Fig 1.
Superposition of two pure sinusoids.
This is a time plot of two plane waves and
of different frequency and wavelength but equal amplitude, as well as their superposition
taken at the point x = 1 in arbitrary units.
Fig 2.
Dissonance between two real notes with a sawtooth timbre.
Plot of the dissonance function D(x) as a function of the pitch difference x. In this case, wc = 0.03, and we see that D is zero for the unison, and zero again at the octaves (x = 1 and x = 2). Vertical dashed lines represent the twelve-tone equal temperament (12 TET) octave division.
Fig 3.
Roughness between two pure sinusoidal tones.
Plot of the two pure tone roughness function as a function of the pitch difference
for wc = 0.03 and wc = 0.22, where we see that as the pitch difference increases, the two pure tone roughness decreases to zero only for the smaller value of wc.
Fig 4.
Plot of D(x) for both (blue line) and
(orange line).
Fig 5.
Total Dissonance between two real notes with a sawtooth timbre.
Plot of the total dissonance function DP(x) as a function of the pitch difference x. In this case, wc = 0.03, and we see that D is zero for the unison, and zero again at the octaves (x = 1 and x = 2). We also note the symmetry about the point x = 0.5
Fig 6.
Fourier spectrum of as a function of
.
Plot of the Fourier representation of DP(x) as a function of k with a sawtooth timbre structure and harmonic amplitudes decreasing as 1/n, for three values of wc, and we see that largest Fourier coefficient is the 19th when wc = 0.018, the 12th when wc = 0.03 and the 5th when wc = 0.07.
Fig 7.
P(x) for selected temperatures for 0.018,
0.03, and
0.07.
Note that as temperature increases, the solution tends toward periodic with a single dominant Fourier coefficient. In the wc = 0.018 case, . In the wc = 0.03 case,
. In the wc = 0.07 case,
.
Fig 8.
Lowest twelve Fourier harmonic amplitudes from the decomposition of P(x) over a range of temperatures using 0.03.
Between the two critical temperatures and
, the dominant component contributing to P(x) is kP = 12, leading to the periodic solution seen in Fig 7.
Fig 9.
Calculated P(x) for four different harmonic timbres at two temperatures, one slightly below critical temperature (left), and another slightly above
(right).
From top to bottom, the timbres used were sawtooth, triangle, square, and human voice.
Fig 10.
P(x) over 3 octaves for the sawtooth waveform and wc = 0.03.
In this case, we do not impose periodicity or symmetry over the octaves, and instead simply define .
Fig 11.
Same conditions as Fig 10, but forcing to be symmetric about the domain midpoint.
Fig 12.
Maximum negative Fourier coefficient, , of
, as a function of wc, and thus a prediction of the periodicity of P(x) over the octave for each harmonic series, bonang and saron.
Fig 13.
Select regions of the three-octave P(x) for for each harmonic series, bonang and saron.
Dashed lines indicate five pitch Slendro scale divisions on the left, and seven pitch Pelog scale divisions on the right.