The “Tritone Paradox” and Spectral Dominance

An unnatural test of a natural model of pitch perception:

The “tritone paradox” and spectral dominance

Richard PARNCUTT, University of Graz Amos Ping TAN, Universal Music, Singapore Octave-complex tone (OCT) on C 2

1 amplitude

0 100 1000 10000 frequency (Hz) Sound demonstrations C A# OCTs: D G# E F# pure tones: F#2 G#2 A#2 C3 D3 E3 F#3 G#3 A#3 C4 D4 E4

F#4 G#4 A#4 C5 D5 E5 F#5 G#5 A#5 C6 D6 E6 Octave-complex tones (OCTs) on C and F# 2

C F#

1 amplitude

0 100 1000 10000 frequency (Hz) Shepard tone on C with bell-shaped amplitude envelope 60

20 SPL (dB) SPL

0 100 1000 10000 frequency (Hz) Studies by Deutsch and by Repp Perceived direction of tritone depends on: Sound: • Absolute frequency • Spectral envelope • Previous context Listener: • Analytic versus synthetic hearing • Bias toward rising or falling • Culture, language or dialect Diana Deutsch (1987)

The tritone paradox: Effects of spectral variables. Perception & Psychophysics.

„...the form of the relationship between pitch class and perceived height can be surpisingly robust in face of substantial differences in both the relative amplitudes of the sinusoidal components of the tones and their overall heights“ Bruno Repp (1997)

Spectral envelope and context effects in the tritone paradox. Perception.

• Lower pitch ~ center of spectral envelope • Big individual differences in envelope dep. • Big context effect Two different „paradoxes“

1. Inter-individual paradox: Some listeners consistently hear tritone rise, others consistently hear it fall.

2. Intra-individual paradox: The same listener hears the same tritone rise or fall on different occasions. Terhardt‘s pitch theory Spectral pitches ~ audible partials SP salience depends on: • level above masked threshold (saturation) • spectral dominance (prox. to 700 Hz)

Virtual pitches ~ fundamentals of harmonic patterns of spectral pitches VP salience depends on: • goodness of fit (harm. template, spectrum) • harmonic number (the lower the better) Terhardt‘s spectral dominance region

1,5

1,0 weight 0,5

0,0 100 1000 10000 log frequency (Hz) Shepard tone on C Predictions of Terhardt‘s pitch algorithm

60 SPL (dB) VP salience (x50) SP salience (x50)

0 100 1000 10000 log frequency (Hz) Ernst Terhardt (1991) Music perception and sensory information acquisition. Music Perception. Pitch(es) of an octave-complex tone: • usually virtual • typically near 300 Hz (D4) Their saliences: • depend on spectral dominance function • VP: almost indep. of spectral envelope Aims of our experiment

• Determine the octave register of the perceived pitch of each tone • Use this to predict perceived direction of tritone Assumptions 1. Pitch, like frequency, is one-dimensional 2. If listeners hear e.g.: C – F#, D – G# and E – A# as rising then they hear: C as the lowest, A# as the highest 3. Pitch of pure tones is clear of complex tones is ambiguous => register of OCT by comparison with pure Experiments 1. Direction („tritone paradox“ paradigm) Stimulus: harmonic tritone of OCTs Response: up or down 2. Height (new) Stimulus: isolated OCT Response: 1 = very low to 5 = very high 3. Distance (new) Stimulus: OCT then pure, same pitch class Response : 1 = very close to 5 = very far Listeners

• 10 male, 10 female • 19-40 years, mean 23 • mostly undergraduates, mostly psychology Equipment

• Standard PC and software • Audiocard Soundblaster 16 • BeyerDynamic DT 100 headphones Stimuli

Octave-complex tones (OCTs): • all tones, partials: ET whole-tone scale • 10 partials per OCT, 20 – 20 000 Hz • equal amplitude before amplification

