Table 1.
Ordering of consonances for two-tone intervals from Helmholtz (1877, pp. 183 and 194) as accepted in the Western musical culture in decreasing order of ‘perfection’ from most consonant to most dissonant. (See also table 5.2 in Roederer (1975, p. 141). The third column lists the frequency ratios of the two tones as set out in Helmholtz (1877). The fourth column lists ΔΩ, the width of the stability interval (see text and figure 4), that is associated with each musical interval as determined numerically using equations (4.1) and (4.3) with ϵ=5, α=100, τ1=τ2=1. Being dissonances, the minor second and tritone have extremely small stability intervals making them difficult to identify.)
| interval's evaluation | interval's name | interval's ratio | ΔΩ | |
|---|---|---|---|---|
| absolute consonances | unison | 1 : 1 | 0.075 | consonance |
| octave | 1 : 2 | 0.023 |  |
|
| perfect consonances | fifth | 2 : 3 | 0.022 | |
| fourth | 3 : 4 | 0.012 | ||
| medial consonances | major sixth | 3 : 5 | 0.010 | |
| major third | 4 : 5 | 0.010 | ||
| imperfect consonances | minor third | 5 : 6 | 0.010 | |
| minor sixth | 5 : 8 | 0.007 | ||
| dissonances | major second | 8 : 9 | 0.006 | |
| major seventh | 8 : 15 | 0.005 | ||
| minor seventh | 9 : 16 | 0.003 | ||
| minor second | 15 : 16 | — | ||
| tritone | 32 : 45 | — | dissonance |