Optimising Computational Measures from Behavioural Data Predicts Perceived Consonance

Edward T. R. Hall; Ran Tamir; Martin Rohrmeier

doi:10.5334/tismir.243

Figures & Tables

Table 1

Interval weights and Pearson’s r correlations for cross‑validated (Dataset A, Bowling et al., 2018) and test datasets (B and C, Johnson‑Laird et al., 2012 and Popescu et al., 2019) for sum and type measures.

Description		Interval												Dataset
		m2	M2	m3	M3	P4	TT	P5	m6	M6	m7	M7	P8	A	B	C
														r [SD]	r	r
Sum method
Binary		−1	−1	+1	+1	+1	−1	+1	+1	+1	−1	−1	+1	.592	.503	.647
Schwartz et al. (2003)		−1.0	−0.4	0.0	+0.4	+0.6	−0.2	+0.8	+0.2	+0.4	−0.4	−0.8	+1.0	.782	.664	.745
Optimised:
Simple		−0.39	−0.04	−0.01	+0.05	+0.14	−0.12	+0.15	−0.08	−0.03	−0.14	−0.20	+0.24	.821 [.033]	.703	.806
1‑split	incl. 8	−0.38	−0.01	−0.02	−0.05	+0.19	−0.09	+0.17	−0.09	−0.09	−0.13	−0.20	+0.40	.824 [.034]	.735	.787
1‑split	excl. 8	−0.39	−0.08	−0.01	+0.13	+0.13	−0.15	+0.14	–	+0.01	−0.18	−0.19	+0.23	.824 [.034]	.735	.787
2‑split	in. 4, in. 8	−0.53	−0.05	−0.12	−0.16	+0.40	−0.02	+0.23	+0.11	−0.18	−0.22	−0.07	+0.35	.810 [.038]	.734	.732
2‑split	in. 4, ex. 8	−0.44	−0.05	−0.01	+0.10	+0.18	−0.15	+0.20	–	−0.01	−0.15	−0.24	+0.22
	ex. 4, in. 8	−0.31	+0.03	+0.09	–	+0.08	−0.14	+0.06	−0.12	−0.09	−0.11	−0.28	+0.13
	ex. 4, ex. 8	−0.32	−0.15	+0.02	–	+0.15	−0.19	+0.13	–	+0.02	−0.20	−0.20	+0.24
Type method
Binary		−1	−1	+1	+1	+1	−1	+1	+1	+1	−1	−1	+1	.645	.494	.725
Schwartz et al. (2003)		−1.0	−0.4	0.0	+0.4	+0.6	−0.2	+0.8	+0.2	+0.4	−0.4	−0.8	+1.0	.804	.640	.740
Optimised:
Simple		−1.90	−0.30	−0.03	+0.26	+0.62	−0.58	+0.68	−0.34	−0.14	−0.71	−0.90	+1.01	.835 [.039]	.676	.837
1‑split	incl. 8	−2.35	−0.17	−0.28	−0.48	+1.03	−0.61	+0.91	+0.01	−0.47	−0.90	−1.27	+2.08	.851 [.036]	.732	.829
1‑split	excl. 8	−1.76	−0.41	−0.03	+0.47	+0.50	−0.67	+0.69	–	+0.03	−0.78	−0.80	+0.85	.851 [.036]	.732	.829
2‑split	in. 4, in. 8	−3.18	−0.26	−0.67	−0.74	+2.44	−0.06	+1.43	+0.26	−0.97	−1.25	−0.31	+2.05	.856 [.034]	.739	.780
	in. 4, ex. 8	−2.55	−0.22	−0.02	+0.51	+1.08	−0.84	+1.15	–	−0.14	−0.70	−1.51	+1.34
	ex. 4, in. 8	−1.95	−0.00	+0.35	–	+0.38	−0.96	+0.23	+0.03	−0.58	−0.88	−1.86	+0.67
	ex. 4, ex. 8	−1.49	−0.54	+0.06	–	+0.44	−0.65	+0.59	–	−0.10	−0.75	−0.77	+0.81

Table 2

Pearson’s r correlations between behavioural datasets and sum and type measures based on the conditional inclusion of a specific interval. Column n–✓ gives the number of chords in Dataset A (Bowling et al., 2018) that contain interval i. Correlations that are higher than the previous ‘simple’ optimised weights, for each method and dataset, are shown in bold.

	Dataset A						B		C
	n		Sum		Type		Sum	Type	Sum	Type
i:	✓	x	r	SD	r	SD	r	r	r	r
1	158	140	.835	.035	.857	.036	.665	.662	.754	.816
2	150	148	.835	.034	.819	.045	.722	.683	.745	.822
3	142	156	.811	.034	.841	.036	.698	.666	.797	.821
4	134	164	.826	.033	.856	.036	.721	.707	.773	.822
5	125	173	.820	.033	.835	.038	.685	.666	.808	.854
6	117	181	.814	.034	.829	.039	.700	.684	.795	.842
7	117	181	.812	.035	.830	.040	.685	.668	.800	.826
8	107	191	.824	.034	.851	.036	.735	.732	.787	.829
9	97	201	.818	.032	.834	.037	.711	.670	.777	.820
10	87	211	.825	.036	.825	.042	.684	.647	.755	.825
11	77	221	.820	.036	.829	.042	.710	.665	.746	.832
12	67	231	.809	.036	.827	.039	.718	.681	.761	.818

‘Simple’ and 1‑split (including/excluding m6) optimised interval weights for *sum* and *type* measures. Error bars display the standard errors of optimised weights.

