Designing musical structures using a constrained optimization approach

Prestresses are purposefully added to an object to improve its performance, such as tuning a guitar string by adding tension. This paper reports how the normal modes of a sheet metal component can be tuned through the prestresses generated by cold-forging small dimples. Finite element analysis showed that the frequencies of specific mode shapes were differentially affected by the location of residual stress fields due to dimple formation in relation to modal stress fields. The frequencies of overtones were most sensitive to the depth of the dimples located near the maxima of modal stresses. Using this approach a series of musical gongs were designed with up to the first five overtones tuned to within 5% of the harmonic series. The balance of harmonic and inharmonic overtones in these gongs that are well resolved by the human cochlea may constitute a set of recognizable musical timbres with sufficient harmonicity to produce an unambiguous pitch for most listeners. Since many other mechanical properties of sheet metal components are affected by residual stresses this manufacturing technique may have broader application in design engineering.

Section: B Musical Gongs and The Perception Of Pitchmentioning

confidence: 98%

Tuning natural modes of vibration by prestress in the design of a harmonic gong

McLachlan

Adams

Burvill

2012

“…stiffness at the bar centre both lower the frequency of these modes. Shape optimization has been applied to FEA models of xylophone bars to alter the undercut arch and tune the first three transverse modes to ratios of 1:4:10 (Petrolito and Legge, 1997;Petrolito and Legge, 2005) and the ratios 1:3:6, 1:4:8-9, and 1:5:10-13 (Orduña-Bustamente, 1991). Similarly up to five transverse modes have been tuned to a variety of frequency ratios for vibraphone bars with complex undercut surface shapes (Henrique and Antunes, 2003).…”

Section: B Metalophone Keysmentioning

confidence: 99%

The design of a harmonic percussion ensemble (L)

McLachlan

2011

The primary oscillators of tuned percussion instruments are rigid structures that exhibit bending waves. Bending waves do not naturally occur at harmonically related frequencies, which may lead to uncertainty associated with the intended pitch of the instrument. Despite millennia of development across many cultures, tuned percussion instruments rarely exhibit harmonic tuning of the first three or more partials. This letter presents three percussion instruments that have three or more harmonically tuned partials, and provides an overview of the methods used to tune the partials and manufacture the instruments at relatively low production costs. These instruments form the basis of new percussion instrument ensembles for educational, recreational, and professional use.

“…Both of these techniques assume plane-wave propagation. Similar techniques have been applied to the optimization of xylophones 27,28 and to leak detection. 29 This section describes briefly the computer optimization technique developed by Braden, further details of which are given elsewhere.…”

Section: Bore Optimization Using Specified Input Impedance Targetsmentioning

confidence: 99%

Trombone bore optimization based on input impedance targets

Braden

Newton

Campbell

2009

Optimization methods based on input impedance target functions have been proposed for the design of brass musical instruments. Criteria for target functions in trombone bore optimization are discussed, drawing on experimental input impedance data from a variety of high-quality trombones of differing sizes. An "inharmonicity plot" is introduced and used to aid the interpretation of impedance curves. An efficient optimization technique is described and is shown to be capable of predicting bore changes which achieve specified modifications to the input impedance curve while maintaining a smoothly-flaring bell contour. Further work is required to clarify the relationship between input impedance targets and the preferences of professional players.