Wedgie/Archived version: Difference between revisions

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To find (a JI interpretation of) the '''generator''': Use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a JI ratio g = q_1^a_1 ... q_n^a_n (equivalently, a linear combination g = a_1 q_1 + ... + a_n q_n) such that W(2, g) = a_1 W(2, q_1) + ... a_n W(2,q_n) = d.

Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[~~Tenney-Euclidean_temperament_measures#TE_error|~~TE error]].

Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]].

[In geometric terms, given JI ratios u and v, and wedgie W, the number W(u,v) is the signed area of the parallelogram spanned by (tempered versions of) u and v. The entries of the wedgie give the values of the wedgie on the basis elements of the JI subgroup that the temperament is on. By the alternating property [i.e. W(u, v) = -W(v, u)] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. This is the determinant of the tempered versions of u and v. The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v.]

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== Technical introduction ==

An alternating [http://en.wikipedia.org/wiki/Multilinear_map multilinear map] which is a multilinear function taking a certain number n of [[~~monzos|~~monzos]] as arguments and returning an integer as a value we may call an '''n-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.

An alternating [http://en.wikipedia.org/wiki/Multilinear_map multilinear map] which is a multilinear function taking a certain number n of [[monzos]] as arguments and returning an integer as a value we may call an '''n-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.

The simplest kind of n-map is the 1-map, or [[~~Vals|~~val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [http://en.wikipedia.org/wiki/Group_homomorphism group homomorphism] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [http://en.wikipedia.org/wiki/Linear_map linear]: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).

The simplest kind of n-map is the 1-map, or [[val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [http://en.wikipedia.org/wiki/Group_homomorphism group homomorphism] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [http://en.wikipedia.org/wiki/Linear_map linear]: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).

One use for such things is as "machines" for measuring complexity. If we consider the 1-map which is the val for 11-limit 31et, we find we have <31 49 72 87 107|. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 72 in this system. Now consider a 2-map "meantone(u,v)" which tells us, roughly speaking, how many generator steps it takes to get to v assuming u is being used as a period in septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, in which case we have meantone(2,3)=1, meantone(2,5)=4, meantone(2,7)=10. With 3 as a period and 3/2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=13. Finally, with if we take 5 as a period we find that four 3/2s give 5, so 5^(1/4) or equivalently 3/2 is the basic period. Using 3/2 as a period and 9/8 as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5^(1/4) gives us meantone(5,7)=12. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.

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where the sum is taken over S(n,m), the set of all [http://en.wikipedia.org/wiki/Permutation permutations] of the first n+m integers which are an [http://en.wikipedia.org/wiki/%28p,q%29_shuffle (n,m) shuffles], and sgn(t) is the [http://en.wikipedia.org/wiki/Parity_of_a_permutation parity of the permutation] t, which is +1 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.

If f and g are both vals (1-maps) then this becomes especially easy: f∧g(u,v) = f(u)g(v) - f(v)g(u). Let's consider a specific example. Suppose E19 = ~~<~~19 30 44 53| is the equal temperament val for septimal 19et, and E31 = ~~<~~31 49 72 87| is the val for septimal 31et. Then writing intervals multiplicatively, we have

If f and g are both vals (1-maps) then this becomes especially easy: f∧g(u,v) = f(u)g(v) - f(v)g(u). Let's consider a specific example. Suppose E19 = {{val| 19 30 44 53 }} is the equal temperament val for septimal 19et, and E31 = {{val| 31 49 72 87 }} is the val for septimal 31et. Then writing intervals multiplicatively, we have

<math>(E19\wedge E31)(2,3) = E19(2)E31(3) - E19(3)E31(2) = 19*49 - 31*30 = 1</math>

We may continue in this way to consider (2,5), (2,7), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields <<1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank_two_temperament|rank two temperament]]s such as [[~~Meantone|~~meantone]], trivals to [[rank_three_temperament|rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19∧E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone.

We may continue in this way to consider (2,5), (2,7), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields <<1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank_two_temperament|rank two temperament]]s such as [[meantone]], trivals to [[rank_three_temperament|rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19∧E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone.

This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [http://en.wikipedia.org/wiki/Greatest_common_divisor GCD] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[Regular_Temperaments|regular temperaments]].

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Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)lb(q)lb(p)) then wedging K = <1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.

More generally, we can reconstitute W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point <1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||(W-Y)+Y||=||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K||=||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[~~Generator_complexity|~~generator complexity]].

More generally, we can reconstitute W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point <1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||(W-Y)+Y||=||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K||=||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[generator complexity]].

In the particular case of the 11-limit in rank three, we have that (W∨2)∧K gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes E ≤ 1/(2√5 q5q7q11), then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.

