Wedgie
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This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. The corresponding expert page for this topic is Plücker coordinates. |
A wedgie is an object that uniquely characterizes a regular temperament regardless of choice of period vs. generator or of equave, which can therefore provide some illuminating information beyond the mapping matrix, which specifies a particular equave. A wedgie takes the form ⟨⟨x y z]], with a number of entries (not necessarily 3; in fact that is the simplest possible case) listed in between multiple val brackets (double brackets for rank-2, triple brackets for rank-3, etc). Wedgies can be thought of as a generalization of vals, called multivals, so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.
How to read a wedgie
Following intuitions from ploidacot, one way to characterize a temperament is how many equal parts it splits the perfect fifth (3/2) into. For an example, meantone doesn't split it at all, so we say it is monocot. We also say it is haploid, since it doesn't split the octave (2/1) at all.
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 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: ⟨⟨1 4 4]]. 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 1, 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 2.5, the procedure generalizes, with the entry, 4, 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 the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we're thinking tritave-equivalently now, so we'll be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2s.
For another example, take father, which has the wedgie ⟨⟨1 -1 -4]].
Here, we again have a 1 as our entry for 2.3, meaning that the temperament is haploid monocot, or in other words that 2/1 is unsplit and 3/2 is one generator.
Thus, going into our second entry, -1 for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect fourth. Since we're thinking octave-equivalently, this can be seen as going down a perfect fifth.
Finally, for 3.5, we have the entry -4. Again, we're tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we'll be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we're in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it - 5/3 is split into four parts, which each contain a negative generator.
For our final example, we will consider blackwood ⟨⟨0 5 8]].
Blackwood's first entry is 0, which means that it reduces 2.3 to a rank-1 structure. This can be seen as 3 being found 0 generators from some ploid (since 3/2 in blackwood is 3\5), since 0 times anything is 0.
The next entry, 5, is simple: in 2.5, 5 (in this case, 5/4) is found by going up one generator, but remember that each entry is where the two primes are found multiplied together. Since 2 is found at 5 ploids, the entry is 1 * 5 = 5. (Technically, there's a hidden 5 in the 2.3 entry that gets multiplied by 0 and vanishes.)
And then the final entry, for 3.5, is 8. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, 1 * 8 = 8.
For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit! If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.
For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.
The form of a wedgie
The notation being used previously, ⟨⟨x y z]], is formally a shorthand for a matrix form, written
[math]\displaystyle{ \begin{bmatrix}
0 & x & y\\
-x & 0 & z\\
-y & -z & 0
\end{bmatrix} }[/math]. For a wedgie on a 4-prime subgroup, the structure of ⟨⟨a b c d e f]] is actually [math]\displaystyle{ \begin{bmatrix}
0 & a & b & c\\
-a & 0 & d & e\\
-b & -d & 0 & f\\
-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.