need definition: wedgie

can someone please write an *understandable* definition
of wedgie, for inclusion into the Encyclopaedia?

i have a copy of a list posting there for right now,
but it just illustrates and doesn't define.

the definitions on Gene's website i'm sure are the
most precise ones, but they're incomprehensible to me.

-monz

monz wrote:

> can someone please write an *understandable* definition
> of wedgie, for inclusion into the Encyclopaedia?
> > i have a copy of a list posting there for right now,
> but it just illustrates and doesn't define.
> > the definitions on Gene's website i'm sure are the
> most precise ones, but they're incomprehensible to me.

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:

[<1, 2, 4, 7|, <0, -1, -4, -10|]

This map is based on using a fourth as a generator; if you use a fifth, the map is different.

[<1, 1, 0, -3|, <0, 1, 4, 10|]

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", multiply the period map of prime "a" by the generator map of prime "b", then subtract the product of the generator map of prime "a" times the period map of prime "b". In other words, if val1 represents the period map (e.g. <1, 2, 4, 7|) and val2 represents the generator map (<0, -1, -4, -10|), the element of the wedgie is given by

val1[a] val2[b] - val2[a] val1[b]

Example: calculating the wedgie of meantone:
[<1, 2, 4, 7|, <0, -1, -4, -10|]

1st element (2, 3): (1 * -1) - (0 * 2) = -1
2nd element (2, 5): (1 * -4) - (0 * 4) = -4
3rd element (2, 7): (1 * -10) - (0 * 7) = -10
4th element (3, 5): (2 * -4) - (-1 * 4) = -4
5th element (3, 7): (2 * -10) - (-1 * 7) = -13
6th element (5, 7): (4 * -10) - (-4 * 7) = -12

Result: <<-1, -4, -10, -4, -13, -12||.

By convention, if the first number is negative, the wedgie is normalized by multiplying each element by -1. So the normalized wedgie for meantone is <<1, 4, 10, 4, 13, 12||.

Note that you get the same result if you start from the fifth-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 representation |-4 4 -1 0> for 81;80 and |1 2 -3 1> for 126;125. 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.]

Herman Miller wrote:

> By convention, if the first number is negative, the wedgie is normalized > by multiplying each element by -1. So the normalized wedgie for meantone > is <<1, 4, 10, 4, 13, 12||.

To be more precise, "if the first nonzero element is negative".

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> By convention, if the first number is negative, the wedgie is
normalized
> by multiplying each element by -1.

Another convention is that if there is a common divisor, it is divided
out.

Thanks, Herman.