Table 3

Pearson’s r correlations between behavioural datasets and best‑performing sum and type measures based on the conditional inclusion of pairs of intervals. Column n–✓–✓ gives the number of chords in Dataset A that contain both intervals i and j. Correlations that are higher than the best‑performing measure of Table 2, for each method and dataset, are shown in bold.

Dataset A										B		C
		n				Sum		Type		Sum	Type	Sum	Type
i:		✓	✓	×	×	r	SD	r	SD	r	r	r	r
	j:	✓	×	✓	×	r	SD	r	SD	r	r	r	r
1	3	77	81	65	75	.818	.040	.858	.035	.660	.660	.712	.774
1	4	71	87	63	77	.833	.034	.871	.032	.674	.690	.745	.817
1	8	59	99	48	92	.854	.031	.873	.030	.639	.701	.686	.797
1	9	54	104	43	97	.842	.033	.848	.038	.652	.639	.721	.789
2	4	57	93	77	71	.821	.036	.831	.043	.746	.696	.737	.813
2	6	62	88	55	93	.823	.036	.812	.043	.736	.706	.741	.808
2	10	49	101	38	110	.838	.037	.807	.049	.699	.648	.688	.788
3	5	63	79	62	94	.808	.035	.840	.036	.659	.646	.806	.852
4	8	43	91	64	100	.810	.038	.856	.034	.734	.739	.732	.780
5	8	36	89	71	102	.819	.035	.856	.032	.705	.699	.786	.851
6	8	35	82	72	109	.810	.037	.845	.034	.717	.727	.769	.795

Perceptual vs. estimated scores for interval measures based on the inclusion/exclusion of intervals 4 and 8. Left: comparison between Datasets A, B and C (Bowling et al., 2018; Johnson‑Laird et al., 2012; Popescu et al., 2019) for the *sum* (top) and *type* methods (bottom). Right: the four clusters of Dataset A, based on the inclusion/exclusion of intervals 4 and 8, for the *sum* (top) and *type* methods (bottom).

Table 4

Interval–class weights and Pearson’s r correlations for cross‑validated (Dataset A, Bowling et al., 2018) and test datasets (B and C, Johnson‑Laird et al., 2012; Popescu et al., 2019) for sum and type measures.

Description		Interval class							Dataset
		P1/P8	m2/M7	M2/m7	m3/M6	M3/m6	P4/P5	TT	A	B	C
									r [SD]	r	r
Sum method
Binary		+1	−1	−1	+1	+1	+1	−1	.587	.503	.647
Schwartz et al. (2003):
Class max.		+1.0	−0.8	−0.4	+0.4	+0.4	+0.8	−0.2	.733	.623	.782
Class min.		+1.0	−1.0	−0.4	0.0	+0.2	+0.6	−0.2	.791	.661	.761
Huron (1994)		–	−1.43	−0.58	+0.59	+0.39	+1.24	−0.45	.703	.680	.739
Optimised:
Simple		+0.25	−0.34	−0.08	−0.02	−0.00	+0.15	−0.11	.796 [.038]	.704	.788
1‑split	incl. 4	+0.34	−0.37	−0.05	−0.01	−0.05	+0.22	−0.09	.808 [.037]	.751	.765
1‑split	excl. 4	+0.25	−0.27	−0.18	+0.02	–	+0.14	−0.20	.808 [.037]	.751	.765
2‑split	in. 4, in. 6	+0.36	−0.32	−0.02	−0.01	−0.06	+0.15	−0.10	.801 [.039]	.757	.760
	in. 4, ex. 6	+0.33	−0.35	−0.07	−0.01	−0.07	+0.25	–
	ex. 4, in. 6	+0.33	−0.36	+0.02	−0.01	–	+0.11	−0.16
	ex. 4, ex. 6	+0.25	−0.24	−0.21	−0.01	–	+0.18	–
Type method
Binary		+1	−1	−1	+1	+1	+1	−1	.645	.494	.725
Schwartz et al. (2003):
Class max.		+1.0	−0.8	−0.4	+0.4	+0.4	+0.8	−0.2	.779	.623	.711
Class min.		+1.0	−1.0	−0.4	0.0	+0.2	+0.6	−0.2	.802	.643	.802
Huron (1994)		–	−1.43	−0.58	+0.59	+0.39	+1.24	−0.45	.722	.656	.771
Optimised:
Simple		+1.03	−1.57	−0.49	−0.09	+0.00	+0.67	−0.56	.804 [.046]	.683	.818
1‑split	incl. 4	+1.64	−2.17	−0.37	−0.10	−0.08	+1.21	−0.56	.835 [.037]	.731	.807
1‑split	excl. 4	+0.83	−1.21	−0.67	+0.00	–	+0.52	−0.68	.835 [.037]	.731	.807
2‑split	in. 4, in. 6	+1.61	−2.01	−0.20	−0.17	−0.44	+0.77	−0.13	.833 [.038]	.747	.812
	in. 4, ex. 6	+1.54	−2.12	−0.44	−0.10	−0.12	+1.34	–
	ex. 4, in. 6	+0.80	−2.04	−0.70	−0.03	–	+0.76	−0.39
	ex. 4, ex. 6	+0.77	−1.08	−0.72	−0.05	–	+0.65	–

