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 (N)Force 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 (dB) Mobility -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 collisions modelling

28 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Collisions in musical instruments

(Walstijn, 2017)

String-string collisions 30 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions String/string collision modelling

Coplanar cross-sections of two strings:

No Penetration

• Absolute coordinates: • Distance between centroids: • Impact angle: () () 푍 − 푍 푌 푥, 푡 = 푦 + 푦 (푥, 푡) 푟 푥, 푡 = (푌 − 푌 ) −(푍 − 푍 ) γ(푥, 푡)= 푎푟푐푡푎푛 푌 − 푌 () () 푍 푥, 푡 = 푧 + 푧 (푥, 푡) 31 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions String/string collision modelling

Two colliding cross-sections:

• Nonlinear model with hysteresis damping (Hunt and Crossley, 1975) Interpenetration Compliance exponent

풊풎풑 풔 ̇ 푭 (푥, 푡) = 퐾δ 푥, 푡 +λ훿 푥, 푡 δ(푥, 푡)

Contact stiffness Damping coefficient

32 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions

V. Sound synthesis of the viola caipira: numerical experiments

32 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Simulation parameters Number of modes • Strings: modes up to 5000 Hz • Body: 20 modes between 0 Hz and 1000 Hz Excitation model Ramp function Maximum amplitude Release time 퐹(푡) 퐹

() Initial time • 푡 − 푡 = 8 푚푠 () • 퐹 = 3 푁

Excitation position: 8.5 cm from the bridge Sampling frequencies (after convergence tests) Impact parameters A. Without collisions: 220.5 kHz Table 1. Impact model parameters B. With collisions: 441.5 kHz

33 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Organization of collisions in space and time

Collisions space-time diagram

Pluck parallel to the soundboard

• Collision point moves along the string length. • Collisions occur only in the immediate transient phase just after the pluck.

35 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Collisions effects on the sound: buzzing and spectral enrichment

Without collision With collision

• Buzzing effect: induced by the repeated collisions in the early transient phase. • Spectral enrichment: spectral rays are broadened during the collisions . 36 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Collisions effects on the sound: redistribution of the spectral components

Without collision With collision

• The filtering related to the plucking position is cancelled since the collision point moves. 37 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Collisions effects on the vibration: polarization change mechanism

Parallel pluck Inclined pluck

y y D3 D4 D3 D4

Polarization remains in the Polarization changes plane

• The plucking angle plays an important role in the polarization change.

38 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Fully coupled system simulations: sympathetic resonances and beating

Sympathetic resonances! Single string (D4)

10 strings (D4) 10 strings (D4)

Pluck parallel to the soundboard Without collisions With collisions

• Strings couplings through the bridge induce sympathetic resonances. Time-history of string 5 (G3)

Parallel displacement

[mm] Normal displacement

0 Time [s] 5

• Beating phenomena are also observed on some strings.

39 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Fully coupled system simulations: influence of collisions on the aftersound Without collisions With collisions

Parallel displacements

Normal displacements

40 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Fully coupled system simulations: effect of the pluck direction

Ten strings (D4) Ten strings (D4)

Parallel pluck Normal pluck

• The sound halo is much more perceptible for an excitation normal to the soundboard.

41 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions

VI. Application to the instrument making: tools for makers

42 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions

Goal • Collaboration with : Music Instrument Making Platform

• http://pafi.univ-lemans.fr

• Integration to the platform i/ Wire technique: low cost technique for measuring mobilities ii/ Sound synthesis tool

43 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions

VII. Conclusions and perspectives

46 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions Conclusions • Specificities of the viola caipira : i) Particular double pluck ii) Sympathetic resonances and beating phenomena iii) Collisions between strings.

• A physical model for sound synthesis including string/string collisions. It reproduces the main sound features of the viola caipira.

