Page 13 - Summer 2010

Fig. 6. Multidimensional scaling solutions produced from the relative pitch judgments of tones created using the algorithm of Deutsch,16 made by an individual subject. The plot on the left shows the solution for tones in the scale based on the A4 tonic, and the plot on the right shows the solution for tones in the scale based on the F#4 tonic.
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Adapted from Deutsch, Dooley, and Henthorn.
module to produce these transformations was created for the Pd Programming environment,15, 17 so that composers and performers can now begin to experiment with this algorithm live and in real time, as well as in recording contexts.
Hypothesized neuroanatomical substrates
What do we know about the neuroanatomical substrates underlying the circular component of pitch? An interesting
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study by J. D. Warren and colleagues sheds light on this issue. These researchers used functional magnetic resonance imag- ing (fMRI) to explore patterns of brain activation in response to two types of tone sequence. In the first type, the harmonic components of the tones were at equal amplitude, but F0 was varied, so that pitch class and pitch height varied together. In the other type of sequence, pitch class was kept constant but the relative amplitudes of the odd and even harmonics were varied, so that only differences in pitch height were produced. Presentation of the first type of sequence resulted in activation specifically in an area anterior to the primary auditory cortex, while the second type of sequence produced activation prima- rily in an area posterior to this region. Based on these findings, the authors concluded that the circular component of pitch is represented in the anterior region.
Pitch circularity might, however, have its origins earlier in the auditory pathway. Gerald Langner has provided evi- dence in the gerbil that the ventral nucleus of the lateral lem- niscus is organized in terms of a neuronal pitch helix, so that pitches are arranged in helical fashion from top to bottom
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This indicates that the lateral lemniscus might be the source of the circular
with one octave for each turn of the helix.
component, and that it is further represented in the cortex.
A paradox within a paradox
There is an additional twist to the paradox of pitch cir- cularity. We have seen that when listeners are presented with ordered pairs of tones that are ambiguous with respect to height, they invoke proximity along the pitch class circle in making judgments of relative pitch. But we can then ask what happens when a pair of such ambiguous tones is presented which are separated by a half-octave (or tritone) so that the same distance along the circle is traversed in either direction.
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circle. Figure 5 shows the percentages of judgments that
were in accordance with proximity, for both scales, and aver- aged across all subjects. As can be seen, when the tones with- in a pair were separated by a semitone, proximity determined their judgments almost entirely. As with Shepard’s experi- ment on octave-related complexes, as the tones within a pair were separated by a larger distance along the pitch class cir- cle, the tendency to follow by proximity lessened. And even when the tones were separated by almost a half-octave, the tendency for judgments to follow the more proximal rela- tionship was very high. When we subjected the data to Kruskal’s nonmetric multidimensional scaling, we obtained strongly circular solutions, as illustrated in those from an individual subject shown in Fig. 6. We also created sound demonstrations based on this algorithm. These included endlessly ascending and descending scales moving in semi- tone steps, and endlessly ascending and descending glissandi, and are presented as Sound Demonstrations 4 - 7.
The finding that circular scales can be obtained from full harmonic series leads to the intriguing possibility that this algorithm could be used to transform banks of natural instrument tones so that they would also exhibit pitch circu- larity. William Brent, then a graduate student at the University of California, San Diego music department, has shown that such transformations can indeed be achieved. He used bassoon samples taken from the Musical Instrument Samples Database at the University of Iowa Electronic Music Studios, ranging in semitone steps from D#2 to D3. Using continuous overlapping Fourier analysis, he transformed the sounds into the frequency domain, and there reduced the amplitudes of the odd harmonics by 3.5 dB per semitone step downward. He then performed inverse Fourier transforms to generate the altered waveforms in the time domain. Circular
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Endlessly ascending and descending scales employing these tones are
banks of bassoon tones were thereby produced.
presented as Sound Demonstrations 8 and 9.
It remains to be determined which types of instrument
sound can be transformed so as to acquire this property. However, Brent has also achieved some success with flute, oboe, and violin samples, and has shown that the effect is not destroyed by vibrato. The Digital Signal Processing (DSP)
12 Acoustics Today, July 2010