Partly supported
“Musical consonance is set by a roughness threshold in the auditory filter bank rather than by the simplicity of frequency ratios.”
Roughness got the beating intervals and timbre dependence right, but ranked the tritone consonant (0.30) and the minor third dissonant (0.90) — where harmonicity/ratio matters.
{
"hypothesis": "Consonance Is Roughness",
"origin": "frontier-theories-10 #5 (Acoustics & Vibration)",
"verdict": "Partially supported",
"model": "Plomp-Levelt (1965) roughness, Zwicker critical bandwidth, harmonic partials 1..8 with 1/n amplitudes; dyad dissonance = sum of amplitude-weighted cross-partial roughness",
"f0": 261.6255653005986,
"n_harmonics": 8,
"dissonance": {
"P1": 0.005343,
"m2": 1.107253,
"M2": 1.266524,
"m3": 0.895795,
"M3": 0.659394,
"P4": 0.414527,
"A4": 0.296584,
"P5": 0.107815,
"m6": 0.316438,
"M6": 0.350884,
"m7": 0.49316,
"M7": 0.716046,
"P8": 0.005343
},
"model_rank": {
"P1": 1,
"P8": 2,
"P5": 3,
"A4": 4,
"m6": 5,
"M6": 6,
"P4": 7,
"m7": 8,
"M3": 9,
"M7": 10,
"m3": 11,
"m2": 12,
"M2": 13
},
"empirical_rank": {
"P1": 1,
"P8": 2,
"P5": 3,
"P4": 4,
"M6": 5,
"M3": 6,
"m3": 7,
"m6": 8,
"M2": 9,
"m7": 10,
"m2": 11,
"M7": 12,
"A4": 13
},
"gradus_suavitatis": {
"P1": 1,
"m2": 11,
"M2": 8,
"m3": 8,
"M3": 7,
"P4": 5,
"A4": 14,
"P5": 4,
"m6": 8,
"M6": 7,
"m7": 9,
"M7": 10,
"P8": 2
},
"gradus_rank": {
"P1": 1,
"P8": 2,
"P5": 3,
"P4": 4,
"M3": 5,
"M6": 6,
"M2": 7,
"m3": 8,
"m6": 9,
"m7": 10,
"M7": 11,
"m2": 12,
"A4": 13
},
"spearman_model_vs_empirical": 0.5879120879120878,
"spearman_gradus_vs_empirical": 0.9725274725274725,
"timbre": {
"P1": {
"sine": 0.0,
"harmonic": 0.005343
},
"m2": {
"sine": 0.591642,
"harmonic": 1.107253
},
"M2": {
"sine": 0.928398,
"harmonic": 1.266524
},
"m3": {
"sine": 0.709738,
"harmonic": 0.895795
},
"M3": {
"sine": 0.479715,
"harmonic": 0.659394
},
"P4": {
"sine": 0.237913,
"harmonic": 0.414527
},
"A4": {
"sine": 0.0,
"harmonic": 0.296584
},
"P5": {
"sine": 0.0,
"harmonic": 0.107815
},
"m6": {
"sine": 0.0,
"harmonic": 0.316438
},
"M6": {
"sine": 0.0,
"harmonic": 0.350884
},
"m7": {
"sine": 0.0,
"harmonic": 0.49316
},
"M7": {
"sine": 0.0,
"harmonic": 0.716046
},
"P8": {
"sine": 0.0,
"harmonic": 0.005343
}
},
"note": "Roughness correctly ranks the beating intervals (seconds and sevenths are dissonant; P1/P8/P5 are consonant) and uniquely explains why timbre shifts consonance at a fixed ratio (P2). But it predicts the tritone is consonant (dissonance 0.30 vs 1.11 for the minor second) and the minor third is dissonant (0.90) - both wrong. Those two cases are where harmonicity/ratio alignment matters, which the ratio-simplicity baseline (Euler gradus, rho=0.97) captures. A complete account needs both roughness and harmonicity (cf. McDermott et al. 2010)."
}