Patent Publication Number: US-10311840-B2

Title: Curved pedal

Description:
FIELD OF THE INVENTION 
     The present invention relates to a curved pedal and to a device employing a curved pedal; in particular, the present invention relates to a curved pedal for a drum or other foot-operated device where dexterity, responsiveness, and/or comfort when operating for an extended period of time are desired. 
     BACKGROUND 
     Many devices employ pedals for foot-actuated operation. Among the many devices capable of foot-actuated operation by way of pedal(s) are automobiles, helicopters, airplanes, backhoes and other such vehicles and heavy equipment, looms, sewing machines, treadles, knitting machines, mills, lathes, pumps, and other such industrial apparatuses, to name just a few examples. 
     Another category of device which may employ pedal(s) for foot-actuated operation is musical instruments such as organs, pianos, and other keyboard instruments, as well as drums, cymbals, and other such percussion instruments. 
     Drum pedals have been used for playing drums for more than a century. Many improvements on the drum pedal have been made, allowing better operability and facilitating various performance styles. 
     One factor still in need of improvement with pedals currently on the market is comfort. Repeated multiple beats, e.g., doublets, triplets, etc., provide an attractive performance but can be difficult and tiring for many players. Many players find that their foot becomes fatigued after performing for an extended period of time, especially when generating repeated multiple beats in rapid succession. 
     Another factor still in need of improvement with pedals currently on the market is ability to accommodate various techniques. 
     To generate a doublet, i.e., two repeated beats, a player might simply repeat the same foot movement twice in rapid succession, or for improved comfort and greater degrees of freedom during playing a player might, for example, employ a sliding technique or a heel-toe technique. 
     In a sliding technique for producing a doublet, a player might first depress one location of the drum pedal with his or her toe to generate a first stroke, slide the foot along the pedal toward the toe or the heel end of the pedal, and then depress a second location of the pedal to generate a second stroke. However, with a conventional flat pedal, many players find foot positioning difficult and find the sliding motion difficult to control or uncomfortable. 
     In a heel-toe technique for producing a doublet, a player might first depress the pedal with his or her heel to generate a first stroke, and then tilt the toe down to depress the pedal with his or her toe to generate a second stroke. This technique can cause fatigue of the ankle when playing for an extended period. 
     Similar techniques may also be employed for producing a triplet, i.e., three repeated beats, which is generally even more difficult than a doublet. 
     Conventional pedals are typically flat, or where such conventional deviate from planar, they may have spiky protrusions, and may employ joggled or stepped surfaces. 
     With a flat drum pedal, techniques such as the sliding technique and the heel-toe technique are tiring and are difficult to master. A flat pedal is generally devoid of features that might assist the player in locating the foot during playing. Unless a player can quickly and reliably locate his or her foot by the “feel” of the pedal, it will be difficult to develop the dexterity required for advanced sliding and heel-toe techniques. 
     Furthermore, a flat pedal is a poor match for the shape of the foot, and a flat pedal requires considerably more movement of the foot and/or ankle than would be necessary if the pedal were a better match for the shape of the foot. 
     Moreover, when using the heel-toe technique with a flat pedal, the heel and/or toe tend to strike the pedal surface at a glancing angle. A pedal shape that would permit the foot—and in particular the heel of the foot and/or the ball of the foot (note that the term “toe” as used herein may include the ball of the foot)—to strike the pedal at an angle more nearly perpendicular to the pedal surface would improve the leverage or efficiency with which force is transferred from the player&#39;s foot to the drum pedal, permitting stronger and/or less tiring performance. 
     Furthermore, a pedal surface that is interrupted by spiky protrusions or sharply stepped surfaces is not conducive to techniques that utilize sliding motion of the foot across the pedal surface. Moreover, a pedal having a smoothly varying contour would be especially desirable for a player who employs bare feet or who wears socks but no shoes or who wears thin shoes or other such foot coverings for improved comfort and sensitivity in locating the foot on a pedal. 
     In addition, whereas conventional pedals tend to be only slightly longer than the foot of the player, a pedal that is substantially longer than the foot of the player would not only increase leverage about the fulcrum of the heel hinge, permitting more powerful and/or less tiring playing, but would also facilitate more sustained sliding along the length direction of the pedal. A pedal substantially longer than the foot of the player may also accommodate multiple striking locations beyond the basic heel-toe striking positions employed conventionally. 
     There is therefore a need for an improved pedal that addresses at least one of the foregoing issues. 
     SUMMARY OF INVENTION 
     One aspect of the present invention is a curved pedal. Another aspect of the present invention is a pedal assembly or other device employing such a curved pedal. One embodiment of the present invention is a curved pedal for a drum or other foot-operated device where dexterity, responsiveness, and/or comfort when operating for an extended period of time are desired. 
     In accordance with one embodiment, a curved pedal may have a pedal reference plane, width direction, and length direction. 
     The curved pedal may comprise an actuatable region for actuation by a foot. The actuatable region may be disposed at a top surface of the curved pedal. 
     The curved pedal may comprise at least one curvature profile in the length direction within at least a portion of actuatable region and/or within the entire actuatable region. 
     Slope at the top surface of the curved pedal relative to the pedal reference plane may vary smoothly within at least a portion of actuatable region and/or within the entire actuatable region. 
     Change in slope as a function of position in the length direction, i.e., the second spatial derivative with respect to position in the length direction, within at least a portion of actuatable region and/or within the entire actuatable region might, for example, be not greater than 30° per inch and/or 11.25° per inch±75%. 
     Radius of curvature of the top surface of the curved pedal within at least a portion of actuatable region and/or within the entire actuatable region may, for example, be not less than one-half of the length of the actuatable region, might be not less than 3″, and/or might be 8″±75%. 
     The at least one curvature profile might be more or less sinusoidal with wavelength 10″±50% and amplitude 0.30″±75%. 
     The at least one curvature profile might be more or less elliptically arcuate with radius of curvature 8″±75% and have an extremum of height 0.30″±75% as measured from the pedal reference plane. 
     The at least one curvature profile might be more or less circularly arcuate with radius of curvature 8″±75% and have an extremum of height 0.30″±75% as measured from the pedal reference plane. 
     The at least one curvature profile might be approximated by a polynomial curve of order not less than three with radius of curvature 8″±75% and have an extremum of height 0.30″±75% as measured from the pedal reference plane. 
     The actuatable region may comprise at least one first convexity, at least one first concavity, at least one second convexity, and/or at least one flat portion. 
     Where at least one first convexity, at least one first concavity, and at least one second convexity are present, the at least one first concavity may be disposed centrally in the length direction between the at least one first convexity and the at least one second convexity. 
     The at least one first convexity and/or the at least one second convexity might be substantially a half-lobe that extends or extend not more than 25% peripherally past an extremum or extrema thereof. 
     Length of actuatable region in the length direction might be not less than 12″. 
     The curved pedal may comprise a heel end having at least one feature permitting mounting to a heel hinge. 
     The curved pedal may comprise a toe end having at least one feature permitting mounting to at least one pivoting linkage arm. 
     The curved pedal may be mounted in a pedal assembly and used to operate a drum or other such percussion instrument, or any of a wide variety of foot-actuated devices. 
     Other embodiments, systems, methods, and features, and advantages of the present invention will be apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, features, and advantages be included within this description, be within the scope of the present invention, and be protected by the accompanying claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Many aspects of the invention can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the present invention. In the drawings, like reference numerals designate corresponding parts throughout the several views, description that would be repetitive being omitted for convenience. 
         FIG. 1  shows drum set  100 , this being an example of a system employing foot-operated device(s) requiring rapid, dexterous, and/or repeated actuation over an extended period of time, and which contains percussion instrument(s)  102 , at least one of which is capable of being actuated by foot by way of pedal assembly  110  in accordance with an embodiment of the present invention. 
         FIG. 2  is a perspective view of a pedal assembly  110  that may be employed at drum set  100  of  FIG. 1  in accordance with an embodiment of the present invention. 
