Calculating geometric complexity and badness

I've just finished computing the coefficients to calculate geometric complexity directly from the wedgie up to the 11 limit; an aggravating but potentially very useful chore.
I'm sorry for the variation in notation, but I took this from various Maple routines which did not have a consistent internal system, having been written at various times. Here are the results.

5 limit

In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma, then [u0, u1, u2]
is the wedgie and

sqrt(log(3)^2 u1^2 + log(3)^2 u1 u2 + log(5)^2 u2^2)

is the geometric complexity.

7 limit

If p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4 are two commas,
then

[u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2,
u1*v3-v1*u3,u1*v4-v1*u4]

is their wedge product p^q.

If u=[u1,u2,u3,u4] and v=[v1,v2,v3,v4] instead are vals, then

[u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4,
u4*v2-u2*v4,u2*v3-v2*u3]

is their wedge product, u^v. These two types of wedge product can be indentified with each other, by "Poincare duality". It does not matter whether the wedgie comes from commas or vals, therefore.

If [l[1], l[2], l[3], l[4], l[5], l[6]] a wedgie, then

sqrt(8.13090525300395*l[1]^2+4.20601082222396*l[2]^2+
2.76216685071565*l[3]^2-3.00701812111698*l[1]*l[2]-
1.56317414960864*l[1]*l[3]-2.39798540221393*l[2]*l[3])

is its geometric complexity. This depends only on l[1], l[2], l[3], which are the number of generator steps times how many intervals of repetition there are in the octave. Hence it can also be computed directly from the mapping.

Planar

If q = 2^u0 3^u1 5^u2 7^u3 11^u4 is a comma, [u0,u1,u2,u3,u4] is the corresponding planar temperament wedgie, and

sqrt(log(3)^2*u1^2+log(5)^2*u2^2+log(7)^2*u3^2
+log(3)^2*u1*u2+log(3)^2*u1*u3+log(5)^2*u2*u3)

is the geometric complexity.

11 limit

Linear

If

p = 2^u0 3^u1 5^u2 7^u3 11^u4
q = 2^v0 3^v1 5^v2 7^v3 11^v4
r = 2^w0 3^w1 5^w2 7^w3 11^w4

are three intervals, then

[u2*v3*w4-u2*v4*w3-v2*u3*w4+v2*u4*w3+w2*u3*v4-w2*u4*v3,
-u1*v3*w4+u1*v4*w3+v1*u3*w4-v1*u4*w3-w1*u3*v4+w1*u4*v3,
u1*v2*w4-u1*v4*w2-v1*u2*w4+v1*u4*w2+w1*u2*v4-w1*u4*v2,
-u1*v2*w3+u1*v3*w2+v1*u2*w3-v1*u3*w2-w1*u2*v3+w1*u3*v2,
u0*v3*w4-u0*v4*w3-v0*u3*w4+v0*u4*w3+w0*u3*v4-w0*u4*v3,
-u0*v2*w4+u0*v4*w2+v0*u2*w4-v0*u4*w2-w0*u2*v4+w0*u4*v2,
u0*v2*w3-u0*v3*w2-v0*u2*w3+v0*u3*w2+w0*u2*v3-w0*u3*v2,
u0*v1*w4-u0*v4*w1-v0*u1*w4+v0*u4*w1+w0*u1*v4-w0*u4*v1,
-u0*v1*w3+u0*v3*w1+v0*u1*w3-v0*u3*w1-w0*u1*v3+w0*u3*v1,
u0*v1*w2-u0*v2*w1-v0*u1*w2+v0*u2*w1+w0*u1*v2-w0*u2*v1]

is their wedge product p^q^r.

Similarly, if u=[u1,u2,u3,u4,u5] and v=[v1,v2,v3,v4,v5]
are two vals, then

[u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u1*v5-v1*u5,
u2*v3-v2*u3,u2*v4-v2*u4,
u2*v5-v2*u5,u3*v4-v3*u4,u3*v5-v3*u5,u4*v5-v4*u5]

is their wedge product, u^v. Both methods can be used to calculate the wedgie of a linear temperament; again by Poincare duality we can identify these two types of wedge products.

