User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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== Chapter III. Power in Proportion == | == Chapter III. Power in Proportion == | ||
The first ever attempt at a systematic tuning solution was Paul Erlich's TOP tuning<ref>"All-Interval Tuning Schemes", ''Dave Keenan & Douglas Blumeyer's Guide to RTT''. Dave Keenan and Douglas Blumeyer. Xenharmonic Wiki. </ref>. This tuning was elegantly explained in his ''Middle Path'' paper in the case of nullity-1 (i.e. single-comma temperaments)<ref>''A Middle Path between Just Intonation and the Equal Temperaments – Part 1''. Paul Erlich. </ref>. In this tuning, every prime makes an effort in the right direction to close out the comma. To illustrate, consider 5-limit meantone, and to simplify it even more, let us start with the constrained equilateral-optimal tuning (CEOP tuning) instead since its effect is the easiest to observe. The CEOP tuning of 5-limit meantone is given in terms of the projection map P as | |||
$$ | |||
P = | |||
\begin{bmatrix} | |||
1 & 4/5 & -4/5 \\ | |||
0 & 1/5 & 4/5 \\ | |||
0 & 1/5 & 4/5 | |||
\end{bmatrix} | |||
$$ | |||
Let us denote the just tuning map in cents by J, the mistuning map E is | |||
$$ | |||
\begin{align} | |||
E &= J(P - I) \\ | |||
&= \left\langle \begin{matrix} 0 & -4.3013 & +4.3013 \end{matrix} \right] | |||
\end{align} | |||
$$ | |||
That is the 1/5-comma tuning, in which harmonics 3 and 5 have an equal magnitude and an opposite sign of error. | |||
TOP tuning works principally the same, except that harmonic 2 is no longer constrained to pure and that the allowed error of ''q'' is log<sub>2</sub> (''q'') times that of prime 2. The TOP mistuning map of 5-limit meantone is | |||
$$ | |||
E = \left\langle \begin{matrix} +1.6985 & -2.6921 & +3.9439 \end{matrix} \right] | |||
$$ | |||
The Euclidean norm we are covering next differs in a very important way. In Manhattan tunings such as TOP, each prime makes equal effort to close out the comma, whereas in Euclidean tunings, how much tempering load is assigned to each prime is proportional to how efficient the prime is to close out the comma. | |||
The effect is most clearly observed in the 5-limit CEE tuning (constrained equilateral-Euclidean tuning), where 2 is constrained to pure and only 3 and 5 placed equally distant in the lattice are under consideration. Here is the CEE projection map of 5-limit meantone: | |||
$$ | |||
P = | |||
\begin{bmatrix} | |||
1 & 16/17 & -4/17 \\ | |||
0 & 1/17 & 4/17 \\ | |||
0 & 4/17 & 16/17 | |||
\end{bmatrix} | |||
$$ | |||
The mistuning map is | |||
$$ | |||
\begin{align} | |||
E &= J(P - I) \\ | |||
&= \left\langle \begin{matrix} 0 & -5.0603 & +1.2651 \end{matrix} \right] | |||
\end{align} | |||
$$ | |||
That is the 4/17-comma tuning, in which prime 3 gets four times the error of prime 5 in the opposite direction. Notice how prime 3 contributes more to close out the comma since it is better at doing it, and the amount of load assigned to it is exactly 4 times that to prime 5. | |||
Understanding Euclidean norms will much simplify the cognitive process to Chebyshevian norms. In short, Chebyshevian norms take it to the other extreme, and can be described as demonstrating the "collapsing effect". In the case of nullity-1, it means the most efficient prime gets all the tempering loads and other primes get no load at all. The CEC (constrained equilateral-Chebyshevian) tuning of 5-limit meantone is | |||
$$ | |||
P = | |||
\begin{bmatrix} | |||
1 & 1 & 0 \\ | |||
0 & 0 & 0 \\ | |||
0 & 1/4 & 1 | |||
\end{bmatrix} | |||
$$ | |||
The mistuning map is | |||
$$ | |||
\begin{align} | |||
E &= J(P - I) \\ | |||
&= \left\langle \begin{matrix} 0 & -5.3766 & 0 \end{matrix} \right] | |||
\end{align} | |||
$$ | |||
That is our familiar 1/4-comma tuning. It is surprising that no interest has yet developed in tunings by the Chebyshevian norm. Compared to the 4/17-comma tuning by the Euclidean norm, The 1/4-comma tuning by the Chebyshevian norm removes all errors in prime 5 at the cost of just a little bit more in prime 3. | |||
To evaluate, the Euclidean tuning turns out advantageous not only because it is easy to compute (the Euclidean norm being the only norm with analytical solutions) but because it is theoretically nice as the more capable are tasked to do proportionately more. Both the Manhattan norm and the Chebyshevian norm show discontinuities when the complexities of the primes are at certain extreme points, and things start to break down as we approach them. The Manhattan norm feels strange when some primes are orders-of-magnitude more complex than the rest. The Chebyshevian norm feels as strange when all primes have near-equal complexities. | |||
== Chapter IV. Art of Compromise == | == Chapter IV. Art of Compromise == | ||