         FIG. 3  is a side view of pedal assembly  110  of  FIG. 2  and shows curved pedal  130  having actuatable region  135  comprising portion(s)  140 ,  150 ,  160  that is or are convex and/or concave relative to pedal reference plane  131  in accordance with an embodiment of the present invention. 
         FIG. 4  is a perspective view of curved pedal  130  of  FIG. 3  in which first convexity  140 , first concavity  150 , and second convexity  160  are arranged in length direction  132  of actuatable region  135  in accordance with an embodiment of the present invention. 
         FIG. 5A  through  FIG. 5J  show various embodiments of the present invention that are variations on curved pedal  130  of  FIG. 3 ,  FIG. 5A  showing curved pedal  230  comprising first concavity  250 ;  FIG. 5B  showing curved pedal  330  comprising first convexity  340  and first concavity  350 ;  FIG. 5C  showing curved pedal  430  comprising first concavity  450  and first convexity  440 ;  FIG. 5D  showing curved pedal  530  comprising first convexity  540 , first concavity  550 , and second convexity  560 ;  FIG. 5E  showing curved pedal  630  comprising first convexity  640 ;  FIG. 5F  showing curved pedal  730  comprising first convexity  740 ;  FIG. 5G  showing curved pedal  830  comprising first convexity  840  and second convexity  860 ;  FIG. 5H  showing curved pedal  930  comprising first convexity  940  and first concavity  950 ;  FIG. 5I  showing curved pedal  1030  comprising first convexity  1040 , first concavity  1050 , and second concavity  1070 ; and  FIG. 5J  showing curved pedal  1130  comprising first convexity  1140 , first concavity  1150 , and second convexity  1160 . 
         FIG. 6  is a side view of arcuately curved pedal  130   a  in an embodiment of the present invention in which first convexity  140   a , first concavity  150   a , and second convexity  160   a  have radii of curvature that are respectively uniform, being circular arcs, and in which arrangement and radii of curvature of first convexity  140   a , first concavity  150   a , and second convexity  160   a  are such as to produce smooth inflection points, without interposition of flat portions, therebetween. 
         FIG. 7  is a side view of curved pedal  130  as it might exist when undepressed in pedal assembly  110  of  FIG. 3 , and shows inclination of extrema  141 ,  151 ,  161  and inflection points  145 ,  165  due to pedal mount angle  128  formed by pedal reference plane  131  and baseboard plane  113 . 
         FIG. 8  is a side view of arcuately curved pedal  130   b  in an embodiment of the present invention in which first convexity  140   b , first concavity  150   b , and second convexity  160   b  have radii of curvature that are respectively uniform, being circular arcs, and in which arrangement and radii of curvature of first convexity  140   b , first concavity  150   b , and second convexity  160   b  are such as to accommodate interposition of horizontal flat portions at inflection points therebetween as a result of the smaller radii of curvature in the embodiment shown in  FIG. 8  as compared with the embodiment shown in  FIG. 6 . 
         FIG. 9  is a side view of arcuately curved pedal  130   c  in an embodiment of the present invention in which first convexity  140   c , first concavity  150   c , and second convexity  160   c  have radii of curvature that are respectively uniform, being circular arcs, and in which arrangement and radii of curvature of first convexity  140   c , first concavity  150   c , and second convexity  160   c  are such as to accommodate interposition of vertical flat portions at inflection points therebetween as a result of the larger radii of curvature in the embodiment shown in  FIG. 9  as compared with the embodiment shown in  FIG. 6 . 
         FIG. 10  is a side view of arcuately curved pedal  130   d , which is identical to arcuately curved pedal  130   a  of  FIG. 6  except that portions peripheral to actuatable region  135   d  have been removed so as to draw attention to first convexity half-lobe  144   d , first concavity half-lobes  154   d , and second convexity half-lobe  164   d  within actuatable region  135   d.    
         FIG. 11  shows a working example in which top surface  136  of curved pedal  130  has a uniform sinusoidal profile of wavelength 11.6″ and amplitude 0.30″ throughout actuatable region  135  in length direction  132 . 
         FIGS. 12A and 12B  show a working example in which top surface  136  of curved pedal  130  has a varying sinusoidal profile within actuatable region  135  in length direction  132 , the combined portion comprising first convexity half-lobe  144  and first concavity heel-side half-lobe  154  having a sinusoidal profile of wavelength 11.6″ and amplitude 0.30″ as shown in  FIG. 12A , and the combined portion comprising first concavity toe-side half-lobe  154  having a sinusoidal profile of wavelength 8.4″ and amplitude 0.21″ as shown in  FIG. 12B . 
         FIG. 13  shows a working example in which top surface  136  of curved pedal  130  has a 5th-order polynomial profile throughout actuatable region  135  in length direction  132 . 
         FIGS. 14A and 14B  show a working example in which top surface  136  of curved pedal  130  has a varying 3rd-order polynomial profile within actuatable region  135  in length direction  132 , the combined portion comprising first convexity half-lobe  144  and first concavity heel-side half-lobe  154  having a 3rd-order polynomial profile as shown in  FIG. 14A , and the combined portion comprising first concavity toe-side half-lobe  154  having a 3rd-order polynomial profile as shown in  FIG. 14B . 
         FIGS. 15A and 15B  show a working example in which top surface  136  of curved pedal  130  has a varying 4th-order polynomial profile within actuatable region  135  in length direction  132 , the combined portion comprising first convexity half-lobe  144  and first concavity heel-side half-lobe  154  having a 4th-order polynomial profile as shown in  FIG. 15A , and the combined portion comprising first concavity toe-side half-lobe  154  having a 4th-order polynomial profile as shown in  FIG. 15B . 
         FIGS. 16A through 16C  are diagrams to assist in describing one example of use of curved pedal  130  in pedal assembly  110  in accordance with an embodiment of the present invention. 
         FIG. 17  shows drum set  100  similar to drum set  100  shown in  FIG. 1  except that drum set  100  of  FIG. 17  contains two bass drums  103 , each of which has an independent pedal assembly  110  in accordance with an embodiment of the present invention. 
         FIG. 18  shows dual pedal linkage  111  which links two pedal assemblies  110  in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     One embodiment of the present invention is a curved pedal. 
     A curved pedal in accordance with an embodiment of the present invention may be employed in any of a wide variety of devices that employ pedals for foot-actuated operation, such as automobiles, helicopters, airplanes, backhoes and other such vehicles and heavy equipment, looms, sewing machines, treadles, knitting machines, mills, lathes, pumps, and other such industrial apparatuses. 
     Although embodiments of the present invention are described in terms of an example in which a curved pedal mounted in a pedal assembly operates a beater to strike a vertical bass drum, it should be understood that the present invention is not limited to the example of a pedal assembly for causing actuation of a beater that strikes a vertical bass drum, but may also be applied to a pedal assembly for causing actuation of a beater that strikes a horizontal bass drum, a pedal assembly for causing actuation of high-hat cymbals, and to a pedal assembly for causing actuation of any of a wide variety of devices in which motion from a foot-actuated pedal can be converted into motion for driving and/or controlling the device or any portion thereof through an appropriate linkage or transmission mechanism, of which the pedal assembly described below is merely one example. 
     Referring to  FIG. 1 , this shows drum set  100 . Drum set  100  is an example of a system employing foot-operated device(s) requiring rapid, dexterous, and/or repeated actuation over an extended period of time. More specifically, drum set  100  includes a number of percussion instruments  102 , two among which, i.e., bass drum  103  and high-hat cymbals  104 , are capable of being actuated by foot by way of respective pedal assemblies  110 . The description that follows is given in terms of an example in which pedal assembly  110  operates a beater that strikes bass drum  103 , but pedal assembly  110  may be applied to actuation of high-hat cymbals  104  or to any of a wide variety of devices that may employ pedals for foot-actuated operation. 
     Referring now to  FIGS. 2 and 3 , these respectively show perspective and side views of a pedal assembly  110  in accordance with one embodiment of the present invention. 
