Musical chord dissonance models
Project description
This package implements various models of perceptual chord dissonance. Given a musical chord composed of individual tones, each composed of partials, a dissonance model provides a score that estimate how dissonant (harsh) does it sound to human listener.
It contains implementation of several models of dissoance with corrections of errors found in the formulas in the papers. See below.
Installation
pip install dissonant
Why dissonant? PyPI package dissonance was already taken.
Usage
Dissonance of a C major chord with harmonic tones in 12TET tuning with base 440 Hz using the Sethares1993 model:
from dissonant import harmonic_tone, dissonance, pitch_to_freq freqs, amps = harmonic_tone(pitch_to_freq([0, 4, 7, 12]), n_partials=10) d = dissonance(freqs, amps, model='sethares1993')
Dissonance curve of a sliding interval of two harmonic tones: see notebooks/dissonance_curve.ipynb.
Papers
1965, Plomp, Levelt  Tonal Consonance and Critical Bandwidth
1979, Hutchinson, Knopoff  The significance of the acoustic component of consonance in Western triad
1993, Sethares  Local Consonance and the Relationship Between Timbre and Scale
2000, Vassilakis  Auditory roughness estimation of complex spectra
PDF is not public, only HTML
2001, Vassilakis  Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance  PhD thesis
2005, Vassilakis  Auditory Roughness as Means of Musical Expression
2006, Benson  David Benson Music: A Mathematical Offering (book page)
2006, Cook, Fujisawa  The Psychophysics of Harmony Perception: Harmony is a ThreeTone Phenomenon
2009, Cook  Harmony Perception: Harmoniousness Is More Than the Sum of Interval Consonance
2013, Dillon  Calculating the Dissonance of a Chord according to Helmholtz theory
Extra
2001, McKinney et al.  Neural correlates of musical dissonance in the inferior colliculus
Kameoka, A. & Kuriyagawa, M. (1969a). Consonance theory, part I: Consonance of dyads. Journal of the Acoustical Society of America, Vol. 45, No. 6, pp. 14511459.
Kameoka, A. & Kuriyagawa, M. (1969b). Consonance theory, part II: Consonance of complex tones and its computation method. Journal of the Acoustical Society of America, Vol. 45, No. 6, pp. 14601469.
Mashinter, Keith. (1995). Discrepancies in Theories of Sensory Dissonance arising from the Models of Kameoka & Kuriyagawa, and Hutchinson & Knopoff. Undergraduate thesis submitted for the double honours degrees in Applied Mathematics and Music, University of Waterloo.
Phil Keenan ([@ingthrumath](https://twitter.com/ingthrumath))
Other resources
http://www.res.kutc.kansaiu.ac.jp/~cook/
http://www.res.kutc.kansaiu.ac.jp/~cook/Harmony.c
N. D. Cook, T. Fujisawa and K. Takami. This program is an improved version of the algorithm reported in Tone of Voice and Mind (Benjamins, Amsterdam, 2002). It calculates the dissonance, tension, instability and modality of chords containing 25 tones and assuming the presence of 15 upper partials.
video: CogMIR 2013  Norman D Cook  Visual Display of the Acoustical Properties of Harmony
http://sethares.engr.wisc.edu/forrestjava/html/TuningAndTimbre.html
http://sethares.engr.wisc.edu/comprog.html  MATLAB, C++, Lisp
http://www.davideverotta.com/A_folders/Theory/m_dissonance.html
Models
There are several acoustic models of dissonance. They may work on different data and provide several various metrics. All of them are based on some perceptual experiments.
Input signals:
two plain sinusoidal tones
two harmonic tones  integer multiples of the base frequency
chord (two or more) of harmonic tones
signals with continuous spectrum
PlompLevelt1965
They measured the perceived dissonance of pairs of sinusoidal tones and of complex tones with 6 harmonics. They provide only experimental data, not a parametric model.
Sethares1993
First, he explicitly parameterizes the data of PlompLevelt1965 (for a pair of sinusoids and for any number of complex tones) with constants found by fitting the curve to the data.
Dissonance d(x) for the difference x between a pair of frequencies. This ignores the absolute frequencies.
d(x) = exp(a * x)  exp(b * x) Constants: a = 3.5 b = 5.75
Dissonance for a pair of frequencies (f_1, f_2) (for f_1 < f_2) and their amplitudes (a_1, a_2). This takes into account the absolute frequencies. d_max is just the maximum of d(x).
d_pair(f_1, f_2, a_1, a_2) = s = d_max / (s_1 * f_1 + s_2) x = s * (f_2  f_1) a_1 * a_2 * (exp(a * x)  exp(b * x)) Constants: a = 3.5 b = 5.75 d_max = 0.24 s_1 = 0.0207 # 0.021 in the paper s_2 = 18.96 # 19 in the paper
Expressed in terms of d(x):
d_pair(f_1, f_2, a_1, a_2) = a_1 * a_2 * d((f_2  f_1) * d_max / (s_1 * f_1 + s_2))
Model of dissonance of a single complex tone (containing various partials) is just the sum of dissonances for all pairs of partials. In case the partials are integer multiples of a base frequency we can call tone “harmonic”, otherwise just a general “timbre”.
freqs = (f_1, f_2, ... f_n) amps = (a_1, a_2, ... a_n) d_complex(freqs, amps) = 0.5 * sum([d(f_i, f_j, a_i, a_j) for i in range(n) for j in range(n)]) or equally: d_complex(freqs, amps) = sum([ d(freqs[i], freqs[j], amps[i], amps[j]) for i in range(n) for j in range(n) if i < j])
Then we can model dissonance of a pair of complex tones (timbres). Basically it’s still a sum of dissonances of all pairs of partials. We can however express freqs_2 = alpha * freqs_1 and plot d_complex() for fixed freqs_1 and varying alpha to get a “dissonance curve”.
