Bell with subharmonic difference tone

Abstract

A bell and method of tuning a bell with its lowest frequency partials at f.sub.1=f and f.sub.2=3f=2. The simultaneous presence of physical tones at these partial frequencies yields a difference tone, perceived by the listener, at f.sub.2f.sub.1=3f=2f=f=2. The difference tone is subharmonic, in that its perceived frequency (f=2) is below the frequency of the fundamental (f). Preferably, the bell has one or more additional partials at frequencies f.sub.n=(n+1)f=2, with n 2 f3; 4; 5: : : g, strengthening the listener's perception of the difference tone at f=2. The bell thus yields a strike tone at f=2 but has a characteristic dimension (e.g. height or diameter) equal to that of conventional bells with a strike tone at f, providing art eightfold savings in bell mass.

Claims

1. A bell with a sequence of partials, comprising: a lowest partial substantially at frequency f, and a second-lowest partial substantially at frequency 3f/2, wherein said lowest partial and said second-lowest partial generate a subharmonic difference tone substantially at frequency f/2, and wherein said subharmonic difference tone is perceived as a strike tone of said bell that is characteristic of a strike tone that is produced by a conventional bell having a characteristic height or diameter dimension that is substantially larger than that of said bell.

2. The bell of claim 1, with one or more additional partials substantially spaced at a frequency interval of f/2 above said second-lowest partial.

3. The bell of claim 1, wherein f=130.8 Hz.

4. A bell with a sequence of partials, comprising: a lowest partial with a frequency within a range fEf, and a second-lowest partial with a frequency within a range 3f/2E(3f/2), wherein E is a fractional tolerance, wherein said lowest partial and said second-lowest partial generate a subharmonic difference tone substantially at frequency f/2, and wherein said subharmonic difference tone is perceived as a strike tone of said bell that is characteristic of a strike tone that is produced by a conventional bell having a characteristic height or diameter dimension that is substantially larger than that of said bell.

5. The bell of claim 4 wherein E has a value between 0.001 and 0.05.

6. A method of tuning a bell having a sequence of partials, comprising the steps of: selecting a desired strike tone frequency f/2, tuning the frequency of the lowest partial substantially at frequency f, and tuning the frequency of the second-lowest partial substantially at frequency 3f/2, wherein said lowest partial and said second-lowest partial generate a subharmonic difference tone substantially at said strike tone frequency, and wherein said subharmonic difference tone is perceived as a strike tone of said bell that is characteristic of a strike tone that is produced by a conventional bell having a characteristic height or diameter dimension that is substantially larger than that of said bell.

7. The method of claim 6, additionally comprising the steps of: tuning one or more additional partials substantially spaced at a frequency interval of f/2 above said second-lowest partial.

8. The bell of claim 6, wherein f=130.8 Hz.

9. A method of tuning a bell having a sequence of partials comprising the steps of: selecting a desired strike tone frequency f/2, tuning the frequency of the lowest partial to be within a range fEf, and tuning the frequency of the second-lowest partial to be within a range 3f/2E(3f/2), wherein E is a fractional tolerance, wherein said lowest partial and said second-lowest partial generate a subharmonic difference tone substantially at said strike tone frequency, and wherein said subharmonic difference tone is perceived as a strike tone of said bell that is characteristic of a strike tone that is produced by a conventional bell having a characteristic height or diameter dimension that is substantially larger than that of said bell.

10. The method of claim 9 wherein E has a value between 0.001 and 0.05.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a flowchart summarizing a method for tuning a bell with a subharmonic difference tone according to a preferred embodiment of the invention according to a preferred embodiment of the invention.

DETAILED DESCRIPTION

(2) The invention is a bell and method of tuning a bell with its lowest frequency partials at f.sub.1=f and f.sub.2=3f/2. The simultaneous presence of physical tones at these partial frequencies yields a difference tone, perceived by the listener, at f.sub.2f.sub.1=3f/2f=f/2. The difference tone is subharmonic, in that its perceived frequency (f/2) is below the frequency of the fundamental (f). Preferably, the bell has one or more additional partials at frequencies f.sub.n=(n+1f/2, with n{3, 4, 5 . . . }, strengthening the listener's perception of the difference tone at f/2. The bell thus yields a strike tone at f/2 but has a characteristic dimension (e.g. height or diameter) equal to that of conventional bells with a strike tone at f, providing an eightfold savings in bell mass.

(3) The design of the bell differs from conventional bells with (as described above) a missing fundamental and a hum tone below the missing fundamental. Rather, the present bell may be described as having a missing hum, a strong fundamental, and a strong perfect fifth. Preferably, the bell also has an octave and additional partials at frequencies spaced at an interval of f/2. Preferably, the fundamental, perfect fifth, and higher frequency partials sound simultaneously upon strike and persist for as long as possible after strike.

