1. NONLINEAR SOURCE-FILTER COUPLING IN SPEECH AND SINGING
MUSIC 318 MINI-COURSE ON SPEECH AND SINGING
NONLINEAR SOURCE-FILTER COUPLING IN SPEECH AND SINGING
“Modeling source-filter interaction in belting and high-pitched operatic male
singing” (I. Titze and A. Worley, JASA 126, 1530 (2009))
“Source-Filter Interaction in Speaking and Singing is Nonlinear” (I.Titze, ECHOES
Summer 2007)_
“The Human Instrument” (I.Titze, Scientific American Jan. 2008)

2. NONLINEAR vs LINEAR MODELS OF PHONATION
THE LINEAR MODEL OF SPEECH AND SINGING ASSUMES THAT THE SOURCE (GLOTTIS), FILTER (VOCAL TRACT) AND RADIATOR (MOUTH, NOSE) ACT INDEPENDENTLY.
THE LINEAR MODEL SUCCESSFULLY EXPLAINS MANY ASPECTS OF SPEECH AND SINGING. AS LONG AS THE DOMINANT SOURCE FREQUENCIES ARE WELL BELOW THE FORMANT FREQUENCIES OF THE VOCAL TRACT (GENERALLY TRUE IN MALE SPEECH), THE SOURCE IS INFLUENCED ONLY SLIGHTLY BY THE FILTER. IN FEMALE AND CHILD SPEECH, HOWEVER, THE INTERACTION IS GREATER.
RECENT RESEARCH SUGGESTS THAT HUMANS CAN OPERATE THEIR SOURCE-FILTER SYSTEMS WITH EITHER LINEAR OR NONLINEAR COUPLING. FOR LINEAR COUPLING THE SOURCE IMPEDANCE (TRANSGLOTTAL PRESSURE DIVIDED BY GLOTTAL FLOW) IS KEPT MUCH HIGHER THAN THE INPUT IMPEDANCE TO THE VOCAL TRACT. THIS IS ACCOMPLISHED BY ADDUCTING THE VOCAL FOLDS FIRMLY AND WIDENING THE EPILARYNX TUBE SO THAT GLOTTAL FLOW IS DETERMINED BY AERODYNAMICS, AND ACOUSTIC PRESSURES ABOVE AND BELOW THE GLOTIS HAVE LITTLE INFLUENCE.

3. NONLINEAR SOURCE-FILTER COUPLING IN PHONATION
IN NONLINEAR COUPLING THE ACOUSTIC AIRWAY PRESSURES CONTRIBUTE TO THE PRODUCTION OF FREQUENCIES AT THE SOURCE. TRANSGLOTTAL PRESSURE INCLUDES A STRONG ACOUSTIC COMPONENT, MUCH AS IN WIND INSTRUMENTS.
FOR NONLINEAR COUPLING, THE GLOTTAL IMPEDANCE IS ADJUSTED TO BE COMPARABLE TO THE VOCAL TRACT INPUT IMPEDNACE, MAKING THE GLOTTAL FLOW HIGHLY DEPENDENT ON ACOUSTIC PRESSURES IN THE VOCAL TRACT. THIS IS ACCOMPLISHED BY SETTING ADDUCTION LEVELS OF THE VOCAL FOLDS THAT MATCH A NARROWER EPILARYNX TUBE.
EVIDENCE OF NONLINEAR COUPLING IS THE PRODUCTION OF NEW FREQUENCIES IN THE FORM OF DISTORTION PRODUCTS, LOWERING OF THE OSCILLATION THRESHOLD PRESSURE, PRODUCTION OF SUBHARMONICS, AND SUDDEN JUMPS AS EITHER VOWEL OR F0 ARE CHANGED.

4. POSITIVE FEEDBACK TO VOCAL FOLDS
POSITIVE FEEDBACK FROM THE VOCAL TRACT TO THE VOCAL FOLDS CAN INCREASE SOUND PRODUCTION. (THIS IS SIMILAR TO THE POSITIVE FEEDBACK FROM A BRASS INSTRUMENT TO THE PLAYER’S LIPS). THE IDEAL TIMING OF THE “KICK” COMES WHEN THE MOVEMENT OF THE AIR IS DELAYED WITH RESPECT TO THE MOVEMENT OF THE VOCAL FOLDS. THE AIR COLUMN THEN HAS NEGATIVE INERTANCE. INERTIVE REACTANCE HELPS TO SUSTAIN THE FLOW-INDUCED OSCILLATION OF THE VOCAL FOLDS. (See Titze, 2008).
THE SINGER’S TASK IS TO ADJUST THE SHAPE OF THE VOCAL TRACT (BY CAREFULLY SELECTING FAVORABLE “SINGING” VOWELS) SO THAT INERTIVE REACTANCE IS EXPERIENCED OVER MOST OF THE PITCH RANGE—NO EASY TASK.

