~
by ela8tlc cords (wh~ch are ateached to the ~oundboard at polnts along a non-central line). ~he soundboa~d ls bxaced by rlbs glued to lts back face, as ln a ~u~tar or harp3i~hord. The distribution of the~e rlbs ~nd the volume of the backing cavity or box are such that there ~s an appropriately ~haped raaiation response over the playin~
range of the instrument. The thlrteen graded ~ize pentangles were "tuned" to enable the instrument to play a full scaie.

2S This ln6trument produces a mellow concordant sound when the pentangles are excited by a blow from a hard mallet. When a soft beater ls usea by the percusslonist, the ln~trument produces a warm, harp-l$ke ~ound. ~ncrea~ln~ the hardness of the beater or mallet increage~ the hlgher frequency W093tl8~3 P~r/A~93/00101 ;
content of the sound produced, 80 that a bell-l~ke qound le . .
produced when the pentangles are struck w~th a hard mallet.
A mor~ percu8~1ve gound ls producQd if a pentangle ls hlt on on~ side The fsct that out-of-plane vlbrat~on modes are not a~usted to harmonic relationshlp ha3 been found to give a u~Qful degree of tonal frQedom to the performer, a po~ibillty that could be enhanced by mak~ng each pentangle from a rectangular steel ~ar instead of from a Gteel rod.
,~
A number of microphoneg hav~ been ~nstalled within the ~oundbox o~ t~e instrument illustratea ~n Flgura 7. These microphones enable the instrument to be used w~th the acoustic resonator, w~th an electron~c amplifiér, or wlth both acoustic ~nd electronic ampllflcat~on of the sound -whlch is produced.

For the sake o completeness, an ex~mple of the scallng of a pentangle (used by the presen~ inventors to te~t the scaling ~pproach sub~equently adopted) wlll now be g~ven, together with iniormation aboùt the use of sound radiators wlth the "pentangles" of the present lnvention. ;~

~cal~ng Example The aim of thls scaling example was to produce an instrument of the Solution I shape ( seQ Figure 5) for the note E1 ~ 41 2 Hz. Solution I was chosen ~or the pentangle shape since $t ls ~ore compact and use~ 1QQS rod material than the 8hap~ shown ~n Flgure 6 (Solution II). H~cause the pentangle is rather lsrge, 14 mm diameter ~teel rod was used to give adequate wei~t and rigidity.

. - : .:: ~- ~: - . -:~

W093/l8~ 2 1 3 1 6 91 rcl ~U93/~D10l ~ .
The original Solutlon I pentangle was rectan~ulax in shape wlth e ~ ~ soo. It had a total rod lengt~ of ~13 mm.
The section lengths, meagured for tha str~ight gectlons, were a~ ~ 85 mm, a2 ~ 212 mm, a3 ~ 89 mm. There was a neutral sectlon bend radius of 26 ~m at egch corner. When made from 12.6 mm diameter rod, thls pent~ngle gave ~n ~nternal bend radius of approxlma~ely 20 mm. ~he pentangle had reasonably good t~ning and a measured fre~uency of about 185 Hz. Th~s pentan~le was to be ~caled to produce l~ the requlred lower pitch for the n~t~ E~

Assuming that the or~glnal instrument had a rod length 1~
a bar dlam~ter d, and a frequency fl, while t~e new ~nstru~ent has a bar diameter d2 and a design fre~uency f2, then the rod length 12 or the new instrument i8 gl~en by the scallng algorithm ~f2 a~

The new pentangle was ~uccessfully constructed uslng this relaff onshlp for the rod length, ~lth an equivalent inner bend radiu~ of 38 mm, and wlth the interior bend angles rema~nlng at 90~

Any one o a number of d~fferent 80und radlators may be matched to the percussion instrument of the present in~ention to enhance sound transmlsslon, but on the basis of experience a resonant structure is preferre~ to a wlde band radiator when the lnstrument comprises a slngle ': :' : '~' ~' .

. . ; : . -.

W093/18~V3 PCT~AU93/00101 p~ntangle. The ~mplest rQ80~nt ~truoture 1R a d~aphragm coupled pipe reson~tor. However, a tube-loaded cavlty re~onator or an alr-loaded re~onant dlaphragm may be used.
These alternatives ar~ considered b~low.