     In the embodiment shown in  FIGS. 2 and 3 , pedal assembly  110  comprises curved pedal  130 , one end of which, hereinafter referred to as the heel end, has hole(s) and/or other features permitting it to be pivotably mounted on heel hinge  114  at a location toward what will be referred to as the heel end of baseboard  112 . The other end of curved pedal  130 , hereinafter referred to as the toe end, is free to pivot about the shaft of heel hinge  114  as curved pedal  130  goes from its raised or undepressed position at which pedal reference plane  131  is more or less inclined at pedal mount angle  128  to its lowered or fully depressed position at which pedal reference plane  131  is more or less parallel (except to the extent limited by a stopper or the like to prevent damage to the drum surface or other parts) with baseboard plane  113 , when curved pedal  130  is depressed by a foot against the restoring force provided by pedal return spring  126 . 
     Having identified one end of curved pedal  130  as the heel end thereof, and having identified the other end of curved pedal  130  as the toe end thereof, these directions, i.e., the heel end or side which is toward the left as seen in  FIG. 3 , and the toe end or side which is toward the right as seen in  FIG. 3 , may be employed herein for convenience of description. 
     Pivoting linkage arms  122  are oriented more or less vertically, the bottom ends of pivoting linkage arms  122  being connected to either side of the toe end of curved pedal  130 , toe end of curved pedal  130  having hole(s) and/or other features permitting connection to the bottom ends of pivoting linkage arms  122 , and the top ends of pivoting linkage arms  122  being connected to either side of the toe end of a rocker  120  on which beater stem  118  terminating in beater  115  is mounted. As the toe end of curved pedal  130  swings through its arc about the pivot of heel hinge  114 , transfer of this rotary motion to rocker  120  via pivoting linkage arms  122  causes rocker  120  to pivot about rocker axle  116  which is supported by bearings held by support posts  124  secured to baseboard  112 . 
     With continued reference to  FIG. 3  and additional reference to  FIG. 4 , curved pedal  130  will now be described.  FIGS. 3 and 4  respectively show side and perspective views of curved pedal  130  of  FIG. 2 ,  FIG. 3  showing curved pedal  130  as mounted in pedal assembly  110  and  FIG. 4  showing curved pedal  130  by itself. Additional reference may also be made to  FIG. 10 , in which like reference numerals indicate like parts. 
     As shown in  FIG. 4 , curved pedal  130  may have a length direction  132  and a width direction  133 . 
     In one embodiment, curved pedal  130  may have an actuatable region  135  at a top surface  136  of curved pedal  130 . Where this is the case, bottom surface  137  may be disposed opposite top surface  136 . Thickness of curved pedal  130 , i.e., the dimension shown in  FIGS. 3 and 4  between top surface  136  and bottom surface  137 , is preferably at least of magnitude sufficient to support and allow actuation by a foot but not so large as to impede movement of curved pedal  130 . For example, where curved pedal  130  is made of 6061 or similar aluminum, thickness of curved pedal  130  might be on the order of 0.375″. Note that there is no objection to employment of a curved pedal  130  of nonuniform thickness; for example, there is no objection to employment of a curved pedal  130  in which thickness varies with position in length direction  132  and/or in width direction  133 . For example, in one embodiment, thickness of curved pedal  130  may vary such that bottom surface  137  is flat, e.g., where convenient for manufacture of curved pedal  130 ; so long as top surface  136  is curved or otherwise has curvature and/or other feature(s) as described herein, there is no particular objection to employment of any arbitrary configuration at bottom surface  137 , provided that this does not impede operation of pedal assembly  110 . 
     Although aluminum has been mentioned by way of example, curved pedal  130  may be made of any suitable material, including steel or other suitable metal, thermoplastic and/or thermosetting resin, wood, glass, ceramic, and/or the like, and may comprise any suitable laminated and/or composite material(s). Curved pedal  130  may be cast, machined, molded, formed in a vice or other such device, or manufactured and/or shaped by any other suitable technique. 
     Length of actuatable region  135  in length direction  132  is preferably at least long enough to permit comfortable actuation by the foot of a typical player, or by the feet of various players who may range in age from child to adult. For example, in one embodiment, length of actuatable region  135  in length direction  132  might be 5 inches to 20 inches. When length of actuatable region  135  is 5 inches to 20 inches, this may provide good but not excessive leverage for comfortable and responsive actuation of curved pedal  130 . In a preferred embodiment, length of actuatable region  135  in length direction  132  is substantially longer than the foot of a typical player so as to permit increased leverage and facilitate various sliding actuation techniques. For example, in one embodiment, length of actuatable region  135  in length direction  132  is preferably not less than 12″, more preferably not less than 14″, and still more preferably not less than 16″. Actuatable region  135  is described further below with reference to  FIG. 10 . 
     There is no particular limitation with respect to width of curved pedal  130  in width direction  133 , it being sufficient that width of curved pedal  130  in width direction  133  be such as to permit comfortable actuation by the foot of a typical player, or by the feet of various players who may range in age from child to adult. Note that there is no objection to employment of a curved pedal  130  of nonuniform width; for example, there is no objection to employment of a curved pedal  130  in which width varies with position in length direction  132 . For example, width of curved pedal  130  in width direction  133  may vary to accommodate the varying width of a typical foot. Furthermore, width of curved pedal  130  may narrow near the heel end and/or toe end of curved pedal  130  for convenience of mounting in pedal assembly  110  and to provide clearance with respect to support posts  124  and/or other parts. 
     In the embodiment shown in  FIGS. 2 through 4 , curved pedal  130  has actuatable region  135  comprising portion(s)  140 ,  150 ,  160  that is or are convex and/or concave relative to pedal reference plane  131 . More specifically, curved pedal  130  in the embodiment shown in  FIGS. 2 through 4  comprises actuatable region  135  having first convexity  140 , first concavity  150 , and second convexity  160 . In the embodiment shown in  FIGS. 2 through 4 , first convexity  140 , first concavity  150 , and second convexity  160  are arranged in length direction  132  of actuatable region  135 . 
     Except where stated otherwise herein, what is referred to herein as curvature of curved pedal  130  is curvature of top surface  136  thereof in length direction  132  as most easily seen in side view such as is shown in  FIG. 3  and  FIGS. 6 through 10 . Except where stated otherwise herein, what is referred to herein as convexity or concavity of curved pedal  130  is convexity or concavity of top surface  136  thereof as viewed from a point above top surface  136  and as most easily seen in side view such as is shown in  FIG. 3  and  FIGS. 6 through 10 . 
     Where curved pedal  130  contains multiple inflection points  145 ,  165 , pedal reference plane  131  is defined as the plane that contains the best-fit line through those multiple inflection points  145 ,  165  as seen in a sectional view taken at a point located approximately centrally in width direction  133  of curved pedal  130  as shown in the side view of  FIG. 3 . Where curved pedal  130  contains less than two inflection points, pedal reference plane  131  is defined as the plane that contains the best-fit line through top surface  136  as seen in a sectional view taken at a point located approximately centrally in width direction  133  of curved pedal  130  as shown in the side view of  FIG. 3 . 
     Thus, in some embodiments, curved pedal  130  may be curved in at least a pedal length direction  132 . Where this is the case, curved pedal  130  is preferably curved within at least a portion of an actuatable region  135  in the pedal length direction  132 . 
     In one embodiment, the profile of top surface  136  of curved pedal  130  in length direction  132  has at least one inflection point  145 ,  165  (see  FIG. 6 through 10 ) where curvature transitions between convex and concave, regardless of order, in length direction  132 . In a preferred embodiment, there are at least two such inflection point  145 ,  165 . 
     In a preferred embodiment, there are no horizontal flat portions (see  FIG. 8 ) within at least a portion of actuatable region  135  and/or within the entire actuatable region  135 . In one embodiment, slope of top surface  136  at inflection point(s)  145 ,  165  where curvature transitions between convex and concave in length direction  132  is preferably not less than 5°, more preferably not less than 10°, and most preferably not less than 15°. 