5. SOURCE-FILTER INTERACTION IN BELTING AND OPERATIC SINGING
BELTERS USE VOCAL TRACT RESONANCES (FORMANTS) DIFFERENTLY FROM CLASSICALLY-TRAINED (OPERA AND ART SONG) SINGERS. THE SECOND HARMONIC, FOR EXAMPLE, RECEIVES STRONG REINFORCEMENT FROM THE FIRST FORMANT IN BELTING. (COMPARE THIS TO PAVAROTTI WHO USES THE SECOND FORMANT TO REINFORCE THE THIRD HARMONIC IN OPERA SINGING).
TITZE AND WORLEY (2009) OBSERVE THAT THIS FORMANT TUNING CAN BE AIDED BY UTILIZING SUPRAGLOTTAL INERTIVE REACTANCE TO REINFORCE VOCAL FOLD VIBRATION BY CHOOSING PITCH-VOWEL COMBINATIONS THAT KEEP SEVERAL HARMONICS IN FAVORABLE REACTANCE SIMULTANEOUSLY.

6. D5 SUNG BY BARBRA STREISAND ON THE WORD “STRONG” SHOWING A STRONG 2ND HARMONIC (Fig. 11.1 in Miller)

7. SOURCE-FILTER INTERACTION
DIAGRAM OF VOCAL FOLDS AND LOWER VOCAL TRACT

8. SOURCE-FILTER INTERACTION IN BELTING AND OPERATIC SINGING
BELTERS USE VOWELS THAT ARE MODIFIED TOWARD A “MEGAPHONE” MOUTH SHAPE. BOTH THE FUNDAMENTAL AND THE SECOND HARMONIC ARE THEN KEPT BELOW THE FIRST FORMANT.
OPERA SINGERS, ON THE OTHER HAND, USE VOWELS MODIFIED TOWARD AN INVERTED MEGAPHONE MOUTH SHAPE FOR TRANSITIONING INTO THE HIGH-PITCH RANGE. THIS ALLOWS ALL THE HARMONICS EXCEPT THE FUNDAMENTAL TO BE “LIFTED” OVER THE FIRST FORMANT.

9. MOUTH AREA OF SINGERS
MOUTH AREA AND SOUND SPECTRUM FOR PAVOROTTI SINGING /α/ VOWEL AT A4
MOUTH AREA AND SOUND SPECTRUM FOR CAB CALLOWAY SINGING /α/ VOWEL AT A4
(Titze and Worley 2009)

10. ACOUSTIC IMPEDANCE
ACOUSTIC IMPEDANCE IS SOUND PRESSURE DIVIDED BY FLOW VELOCITY
ACOUSTIC ADMITTANCE (1/IMPEDANCE) IS FLOW VELOCITY DIVIDED BY SOUND PRESSURE. CONDUCTANCE IS THE IN-PHASE (REACTIVE) PART OF ADMITTANCE; INERTANCE IS THE OUT-OF-PHASE PART

11. VOCAL TRACT INPUT IMPEDANCE
VOCAL TRACT CARICATURES (left) AND CORRESPONDING INPUT IMPEDANCES (right) AS A FUNCTION OF FREQUENCY (THICK LINES ARE REACTANCES AND THIN LINES ARE RESISTANCES) (Titze and Worley, 2009)

12. COMPUTER SIMULATION OF GLOTTAL AIRFLOW
COMPUTER SIMULATION OF GLOTTAL AIRFLOW WITH A SELF-SUSTAINED OSCILLATION VOCAL-FOLD MODEL THAT INTERACTS WITH: A UNIFORM TUBE OF DIFFERENT AREAS (left) AND A NEUTRAL TUBE WITH DIFFERENT EPILARYNX AREAS (right) (Titze and Worley, 2009)

13. INERTANCE
(INERTANCE IS THE REACTIVE PART OF ADMITTANCE; IT INDICATES ENERGY STORAGE DURING OSCILLATION)
INERTOGRAMS (INERTANCE vs FREQUENCY FOR SIX TUBE SHAPES (Titze and Worley, 2009)

14. INERTOGRAMS OF SINGERS
VOCAL TRACT SHAPES DERIVED FROM MRI DATA OF A BARITONE SINGER (left) AND CORRESPONDING INERTOGRAMS (Titze and Worley, 2009)

16. MOUTH-TO-HEAD AREA RATIOS
Table I. Mouth-to-head area ratios.
Note
Ratio
Vowel
Male operatic (Luciano Pavarotti)
D#4
0.0137
/e/ E4
0.0205 //
F#4
0.0288 G4
0.0290 A4
0.0291
Male belt (Cab Calloway) D
0.0170
/u/
0.0364
/o/ F
0.0614
/a/-/o/ (diphthong)
0.0662
/a/
0.0840