User:Vector/Wedgie
The wedgie of a regular temperament is a mathematical object that uniquely characterizes the temperament independently of choice of generator or equave. A wedgie takes the form ⟨⟨…⟨ w1 w2 … wn ]]…], with n entries listed in between multiple val brackets (double brackets for rank-2, triple brackets for rank-3, and so on). 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.
Rank-1 temperaments
The wedgie of a rank-1 temperament is its val. As such, wedgies can be thought of as a generalization of vals, called multivals.
Rank-2 temperaments
Each wedgie entry for a rank-2 temperament contains information about a pair of primes. In the rank-2 case, these provide useful information about the MOS scales a regular temperament provides:
- the number of imperfect instances of the ratio between the two primes, conveyed by the absolute value
- the direction of alteration of the imperfect interval, relative to some standard reference (usually the direction of alteration of the imperfect 3/2), conveyed by the sign
These two properties are, as it turns out, invariant with MOS size given the same temperament. We may step through a few examples.
Meantone
Meantone has one imperfect fifth (ratio 3/2). This is true regardless of whether you take meantone[5], meantone[7], meantone[12] or another meantone MOS. As such, the first entry of meantone's wedgie is ±1 (1 by convention).
Meantone additionally has four imperfect major thirds (ratio 5/2, or equivalently 5/4). Again, this is true regardless of which meantone scale you use. (In diatonic, the "imperfect major third" is the minor third.) As such, the second entry of meantone's wedgie is ±4. Imperfect major thirds are always flat when imperfect fifths are flat, and likewise for when they are sharp. For instance, in diatonic, we see a minor third and a diminished fifth, while in pentic, the perfect fourth is the imperfect major third, and the minor sixth is the imperfect fifth. Thus, the second entry is 4.
For the third entry, we examine 5/3, but tritave-equivalently, so that instead of the octave-equivalent meantone scales we instead use the tritave-equivalent ones. In this context it can be seen that there are four imperfect 5/3s in a tritave-equivalent scale. So the third entry is ±4, but how do we determine whether it is positive or negative? Well, every 2/1-equivalent meantone MOS has a corresponding 3/1-equivalent MOS, and the tritave version of meantone[7] is meantone[11]. In this scale, 5/3's imperfect counterpart is flat of it (and in meantone[8], the tritave counterpart of pentic, 5/3's imperfect counterpart is sharp of it), so the third entry is also 4.
Therefore, meantone's wedgie is ⟨⟨ 1 4 4 ]].
Diaschismic
Diaschismic has two imperfect fifths, so its first wedgie entry is 2. There are four imperfect major thirds. However, when imperfect fifths are flat, imperfect major thirds are sharp (as in diaschismic[12]), and vice versa for diaschismic[10]. So, the second entry is actually -4.
The third entry requires examining tritave-equivalent diaschismic, which has a generator of 600 cents and a period of a tritave. It turns out that in this system there are eleven imperfect 5/3s, and the imperfect 5/3s alter in the same direction as the 5/4s (flatwards in diaschismic[16], the tritave counterpart of diaschismic[10]), so the third entry is -11.
Therefore, diaschismic's wedgie is ⟨⟨ 2 -4 -11 ]].
Blackwood
Blackwood has *zero* imperfect fifths, since the fifth is an even multiple of the period of 1/5 an octave. So, the first entry of its wedgie is 0. In any given blackwood MOS (though most notably blackwood[10]), there are five imperfect 5/4s; this is because there are five periods per octave. Tritave-equivalent blackwood divides the tritave into eight equal parts, so there are eight imperfect 5/3s and eight periods per tritave.
Therefore, blackwood's wedgie is ⟨⟨ 0 5 8 ]].
Extensions
Temperaments of more than 3 primes proceed in a similar way. Every ratio between primes is examined (for 2.3.5.7, this would be 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7). The 7-limit wedgie for meantone is ⟨⟨ 1 4 10 4 13 12 ]].
Rank-3 temperaments
For higher-rank temperaments, we must think more abstractly - for any prime subgroup with rank equivalent to the rank of the temperament, how many copies of it exist in the temperament? In other words, how many "universes" representing that subgroup and that can be traveled between by intervals outside of it are there? These wedgies are much less directly useful for composition theory, but they are still being briefly covered for completeness. A four-prime rank-3 temperament has four entries; a five-prime rank-3 temperament has ten entries.
Relationship to edo joins
A temperament's wedgie can be derived from its edo join by applying an operation called a wedge product to the vals representing the equal temperaments. To calculate the entry of a wedgie corresponding to the a.b subgroup in a given temperament represented by edos N and M, you take M(a)*N(b) - N(a)*M(b), where the function notation represents mapping the interval in the val. This operation, applied to all possible combinations of a and b in the temperament's subgroup, is called the wedge product. For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨ (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) ]], which simplifies to ⟨⟨ (55 - 56) (80 - 84) (128 - 132) ]] and thus to ⟨⟨ -1 -4 -4 ]]. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain ⟨⟨ 1 4 4 ]], which is the wedgie for 5 & 7, a.k.a. meantone.
More than two vals can be combined into a higher-rank wedgie by an analogous method.