(4) FIG. 1 shows a flowchart summarizing a method for tuning a bell with a subharmonic difference tone according to a preferred embodiment of the invention according to a preferred embodiment of the invention. To perform the method, a bell designer begins 100 by choosing 200 a desired strike tone frequency f/2. The designer then tunes 300 the frequency of the lowest, fundamental partial at frequency f and tunes 400 the frequency of the second-lowest, perfect fifth partial at frequency 3f/2. If additional partials 500 are not desired, the designer finishes 700. If additional partials are desired, the designer tunes the frequency of the next partial at frequency interval of f/2 above the previous partial 600. The designer then considers whether additional partials are desired.

(5) The method of FIG. 1 yields a sequence of frequencies {f.sub.0, f.sub.1, f.sub.2 . . . f.sub.n}, where f.sub.0 is the frequency of the strike tone, f.sub.1 is the frequency of the fundamental partial, f.sub.2 is the frequency of the perfect fifth partial, and f.sub.3 . . . f.sub.n are the frequencies of any additional desired partials. For example, if one additional partial is desired, the resulting sequence of frequencies is {f, 3f/2, 2f}. In one embodiment of the invention, a fundamental partial at f.sub.1=C.sub.3=130.8 Hz and a perfect fifth partial at f.sub.2=G.sub.3=196.2 Hz yield a difference tone at f.sub.0=C.sub.2=65.4 Hz. The additional partial, if desired, would lie at C.sub.4=261.6 Hz.

(6) The precise size and shape of the bell is tuned to yield vibrational modes generating partials of the desired frequencies. In the preferred embodiment of the invention, the lowest partial is generated by the (2, 0) mode of vibration of the bell and the second-lowest partial is generated by the (3, 0) mode of vibration of the bell. In the preferred embodiment of the invention, the bell is tuned using an iterative optimization procedure in which the frequencies of the vibrational modes of each candidate design are calculated using a finite element analysis.

(7) Specific techniques for iterative optimization of the bell size and shape are well known in the art. In the preferred embodiment of the invention, the sequence of frequencies described for FIG. 1 is tuned based on the method outlined in U.S. Pat. No. 6,915,756 to McLachlan et al., which patent is incorporated herein in its entirety this reference thereto. The final bell design is determined by iterative exploration of candidate designs. Each candidate bell design (i.e. a particular bell size and shape) of an axisymmetric, generally conical bell is defined by a point within a parameter space with dimensions of cone angle, side length, wall thickness, wall taper, and wall curvature.

(8) The iterative optimization procedure proceeds by

(9) 1. setting the current bell design to an initial bell design;

(10) 2. selecting one of the partial frequencies to be tuned as a current objective;

(11) 3. selecting a desired value for the current objective;

(12) 4. modifying the current bell design in accordance with an optimisation method that moves the current value of the current objective towards the desired value;

(13) 5. repeating Step 4 until the current value of the current objective is substantially equal to the desired value (e.g. is within an allowable tolerance);

(14) 6. if the frequencies to be tuned do not match the desired sequence, selecting another one of the frequencies to be tuned as the current objective; and

(15) 7. repeating Steps 3-6 until the frequencies to be tuned are in the desired sequence.

(16) In Step 4, the current value of the current objective (i.e. the frequency of the partial that is the current optimization target) is evaluated using a finite element analysis. Modification of the current bell design proceeds according to a method of gradient descent through the parameter space of candidate bell designs.

(17) While the bell of the preferred embodiment of the invention is based on an axisymmetric, generally conical bell design, the invention is not limited to such bell geometries. Other bell geometries may be constructed using different parameter spaces without departing from the scope of the invention.

(18) One skilled in the art will appreciate that it is impractical to tune a bell with infinite precision. In practice, each partial frequency within the desired series of partial frequencies can be attained to only a reasonable degree of precision. Herein, when a partial is stated to be at a frequency f.sub.0, one skilled in the art will appreciate that this indicates that the frequency of the partial is substantially at f.sub.0, falling within a range of possible values about a nominal value of f.sub.0. For example, the actual value of the partial frequency f may lie within a range, f.sub.0f.sub.0 defined by a fractional tolerance . For example, for a tolerance of =1%, the actual value of the partial frequency fall in the range f.sub.00.01 f.sub.0. Tolerances of 0.1%, 1%, 5%, and other values are possible without departing from the scope of the invention.

(19) Although the invention is described herein with reference to several embodiments, including the preferred embodiment, one skilled in the art will readily appreciate that other applications may be substituted for those set forth herein without departing from the spirit and scope of the invention.

(20) Accordingly, the invention should only be limited by the following Claims.