The physics of the viola caipira Guilherme O. Paiva [email protected] Acoustics Laboratory of the University of Maine, Le Mans, France School of Mechanical Engineering, DMC, Unicamp, Campinas, Brazil Turbonius – Research & Development, Campinas, Brazil June, 2020. I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions I.Introduction 2 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions What is a viola caipira ? • It is a typical Brazilian guitar • Instrument little studied in musical acoustics Not every BRAZILIAN GUITAR is aVIOLA CAIPIRA Examples of Brazilian guitars: Viola Machete Viola Nordestina Viola de Buriti Viola de Fandango Viola de Cocho • Variations of geometries, materials, string arrangements, tuning types, etc… The viola caipira (or the violas caipiras? ) 3 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Fernando Sodré performing Luzeiro from Almir Sater 4 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Musical acoustics context Viola caipira Folk guitar Classical guitar Sounds Structural characteristics RELATIONSHIP ? Objective approach: Perceptive approach: Physical parameters to describe Sensory experience of the individual. the instrument and its sounds. • This research: objective approach to study the viola caipira. 5 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Objectives of the thesis • Identification and study of the vibrational and acoustical specificities of the viola caipira. 1. Experimental study A. High-speed camera analysis B. Vibration analysis C. Sound analysis • Development of a sound synthesis model able to reproduce the specificities of the viola caipira. 2. Numerical study A. Physical modelling B. Sound simulations 6 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions II. Experimental study of the viola caipira 7 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions The studied viola caipira: a representative exemplar • Rozini brand, Ponteio Profissional model. • Smaller body with a narrower waist than those of classical guitars. • Tuning: Rio Abaixo • Wood types • Soundboard: Sitka Spruce • Back and sides: Indian Rosewood • Neck: Indian Rosewood • Fretboard: Ebony 8 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions A. High speed camera analysis: experimental setup High speed camera: Photron, FASTCAM SA-X2 • 1024 × 768 pixels of resolution • 5000 frames/sec 9 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions A. High speed camera analysis • Particular double pluck • Non-planar motions • Collisions between strings 12String/stringNon-planarstndpluckpluck after motions collisions ~14 ms 10 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions A. High speed camera analysis • String sympathetic resonances due to the bridge motion. 11 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions B. Sound analysis • Sympathetic resonances and beating due to the strings coupling through the bridge. 1 free string 10 free strings Beating associated to the two orthogonal transverse motions. Sound halo: Effect due to the string sympathetic vibrations Only one string is plucked Beating associated to the interaction of multiple strings. 12 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions C. Vibration analysis of the body • Global vibration map: Operating deflection shapes (ODSs) f = 129.7 Hz f = 264.1 Hz f = 356.3 Hz f = 751.6 Hz A0 T T T 1 2 3 Scanning laser vibrometer Measured receptance 13 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions C. Vibration analysis of the body • ODSs along the bridge using a laser vibrometer 1 2 3 4 5 6 7 8 9 10 z y 10 string/body coupling points f = 134 Hz, A0 f = 267 Hz, T1 f = 351 Hz, T2 • At some frequencies, the mobility at the 10 points are significantly different. f = 751 Hz, T3 • These ODSs reveal the coupling between strings due to the bridge motion. f = 1794 Hz 14 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions III. Mobility measurements using the wire-breaking method 15 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Mobility: definition and why measuring • Definition: transfer function (ω) (ω) = (ω) • The mobility measured at the bridge quantifies the conversion of string force into bridge velocity (degree of coupling between strings and body ) • Powerful sound Large Mobility Strong coupling • Fast decay • Less powerful sound Small Mobility Weak coupling • Slower decay 16 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Mobility measurement using the hammer method (classical method) 17 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Mobility measurement using the wire-breaking method 18 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Mobility: Replaces the impact hammer method: 퐴 ω Wire breaking force 푌 ω = Excitation at the coupling point 푓 Excitation in different directions Low cost (suitable for makers) Measurement of 푓 is required 19 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Wire-breaking method vs. Hammer method 20 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Calibration of mobility from ퟎ measurement • Wire diameter: 100 μm 10 measures 5 Wire 4 Wire 3 2 1 Force Force (N) 0 -1 Fixed force transductor -2 0.44 0.46 0.48 0.5 0.52 0.54 Time (s) ퟎ -10 calibrated hammer -20 calibrated wire -30 -40 -50 ퟎ Mobility (dB) -60 -70 -80 0 500 1000 1500 2000 2500 3000 Frequency (Hz) 21 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions The “Roving Wire-Breaking Technique”: modal analysis procedure • Excitation positions: 0 1 to 5 2 1 5 4 3 • Excitation normal and parallel to the soundboard • Response position: 0 • Wire diameter: 100 μm x • Instrument on a guitar stand Accelerometer z position 0 y Wire holder • 12 inertance curves • Single-Input-Multiple-Output High-resolution modal analysis 1st step. Estimation of poles: ESPRIT (Roy & Kailath, 1989) 2nd step. Estimation of the mode shapes components • Time domain method • Fit in the frequency domain (collocated mobility) Signal subspace 푗ω 푲 Noise subspace 푌, ω = 퐴 ω +ω + 푗2ξωω () 푠 푛 = 푎푒 푒 + 푤 푛 퐴 =ϕ,ϕ, 퐻 ω , computed from poles! Constraint 1: 퐴 is positive • The signal subspace verifies the rotational invariance property: Constraint 2: 퐴 is real 푾↑ 2퐾 = 푾↓ 2퐾 푹(2퐾) • Non-Negative Least Square Procedure (NNLS) 푻 푾 2퐾 : basis of the signal subspace min 푪풙 − 풅 풙 =[퐴 … 퐴] , 푥 ≥ 0 ∀푘 풙 푹(2퐾): matrix whose eigenvalues are the poles 푧 = 푒 - Provides intrinsically the modal order • Other modal amplitudes: Standard Least Square 23 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions The “Roving Wire-Breaking Technique”: modal analysis procedure Modal fit Modal identification • 47 modes between 50 Hz and 3000 Hz Mode shapes at the string/body coupling points A0 T1 T2 f = 133.9 Hz f = 264.4 Hz f = 355.4 Hz 23 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions VI. Physical modelling 24 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Principle of the model: hybrid modal approach 25 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Simplified case: 1 string,1 polarization String displacement String motion equation (modal) 풆 풄 At the string/bridge coupling point, x = L: Bridge displacement Bridge motion equation (modal) 풃 풄 27 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Fully coupled system:10 strings, 2 polarizations Strings 푧 (푥, 푡) 푦 (푥, 푡) 휱(푥) Bridge 푧(푝 , 푡) 푦(푝 , 푡) 휱(푥) • Resolution in time domain • Centered finite differences • Explicit matricial formulation 28 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions String/string