Video 1. Integer ratios chosen for equal-tempered (ET) intervals by the neurodynamic model. This video shows that out of an infinite number of natural resonances within an octave (i.e., between 1:1 and 2:1), the neurodynamic model chooses the strongest resonance that is close enough to each ET ratio. For each ET ratio from the unison to the octave, a pure tone dyad tuned to the ratio is played, two ET-tuned oscillators are set into motion, mode-locking in the chosen integer ratio (governed by Equation 2; the bottom plot shows the real parts), and the resonance region (Arnol’d tongue) for the modelocking oscillators is displayed (the top plot). For human eyes, oscillations are slowed down so that the lower C becomes 1 Hz.

 

 

Video 2. Training an oscillator network with Western chord progressions. This video shows the training portion of the simulation described in the Appendix of the paper. The network was trained on cadences in the key of C (IV-V7-I), repeated until the strengths of plastic connections appeared to reach stable values (35 training iterations). See the Appendix and listen to the commentary in the video for more details.

 

 

Video 3. Testing the trained oscillator network with Western and Indian musical sequences. The trained network was tested with Twinkle Twinkle Little Star and a theme in rāga Bilāval. The amplitude patterns in the network after stimulation with the melodies resemble the probe-tone ratings by Western listeners. See the Appendix and listen to the commentary in the video for more details.

Reference: Large, E. W., Kim, J. C., Flaig, N., Bharucha, J., & Krumhansl, C. L. (2016). A neurodynamic account of musical tonality. Music Perception. 33 (3), 319-331.