Figure 1
An example of a single resonance in the time domain and frequency domain . The real part is shown as a solid line and the imaginary part is shown as the dotted line. (a) A resonance in the time domain is a complexvalued damped oscillator. (b) A resonance in the frequency domain is a complexvalued Lorentzian peak.
Figure 2
An example of a harmonic signal and spectrum generated from a single seed resonance with |dk| = 2, ψk = 0, ϕk = 10, and γk = –1, the same as shown in Figure 1, with attenuation A(ϕ, n) = n–1 and N → ∞. The Re[Hf] is shown as the solid line and the Im[Hf] is shown as the dotted line. (a) A harmonic resonance in the time domain is a complex-valued damped train of spikes. (b) A harmonic resonance in the frequency domain is a complex-valued train of Lorentzian peaks.
Figure 3
Correlation of best fit models over a range of N, the total number of overtones in the harmonic spectrum. The blue line is fit using the key affinity profile data. The orange line is fit using the inter-key distance profile data, and the green line is the mean of those two curves. The maximum correlation occurs at N = 12, with a close second at N = 4.
Figure 4
Key affinity sc(Hf, Hg) profiles between the C-major key and all frequencies ranging from C to C’, the octave above. Each equal-tempered semitone is located on the horizontal axis corresponding to its fundamental frequency, and the log-scale vertical axis indicates the cosine similarity between C-major and the tone specified on the horizontal axis. Note that the peaks corresponding with high-affinity tones have non-zero width, which accounts for why different tuning systems, including the modern equal-temperament system, are tolerable to listeners. The two profiles shown here are unnormalized and displayed on a log scale to highlight the effect of the choice of the attenuation and peak width on the resulting profiles. (a) Our model parameterized by A(ϕ,n) = n–1, γ0 = 0.01, and N = 50. (b) Our best fit model to the data of Krumhansl and Kessler (1982).
Figure 5
Comparison of the empirically measured key profiles of Krumhansl and Kessler (1982) (in orange) with the key affinity profile of our model (in blue). Across both profiles the correlation ρ = 0.950. (a) Key affinity profile of C-major. ρ = 0.953. (b) Key affinity profile of C-minor. ρ = 0.954.
Table 1
Comparison of key affinity and inter-chord distance between our model and models reported elsewhere. (a) Correlation of selected models of key affinity with the major and minor profiles of Krumhansl and Kessler (1982), reproduced from Milne et al. (2015). (b) Correlation of selected models of inter-chord distance with empirical profiles of perceived triadic distance, reproduced from Milne and Holland (2016).
| (a) | |||
|---|---|---|---|
| KEY AFFINITY | BOTH | MAJOR | MINOR |
| Milne15c | .96 | .98 | .97 |
| Lerdahl88 | .95 | .98 | .95 |
| Parncutt89 | .95 | .99 | .94 |
| This paper | .95 | .95 | .95 |
| Parncutt11a | .93 | .94 | .95 |
| Milne15b | .92 | .98 | .97 |
| Milne15a | .91 | .96 | .93 |
| (b) | |||
| INTER-CHORD DISTANCE | CORRELATION | ||
| Tonnetz | .92 | ||
| Spectral Pitch Class | .91 | ||
| This paper | .90 | ||
| Transformational | .83 | ||
| Minimal Voice Leading | .72 | ||
| Standard Voice Leading | .62 | ||
| Hamming | .88 | ||
Figure 6
Comparison of inter-key distances derived from the 4-dimensional empirical model of Krumhansl and Kessler (1982) (in orange) with the inter-key distances of our model (in blue). Across all profiles correlation ρ = 0.916. The non-zero intercept seen in Figures 6a and 6d is an artifact of the normalization used to visualize the profiles, and so has no effect on the correlation. (a) Inter-key distance profile between C major and all major keys. ρ = 0.944. (b) Inter-key distance profile between C major and all minor keys. ρ = 0.900. (c) Inter-key distance profile between C minor and all major keys. ρ = 0.872. (d) Inter-key distance profile between C minor and all minor keys. ρ = 0.967.