S At 41.2 Hz, the normal sound wa~elength l~ about 8.25 metres, so that a quarter wa~e pipe re~onator, driven a~ a hlgh lmpeda~ce, will have an acou3tic length of 2060 mm.
This is long enough to require fold~ng, ~ut may st~ll be practicable. Suppoce the resonator ~s made from pipe of internal diameter d, then lts phy~ical length L shoul~ be ~=2060-0.3d mm where d l~ also in m~llimet~es. If th~ pipe i-~ bent ~ack along itself uslng two ri~ht angle bends, then the len~th should be mea~ured roughly along the centre llne of the plpe. It is be~t to make the pip~ too long by perhaps 300 mm ana to cut a slot abou~ on~-thlrd of the dlameter ln wldth along the length of the excess ~ection. ~h~s slot can then be ~overed over pro~ress~vely to tune the pipe (as in some or~an p~pes). Alternatively a tunlng sleeve could be fitted outQide the pipe for slidlng over the plpe to increase its length. The diaphragm cover~ng the drl~en end of the plpe should be only moderately taut, ~nce the frequency at whlch ~t must vlbra~e i~ quite low. An optlmal comblnation of d~aphragm thickness and tenslon re~ulre~ experlmentatlon.
.~, ,: .
2S ~ design a Helmholtz cavity resonator con~l5tlng of a relatlvely l~r~e volume, ~he 3hape of whIch is not .

: , : . : , , -2131691 :

WO93~18~3 PCJJAU93/00101 :~ ' .

important, coupled to the envlronment through a tunlng plpe and driven by meanQ of a membran~ in lts base or side wall, ~;
take the volume of the cavity as V (in cubic metres) and a~sume that the coupling is effected by a pipe of length ~
and diameter d (both in metre8). At the chosen frequency of 41.2 Hz, ~he pipe length of guch a re~onator ie given by the r~lationshlp -~

L-2d -06d For a cavlty with a volume of about 0.6 m3 and a p~pe dlameter of 100 mm, the required pip~ length is about Z70 10 mm. Some of thi~ length could protrude into the interior - ~ -of the cavlty. ~ain, lt might be advantageous to provlde a means for tunlng the length of the pipe.

Another type of re~onant radia~or that may be used with a - ~-single pentangle 18 analogous to the membrane and ~ettle of the tympani or, in a ~impler form, to the ~e~br~n~ of a ba~s dr~m. The membrane would be tuned, as ln the tympan~
to the nominal pltch of the pentangle. The attachment cord of the pentangle instrument ~hould meet the membrane at about the point chosen for strlking the tympani - that is, about one thi~d of the way in from the edge - and not at the centre of the membrane. ;~
- - ::~ :,: ,:
The ba~s drum resonator 1~ rather sim~lar, though the tun~ng i8 much le3s critical and the attachment polnt mlght well be in the centre of the membrane, rather than off to ' ' .
' '~ .
' : ' , ~:

W093~1~03 PCT/AU93~00101 one side. However, it would give a much le~ resonant sound ~han the tympani-type resonator, wh~c~ ha~ s~veral modes ~n nearly harmonlc relatlon.

~ hose skilled in t~e art of mu~lcal instrument manufacture and mathematical modelllng will appre~iate that the adoptlon of the term ~pentangle" is perhaps inappropriate when the structure has the shape shown in Figure 6, with the fifth angle (of the "open corner") not readily apparent to non-mathematicians. However, the present inventors prefer to use thls ter~ in view of ~a~ the mathemat~cal modelling approach used and (b) the relationship of the present invent~on to the orches~ral triangle. I~ wlll also be appreciated that the ~nventlon descrlbed above ls susceptible to varia~ions and modificatlons other than those speclflcally descrlbed and lllustrated ~n this speclfication, and ~t is to be understooa that the invention includes all such varlatlons and modlfications that fall withln the sCope of the followlng claim~
Included in those variations and modificatlons are the use of hard plast~c and advanced materials to form the sections of the instrument, the provis~on of one or more arcuate or curved sections, tubular constructlon of the sections, production of several connected pentangles (or other mult~-seetion ins~ruments) by a multiple castl~g technique, and - in the case of an instr~men~ of the type illus~rated ln Figure 7 - the inclusion of mechanlcal beaters, attached to or separate from the instrument. This list is not lntended to be exhaustive.

