User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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== Chapter II. Divisive and Multiplicative Ratios == | == Chapter II. Divisive and Multiplicative Ratios == | ||
Divisive ratios and multiplicative ratios are always said relative to each other. If a divisive ratio is of the form ''n''/''d'', where ''n'' and ''d'' are integers, then a multiplicative ratio is of the form ''nd''. For example, 5/3 is a | Divisive ratios and multiplicative ratios are always said relative to each other. If a divisive ratio is of the form ''n''/''d'', where ''n'' and ''d'' are integers, then a multiplicative ratio is of the form ''nd''. For example, 5/3 is a divisive ratios; 15/1 is a multiplicative ratio. The question is, thus, if ratios of the form ''n''/''d'' are more important than those of the form ''nd''. | ||
The problem is hard because it is not clear what is implied by importance and what context it can be applied to. Of course, importance means simplicity. But simplicity of ratios is used in two major contexts: chord construction and tuning optimization, and they correspond to distinct psychoacoustic effects. Chord construction has to do with the revelation of harmonic identities due to timbral fusion to a virtual fundamental as discussed above, whereas tuning optimization has to do with percept formation and excitation, and to the better end, minimization of mistuned beating. These are fundamentally different effects – this essay takes the liberty of being the first to treat them separately. | The problem is hard because it is not clear what is implied by importance and what context it can be applied to. Of course, importance means simplicity. But simplicity of ratios is used in two major contexts: chord construction and tuning optimization, and they correspond to distinct psychoacoustic effects. Chord construction has to do with the revelation of harmonic identities due to timbral fusion to a virtual fundamental as discussed above, whereas tuning optimization has to do with percept formation and excitation, and to the better end, minimization of mistuned beating. These are fundamentally different effects – this essay takes the liberty of being the first to treat them separately. | ||
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Regarding 5/3, it is the opposite situation. E<sub>2</sub> comes out superior to E<sub>1</sub> as it perfectly hits 5/3 whereas E<sub>1</sub>'s ~5/3 is off by +2''ε''. | Regarding 5/3, it is the opposite situation. E<sub>2</sub> comes out superior to E<sub>1</sub> as it perfectly hits 5/3 whereas E<sub>1</sub>'s ~5/3 is off by +2''ε''. | ||
However, the beating occurs at ~15/1 and multiples thereof, not at ~5/3. The ~5/3, played as a nonrooted dyad, is free from a real reference point (e.g. harmonic series) for it to beat against, so it lacks relevance in tuning optimization. The only scenario to account for its accuracy is where it is played on the chord's formal root, in which case its 3rd harmonic beats against the formal root's 5th harmonic, for example. That is still not a good argument for its relative importance since we would have manipulated the chord structure just in order to obtain this result. A chord with ~15/1 played on the formal root would call for an accurate ~15/1 and then neutralize the demand for an accurate ~5/3 as previously posed. For example, the just major sixth chord 1–5/4–3/2–5/3 and the just major seventh chord 1–5/4–3/2–15/8 cancel each other out up to octave equivalence. More generally, for any chord featuring a | However, the beating occurs at ~15/1 and multiples thereof, not at ~5/3. The ~5/3, played as a nonrooted dyad, is free from a real reference point (e.g. harmonic series) for it to beat against, so it lacks relevance in tuning optimization. The only scenario to account for its accuracy is where it is played on the chord's formal root, in which case its 3rd harmonic beats against the formal root's 5th harmonic, for example. That is still not a good argument for its relative importance since we would have manipulated the chord structure just in order to obtain this result. A chord with ~15/1 played on the formal root would call for an accurate ~15/1 and then neutralize the demand for an accurate ~5/3 as previously posed. For example, the just major sixth chord 1–5/4–3/2–5/3 and the just major seventh chord 1–5/4–3/2–15/8 cancel each other out up to octave equivalence. More generally, for any chord featuring a divisive ratio on the formal root, there is a counterpart featuring a multiplicative ratio alike. | ||
We should also note the just minor triad is of equal complexity as the just major triad by the principle of invertibility. The just major triad is sometimes considered to be more important by being isodifferential and thus having a common beating rate. The just minor triad is also isodifferential, though not with respect to frequency but to its inverse, the length of a virtual vibrating string. Optimizing for the just minor triad requires us to put it in the context of negative harmony. Starting atop and step downwards, the optimization targets are first 1/3 and then 1/5, which are analytically equivalent to 3/1 and 5/1 respectively in | We should also note the just minor triad is of equal complexity as the just major triad by the principle of invertibility. The just major triad is sometimes considered to be more important by being isodifferential and thus having a common beating rate. The just minor triad is also isodifferential, though not with respect to frequency but to its inverse, the length of a virtual vibrating string. Optimizing for the just minor triad requires us to put it in the context of negative harmony. Starting atop and step downwards, the optimization targets are first 1/3 and then 1/5, which are analytically equivalent to 3/1 and 5/1 respectively in positive harmony. | ||
That all but suggests practically equal importance of divisive ratios and multiplicative ratios in tuning optimization. | That all but suggests practically equal importance of divisive ratios and multiplicative ratios in tuning optimization. | ||