If [l[1],l[2],l[3],l[4],l[5],l[6],l[7],l[8],l[9],l[10]] is a linear temperament wedgie, then its geometric complexity is given by

sqrt(37.4669556799257*l[1]^2+19.8578149236787*l[2^2
+13.8576504872655*l[3]^2+8.43459044261178*l[4]^2
-9.24811506924099*l[2]*l[3]-12.9637247061659*l[1]*l[2]
-6.02856647730047*l[1]*l[3]-6.41005295068753*l[3]*l[4]
-2.95953128437572*l[1]*l[4]-4.54006536578504*l[2]*l[4])

Again, this depends only on l[1] through l[4] which are the number of generator steps times the number of intervals of equivalence in an octave, so this is also computable directly from the mapping.

Planar

If p = 2^u0 3^u1 5^u2 7^u3 11^u4 and q = 2^v0 3^v1 5^v2 7^v3 11^v4
are two intervals, then

[u3*v4-u4*v3, u2*v4-u4*v2, u2*v3-u3*v2, u1*v4-u4*v1, u1*v3-u3*v1,
u1*v2-u2*v1, u0*v4-v0*u4, u0*v3-u3*v0, u0*v2-v0*u2, u0*v1-u1*v0]

is their wedge product. If u=[u0,u1,u2,u3,u4], v=[v0,v1,v2,v3,v4],
and w=[w0,w1,w2,w3,w4] are three vals, then

[-u0*v1*w2+u0*v2*w1+v0*u1*w2-v0*u2*w1-w0*u1*v2+w0*u2*v1,
u0*v1*w3-u0*v3*w1-v0*u1*w3+v0*u3*w1+w0*u1*v3-w0*u3*v1,
-u0*v1*w4+u0*v4*w1+v0*u1*w4-v0*u4*w1-w0*u1*v4+w0*u4*v1,
-u0*v2*w3+u0*v3*w2+v0*u2*w3-v0*u3*w2-w0*u2*v3+w0*u3*v2,
u0*v2*w4-u0*v4*w2-v0*u2*w4+v0*u4*w2+w0*u2*v4-w0*u4*v2,
-u0*v3*w4+u0*v4*w3+v0*u3*w4-v0*u4*w3-w0*u3*v4+w0*u4*v3,
u1*v2*w3-u1*v3*w2-v1*u2*w3+v1*u3*w2+w1*u2*v3-w1*u3*v2,
-u1*v2*w4+u1*v4*w2+v1*u2*w4-v1*u4*w2-w1*u2*v4+w1*u4*v2,
u1*v3*w4-u1*v4*w3-v1*u3*w4+v1*u4*w3+w1*u3*v4-w1*u4*v3,
-u2*v3*w4+u2*v4*w3+v2*u3*w4-v2*u4*w3-w2*u3*v4+w2*u4*v3]

is their wedge product. Again by Poincare duality, we identify these two types of wedge product, and an 11-limit planar temperament wedgie may be calculated from either.

If [l[1],l[2],l[3],l[4],l[5],l[6],l[7],l[8],l[9],l[10]] is a planar temperament wedgie, then its geometric complexity is given by

sqrt(18.1878630999135*l[1]^2
+13.2165141603747*l[2]^2
+8.13090525300357*l[3]^2
+6.57565648053141*l[4]^2
+4.20601082222387*l[5]^2
+2.76216685071556*l[6]^2
+9.98976207451545*l[1]*l[2]
-4.90415316714511*l[1]*l[3]
+4.65474179366984*l[1]*l[4]
-2.28509613536279*l[1]*l[5]
-.721921985754273*l[1]*l[6]
+6.45350417171744*l[2]*l[3]
+5.37666377942441*l[2]*l[4]
+1.44384397150855*l[2]*l[5]
-1.56317414960855*l[2]*l[6]
+3.00701812111700*l[3]*l[5]
-1.56317414960855*l[3]*l[6]
+3.84182937372243*l[4]*l[5]
+2.39798540221390*l[4]*l[6]
+2.39798540221390*l[5]*l[6])

This depends only on l[1] through l[6].