     In a preferred embodiment, there are no vertical flat portions (see  FIG. 9 ) within at least a portion of actuatable region  135  and/or within the entire actuatable region  135 . In one embodiment, slope of top surface  136  at inflection point(s)  145 ,  165  where curvature transitions between convex and concave in length direction  132  is preferably not greater than 85°, more preferably not greater than 80°, and most preferably not greater than 75°. 
     Where horizontal, vertical, and/or inclined flat portion(s) exist within actuatable region  135 , these are preferably beveled or rounded so as to prevent occurrence of sharp corners  139  (see  FIGS. 8 and 9 ) at transition(s) between flat portion(s) and convex and/or concave portion(s). 
     In one embodiment, local radius of curvature along top surface of curved pedal  130  within at least a portion of actuatable region  135  and/or within the entire actuatable region  135  is preferably not less than one-quarter of, more preferably not less than one-third of, and most preferably not less than one-half of the length of actuatable region  135 . In a preferred embodiment, local radius of curvature along top surface of curved pedal  130  within at least a portion of actuatable region  135  and/or within the entire actuatable region  135  is preferably not less than 3″, more preferably not less than 5″, and most preferably not less than 7″. In one embodiment, local radius of curvature along top surface of curved pedal  130  within at least a portion of actuatable region  135  and/or within the entire actuatable region  135  is preferably 8″±75%, more preferably is 8″±50%, and most preferably is 8″±25%. 
     In one embodiment, curved pedal  130  has smoothly varying slope within at least a portion of actuatable region  135  and/or within the entire actuatable region  135 . 
     In one embodiment, the change in slope as a function of position along length direction  132 , i.e., the second spatial derivative with respect to position in length direction  132 , within at least a portion of actuatable region  135  and/or within the entire actuatable region  135  is preferably not greater than 30° per inch, more preferably not greater than 18° per inch, and most preferably not greater than 13° per inch. In one embodiment, the second spatial derivative with respect to position in length direction  132  within at least a portion of actuatable region  135  and/or within the entire actuatable region  135  is preferably 11.25° per inch±75%, more preferably is 11.25° per inch±50%, and most preferably is 11.25° per inch±25%. 
     In some embodiments, the profile of curved pedal  130  may be or approximate a sinusoidal curve in length direction  132  over at least a portion of actuatable region  135 . 
     Where curved pedal  130  has such a sinusoidal profile, wavelength in length direction  132  is preferably on the order of or longer than the length of the foot of a typical player. For example, in one embodiment, wavelength of curved pedal  130  in length direction  132  is preferably 10″±50%, more preferably is 10″±25%, and most preferably is 10″±10%. 
     Where curved pedal  130  has such a sinusoidal profile, amplitude as measured from pedal reference plane  131  is preferably on the order of the height of the arch of the foot of a typical player. For example, in one embodiment, amplitude is preferably 0.30″±75%, more preferably is 0.30″±50%, and most preferably is 0.30″±25%. 
     In some embodiments, the profile of curved pedal  130  may be or may approximate a circular or elliptical arc in length direction  132  over at least a portion of actuatable region  135 . Where curved pedal  130  has such an arcuate profile, radius of curvature is preferably 8″±75%, more preferably is 8″±50%, and most preferably is 8″±25% 
     Where curved pedal  130  has such an arcuate profile, distance between extrema  141 ,  161  (see  FIG. 6  and  FIGS. 8 through 10 ) of similar curvature, e.g., between successive convexities  140 ,  160 , in length direction  132  is preferably on the order of or longer than the length of the foot of a typical player. For example, in one embodiment, interpeak distance, e.g., between first convexity extremum  141  and second convexity extremum  161 , in length direction  132  is preferably 10″±50%, more preferably is 10″±25%, and most preferably is 10″±10%. 
     Where curved pedal  130  has such an arcuate profile, height of extrema  141 ,  151 ,  161  (see  FIG. 6  and  FIGS. 8 through 10 ) as measured from pedal reference plane  131  is preferably on the order of the height of the arch of the foot of a typical player. For example, in one embodiment, height of first convexity extremum  141 , first concavity extremum  151 , and/or second convexity extremum  161  as measured from pedal reference plane  131  is preferably 0.30″±75%, more preferably is 0.30″±50%, and most preferably is 0.30″±25%. 
     In some embodiments, the profile of curved pedal  130  may be or may approximate a polynomial curve in length direction  132  over at least a portion of actuatable region  135 . 
     Where curved pedal  130  has such a polynomial profile, the order of the polynomial is preferably at least three, more preferably at least four, and most preferably at least five. 
     Where curved pedal  130  has such a polynomial profile, distance between extrema  141 ,  161  (see  FIG. 6  and  FIGS. 8 through 10 , which, though not of polynomial profile, show analogous extrema  141 ,  161  of arcuately curved pedal  130   a ) of similar curvature, e.g., between successive convexities  140 ,  160 , in length direction  132  is preferably on the order of or longer than the length of the foot of a typical player. For example, in one embodiment, interpeak distance, e.g., between first convexity extremum  141  and second convexity extremum  161 , in length direction  132  is preferably 10″±50%, more preferably is 10″±25%, and most preferably is 10″±10%. 
     Where curved pedal  130  has such a polynomial profile, height of extrema  141 ,  151 ,  161  (see  FIG. 6  and  FIGS. 8 through 10 , which, though not of polynomial profile, show analogous extrema  141 ,  151 ,  161  of arcuately curved pedal  130   a ) as measured from pedal reference plane  131  is preferably on the order of the height of the arch of the foot of a typical player. For example, in one embodiment, height of first convexity extremum  141 , first concavity extremum  151 , and/or second convexity extremum  161  as measured from pedal reference plane  131  is preferably 0.30″±75%, more preferably is 0.30″±50%, and most preferably is 0.30″±25%. 
     In some embodiments, curved pedal  130  may additionally be curved in pedal width direction  133 . Where this is the case, curvature of top surface  136  in pedal width direction  133  may in some embodiments be convex, or curvature of top surface  136  in pedal width direction  133  may in other embodiments be concave. There is no particular objection to a saddle-shaped or similarly contoured curved pedal  130  in which curvature in length direction  132  may be locally opposite to curvature in width direction  133 . 
     Although curved pedal  130  has been described with reference to  FIGS. 2 through 4  in terms of an example in which actuatable region  135  is divided into three curved portions  140 ,  150 ,  160  without interposition of flat portion(s), e.g., horizontal or vertical flat portions (see  FIGS. 8 and 9 ), at inflection points  145 ,  146  therebetween, actuatable region  135  may be divided into greater or fewer than three curved portion(s), and there is no particular objection to presence of flat portion(s); e.g., interposition of noncurved or flat portion(s) between respective curved portions  140 ,  150 ,  160 . Although  FIGS. 8 and 9  respectively show embodiments in which horizontal and vertical flat portions intervene between curved portions  140 ,  150 ,  160 , in an embodiment in which flat portion(s) are present note that there is no objection to employment of flat portion(s) that are inclined with respect to pedal reference plane  131 ; i.e., flat as used in this context means noncurved and not necessarily that such flat portion(s) need be parallel to (horizontal) or perpendicular to (vertical) pedal reference plane  131 . Where horizontal, vertical, and/or inclined flat portion(s) exist within actuatable region  135 , these are preferably beveled or rounded so as to prevent occurrence of sharp corners  139  (see  FIGS. 8 and 9 ) at transition(s) between flat portion(s) and convex and/or concave portion(s). 
     Referring to  FIG. 5A  through  FIG. 5J , these show various embodiments in which actuatable region  135  has been subdivided into three portions, each of which may respectively contain a convex portion  140 ,  160 ; a concave portion  150 ; or a noncurved or flat portion. 
     In the embodiment shown in  FIG. 5A , curved pedal  230  comprises first concavity  250 . 
     In the embodiment shown in  FIG. 5B , curved pedal  330  comprises first convexity  340  and first concavity  350 . 
     In the embodiment shown in  FIG. 5C , curved pedal  430  comprises first concavity  450  and first convexity  440 . 
