User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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\mathcal{E}_1 = \left\langle \begin{matrix} 0 & -\varepsilon & +\varepsilon \end{matrix} \right] | |||
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\mathcal{E}_2 = \left\langle \begin{matrix} 0 & -\varepsilon & -\varepsilon \end{matrix} \right] | |||
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the ~15/1 will be a combination of harmonics with pitch errors of -''ε'', -''ε'', and 0, but the played harmonic is at -2''ε''. So we see this harmonic will get pretty off the track whenever played. | the ~15/1 will be a combination of harmonics with pitch errors of -''ε'', -''ε'', and 0, but the played harmonic is at -2''ε''. So we see this harmonic will get pretty off the track whenever played. | ||
Regarding 5/3, it is the opposite situation. | Regarding 5/3, it is the opposite situation. Ɛ<sub>2</sub> comes out superior to Ɛ<sub>1</sub> as it perfectly hits 5/3 whereas Ɛ<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 divisive ratio on the formal root, there is a counterpart featuring a multiplicative ratio alike. | 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. | ||
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Let us denote the just tuning map in cents by J, the mistuning map | Let us denote the just tuning map in cents by T<sub>J</sub>, the mistuning map Ɛ is | ||
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\begin{align} | \begin{align} | ||
E &= | \mathcal{E} &= T_J(P - I) \\ | ||
&= \left\langle \begin{matrix} 0 & -4.3013 & +4.3013 \end{matrix} \right] | &= \left\langle \begin{matrix} 0 & -4.3013 & +4.3013 \end{matrix} \right] | ||
\end{align} | \end{align} | ||
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E = \left\langle \begin{matrix} +1.6985 & -2.6921 & +3.9439 \end{matrix} \right] | \mathcal{E} = \left\langle \begin{matrix} +1.6985 & -2.6921 & +3.9439 \end{matrix} \right] | ||
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\begin{align} | \begin{align} | ||
E &= | \mathcal{E} &= T_J(P - I) \\ | ||
&= \left\langle \begin{matrix} 0 & -5.0603 & +1.2651 \end{matrix} \right] | &= \left\langle \begin{matrix} 0 & -5.0603 & +1.2651 \end{matrix} \right] | ||
\end{align} | \end{align} | ||
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$$ | $$ | ||
\begin{align} | \begin{align} | ||
E &= | \mathcal{E} &= T_J(P - I) \\ | ||
&= \left\langle \begin{matrix} 0 & -5.3766 & 0 \end{matrix} \right] | &= \left\langle \begin{matrix} 0 & -5.3766 & 0 \end{matrix} \right] | ||
\end{align} | \end{align} | ||
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© 2023 Flora Canou | © 2023 Flora Canou | ||
Version Stable | Version Stable 1 | ||
This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License]. | This work is licensed under the [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution-ShareAlike 4.0 International License]. | ||