Spacial

If q = 2^u0 3^u1 5^u2 7^u3 11^u4 is a comma, [u0,u1,u2,u3,u4] is the corresponding spacial temperament wedgie, and

sqrt(log(3)^2*u1^2+log(5)^2*u2^2+log(7)^2*u3^2+log(11)^2*u4^2+
log(3)^2*u1*u2+log(3)^2*u1*u3+log(3)^2*u1*u4
+log(5)^2*u2*u3+log(5)^2*u2*u4+log(7)^2*u3*u4)

is the geometric complexity.

Badness

If r is error (usually rms error), c is complexity, n is the number of primes in the p-limit (n=pi(p)) and d is the codimension, which is to say the number of commas used to define the temperament, then

badness = r c^(n/d)

is the log-flat badness measure.

> 5 limit
>
> In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma,
> then [u0, u1, u2] is the wedgie

That's all the wedgie is for a single comma? I
take it this would be the chromatic uv in the
linear case?

> sqrt(log(3)^2 u1^2 + log(3)^2 u1 u2 + log(5)^2 u2^2)
>
> is the geometric complexity.

Cool. I can do that. ;)

> 11 limit
>
> Linear
>
> If
>
> p = 2^u0 3^u1 5^u2 7^u3 11^u4
> q = 2^v0 3^v1 5^v2 7^v3 11^v4
> r = 2^w0 3^w1 5^w2 7^w3 11^w4
>
> are three intervals,

You mean commas, right?

> Spacial

You the man, Gene!

> Badness
>
> If r is error (usually rms error), c is complexity, n is the
> number of primes in the p-limit (n=pi(p)) and d is the
> codimension, which is to say the number of commas used to
> define the temperament,

Does it matter how many we temper out here?

> then
>
> badness = r c^(n/d)
>
> is the log-flat badness measure.

Great! I always thought the critical exponent had to be
determined empirically.

-Carl

I wrote..
> > In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma,
> > then [u0, u1, u2] is the wedgie
>
> That's all the wedgie is for a single comma? I
> take it this would be the chromatic uv in the
> linear case?

D'oh, I meant commatic.

-C.

I get a slightly different result calculating geometric complexity for
meantone using the 5 limit comma formula. Should it be 4.132030727? Thanks.
Also I was wondering why complexity for a planar temperment doesn't include
u4 (power of 11). Thanks, its neat to get the formulas for complexity and
badness. If its not asking too much, could I also get the formula for rms
(in the general sense, of course, I know the formula for the 2-optimal
generator )

"Gene Ward Smith
<genewardsmith@ju To: tuning-math@yahoogroups.com
no.com>" cc: (bcc: Paul G Hjelmstad/US/AMERICAS)
<genewardsmith Subject: [tuning-math] Calculating geometric complexity and badness

01/19/2003 08:48
AM
Please respond to
tuning-math

I've just finished computing the coefficients to calculate geometric
complexity directly from the wedgie up to the 11 limit; an aggravating but
potentially very useful chore.
I'm sorry for the variation in notation, but I took this from various Maple
routines which did not have a consistent internal system, having been
written at various times. Here are the results.

5 limit

In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma, then [u0, u1, u2]
is the wedgie and

sqrt(log(3)^2 u1^2 + log(3)^2 u1 u2 + log(5)^2 u2^2)

is the geometric complexity.

7 limit

If p = 2^u1 3^u2 5^u3 7^u4 and q = 2^v1 3^v2 5^v3 7^v4 are two commas,
then

[u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2,
u1*v3-v1*u3,u1*v4-v1*u4]

is their wedge product p^q.