     In the embodiment shown in  FIG. 5D , curved pedal  530  comprises first convexity  540 , first concavity  550 , and second convexity  560 . 
     In the embodiment shown in  FIG. 5E , curved pedal  630  comprises first convexity  640 . 
     In the embodiment shown in  FIG. 5F , curved pedal  730  comprises first convexity  740 . 
     In the embodiment shown in  FIG. 5G , curved pedal  830  comprises first convexity  840  and second convexity  860 . 
     In the embodiment shown in  FIG. 5H , curved pedal  930  comprises first convexity  940  and first concavity  950 . 
     In the embodiment shown in  FIG. 5I , curved pedal  1030  comprises first convexity  1040 , first concavity  1050 , and second concavity  1070 . 
     In the embodiment shown in  FIG. 5J , curved pedal  1130  comprises first convexity  1140 , first concavity  1150 , and second convexity  1160 . 
     Similar variations, included within the scope of the claims appended hereto, are possible when actuatable region  135  of curved pedal  130  is subdivided into greater or fewer than three portions. 
     Note that there is no objection to an embodiment in which convex portion(s)  140 ,  160 , concave portion(s)  150 , and/or noncurved or flat portion(s) occupy two or more of the portions into which actuatable region  135  is divided. For example, where actuatable region  135  is subdivided into three portions as shown in  FIG. 5A through 5J , there is no objection to an embodiment in which first convexity  140  occupies two of the portions, and first concavity  150  occupies the remaining portion, or vice-versa. Such a variation is indicated by way of example at  FIG. 5I , where second concavity  1070  occupies two of the portions into which actuatable region  135  is divided. 
     Note that there is no objection to combination of convex portion(s) and/or concave portion(s) with noncurved or flat portion(s), some examples of which are shown at  FIGS. 5A through 5J . 
     Furthermore, there is no particular objection to use of angled flat portion(s) to form convex and/or concave portion(s), some examples of which are shown in  FIGS. 5A through 5J . Where such angled flat portion(s) exist within actuatable region  135 , these are preferably beveled or rounded so as to prevent occurrence of sharp corners  139  (see  FIGS. 8 and 9 ) at transition(s) between flat portion(s) and convex, concave portion(s) and/or other flat portion(s). 
     In a preferred embodiment, at least one concave portion  150  is disposed more or less centrally in length direction  132  and/or is disposed between two convex portions  140 ,  160  in length direction  132 . 
     For example, curved pedal  130  shown in  FIGS. 2 through 4  and  FIGS. 6 through 10  has first convexity  140 , first concavity  150 , and/or second convexity  160 , first concavity  150  being disposed centrally between first convexity  140  and second convexity  160  along length direction  132  of actuatable region  135 . 
     Referring to  FIG. 6 , this is a side view of arcuately curved pedal  130   a  in an embodiment of the present invention in which first convexity  140   a , first concavity  150   a , and second convexity  160   a  have radii of curvature  142   a ,  152   a ,  162   a  that are respectively uniform, being circular arcs, and in which arrangement and radii of curvature  142   a ,  152   a ,  162   a  of first convexity  140   a , first concavity  150   a , and second convexity  160   a  are such as to produce smooth inflection points  145   a ,  165   a , without interposition of flat portions, therebetween. 
     In the embodiment shown in  FIG. 6 , first arcuately curved convexity  140   a  has radius of curvature  142   a , first arcuately curved concavity  150   a  has radius of curvature  152   a , and second arcuately curved convexity  160   a  has radius of curvature  162   a.    
     In the embodiment shown in  FIG. 6 , first arcuately curved convexity  140   a  has height (i.e., amplitude)  143   a  at extremum  141   a  as measured from pedal reference plane  131   a . First arcuately curved concavity  150   a  has height (i.e., amplitude)  153   a  at extremum  151   a  as measured from pedal reference plane  131   a . Second arcuately curved convexity  160   a  has height (i.e., amplitude)  143   a  at extremum  141   a  as measured from pedal reference plane  131   a.    
     In the embodiment shown in  FIG. 6 , first convexity inflection point  145   a  is present where curvature transitions between convex and concave between first arcuately curved convexity  140   a  and first arcuately curved concavity  150   a  in length direction  132 , and second convexity inflection point  165   a  is present where curvature transitions between concave and convex between first arcuately curved concavity  150   a  and second arcuately curved convexity  160  in length direction  132 . 
     In the embodiment shown in  FIG. 6 , arrangement of first arcuately curved convexity  140   a , first arcuately curved concavity  150   a , and second arcuately curved convexity  160   a , i.e., respective distances between extrema  141   a ,  151   a ,  161   a  and respective heights of extrema  141   a ,  151   a ,  161   a  as measured from pedal reference plane  131 , and respective radii of curvature  142   a ,  152   a ,  162   a , are chosen such that adjacent arcs of opposite curvature more or less exactly meet at inflection points  145   a ,  165   a  as to produce smooth inflection points  145   a ,  165   a  without interposition of flat portions therebetween. 
     Referring to  FIG. 7 , this is a side view showing in schematic fashion how curved pedal  130 , e.g., arcuately curved pedal  130   a  of the embodiment shown in  FIG. 6 , might appear when mounted in pedal assembly  110  of  FIG. 3 . In the schematic diagram of  FIG. 7 , curved pedal  130  is in its raised or undepressed position, being inclined more or less at pedal mount angle  128  (see  FIG. 3 ). As indicated in the graph shown in  FIG. 7 , respective positions in the x and y axes of first convexity extremum  141   a , first convexity inflection point  145   a , first concavity extremum  151   a , second convexity inflection point  165   a , and second convexity extremum  161   a -respectively indicated by indices 1, 2, 3, and 4—are inclined at pedal mount angle  128  formed by pedal reference plane  131  and baseboard plane  113 . 
     Referring to  FIG. 8 , this is a side view of arcuately curved pedal  130   b  in an embodiment of the present invention in which first convexity  140   b , first concavity  150   b , and second convexity  160   b  have radii of curvature  142   b ,  152   b ,  162   b  that are respectively uniform, being circular arcs, and in which arrangement and radii of curvature  142   b ,  152   b ,  162   b  of first convexity  140   b , first concavity  150   b , and second convexity  160   b  are such as to accommodate interposition of horizontal flat portions at inflection points  145   b ,  165   b  therebetween as a result of the smaller radii of curvature  142   b ,  152   b ,  162   b  in the embodiment shown in  FIG. 8  as compared with the radii of curvature  142   a ,  152   a ,  162   a  employed in the embodiment shown in  FIG. 6 . 
     Note that where corner(s)  139   b  are produced at transition(s) between flat portion(s) and convex and/or concave portion(s), it is preferred that these be beveled or rounded so that local radius of curvature is not substantially smaller than radius of curvature at other locations along the curved profile at top surface  136  of curved pedal  130 . In a preferred embodiment, radii of curvature at corner(s)  139   b  at transition(s) between flat portion(s) and convex and/or concave portion(s) are preferably not less than 3″, more preferably not less than 5″, and most preferably not less than 7″. 
     Referring to  FIG. 9 , this is a side view of arcuately curved pedal  130   c  in an embodiment of the present invention in which first convexity  140   c , first concavity  150   c , and second convexity  160   c  have radii of curvature  142   c ,  152   c ,  162   c  that are respectively uniform, being circular arcs, and in which arrangement and radii of curvature  142   c ,  152   c ,  162   c  of first convexity  140   c , first concavity  150   c , and second convexity  160   c  are such as to accommodate interposition of vertical flat portions at inflection points  145   c ,  165   c  therebetween as a result of the larger radii of curvature  142   c ,  152   c ,  162   c  in the embodiment shown in  FIG. 9  as compared with the radii of curvature  142   a ,  152   a ,  162   a  employed in the embodiment shown in  FIG. 6 . 