Table 5

Pearson’s r correlations between behavioural datasets and sum and type measures based on the conditional inclusion of a specific interval class. Correlations that are higher than the previous ‘simple’ optimised weights, for each method and dataset, are shown in bold.

	Dataset A						B		C
	n		Sum		Type		Sum	Type	Sum	Type
i:	✓	×	r	SD	r	SD	r	r	r	r
0	67	231	.787	.041	.800	.046	.721	.694	.744	.817
1	188	110	.813	.041	.843	.038	.715	.693	.736	.782
2	188	110	.828	.037	.794	.048	.738	.687	.723	.816
3	187	111	.787	.039	.816	.039	.711	.692	.781	.822
4	198	100	.808	.037	.835	.037	.751	.731	.765	.807
5	188	110	.796	.038	.816	.041	.713	.679	.785	.811
6	117	181	.795	.037	.807	.042	.712	.706	.782	.829

‘Simple’ and 1‑split (including and excluding M3/m6) optimised interval–class weights for *sum* and *type* measures. Error bars display the standard errors of optimised weights.

Table 6

Pearson’s r correlations between behavioural datasets and best‑performing sum and type measures based on the conditional inclusion of pairs of interval classes. Correlations that are higher than the best‑performing measure of Table 5, for each method and dataset, are shown in bold.

Dataset A										B		C
		n				Sum		Type		Sum	Type	Sum	Type
		✓	✓	×	×	r	SD	r	SD	r	r	r	r
i:	j:	✓	×	✓	×	r	SD	r	SD	r	r	r	r
0	2	29	38	159	72	.830	.041	.790	.049	.750	.693	.610	.774
0	4	30	37	168	63	.794	.045	.826	.041	.760	.730	.628	.794
1	3	128	60	59	51	.824	.036	.857	.033	.740	.712	.699	.762
1	4	132	56	66	44	.822	.038	.857	.035	.737	.726	.729	.800
1	5	130	58	58	52	.818	.036	.865	.030	.731	.705	.737	.800
2	4	124	64	74	36	.825	.036	.830	.040	.765	.730	.740	.820
2	5	114	74	74	36	.827	.039	.805	.045	.739	.686	.728	.798
3	4	132	55	66	45	.816	.036	.833	.038	.755	.733	.771	.815
3	5	128	59	60	51	.797	.037	.833	.035	.717	.691	.786	.819
3	6	55	132	62	49	.783	.042	.818	.036	.735	.711	.773	.823
4	6	82	116	35	65	.801	.039	.833	.038	.757	.747	.760	.812

Perceptual vs. estimated scores for interval–class measures based on the inclusion/exclusion of classes 4 and 6. Left: comparison between the three datasets (Bowling et al., 2018; Johnson‑Laird et al., 2012; Popescu et al., 2019) for the *sum* (top) and *type* methods (bottom). Right: the four clusters of dataset Bowling et al. (2018), based on the inclusion/exclusion of intervals 4 and 8, for the *sum* (top) and *type* methods (bottom).

Table 7

Pearson r correlations of perceptual models of consonance to behavioural datasets. Highest correlations for each dataset are shown in bold for perceptual models and for the measures of the present paper displayed.

Model	Description	Dataset r
		A	B	C
Perceptual models
Harrison and Pearce (2018)	harmonicity	.620	.496	.706
Stolzenburg (2015)	harmonicity	.740	.723	.785
Huron (1994)	roughness	.685	.680	.650
Hutchinson and Knopoff (1978)	roughness	.713	.440	.550
Johnson‑Laird et al. (2012)	cultural	.634	.719	.725
Harrison and Pearce (2020)	cultural	.722	.568	.850
Harrison and Pearce (2020)	composite	.801	.628	.830
Optimised models
Simple:
Interval	sum	.821	.703	.806
	type	.835	.676	.837
Interval class	sum	.796	.704	.788
	type	.804	.683	.818
1‑split:
Interval (incl./excl. 8)	sum	.824	.735	.787
	type	.851	.732	.829
Interval class (incl./excl. 4)	sum	.808	.751	.765
	type	.835	.731	.807

Optimising Computational Measures from Behavioural Data Predicts Perceived Consonance

Figures & Tables

Table 1

Table 2

Figure 1

Table 3

Figure 2

Table 4

Table 5

Figure 3

Table 6

Figure 4

Table 7

Paradigm

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