Note that this model can be used to model dissonance of intervals of complex tones, as well as chords (any number of tones).
Note the constants s_1, s_2 are defined in the paper with low precision. Better precision is provided in the code by Mr. Sethares: http://sethares.engr.wisc.edu/comprog.html.
Questions?
Why we just sum the dissonance and not compute mean? Which aggregation operation makes more sense?
Vassilakis2001
There’s a modification to the d_pair() function which should make it depend more reliably on “SPL” and “AFdegree”, in particular put more accent on the relative amplitudes of interfering sinusoids rather than on thir absolute amplitudes.
Defined in Eq.(6.23) on page 197 (219 in the PDF).
a_2 should be >= a_1 d_pair(f_1, f_2, a_1, a_2) = (a_1 * a_2) ^ 0.1 * 0.5 * ((2 * a_2) / (a_1 + a_2)) ^ 3.11 * (exp(a * s * (f_2  f_1))  exp(b * s * (f_2  f_1))) Where: spl = a_1 * a_2 af_degree = (2 * a_2) / (a_1 + a_2) a = 3.5 b = 5.75 d_max = 0.24 s_1 = 0.0207 s_2 = 18.96 It can be expressed in terms of d(x): d_pair(f_1, f_2, a_1, a_2) = spl = a_1 * a_2 af_degree = (2 * a_2) / (a_1 + a_2) s = d_max / (s_1 * f_1 + s_2) x = s * (f_2  f_1) spl ^ 0.1 * 0.5 * af_degree ^ 3.11 * d(x)
In comparison in the Sethares1993 model it is:
d_pair(f_1, f_2, a_1, a_2) = spl = a_1 * a_2 s = d_max / (s_1 * f_1 + s_2) x = (f_2  f_1) * x spl * d(x)
Note that in order to handle the case if a_1 < a_2 we could define:
af_degree(a_1, a_2) = (2 * min(a_1, a_2)) / (a_1 + a_2)
Looking at Vassilakis2010 this is exactly what they do, extending Vassilakis2001, otherwise the model is the same.
Extending this to a set of complex tones is the same as in Sethares1993  just aggregate d_pair() for all pairs via a sum.
Note there’s an additional 0.5 factor in the Vassilakis2001 model compared to Sethares1993. IMHO the meaning is to compensate for pairs of partials being summed twice. In order to make the models comparable we should remove this term and take only pairs of partials only once in all models.
Cook2002
Dissonance model (Cook2002, Appendix 2, page 276 and on, eg. A21):
Original  wrong: d_pair(x, a_1, a_2) = mu_a = (a_1 + a_2) / 2 mu_a * (exp(a * x)  exp(b * x)) Fixed: d_pair(x, a_1, a_2) = mu_a = (a_1 + a_2) / 2 mu_a * c * (exp(a * x)  exp(b * x)) Where: mu_a ... mean amplitude, within [0.0; 1.0] x ... interval between the frequencies (in semitones) a = 1.2 b = 4.0 c = 3.5351 = 1 / (np.exp(1.2)  np.exp(4))
Parameters were “chosen to give a maximal dissonance value at roughly one quartertone, a dissonance of 1.00 at an interval of one semitone, and smaller values for larger intervals”.
The constants a and b are quite different than in the Sethares1993 model. Also the amplitudes are aggregated by mean, instead of product or minimum. The constant c is missing in the paper but needs to be on the place as in the formula above.
Cook2006
Other metrics: dissonance, tension, modality, tonal instability. Based on three tones, not just an interval of two.
Dissonance model (Cook2006, eq. 3).
Note that in the article it’s actually defined in a wrong way:  there’s one more minus sign when applying beta_1, beta_2  logarithm should be of base 2  log2(f_2 / f_1) needs to be multiplied by 12 to get the 12TET semitone interval  there should be bracket, not floor around the difference of exponentials (probably a printing error)
Note: for the sake of readability we replace here original greek nu with v.
Original  wrong: d_pair(f_1, f_2, a_1, a_2) = x = log(f_2 / f_1) v = a_1 * a_2 v * beta_3 * floor( exp(beta_1 * x ^ gamma)  exp(beta_2 * x ^ gamma)) Fixed: d_pair(f_1, f_2, a_1, a_2) = x = 12 * log2(f_2 / f_1) v = a_1 * a_2 v * beta_3 * (exp(beta_1 * x ^ gamma)  exp(beta_2 * x ^ gamma)) Constants: beta_1 = 0.8 # "interval of maximal dissonance" beta_2 = 1.6 # "steepness of the fall from maximal dissonance" beta_3 = 4.0 gamma = 1.25
Total dissonance of a chords is the sum of dissonances of all pairs of partials.
Cook2009
Simplified version of Cook2006. Still the definition is wrong (see above).
There’s no gamma exponent.
Original  wrong: d_pair(f_1, f_2, a_1, a_2) = x = log(f_2 / f_1) v = a_1 * a_2 v * beta_3 * (exp(beta_1 * x)  exp(beta_2 * x)) Fixed: d_pair(f_1, f_2, a_1, a_2) = x = 12 * log2(f_2 / f_1) v = a_1 * a_2 v * beta_3 * (exp(beta_1 * x)  exp(beta_2 * x)) Constants: beta_1 = 0.8 # "interval of maximal dissonance" beta_2 = 1.6 # "steepness of the fall from maximal dissonance" beta_3 = 4.0
Tension of a triad (not implemented yet):
tension(f_1, f_2, f_3) = x = log(f_2 / f_1) y = log(f_3 / f_2) v = a_1 * a_2 * a_3 v * exp(((y  x) / alpha)^2) alpha = ~0.6
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