[[Category:~~math~~]]

[[Category:Math]]

[[Category:~~theory~~]]

[[Category:Theory]]

@@ Line 16: / Line 16: @@
 To find (a JI interpretation of) the '''generator''': Use the [https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm extended Euclidean algorithm] to find a JI ratio g = q_1^a_1 ... q_n^a_n (equivalently, a linear combination g = a_1 q_1 + ... + a_n q_n) such that W(2, g) = a_1 W(2, q_1) + ... a_n W(2,q_n) = d.
-Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[Tenney-Euclidean_temperament_measures#TE_error|TE error]].
+Now choosing an optimal tuning for the temperament is a matter of choosing a way to measure error from JI and minimizing the error with linear algebra (read: a program such as the x31eq temperament finder). For example, the [[TE tuning|TE]] and [[POTE tuning|POTE]] tunings are based on minimizing [[TE error]].
 [In geometric terms, given JI ratios u and v, and wedgie W, the number W(u,v) is the signed area of the parallelogram spanned by (tempered versions of) u and v. The entries of the wedgie give the values of the wedgie on the basis elements of the JI subgroup that the temperament is on. By the alternating property [i.e. W(u, v) = -W(v, u)] and bilinearity [W is linear in each argument separately], specifying the values on basis elements of the JI subgroup is enough to define W as an alternating bilinear form on all of the JI subgroup. This is the determinant of the tempered versions of u and v. The musical interpretation of the parallelogram spanned by u and v is: If you want to consider intervals that are multiples of u apart the same note (for example, if you want an octave-equivalent scale), W(u, v) tells you how many generators it take to get to v.]
@@ Line 44: / Line 44: @@
 == Technical introduction ==
-An alternating [http://en.wikipedia.org/wiki/Multilinear_map multilinear map] which is a multilinear function taking a certain number n of [[monzos|monzos]] as arguments and returning an integer as a value we may call an '''n-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.
+An alternating [http://en.wikipedia.org/wiki/Multilinear_map multilinear map] which is a multilinear function taking a certain number n of [[monzos]] as arguments and returning an integer as a value we may call an '''n-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.
-The simplest kind of n-map is the 1-map, or [[Vals|val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [http://en.wikipedia.org/wiki/Group_homomorphism group homomorphism] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [http://en.wikipedia.org/wiki/Linear_map linear]: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).
+The simplest kind of n-map is the 1-map, or [[val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [http://en.wikipedia.org/wiki/Group_homomorphism group homomorphism] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [http://en.wikipedia.org/wiki/Linear_map linear]: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).
 One use for such things is as "machines" for measuring complexity. If we consider the 1-map which is the val for 11-limit 31et, we find we have &lt;31 49 72 87 107|. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 72 in this system. Now consider a 2-map "meantone(u,v)" which tells us, roughly speaking, how many generator steps it takes to get to v assuming u is being used as a period in septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, in which case we have meantone(2,3)=1, meantone(2,5)=4, meantone(2,7)=10. With 3 as a period and 3/2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=13. Finally, with if we take 5 as a period we find that four 3/2s give 5, so 5^(1/4) or equivalently 3/2 is the basic period. Using 3/2 as a period and 9/8 as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5^(1/4) gives us meantone(5,7)=12. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.
@@ Line 56: / Line 56: @@
 where the sum is taken over S(n,m), the set of all [http://en.wikipedia.org/wiki/Permutation permutations] of the first n+m integers which are an [http://en.wikipedia.org/wiki/%28p,q%29_shuffle (n,m) shuffles], and sgn(t) is the [http://en.wikipedia.org/wiki/Parity_of_a_permutation parity of the permutation] t, which is +1 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.
-If f and g are both vals (1-maps) then this becomes especially easy: f∧g(u,v) = f(u)g(v) - f(v)g(u). Let's consider a specific example. Suppose E19 = &lt;19 30 44 53| is the equal temperament val for septimal 19et, and E31 = &lt;31 49 72 87| is the val for septimal 31et. Then writing intervals multiplicatively, we have
+If f and g are both vals (1-maps) then this becomes especially easy: f∧g(u,v) = f(u)g(v) - f(v)g(u). Let's consider a specific example. Suppose E19 = {{val| 19 30 44 53 }} is the equal temperament val for septimal 19et, and E31 = {{val| 31 49 72 87 }} is the val for septimal 31et. Then writing intervals multiplicatively, we have
 <math>(E19\wedge E31)(2,3) = E19(2)E31(3) - E19(3)E31(2) = 19*49 - 31*30 = 1</math>
-We may continue in this way to consider (2,5), (2,7), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields &lt;&lt;1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank_two_temperament|rank two temperament]]s such as [[Meantone|meantone]], trivals to [[rank_three_temperament|rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19∧E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone.
+We may continue in this way to consider (2,5), (2,7), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields &lt;&lt;1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank_two_temperament|rank two temperament]]s such as [[meantone]], trivals to [[rank_three_temperament|rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19∧E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone.
 This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [http://en.wikipedia.org/wiki/Greatest_common_divisor GCD] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[Regular_Temperaments|regular temperaments]].
@@ Line 91: / Line 91: @@
 Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)lb(q)lb(p)) then wedging K = &lt;1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.
-More generally, we can reconstitute W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||(W-Y)+Y||=||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K||=||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[Generator_complexity|generator complexity]].
+More generally, we can reconstitute W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||(W-Y)+Y||=||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K||=||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[generator complexity]].
 In the particular case of the 11-limit in rank three, we have that (W∨2)∧K gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes E ≤ 1/(2√5 q5q7q11), then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.
-[[Category:math]]
+[[Category:Math]]
-[[Category:theory]]
+[[Category:Theory]]