If u=[u1,u2,u3,u4] and v=[v1,v2,v3,v4] instead are vals, then

[u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u3*v4-v3*u4,
u4*v2-u2*v4,u2*v3-v2*u3]

is their wedge product, u^v. These two types of wedge product can be
indentified with each other, by "Poincare duality". It does not matter
whether the wedgie comes from commas or vals, therefore.

If [l[1], l[2], l[3], l[4], l[5], l[6]] a wedgie, then

sqrt(8.13090525300395*l[1]^2+4.20601082222396*l[2]^2+
2.76216685071565*l[3]^2-3.00701812111698*l[1]*l[2]-
1.56317414960864*l[1]*l[3]-2.39798540221393*l[2]*l[3])

is its geometric complexity. This depends only on l[1], l[2], l[3], which
are the number of generator steps times how many intervals of repetition
there are in the octave. Hence it can also be computed directly from the
mapping.

Planar

If q = 2^u0 3^u1 5^u2 7^u3 11^u4 is a comma, [u0,u1,u2,u3,u4] is the
corresponding planar temperament wedgie, and

sqrt(log(3)^2*u1^2+log(5)^2*u2^2+log(7)^2*u3^2
+log(3)^2*u1*u2+log(3)^2*u1*u3+log(5)^2*u2*u3)

is the geometric complexity.

11 limit

Linear

If

p = 2^u0 3^u1 5^u2 7^u3 11^u4
q = 2^v0 3^v1 5^v2 7^v3 11^v4
r = 2^w0 3^w1 5^w2 7^w3 11^w4

are three intervals, then

[u2*v3*w4-u2*v4*w3-v2*u3*w4+v2*u4*w3+w2*u3*v4-w2*u4*v3,
-u1*v3*w4+u1*v4*w3+v1*u3*w4-v1*u4*w3-w1*u3*v4+w1*u4*v3,
u1*v2*w4-u1*v4*w2-v1*u2*w4+v1*u4*w2+w1*u2*v4-w1*u4*v2,
-u1*v2*w3+u1*v3*w2+v1*u2*w3-v1*u3*w2-w1*u2*v3+w1*u3*v2,
u0*v3*w4-u0*v4*w3-v0*u3*w4+v0*u4*w3+w0*u3*v4-w0*u4*v3,
-u0*v2*w4+u0*v4*w2+v0*u2*w4-v0*u4*w2-w0*u2*v4+w0*u4*v2,
u0*v2*w3-u0*v3*w2-v0*u2*w3+v0*u3*w2+w0*u2*v3-w0*u3*v2,
u0*v1*w4-u0*v4*w1-v0*u1*w4+v0*u4*w1+w0*u1*v4-w0*u4*v1,
-u0*v1*w3+u0*v3*w1+v0*u1*w3-v0*u3*w1-w0*u1*v3+w0*u3*v1,
u0*v1*w2-u0*v2*w1-v0*u1*w2+v0*u2*w1+w0*u1*v2-w0*u2*v1]

is their wedge product p^q^r.

Similarly, if u=[u1,u2,u3,u4,u5] and v=[v1,v2,v3,v4,v5]
are two vals, then

[u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4,u1*v5-v1*u5,
u2*v3-v2*u3,u2*v4-v2*u4,
u2*v5-v2*u5,u3*v4-v3*u4,u3*v5-v3*u5,u4*v5-v4*u5]

is their wedge product, u^v. Both methods can be used to calculate the
wedgie of a linear temperament; again by Poincare duality we can identify
these two types of wedge products.