     Note that where corner(s)  139   c  are produced at transition(s) between flat portion(s) and convex and/or concave portion(s), it is preferred that these be beveled or rounded so that local radius of curvature is not substantially smaller than radius of curvature at other locations along the curved profile at top surface  136  of curved pedal  130 . In a preferred embodiment, radii of curvature at corner(s)  139   c  at transition(s) between flat portion(s) and convex and/or concave portion(s) are preferably not less than 3″, more preferably not less than 5″, and most preferably not less than 7″. 
     Referring to  FIG. 10 , this is a side view of arcuately curved pedal  130   d , which is identical to arcuately curved pedal  130   a  of  FIG. 6  except that portions peripheral to actuatable region  135   d  have been removed, leaving substantially first convexity half-lobe  144   d , first concavity half-lobes  154   d , and second convexity half-lobe  164   d  within actuatable region  135   d.    
     Whereas curved pedal  130  shown in  FIGS. 2 through 4  and  FIGS. 6 through 9  is divided into three curved portions  140 ,  150 ,  160 , actuatable region  135 , i.e., the region contacted by the foot during playing, may in some embodiments not extend all the way to the peripheral ends of first convexity  140  and second convexity  160 . 
     That is, in embodiments in which there is a central concavity  150  and/or a concavity  150  disposed between two convexities  140 ,  160 , it may primarily be the central concavity  150  that serves to locate or orient the foot, while the convexities  140 ,  160  to either side thereof might typically primarily serve to receive striking force from the heel and/or toe. This being the case, in such an embodiment, it may be that it is primarily only the central or interior first convexity half-lobe  144   d  which is disposed between extremum  141   d  and inflection point  145   d  of first convexity  140   d  that is required for actuation, and it may be that it is primarily only the central or interior second convexity half-lobe  164   d  which is disposed between extremum  161   d  and inflection point  165   d  of second convexity  160   d  that is required for actuation. 
     For this reason, actuatable region  135   d  of curved pedal  130   d  is shown in  FIG. 10  as extending only slightly peripherally past first convexity extremum  141   d  at the heel side (left side in  FIG. 10 ) of curved pedal  130   d , and as extending only slightly peripherally past second convexity extremum  161   d  at the toe side (right side in  FIG. 10 ) of curved pedal  130   d.    
     That is, actuatable region  135   d  of curved pedal  130   d  in the embodiment shown in  FIG. 10  comprises the two half-lobes  154   d  of central concavity  150   d  but only substantially the interior half-lobe  144   d  of first convexity  140   d  and only substantially the interior half-lobe  164   d  of second convexity  160   d . Note that in a preferred embodiment, actuatable region  135   d  extends peripherally slightly past first convexity extremum  141   d  to comprise a small portion of what would be the exterior half-lobe of first convexity  140   d , and extends peripherally slightly past second convexity extremum  161   d  to comprise a small portion of what would be the exterior half-lobe of second convexity  160   d.    
     In one embodiment, actuatable region  135   d  preferably extends peripherally not more than 25%, more preferably not more than 15%, and most preferably not more than 10%, past first convexity extremum  141   d . In one embodiment, actuatable region  135   d  preferably extends peripherally not more than 25%, more preferably not more than 15%, and most preferably not more than 10%, past second convexity extremum  161   d.    
     And in an embodiment in which it is desirable that that convex portion(s)  140 ,  160  be at least minimally well-defined, actuatable region  135   d  in such an embodiment preferably extends peripherally not less than 15%, more preferably not less than 10%, and most preferably not less than 5%, past first convexity extremum  141 , and/or actuatable region  135   d  in such an embodiment preferably extends peripherally not less than 15%, more preferably not less than 10%, and most preferably not less than 5%, past second convexity extremum  161   d.    
     Here, the degree to which actuatable region  135   d  extends peripherally past an extremum is measured as the distance from the projection of the extremum onto pedal reference plane  131   d  to the projection of the most peripheral point of actuatable region  135   d  onto pedal reference plane  131   d.    
     Although the profiles of first convexity  140   a ,  140   b ,  140   c ,  140   d ; first concavity  150   a ,  150   b ,  150   c ,  150   d ; and second convexity  160   a ,  160   b ,  160   c ,  160   d  in the embodiments shown in  FIGS. 6, 8, 9 and 10  are circular arcs, there is no objection to employment of elliptical arc(s), conic section(s), and/or any suitable portion(s) of Bezier curve(s) at one or more of first convexity  140   a , first concavity  150   a , and second convexity  160   a , or at any suitable portion(s) thereof. 
     Furthermore, as described with reference to  FIGS. 11 through 15 , any of various sinusoidal and/or polynomial profiles may be employed at one or more of first convexity  140   a ,  140   b ,  140   c ,  140   d ; first concavity  150   a ,  150   b ,  150   c ,  150   d ; and second convexity  160   a ,  160   b ,  160   c ,  160   d , or at any suitable portion(s) thereof. 
     Moreover, curvature profile need not be uniform throughout actuatable region  135  along length direction  132  of curved pedal  130  it being possible, for example, to employ respectively different curvature profiles at curved portions  140 ,  150 ,  160 . Furthermore, curvature profile need not be uniform within each of respective curved portions  140 ,  150 ,  160 , it being possible, for example, to employ different curvature profiles at respective half-lobe(s)  144 ,  154 ,  164  therewithin. 
       FIG. 11  shows a working example in which top surface  136  of curved pedal  130  has a uniform sinusoidal profile of wavelength 11.6″ and amplitude 0.30″ throughout actuatable region  135  in length direction  132 . The sinusoidal profile shown in  FIG. 11  was derived by curvefitting a sinusoidal function to data measured from a prototype constructed by the inventor. More specifically, the curvature profile shown in  FIG. 11  is a graph of the equation y=a+b*cos(cx+d), where coefficients a through d are: a=5.507468819E-01; b=2.959381106E-01; c=5.435591030E-01; and d=−4.978423078E-01. 
       FIGS. 12A and 12B  show a working example in which top surface  136  of curved pedal  130  has a varying sinusoidal profile within actuatable region  135  in length direction  132 , the combined portion comprising first convexity half-lobe  144  and first concavity heel-side half-lobe  154  having a sinusoidal profile of wavelength 11.6″ and amplitude 0.30″ as shown in  FIG. 12A , and the combined portion comprising first concavity toe-side half-lobe  154  having a sinusoidal profile of wavelength 8.4″ and amplitude 0.21″ as shown in  FIG. 12B . The sinusoidal profiles shown in  FIGS. 12A and 12B  were derived by curvefitting sinusoidal functions to data measured from a prototype constructed by the inventor. More specifically, the curvature profile shown in  FIG. 12A  is a graph of the equation y=a+b*cos(cx+d), where coefficients a through d are: a=5.507468819E-01; b=2.959381106E-01; c=5.435591030E-01; and d=−4.978423078E-01. Likewise, the curvature profile shown in  FIG. 12B  is a graph of the equation y=a+b*cos(cx+d), where coefficients a through d are: a=4.703228952E-01; b=2.122825994E-01; c=7.515261318E-01; and d=−1.719790992E+00. 
       FIG. 13  shows a working example in which top surface  136  of curved pedal  130  has a 5th-order polynomial profile throughout actuatable region  135  in length direction  132 . The 5th-order polynomial profile shown in  FIG. 13  was derived by curvefitting a 5th-order polynomial function to data measured from a prototype constructed by the inventor. More specifically, the curvature profile shown in  FIG. 13  is a graph of the equation y=a+bx+cx^2+dx^3+ex^4+fx^5, where coefficients a through f are: a=2.788918668E-01; b=7.270160318E-01; c=−3.118881062E-01; d=4.594107675E-02; e=−2.652644591E-03; and f=4.877817180E-05. 
