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{{Beginner|Plücker coordinates}} | {{Beginner|Plücker coordinates}} | ||
A '''wedgie''' is a primarily mathematical object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] | A '''wedgie''' is a primarily mathematical object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] and [[generator]], which can therefore provide some illuminating information beyond the [[mapping|mapping matrix]]. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). 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. | ||
While wedgies | While wedgies have served as a canonical way to represent temperaments mathematically due to their generator-agnostic nature, their musical utility is not obvious – [[mapping|mappings]] would often be of more immediate use to musicians. | ||
== How to read a wedgie == | == How to read a wedgie == | ||
Following intuitions from [[ploidacot]], one way to characterize temperaments is by how many parts they split important intervals into | Following intuitions from [[ploidacot]], one way to characterize temperaments is by how many parts they split important intervals into – for example, the perfect fifth and the octave. | ||
For any n-prime subgroup of a rank-n temperament, there exist a finite number (1 or more) of copies of that subgroup within the temperament. Think of these as "universes" that are connected exclusively by intervals of that subgroup and may be travelled between by using intervals outside the subgroup. For example, a temperament that is [[Ploidacot/Diploid dicot|diploid dicot]] | For any ''n''-prime subgroup of a rank-''n'' temperament, there exist a finite number (1 or more) of copies of that subgroup within the temperament. Think of these as "universes" that are connected exclusively by intervals of that subgroup and may be travelled between by using intervals outside the subgroup. For example, a temperament that is [[Ploidacot/Diploid dicot|diploid dicot]] – dividing the octave into two parts and also dividing the perfect fifth into two parts – will have a 2.3 wedgie entry of 4 (since there are four distinct copies of the 3-limit – the basic 3-limit, offset by a neutral third, offset by a semioctave, and offset by both). Each wedgie entry counts the number of copies of its corresponding subgroup. Because any temperament can be defined by splitting some interval and assigning the parts just interpretations, this is enough to uniquely characterize the temperament. | ||
=== Example 1: | === Example 1: meantone === | ||
The wedgie for meantone is {{multival| 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, because 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 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. | ||
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For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will 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 3/2's. | For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will 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 3/2's. | ||
=== Example 2: | === Example 2: father === | ||
For another example, take father, which has the wedgie {{multival| 1 -1 -4 }}. | For another example, take father, which has the wedgie {{multival| 1 -1 -4 }}. | ||
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Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will 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 are 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. | Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will 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 are 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. | ||
=== Example 3: | === Example 3: blackwood === | ||
For our final example, we will consider blackwood {{multival| 0 5 8 }}. | For our final example, we will consider blackwood {{multival| 0 5 8 }}. | ||
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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. | 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. | ||
== | == Form of a wedgie == | ||
The notation being used previously, {{multival| x y z }}, is formally a shorthand for a matrix form, written | The notation being used previously, {{multival| ''x'' ''y'' ''z'' }}, is formally a shorthand for a matrix form, written | ||
$$ | $$ | ||
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$$ | $$ | ||
For a wedgie on a 4-prime subgroup, the structure of {{multival| a b c d e f }} is actually | For a wedgie on a 4-prime subgroup, the structure of {{multival| ''a'' ''b'' ''c'' ''d'' ''e'' ''f'' }} is actually | ||
$$ | $$ | ||