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The wedgie is a set of numbers notated in the form <a b c| , which is the list of coefficients found by calculating the wedge-product of a certain number of vals (or breeds), with the GCD (Greatest Common Divisor) divided out if necessary.
The enclosing brackets < | are for the trivial 1-dimensional case where the wedgie is a val (or in the case of an ET, a breed); The enclosing brackets << || are for a 2-dimensional set of vals, and describe a bival (bibreed); <<< ||| would describe a 3-dimensional set of vals and its trival (tribreed); etc. The number of brackets on both sides is the same, and is equal to the dimensionality of the sets of vals being wedged.
The wedgie is the complement of the multimonzo. The wedgie may also be called "multival" or "multibreed"; the latter terms may have a numerical prefix prepended to specify the cardinality of the wedgie, as in bival, bibreed, trival, tribreed, etc.
There is one element in the wedgie for every possible pair of basis vals.
The wedgie shows the common vanishing unison-vectors shared by any set of ETs, thus its use is valuable in grouping temperaments into families.
Wedgie is a term coined by Gene Ward Smith; see Smith's definition of wedgie. His concise definition is:
The easiest way to get these is to start with a pair of vals, such as a tuning map. Take meantone as an example. Here's a map that represents the meantone temperament:
This map is based on using a ">[perfect] fourth as a generator; if you use a [perfect] fifth, the map is different [like this]:
To form the wedgie, take each possible combination of two prime numbers:
2 and 3, 2 and 5, 2 and 7, 3 and 5, 3 and 7, 5 and 7.
By convention, the elements of the wedge product are calculated in this order: all the combinations with 2 first, in numerical order, then the remaining ones with 3, 5, 7, and so on.
To calculate an element of the wedge product involving primes a and b:
In other words:
then the elements of the wedgie are given by:
calculating the wedgie of
By convention, if the first non-zero number is negative, the wedgie is normalized by multiplying each element by -1. Also, if there is a common divisor, it is divided out. So the normalized wedgie for meantone is
Note that you get the same result if you start from the 5th-based map. The nice thing about wedge products is that all tuning maps that temper out the same commas end up with the same wedgie to represent them.
You can also wedge commas together, but the resulting wedgie needs to be reversed and some of the signs negated.
Say you want to wedge the meantone comma 81;80 with the starling comma 126;125. You start with the monzo representations for 81;80 and for 126;125:
|-4 4 -1 0> , |1 2 -3 1>>.
After wedging them together, the result is
||12, -13, 4, 10, -4, 1>>.
In order to convert this to the standard form, first reverse it:
<<1, -4, 10, 4, -13, 12||,
then multiply the 2nd and 5th elements by -1:
<<1, 4, 10, 4, 13, 12||.
[I don't recall the exact rules for determining which signs to negate, and since it's easier to start with a pair of vals, I usually don't bother with wedging commas.]
Yes, you could use the triangular arrangement:
1 4 10 4 13 12 (note that <<1, 4, 4|| is the wedgie for 81;80)
Gene explained a couple of days ago about how the first row of the wedgie in this form is related to the mapping (depending on how the generators are defined, you might need to negate them, as in the case of meantone with a fourth as the generator, and divide by the number of periods in an octave).
Looking at an 11-limit version of meantone, <<1, 4, 10, -13, 4, 13, -24, 12, -44, -71||, you can see the 7-limit meantone wedgie in the triangular arrangement:
1 4 10 -13 4 13 -24 12 -44 -71
Since meantone on the chart is shown as a line between 12 and 19, you can get any other information you need from Graham Breed's temperament finder (http://www.microtonal.co.uk/temper/twoet.html): put in "12", "19", and "7" into the boxes, and this is what you get:
|generator:||13/31, 503.4 cents|
|mapping by period and generator:||[(1, 0), (2, -1), (4, -4), (7, -10)]|
|mapping by steps:||[(19, 12), (30, 19), (44, 28), (53, 34)]|
|highest interval width:||10|
|complexity measure:||10 (12 for smallest MOS)|
|highest error:||0.004480 (5.377 cents)|
Similarly, "mothra" <<3, 12, -1, 12, -10, -36|| is shown as a line from 5 to 26. Follow this link for the stats: http://x31eq.com/cgi-bin/temperament.cgi?et1=5&et2=26&limit=7.
You can get the wedgie by:
* calculating the wedge product of the period and generator mappings;
* calculating the wedge product of the vals of two ETs which support the temperament, and removing the GCD;
* calculating the wedge product of any basis set of vanishing commas, removing the GCD, and taking the complement.
From the wedgie, you can extract temperament mappings (how the primes are constructed from periods and generators, given any choice of whether you want your scales to repeat at [ratios] 2:1, 3:1, 5:1 ...), as well as subgroup commas (which include a basis for the infinitude of vanishing commas).
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