       FIGS. 14A and 14B  show a working example in which top surface  136  of curved pedal  130  has a varying 3rd-order polynomial profile within actuatable region  135  in length direction  132 , the combined portion comprising first convexity half-lobe  144  and first concavity heel-side half-lobe  154  having a 3rd-order polynomial profile as shown in  FIG. 14A , and the combined portion comprising first concavity toe-side half-lobe  154  having a 3rd-order polynomial profile as shown in  FIG. 14B . The 3rd-order polynomial profiles shown in  FIGS. 14A and 14B  were derived by curvefitting 3rd-order polynomial functions to data measured from a prototype constructed by the inventor. More specifically, the curvature profile shown in  FIG. 14A  is a graph of the equation y=a+bx+cx^2+dx^3, where coefficients a through d are: a=2.255092825E-01; b=6.706921138E-01; c=−2.258203518E-01; and d=1.920400372E-02. Likewise, the curvature profile shown in  FIG. 14B  is a graph of the equation y=a+bx+cx^2+dx^3, where coefficients a through d are: a=5.392718044E+00; b=−2.018015119E+00; c=2.522924901E-01; and d=−9.781917019E-03. 
       FIGS. 15A and 15B  show a working example in which top surface  136  of curved pedal  130  has a varying 4th-order polynomial profile within actuatable region  135  in length direction  132 , the combined portion comprising first convexity half-lobe  144  and first concavity heel-side half-lobe  154  having a 4th-order polynomial profile as shown in  FIG. 15A , and the combined portion comprising first concavity toe-side half-lobe  154  having a 4th-order polynomial profile as shown in  FIG. 15B . The 4th-order polynomial profiles shown in  FIGS. 15A and 15B  were derived by curvefitting 4th-order polynomial functions to data measured from a prototype constructed by the inventor. More specifically, the curvature profile shown in  FIG. 15A  is a graph of the equation y=a+bx+cx^2+dx^3+ex^4, where coefficients a through e are: a=2.758305230E-01; b=7.484228120E-01; c=−3.307662679E-01; d=5.000334014E-02; and e=−2.605492952E-03. Likewise, the curvature profile shown in  FIG. 15B  is a graph of the equation y=a+bx+cx^2+dx^3+ex^4, where coefficients a through e are: a=7.459645128E+00; b=−2.888161825E+00; c=3.872523953E-01; d=−1.892717400E-02; and e=2.286314246E-04. 
     Note that the present invention is not limited to the working examples described with reference to  FIGS. 11 through 15 , these merely being exemplary profiles within the ranges of the various parameters—e.g., wavelength, amplitude, interpeak distance and/or distance between extrema, extrema amplitude and/or height as measured from pedal reference plane  131 , and radii of curvature—as claimed and/or as described elsewhere in this specification. 
     Note further that although working examples shown in  FIGS. 12 through 15  employ different or asymmetric amplitudes or gains at first convexity  140  and second convexity  160 , while the working example shown in  FIG. 11  and in the embodiments described with reference to  FIGS. 3 through 10  generally employed symmetric amplitudes or gains at first convexity  140  and second convexity  160 , there is in general no objection to employment of symmetric or asymmetric amplitudes or gains and/or symmetric or asymmetric values for any of the various other parameters at first convexity  140 , first concavity  150 , and second convexity  160  within the ranges of the various parameters—e.g., wavelength, amplitude, interpeak distance and/or distance between extrema, extrema amplitude and/or height as measured from pedal reference plane  131 , and radii of curvature—as claimed and/or as described elsewhere in this specification. 
     As described above, curved pedal  130  of various embodiments of the present invention may be mounted in pedal assembly  110  for use in drum set  100 , for example. 
     Curved pedal  130  mounted in pedal assembly  110  for use in drum set  100  in accordance with embodiments of the present invention may facilitate pedal-actuated drumming and/or may make pedal-actuated drumming less tiring or more comfortable, especially when employing techniques such as the sliding technique and/or the heel-toe technique. 
     Furthermore, the curved shape of curved pedal  130  in accordance with some embodiments may allow a player to quickly and reliably locate his or her foot by the “feel” of curved pedal  130 . 
     Moreover, because curved pedal  130  in accordance with some embodiments may be a good match for the shape of the foot, employment of curved pedal  130  may make it possible to achieve more rapid and powerful striking of the drum with less movement of the foot and/or ankle than is the case conventionally. 
     In addition, the curved shape of curved pedal  130  in accordance with some embodiments may allow the foot—and in particular the heel of the foot and/or the ball of the foot—to strike curved pedal  130  at an angle more nearly perpendicular to top surface  136  thereof, making it possible to improve the leverage or efficiency with which force is transferred from the player&#39;s foot to curved pedal  130 , and/or permitting stronger and/or less tiring performance. 
     Furthermore, the smoothly varying contour of curved pedal  130  in some embodiments may be advantageous for players who employ bare feet or who wears socks but no shoes or who wears thin shoes or other such foot coverings for improved comfort and sensitivity in locating the foot on curved pedal  130 . 
     Moreover, because actuatable region  135  of curved pedal  130  in some embodiments is substantially longer than the foot of the player, this may not only permit increase in leverage about the fulcrum of heel hinge  114 , permitting more powerful and/or less tiring playing, but may also facilitate more sustained sliding along length direction  132  of curved pedal  130 . In addition, a pedal substantially longer than the foot of the player may also accommodate multiple striking locations beyond the basic heel-toe striking positions employed conventionally. 
     Referring now to  FIGS. 16A through 16C , description will be given of how curved pedal  130  in pedal assembly  110  at drum set  100  might be used accordance with an embodiment of the present invention. 
     At drum set  100 , pedal assembly  110  may be used to play a drum  103  or high-hat cymbals  104 , for example, in any suitable manner. For example, where pedal assembly  110  is used to operate bass drum  103 , pedal assembly  110  may be assembled in such fashion as to permit pedal assembly  110  to cause beater  115  to strike vertically standing drum  103  or a horizontally standing drum when curved pedal  130  is depressed. 
     In some embodiments, a player may use pedal assembly  110  to generate a single drum beat. At such time, when the player uses his or her foot to operate pedal assembly  110 , the foot may in general be positioned at any arbitrary location along top surface  136  of curved pedal  130  at the time that curved pedal  130  is depressed. For example, the foot may be positioned as shown in  FIG. 16A . In another example, the foot may be positioned as shown in  FIG. 16B . In yet another example, the foot may be positioned as shown in  FIG. 16C . Possible foot positions are not limited to those shown in  FIG. 16A  through  FIG. 16C . 
     In some embodiments, a player may use pedal assembly  110  to generate a doublet, or two consecutive drum beats. A doublet may be generated in various ways. For example, a player may simply repeat one of the foot movements mentioned above to generate a single drum beat twice in rapid succession. One advantage of some embodiments of the present invention is that it facilitates production of two consecutive drum beats in one foot motion cycle. When two consecutive beats are produced by one foot motion cycle, rapid consecutive beats may be easily achieved. 
     For example, in accordance with one or more embodiments of the present invention, a player may use any of various sliding techniques. In accordance with one such sliding technique, a player might first depress curved pedal  130  using his or her toe to generate a first stroke, slide his or her foot along length direction  132  of curved pedal  130 , and then depress curved pedal  130  again using his or her toe to generate a second stroke. For example, a foot may be positioned for a first toe stroke as shown in  FIG. 16B  and then for a second toe stroke as shown in  FIG. 16C . Alternatively, a foot may be positioned for a first toe stroke as shown in  FIG. 16C  and then for a second toe stroke as shown in  FIG. 16B . Possible foot positions are not limited to those described in  FIG. 16B  and  FIG. 16C . 
     One advantage of at least some embodiments of the present invention is that the curved top surface  136  of curved pedal  130  may be better suited for foot sliding motion and therefore permit easier and less tiring generation of doublets, for example, as compared with a conventional flat pedal. 
     For example, when toe positions for two consecutive toe strokes are in a curved region of curved pedal  130 , e.g., within first concavity  150 , the curved shape of top surface  136  of curved pedal  130  may allow a player to more easily slide his or her toe forward or backward along length direction  132  as the toe depresses curved pedal  130 . 