• Collision effects on the sound: • Buzzing effect/spectral enrichment • Redistribution of spectral components • Polarization change

• The Roving Wire-Breaking Technique: • Novel procedure for modal analysis • Low cost and suitable for instrument makers

47 I. Introduction II. Experimental study III. Wire-breaking method IV. Modelling V. Sound synthesis VI. Application VII. Conclusions

Perspectives • Parameters adjustment and experimental validation of the sound synthesis model

• Inclusion of string non-linearities

• Sound radiation model

48 R&D Projects Computer-aided musical instrument manufacturing - viability

• URUTAU PROJECT: “Cloud environment for design, analysis and simulation of musical instruments”.

• XURI PROJECT: “Small room acoustic correction – web application for optimal treatment”.

Acoustic consulting ...

Acoustic panels manufacturing ...

Contact: [email protected]

48 The synthesized viola caipira!

50 Thank you for your attention.

O Violeiro by Almeida Júnior, 1899. Scientific production Articles • G. Paiva, F. Ablitzer, F. Gautier and J. M. C. dos Santos, “Collisions in double string plucked instruments: physical modelling and sound synthesis of the viola caipira”, Journal of Sound and Vibration. • G. Paiva, F. Ablitzer, F. Gautier and J. M. C. dos Santos, “The Roving Wire-Breaking Technique: a low cost mobility measurement pro cedure for string musical instruments”, Applied Acoustics. Yamaha Music Internship • Numerical Analysis of Electric Guitars, March/2017 – July/2017. Conference papers • G. Paiva, J. M. C. dos Santos, Modelling Fluid-Structure Interaction in a Brazilian Guitar Body. In: International Congress on Mechanical Engineering, Ribeirão Preto, Brazil, 2013. • G. Paiva, J. M. C. dos Santos, Vibroacoustic Modal Analysis of a Brazilian Guitar Resonance Box. In: International Conference on Structural Engineering Dynamics, Sesimbra, Portugal, 2013. • G. O. Paiva, J. M. C. dos Santos, Modal Analysis of a Brazilian Guitar Body. In: International Symposium on Musical Acoustics, 2014, Le Mans, France, 7-12 July 2014. • G.O. Paiva, F.Gautier, F.Ablitzer, J.M.C dos Santos, M. Sécail-Géraud, Measuring the Mobility Matrix at the Bridge of Stringed Instruments by the Wire Breaking Method, 2016, Le Mans, France, 11-15 April 2016. • F. Gautier, F. Ablitzer, G. Paiva, B. David, M. Curtit, M. Sécail, E. Brasseur, G. Michelin, Methodologies and tools for characterising stringed musical instruments in the maker’s workshop, Making wooden musical instruments: an integration of different forms of knowledge, 7-9 September, Barcelona, 2016. Poster presentations • G.O. Paiva, F.Gautier, F.Ablitzer, J.M.C dos Santos, Modélisation physique et synthèse sonore d’une guitare brésilienne Part. 1, Forum Jeunes Recherche, Université du Maine, Le Mans, France, 2016. • G.O. Paiva, F.Gautier, F.Ablitzer, J.M.C dos Santos, Modélisation physique et synthèse sonore d’une guitare brésilienne Part. 2, JJCAB2016 : Journées Jeunes Chercheurs en vibration, Acoustique et Bruit 17-18 nov. Marseille, France, 2016. • G. O. Paiva, J. M. C. dos Santos, Modal Analysis of a Brazilian Guitar Body. In: International Symposium on Musical Acoustics, 2014, Le Mans, France, 7-12 July 2014. Oral presentations • G. O. Paiva, J. M. C. dos Santos, Vibroacoustic Numerical Analysis of a Brazilian Guitar Resonance Box. In: ESSS CONFERENCE ANSYS USERS MEETING, Atibaia, Brazil, 2013. • G. O. Paiva, F. Gautier, F. Ablitzer, J. M. C. dos Santos, Time Domain Simulations of a Ten-String Brazilian Guitar, ViennaTalk2015, Vienna, Austria, 16-19 September 2015. • G. O. Paiva, F. Ablitzer, F. Gautier, M. Sécail-Géraud and J. M. Campos Dos Santos, Physical Modelling and Sound Synthesis of a Viola Caipira, In: International Symposium on Musical Acoustics, Montreal, Canada, 18-22 June 2017.