Future work

Replacingfiddle˜in the implementation by a custom object designed for the spe-cific purpose of retrieving from incoming signals only information about fundamental, partials and their amplitudes could mean a significative reduction in calculation time.

Besides that, although the Hanning window used in fiddle˜ reduces the effects of

FFT leakage, it can also reduce the frequency resolution. The reasons for choosing a Hanning window in fiddle˜ are unknown to the author, but the effect of others windowing artifacts (like Blackman, Flat top, Hamming, etc.) should be studied to compare which of them offers better frequency resolution in a real-time implementa-tion.

The audio signal concurrency problem suggests the implementation of a program using parallel computing techniques, such as POSIXthreads standard (Pthreads) [53].

Flext [54] is a C++ layer for cross-platform development of Max/MSP and pd exter-nals. This tool brings mechanisms to access threads encapsulating low-level imple-mentation details, so it can be used for parallelize the patch presented in this thesis. In a very simplified scenario, on the event of a new signal, one thread could write it into the matrices (Amplitudes and Timbre), and raise a flag for another thread in charge of calculating the dissonance-minimized versions of the incoming signals. This thread in turns should drop whatever partial result it had at the moment the flag is raised and start to calculate it again with the newly arrived data. The scheduling for this parallel ver-sion is not trivial, since it has to control the release time of the processes, commit the deadlines for audio outputs, assure that threads are started in a precise order, manage exceptions such as dead threads, prevent dead-locks, etc.

Another source of parallelism is the calculation of the dissonance itself. The pre-sented problems, calculating the frequencies that belong to the vicinity of the originals offer a minimum dissonance by using brute-force, can be easily circumvented using parallel processes (preferable in several processors), one for each timbre.

The values of the damping constants were calculated by trial and error, and al-though this is a valid technique, a more sophisticated method for tuning them could be employed. In the same way that genetic algorithms are used to train neural net-works, using such meta-algorithms to find more appropriated constant values inSPSA

were not included in the present work, and this exploration can be addressed in further investigations.

In the SPSA implementation, the Bernoulli distribution was used for simplicity,

but other probability distributions might produce vectors with the same properties than those presented in the current implementation (vector entries independent and symmet-rically distributed about 0 with finite inverse moment) could produce different results, perhaps reducing the convergence time.

As explained by Watkinson [55] “The [excited] area of the [basilar] membrane in-volved will increase as the sound level rises,” However, in the model ofERB used in our implementation, the intensity of the aural stimulus is not considered in the expres-sion. Such factor could presumably increase the accuracy of the system, although this extra complication should be weighted against the real-time constraints to determine if it should to be included.

The correctness of the model from a physiological perspective (i.e. that it effec-tively reduces the perceived dissonance of the input signals) was not proven in the present thesis as such a proof is a complicated problem. No formal validation was offered; instead, our results prove that the proposed system is self-consistent and also consistent with the consonance paradigms selected for the implementation. Among other possible tests, one could conduct subjective tests in which non-processed audio is compared with the output of the proposed system. This experiment, though, could potentially be biased by other factors, such the inability of the listener to distinguish between different tuning systems.

Other explorations include the application of the system to non-harmonic sounds, as presented earlier. Only sounds with high energy partials and small overall duration lead the system to unpredictable results, so the dissonance of combinations of some non-western instruments (like the Indonesian gamelan), artificially elaborated sounds,¹ and traditional instruments, can be minimized.

1such the harmonically stretched versions produced by the Helical Keyboard when operating with its internal synthesizer [56]

Chapter 9 Conclusions

A novel approach to generate dissonance minimized version of concurrent sounds in realtime based on the tonotopic theory was presented and analyzed. Goldenear re-duces heuristically the dissonance of sound sources in realtime conserving their origi-nal character linked to a vicinity in ¢ which can defined by the user in the interface.

The proposed model can be applied with no modifications to any kind of sound-sources (harmonic and non-harmonic spectra).

Although no subjective tests were conducted to validate the presented model from a perceptual approach (linking estimation of dissonance with experimental phenomena), the results are in agreement with previous researches, and the convergence time of the algorithm for the chosen platform make it adequate for several real-time applications.

It would be na¨ıve to conclude that such a complex phenomenon as the subjective degree of pleasantness experienced by music perception can be explained by a sin-gle theory; some of the examples presented in this thesis refer almost exclusively to harmonic music, and their results cannot be extended to other genres. There is some evidence that tonotopic dissonance could explain consonance in some ethnic expres-sions, so the presented application could be used to corroborate it. Nevertheless, there are still several unaddressed aspects that should be studied to complement the approach presented here.

Practical application of the model presented in this thesis include using it to process the performance of traditional instruments (with harmonic spectra) in conjunction with instrument with no harmonic spectra like bonang and saron from the Indonesian tradi-tion, as a ‘tutor’ to learn ‘better’ positions in instruments with no fix tuning (strings, trombone, etc.), for learning harmony and the origin of musical scales, as replacement of tunings so instruments can play in whatever tuning they have letting the software to adjust their pitches and achieve better results in terms of consonance, as a musical effect by using non-usual vicinities, etc.

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