Optimising computational measures from behavioural data predicts perceived consonance

This package contains tools for estimating consonance and dissonance based on pairwise intervals within a chord, as reported in the article:

Hall, E. T. R., Tamir, R., & Rohrmeier, M. (2025). Optimising Computational Measures from Behavioural Data Predicts Perceived Consonance. Transactions of the International Society for Music Information Retrieval, 8(1). https://doi.org/10.5334/tismir.243

Calculating consonance

Two functions are provided: (1) for calculating consonance using intervals (minor 2nd to octave) interval_consonance, and (2) using interval classes (0-6) interval_class_consonance.

Both accept:

1. A chord (i.e., a list of MIDI pitches)
2. A list of weights (length 12 or 7 for interval and interval-class measure, respectively), or list of multiple weights' lists if conditionally excluding certain intervals (see below)
3. An aggregation method, either: "sum" where weights are combined by addition, or "type" where interval counts are first normalised.

For example, we can create a simple measure of consonance using binary consonance/dissonance classifications for each interval as weights:

from consonance.consonance import interval_consonance

chord1 = [60, 64, 67]  # i.e., a C major chord
chord2 = [60, 61, 62]  # a dissonant cluster chord

#                 m2  M2  m3  M3  P4  TT  P5  m6  M6  m7  M7  P8
binary_weights = [-1, -1,  1,  1,  1, -1,  1,  1,  1, -1, -1,  1]

interval_consonance(chord1, binary_weights, agg_method="sum")
# 3.0

interval_consonance(chord2, binary_weights, agg_method="sum")
# -3.0

We can do the same using interval classes:

from consonance.consonance import interval_class_consonance

#                     P1/P8  m2/M7  M2/m7  m3/M6  M3/m6  P4/P5  TT
binary_class_weights = [1,    -1,    -1,     1,     1,     1,   -1]

interval_class_consonance(chord1, binary_class_weights, agg_method="sum")
# 3.0

Optimising weights

Given a dataset of chords and ratings of consonance, optimal weights can be obtained. Again functions are provided to optimise weights for intervals (optimise_interval_weights) and interval classes (optimise_interval_class_weights).

The three datasets using in this paper are given in ./data:

1. Bowling et al. (2018): 298 chords (all 2, 3 and 4 combinations in one octave)
2. Johnson-Laird et al. (2012): 55 three-note chords and 43 four-note chords
3. Popescu et al. (2019): 60 chords from Classical, Jazz and Avant-garde repertoire

(see Peter Harrison's inconData for A, B and further datasets; see source for C)

Here, we also scale ratings, where -1 is the most dissonant value on the available rating scale, and +1 the most consonant.

import pandas as pd
from optimise_weights import optimise_interval_weights

def parse_pitches(pitches_str):
    return [int(x) for x in pitches_str.split(",")]
    
def scale_ratings(rating, r_con, r_dis):
    return ((2 * (rating - r_dis)) / (r_con - r_dis)) - 1

bowling2018 = pd.read_table("data/bowling2018_consonance.tsv")

pitches = bowling2018.pitches.apply(parse_pitches)
ratings = bowling2018.rating.apply(scale_ratings, args=(4, 1))

optimise_interval_weights(pitches, ratings, agg_method="sum")
# [-0.39315701025321464,
#  -0.036022745929167974,
#  -0.005999857211278825,
#  0.047937002169191084,
#  0.13833299878214417,
#  -0.117640706624426,
#  0.15345017924359666,
#  -0.07721218728660789,
#  -0.029456392748971896,
#  -0.14381589959165952,
#  -0.19649184453467064,
#  0.24340298910163627]

Conditional inclusion/exclusion of intervals

Different sets of weights can also be optimised based on the conditional inclusion/exclusion of interval(s). Intervals to exclude can be supplied as a list using exclude_ivls. Weights for all combinations will be optimised.

weights_ex8 = optimise_interval_weights(pitches, ratings, exclude_ivls=[8], agg_method="sum")
# [[-0.3856, -0.0767, -0.0108,  0.1334, 0.1256, -0.1476, 0.1439,    None,  0.0080, -0.1758, -0.1884, 0.2332],
#  [-0.3777, -0.0068, -0.0232, -0.0548, 0.1900, -0.0887, 0.1709, -0.0933, -0.0852, -0.1306, -0.1962, 0.3980]]

Weights supplied as list of lists can be used in the consonance function. Weights lists must be supplied for all combinations of excluded or included intervals. Excluded weights have value of None.

interval_consonance(chord1, weights_ex8, agg_method="sum")
# 0.2664685521562091

interval_consonance(chord2, weights_ex8, agg_method="sum")
# -0.8479383716037261

Pre-computed weights

This package also contains all the weights given in Tables 1 (interval_weights) and 4 (interval_class_weights) of the paper. They can be imported as follows:

from consonance.interval_weights import simple_type

simple_type
# [-1.9015876909779474,
#  -0.299102543610847,
#  -0.028932965712551884,
#  0.2573540145134221,
#  0.6162669639189333,
#  -0.5822948481506369,
#  0.676379372110411,
#  -0.3386279468563853,
#  -0.14221773986822045,
#  -0.7119604574442283,
#  -0.8988249813772922,
#  1.0092873128124524]

References

Bowling, D. L., Purves, D., & Gill, K. Z. (2018). Vocal similarity predicts the relative attraction of musical chords. Proceedings of the National Academy of Sciences, 115(1), 216–221. https://doi.org/10.1073/pnas.1713206115

Hall, E. T. R., Tamir, R., & Rohrmeier, M. (2025). Optimising Computational Measures from Behavioural Data Predicts Perceived Consonance. Transactions of the International Society for Music Information Retrieval, 8(1). https://doi.org/10.5334/tismir.243

Johnson-Laird, P. N., Kang, O. E., & Leong, Y. C. (2012). On musical dissonance. Music Perception, 30(1), 19–35. https://doi.org/10.1525/mp.2012.30.1.19

Popescu, T., Neuser, M. P., Neuwirth, M., Bravo, F., Mende, W., Boneh, O., Moss, F. C., & Rohrmeier, M. (2019). The pleasantness of sensory dissonance is mediated by musical style and expertise. Scientific Reports, 9(1), 1070. https://doi.org/10.1038/s41598-018-35873-8