Pure comparison tones : • Same amplitude as partials of OCTs

Duration of all tones and pauses: 250 ms Design

Each experiment: • A few practice trials with feedback • Main trials determined by symmetrical combination of IVs ⇒ • Different random order for each listener Grouping of listeners based only on Expt 1 • Does C sound lower or higher than F#? • Combine data from 2 orders; t-test • Repeat for D-G# and E#-A#

• Groups: RRR (N=5), URR (3), UUU (3), RUU (2), RFF (2), 7 others (1) (R = rising, F = falling, U = unsure = diff. not sig) • Combine RRR, URR, RUU, UUR Listener groups

Group NNN CCC-C---F#F#F#F# DDD-D---G#G#G#G# EEE-E---A#A#A#A# lowlowlow high OCT OCT 111 111111 R (U) R (U) R (U) CCC A#A#A#

222 333 UUU UUU UUU ------

333 222 RRR FFF FFF G#G#G# F#F#F# 444 111 FFF RRR RRR DDD CCC

555 111 UUU FFF FFF G#G#G# EEE

666 111 FFF FFF RRR EEE DDD

777 111 RRR UUU FFF A#A#A# F#F#F#

R=rising, F=falling, U=unsure Results: Group 1, Expt 1 Does the melodic tritone rise or fall?

25 20 15 rising falling 10 unsure 5 0 no. of responses of no. C-F# F#-C D-G# G#-D E-A# A#-E pitch classes of OCTs Results: Group 1, Expt 2 How high is the OCT?

5 A# 4 G# F# 3 E C 2 D perceived height perceived 1 0 2pitch class 4 of OCT 6 8 Results: Group 1, Expt 3 Perceived distance between OCT and pure comparison tone

5 E6 C6 4 D6 A#5 G#5 F#2 3 E5 D5 F#5 G#2 C5 A#2 E4 C3 A#4 E3 D3 F#3

2 G#4 F#4 D4 A#3 C4 G#3 perceived distance perceived 1 0 5 10 15 20 25 pitch of pure tone Results: Group 2, Expt 1 Does the melodic tritone rise or fall?

20 rising falling 15 unsure 10

5 no. of responses no. 0 C-F# F#-C D-G# G#-D E-A# A#-E pitch classes of OCTs Results: Group 2, Expt 2 How high is the OCT?

4 G# A# 3 C F# D E 2 perceived height perceived 1 0 2pitch class 4 of OCT 6 8 Results: Group 2, Expt 3 Perceived distance between OCT and following pure tone

5 4 F#2 A#2 G#2 C3 D3 3 E3 C4 G#3 F#3 A#3 D4 D6 F#5 E4 A#5

2 E6 C6 C5 D5 G#5 A#4 E5 G#4 F#4 1 perceived distance perceived 0 5musical 10 pitch of 15pure tone 20 25 Summary data for all groups

lowest & register assume group N highest OCTs of OCTs listener no. type Expt 1 Expt 2 Expt 3 1 11 C - A# D – A# 3, 4 holistic 2 3 - - 4, 5, 6 analytic 3 2 A# - F# - 3, 4 holistic 4 1 D -C E – A# 3, 4, 5 mixed 5 1 G# - E - 5? mixed 6 1 E - C F# - D 4 holistic 7 1 A# - F# - 4? mixed Conclusions

Holistic listeners (Group 1, N=11): • hear virtual pitch in registers 3 and/or 4 • pitch register predicts tritone direction

Analytic listeners (Group 2, N=3): • hear spectral pitches in registers 4-6 • tritone direction unclear • tendency to hear all intervals rise Paradox? What paradox? 1. Some listeners consistently hear tritone rise, others consistently hear it fall. Interindividual differences in centre and shape of spectral dominance region or of relationship between spectral and virtual pitch

2. The same listener hears the same tritone rise or fall on different occasions. The pitch of ALL complex tones is ambiguous relative to a 1-D pitch scale An unnatural test of a natural model of pitch perception:

The “tritone paradox” and spectral dominance

Richard PARNCUTT, University of Graz Amos Ping TAN, Universal Music, Singapore