If [l[1],l[2],l[3],l[4],l[5],l[6],l[7],l[8],l[9],l[10]] is a linear
temperament wedgie, then its geometric complexity is given by

sqrt(37.4669556799257*l[1]^2+19.8578149236787*l[2^2
+13.8576504872655*l[3]^2+8.43459044261178*l[4]^2
-9.24811506924099*l[2]*l[3]-12.9637247061659*l[1]*l[2]
-6.02856647730047*l[1]*l[3]-6.41005295068753*l[3]*l[4]
-2.95953128437572*l[1]*l[4]-4.54006536578504*l[2]*l[4])

Again, this depends only on l[1] through l[4] which are the number of
generator steps times the number of intervals of equivalence in an octave,
so this is also computable directly from the mapping.

Planar

If p = 2^u0 3^u1 5^u2 7^u3 11^u4 and q = 2^v0 3^v1 5^v2 7^v3 11^v4
are two intervals, then

[u3*v4-u4*v3, u2*v4-u4*v2, u2*v3-u3*v2, u1*v4-u4*v1, u1*v3-u3*v1,
u1*v2-u2*v1, u0*v4-v0*u4, u0*v3-u3*v0, u0*v2-v0*u2, u0*v1-u1*v0]

is their wedge product. If u=[u0,u1,u2,u3,u4], v=[v0,v1,v2,v3,v4],
and w=[w0,w1,w2,w3,w4] are three vals, then

[-u0*v1*w2+u0*v2*w1+v0*u1*w2-v0*u2*w1-w0*u1*v2+w0*u2*v1,
u0*v1*w3-u0*v3*w1-v0*u1*w3+v0*u3*w1+w0*u1*v3-w0*u3*v1,
-u0*v1*w4+u0*v4*w1+v0*u1*w4-v0*u4*w1-w0*u1*v4+w0*u4*v1,
-u0*v2*w3+u0*v3*w2+v0*u2*w3-v0*u3*w2-w0*u2*v3+w0*u3*v2,
u0*v2*w4-u0*v4*w2-v0*u2*w4+v0*u4*w2+w0*u2*v4-w0*u4*v2,
-u0*v3*w4+u0*v4*w3+v0*u3*w4-v0*u4*w3-w0*u3*v4+w0*u4*v3,
u1*v2*w3-u1*v3*w2-v1*u2*w3+v1*u3*w2+w1*u2*v3-w1*u3*v2,
-u1*v2*w4+u1*v4*w2+v1*u2*w4-v1*u4*w2-w1*u2*v4+w1*u4*v2,
u1*v3*w4-u1*v4*w3-v1*u3*w4+v1*u4*w3+w1*u3*v4-w1*u4*v3,
-u2*v3*w4+u2*v4*w3+v2*u3*w4-v2*u4*w3-w2*u3*v4+w2*u4*v3]

is their wedge product. Again by Poincare duality, we identify these two
types of wedge product, and an 11-limit planar temperament wedgie may be
calculated from either.

If [l[1],l[2],l[3],l[4],l[5],l[6],l[7],l[8],l[9],l[10]] is a planar
temperament wedgie, then its geometric complexity is given by

sqrt(18.1878630999135*l[1]^2
+13.2165141603747*l[2]^2
+8.13090525300357*l[3]^2
+6.57565648053141*l[4]^2
+4.20601082222387*l[5]^2
+2.76216685071556*l[6]^2
+9.98976207451545*l[1]*l[2]
-4.90415316714511*l[1]*l[3]
+4.65474179366984*l[1]*l[4]
-2.28509613536279*l[1]*l[5]
-.721921985754273*l[1]*l[6]
+6.45350417171744*l[2]*l[3]
+5.37666377942441*l[2]*l[4]
+1.44384397150855*l[2]*l[5]
-1.56317414960855*l[2]*l[6]
+3.00701812111700*l[3]*l[5]
-1.56317414960855*l[3]*l[6]
+3.84182937372243*l[4]*l[5]
+2.39798540221390*l[4]*l[6]
+2.39798540221390*l[5]*l[6])

This depends only on l[1] through l[6].