     Furthermore, employment of a curved pedal  130  having smoothly varying slope within at least a portion of actuatable region  135  and/or within the entire actuatable region  135  may make it possible for a player to be able to feel on his or her foot a gradual local angle shift, i.e., slope change, of curved pedal  130  during foot sliding motion, and a player may use this shift as an indicator to understand where his or her toe is positioned during a foot motion cycle. The shift that may be felt on a player&#39;s foot may make reproducing a foot motion cycle easier for the player. In particular, where curved pedal  130  has second convexity  160 , the curvature of second convexity  160  may provide further toe positioning guidance. Thus, the smoothly varying slope of curved pedal  130  may allow a player to better rely on the feel of the foot and to eliminate or reduce the need to focus on how far the foot should slide, which may make generating a doublet, for example, more reproducible, less tiring, and more enjoyable. 
     Thus, one advantage of at least some embodiments of the present invention is that curved pedal  130  may make foot tilting motion and hence doublet generation easy and less tiring as compared with a conventional flat pedal. For example, when curved pedal  130  has at least one convexity  140 ,  160 , this may permit a player to be better able to feel on his or her toe a gradual local angle shift, slope change, within first convexity  140 , so as to allow easy positioning of a heel for a heel stroke, for example. 
     As another example of a technique that may be employed, a player may use a heel-toe technique and/or toe-heel technique. 
     In one such heel-toe technique, a player may first depress curved pedal  130  with his or her heel to generate a first stroke, tilt his or her toe down, and then depress curved pedal  130  with his or her toe to generate a second stroke. For example, a heel may be positioned for a first stroke as shown in  FIG. 16A  and then for a second stroke as shown in  FIG. 16B  or  FIG. 16C . 
     In one such toe-heel technique, a toe stroke may be a first stroke and a heel stroke may be a second stroke. For example, the toe may be positioned for a first stroke as shown in  FIG. 16B or 16C , and then the heel may be positioned for a second stroke as shown in  FIG. 16A . Possible foot positions are not limited to those described in  FIG. 16B  and  FIG. 16C . 
     In some embodiments, a player may use pedal assembly  110  to generate triplets, or three consecutive drum beats. Triplets may be generated in any of various ways. For example, a player may simply repeat the foot movement mentioned above for generating a single drum beat three times. One advantage of some embodiments of the present invention is that it facilitates production of three consecutive drum beats in one foot motion cycle. When three consecutive beats are produced by one foot motion cycle, very rapid consecutive beats may be easily achieved. Furthermore, such a foot motion cycle may be repeated as many times as desired to generate more than three consecutive beats. 
     It was unexpectedly found by the present inventor that pedal assembly  110  comprising curved pedal  130  makes it possible to easily combine heel-toe techniques (or toe-heel techniques) with sliding techniques for easy generation of triplets, for example. 
     In accordance with such a combined technique, a player may first depress curved pedal  130  with his or her toe to generate a first stroke, tilt his or her toe down, depress curved pedal  130  with his or her toe to generate a second stroke, slide his or her foot in length direction  132 , and then depress curved pedal  130  again with his or her toe to generate a third stroke. For example, a foot may be positioned for a first stroke as shown in  FIG. 16A , then for a second stroke as shown in  FIG. 16B , and then for a third stroke as shown in  FIG. 16C . Alternatively, a foot may be positioned for a first stroke as shown in  FIG. 16A , then for a second stroke as shown in  FIG. 16C , and then for a third stroke as shown in  FIG. 16B . Possible foot positions are not limited to those described in  FIG. 16A  through  FIG. 16C . 
     One advantage of at least some embodiments of the present invention is that curved pedal  130  may make generating triplets easier and less tiring as compared with a conventional flat pedal. 
     Presence of first convexity  140 , first concavity  150 , and/or second convexity  160  at curved pedal  130  may facilitate utilization of various sliding and/or heel-toe techniques. 
     Furthermore, actuatable region  135  of curved pedal  130  may be longer than the corresponding length in a conventional flat pedal. Where this is the case, the greater length of curved pedal  130  may provide space sufficient to allow a player&#39;s foot to perform ankle tilting and/or foot sliding motions in sequence, allowing greater degrees of freedom in combining toe-heel techniques and sliding techniques, and making it possible to more easily generate triplets, for example. 
     Although various foot positions have been shown in  FIGS. 16A through 16B , there is of course no limitation on the manner in which curved pedal  130  or pedal assembly  110  is used, the exact foot positions with respect to curved pedal  130  for generation of one or multiple drum beats being freely chosen depending, for example, on the player&#39;s preference, the player&#39;s foot shape and/or size, whether or not the player is wearing socks, shoes, and/or or other such foot coverings or is playing with bare feet, for example. 
     Where pedal assembly  110  is used to play high-hat cymbals  104 , there is no particular limitation on the manner in which this may be carried out; for example, use of pedal assembly  110  to play high-hat cymbals  104  may be generally similar to use of pedal assembly  110  to play a drum  103  as described above. 
     Because bass drum  103  in accordance with embodiments of the present invention may permit faster playing than would be possible with a conventional flat pedal, this may allow more versatility in playing than was conventionally possible. 
     For example, whereas with a conventional flat pedal a player might have been forced to employ two pedals on one drum to achieve a certain degree of frequency of repetitions in striking the drumhead, curved pedal  130  in accordance with embodiments of the present invention may allow such a player to achieve comparable frequency of repetitions with a single curved pedal  130 , thus freeing up the other foot to play another drum  103  and/or high-hat cymbals  104 . An arrangement suitable for such manner of playing is shown in  FIG. 17 , which shows a drum set  100  similar to that of  FIG. 1  except that drum set  100  of  FIG. 17  contains two bass drums  103 , each of which has an independent pedal assembly  110  as described above. 
     Note that a singled curved pedal  130  in accordance with embodiments of the present invention may be used to play multiple instruments through use of a pedal assembly  110  in combination with various linkages may permit tandem and/or parallel playing. Similarly, multiple curved pedals  130  in accordance with embodiments of the present invention may be used in pedal assemblies  110  in combination with various linkages to strike the same and/or different instruments. One such arrangement is shown in  FIG. 18 , but it should be understood that all such variations are intended to be within the scope of the claims. 
     Note that curved pedal  130  and pedal assembly  110  are not limited to employment in bass drums  103 , percussion instruments  102 , drum sets  100 , or musical instruments, but may be applied for use in any of a wide variety of applications where dexterity, responsiveness, and comfort are desired, especially when a pedal is to be operated for an extended period of time. Curved pedals  130  and pedal assemblies  110  in accordance with various embodiments of the present invention are particularly useful for generating rapid and/or repeated mechanical motions. 
     In some embodiments, such mechanical motion may be employed for playing a percussion instrument or non-percussion musical instrument. In one embodiment, such mechanical motions may be transmitted to an instrument directly when part of a pedal assembly physically comes in contact with an instrument. In another embodiment, such mechanical motions may be transformed into another form of signal, for example an electrical signal, and transmitted to an instrument indirectly. 
     In some embodiments, such mechanical motion may be employed to operate any of various devices and/or machines. Devices and/or machines in which curved pedals  130  and pedal assemblies  110  in accordance with various embodiments of the present invention may be employed include, without limitation, musical instruments, games, video games, toys, playground equipment, automobiles, helicopters, airplanes, backhoes and other such vehicles, construction equipment, and/or heavy equipment, looms, sewing machines, treadles, knitting machines, saws and/or mills, lathes, pumps, and/or other such manufacturing equipment and industrial apparatuses, as well as any of various devices employed in agriculture, forestry, robotics, and/or aerospace, for example. Regardless of field in which the present invention is applied, mechanical motion of foot-actuated curved pedal  130  may be transmitted by way of an assembly similar to pedal assembly  110  to a target device or machine directly or indirectly. Indirect transmission may include, without limitation, electrical transmission. Although the various embodiments of the present invention have been described in terms of an example in which the operator of curved pedal  130  is human, there is no particular limitation to use of curved pedal  130  or to pedal assembly  110  by a non-human, such as a pet or other animal, or by a non-animal such as a robot, for example. 
     While embodiments of the present invention have been described above, modes of carrying out the present invention are not limited to the foregoing embodiments, a great many further variations being possible without departing from the gist of the present invention.