Wedgie: Difference between revisions
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The elements of a wedgie each represent the number of parts into which a particular [[subgroup]] is split by the temperament, or the number of distinct sets of notes within the temperament linked by motions within that subgroup. They can be thought of as a generalization of the ploidacot information to all possible combinations of [[prime harmonic|primes]] within a temperament, and format that information in a concise manner; as it turns out, this is enough to uniquely characterize the temperament. | The elements of a wedgie each represent the number of parts into which a particular [[subgroup]] is split by the temperament, or the number of distinct sets of notes within the temperament linked by motions within that subgroup. They can be thought of as a generalization of the ploidacot information to all possible combinations of [[prime harmonic|primes]] within a temperament, and format that information in a concise manner; as it turns out, this is enough to uniquely characterize the temperament. | ||
For example, take the wedgie for meantone: <code>⟨⟨1 4 4]]</code>. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is <code>1</code>, telling us that the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1). | For example, take the wedgie for meantone: <code>⟨⟨1 4 4]]</code>. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is <code>1</code>, telling us that the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1). | ||
For an intuition as to why multiplication is performed, consider hemipyth (or ''diploid dicot''), which divides both the octave and perfect fifth into two parts. In hemipyth, you can use radical steps to traverse between four different "universes" connected by 3-limit intervals: base Pythagorean, offset by a semioctave, offset by a neutral third, and offset by both. The wedgie entry essentially counts these universes. | |||
For 2.5, the procedure generalizes, with the entry, <code>4</code>, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up. | For 2.5, the procedure generalizes, with the entry, <code>4</code>, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up. | ||
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-c & -e & -f & 0 | -c & -e & -f & 0 | ||
\end{bmatrix}</math>. This means that a wedgie can easily be "clipped" by removing columns and corresponding rows to produce a restriction of the temperament, and thus wedgies which have a set of entries produced this way in common are strong extensions of some common structure. | \end{bmatrix}</math>. This means that a wedgie can easily be "clipped" by removing columns and corresponding rows to produce a restriction of the temperament, and thus wedgies which have a set of entries produced this way in common are strong extensions of some common structure. | ||
== See also == | == See also == | ||