Spacial

If q = 2^u0 3^u1 5^u2 7^u3 11^u4 is a comma, [u0,u1,u2,u3,u4] is the
corresponding spacial temperament wedgie, and

sqrt(log(3)^2*u1^2+log(5)^2*u2^2+log(7)^2*u3^2+log(11)^2*u4^2+
log(3)^2*u1*u2+log(3)^2*u1*u3+log(3)^2*u1*u4
+log(5)^2*u2*u3+log(5)^2*u2*u4+log(7)^2*u3*u4)

is the geometric complexity.

Badness

If r is error (usually rms error), c is complexity, n is the number of
primes in the p-limit (n=pi(p)) and d is the codimension, which is to say
the number of commas used to define the temperament, then

badness = r c^(n/d)

is the log-flat badness measure.

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--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> I get a slightly different result calculating geometric complexity
for
> meantone using the 5 limit comma formula. Should it be 4.132030727?
Thanks.
> Also I was wondering why complexity for a planar temperment doesn't
include
> u4 (power of 11).

probably gene's typos . . .

> Thanks, its neat to get the formulas for complexity and
> badness. If its not asking too much, could I also get the formula
for rms
> (in the general sense, of course, I know the formula for the 2-
optimal
> generator )

you mean p=2? that *is* rms -- "root mean square". are you looking
for the formula for the p-norm? it's

(|x1|^p + |x2|^p + |x3|^p + . . . + |xn|^p)^(1/p),

where n is the number of consonant intervals and xi is the error in
interval i.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> probably gene's typos . . .

Here are my Maple routines:

qw11 := proc(q)
# 11-limit weighted quadratic form

local a,b,c,d;
a := ordp(q,3);
b := ordp(q,5);
c := ordp(q,7);
d := ordp(q,11);
ln(3)^2*a^2+ln(5)^2*b^2+ln(7)^2*c^2+ln(11)^2*d^2+
ln(3)^2*a*b+ln(3)^2*a*c+ln(3)^2*a*d+ln(5)^2*b*c+ln(5)^2*b*d+ln(7)^2*c*d end:

gc11 := proc(l)
# complexity of 11-limit linear wedgie

sqrt(37.4669556799257*l[1]^2+19.8578149236787*l[2]^2+13.8576504872655*l[3]^2+
8.43459044261178*l[4]^2-9.24811506924099*l[2]*l[3]-
12.9637247061659*l[1]*l[2]-6.02856647730047*l[1]*l[3]-
6.41005295068753*l[3]*l[4]-2.95953128437572*l[1]*l[4]-
4.54006536578504*l[2]*l[4]) end:

gpc11 := proc(l)
# complexity of 11-limit planar wedgie

sqrt(18.1878630999135*l[1]^2
+13.2165141603747*l[2]^2
+8.13090525300357*l[3]^2
+6.57565648053141*l[4]^2
+4.20601082222387*l[5]^2
+2.76216685071556*l[6]^2
+9.98976207451545*l[1]*l[2]
-4.90415316714511*l[1]*l[3]
+4.65474179366984*l[1]*l[4]
-2.28509613536279*l[1]*l[5]
-.721921985754273*l[1]*l[6]
+6.45350417171744*l[2]*l[3]
+5.37666377942441*l[2]*l[4]
+1.44384397150855*l[2]*l[5]
-1.56317414960855*l[2]*l[6]
+3.00701812111700*l[3]*l[5]
-1.56317414960855*l[3]*l[6]
+3.84182937372243*l[4]*l[5]
+2.39798540221390*l[4]*l[6]
+2.39798540221390*l[5]*l[6]) end:

Thanks. Sorry to be lagging so far behind, but I am making progress. Now,
could you just show me how complexity is calculated for meantone (the
easiest example I can think of) using

sqrt(log(3)^2 u1^2 + log(3)^2 u1 u2 + log(5)^2 u2^2)

I know this should equal 4.132030727, I get a different result. I am using
u1=4 and u2=1. I hope I am not being irritating!

Paul