Claims (18)

1. A percussion instrument comprising a plurality of more than three elongate sections, said sections being substantially coplanar non-colinear, and formed integrally from a material which, at room temperature, is rigid and has vibrational properties such that, when one of the sections is struck with a mallet, the instrument emits a musically concordant sound.

2. A percussion instrument as defined in claim 1, having five sections.

3. A percussion instrument as defined in claim 1 or claim 2, in which the lengths of the sections and the angles includes by adjacent sections are such that, when one of the sections is struck by a striker to excite vibrational modes lying in the plane of the sections of the instrument, the sound that is emitted by the instrument has a series of harmonically related frequencies.

4. A percussion instrument as defined in any preceding claim, in which each section is of substantially uniform cross-section, and the sections of the instrument have essentially the same cross-sectional shape and dimensions.

5. A percussion instrument as defined in claim 4, in which said sections are sections of a metal or metal alloy rod or bar, and the instrument is formed by bending the metal or metal-alloy rod or bar.

6. A percussion instrument as defined in claim 5, in which the metal is mild steel.

7. A percussion instrument as defined in claim 1, claim 2, claim 3 or claim 4, in which said sections are formed by casting a molten metal.

8. A percussion instrument as defined in claim 7, in which the metal is bronze.

9. A percussion instrument as defined in claim 1, claim 2, claim 3 or claim 4, in which said instrument is formed by (a) pressure moulding a mixture of particles of at least one ceramic material and a fugitive binder to form a green body having the required shape of the instrument, (b) firing the green body at an elevated temperature, then (c) allowing the fired body to cool to room temperature at a rate sufficient to avoid the formation of large cracks in the fired body, thereby producing a percussion instrument of a mechanically strong ceramic material.

10. A percussion instrument as defined in any preceding claim, said instrument being made by bending a steel rod of radius 26 mm and having five sections, characterised in that (a) the five sections form a shape which is mirror-symmetric relative to the central point of the central section;
(b) the central section has a length a1;
(c) each section adjacent to the central section has a length a2;
(d) each section remote from the central section has a length a3;
(e) the included angle between the central section and each adjacent section is .theta.;
(f) the included angle between the sections of length a2 and a3 is .PHI.;
(g) the ratios a1:a2:a3 are substantially 1.00:2.50:1.06; and (h) .theta. and .PHI. are substantially 90°.

11. A percussion instrument as defined in any one of claims 1 to 9, said instrument being made by bending a steel rod of radius 26 mm and having five sections, characterised in that (a) the five sections form a shape which is mirror-symmetric relative to the central point of the central section;
(b) the central section has a length a1;
(c) each section adjacent to the central section has a length a2;
(d) each section remote from the central section has a length a3;
(e) the included angle between the central section and each adjacent section is .theta.;

(f) the included angle between the sections of length a2 and a3 is .PHI.;
(g) the ratios a1:a2:a3 are substantially 1.00:2.21:0.88; and (h) .theta. is substantially 136° and .PHI. is substantially 19°.

12. A percussion instrument as defined in any preceding claim, including means to support said plurality of sections.

13. A percussion instrument as defined in any preceding claim, in which said sections are acoustically coupled to a sound radiator, to increase the efficiency of sound production by the instrument.

14. A percussion instrument as defined in claim 13, in which said sound radiator is a resonant radiator.

15. A percussion instrument as defined in claim 13, in which said sound radiator comprises electronic amplification means.

16. A percussion instrument comprising an assembly of a plurality of percussion instruments as defined in any one of claims 1 to 11, each percussion instrument in said assembly having a respective characteristic pitch which is different from the characteristic pitch of each other percussion instrument in said assembly; each percussion instrument being mechanically and acoustically coupled to an acoustic radiator or to a respective individual acoustic radiator.

17. A percussion instrument as defined in claim 16, including electronic amplification means associated with the or each acoustic radiator.

18. A percussion instrument as defined in claim 1 or claim 16, substantially as hereinbefore described with reference to the accompanying drawings.