Wedgie/Archived version: Difference between revisions
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For the ''p''<sub>''n''</sub>-prime limit, the entries of W are conventionally listed in the order | For the ''p''<sub>''n''</sub>-prime limit, the entries of W are conventionally listed in the order | ||
<math>\ | <math>\bival{\operatorname{W}\left(\mathbf{2}, \mathbf{3}\right) \ \ldots \ \operatorname{W}\left(\mathbf{2}, \mathbf{p}_n\right) & \operatorname{W}\left(\mathbf{3}, \mathbf{5}\right) \ \ldots \ \operatorname{W}\left(\mathbf{3}, \mathbf{p}_n\right) \ldots \operatorname{W}\left(\mathbf{p}_{n-2}, \mathbf{p}_{n-1}\right) & \operatorname{W}\left(\mathbf{p}_{n-2}, \mathbf{p}_n\right) & \operatorname{W}\left(\mathbf{p}_{n-1}, \mathbf{p}_n\right)}.</math> | ||
For example, a 5-limit wedgie is of the form | For example, a 5-limit wedgie is of the form | ||
<math>\ | <math>\bival{\operatorname{W}\left(\mathbf{2}, \mathbf{3}\right) \ \operatorname{W}\left(\mathbf{2},\mathbf{5}\right) \ \operatorname{W}\left(\mathbf{3}, \mathbf{5}\right)},</math> | ||
and a 7-limit wedgie is of the form | and a 7-limit wedgie is of the form | ||
<math>\ | <math>\bival{\operatorname{W}\left(\mathbf{2}, \mathbf{3}\right) & \operatorname{W}\left(\mathbf{2},\mathbf{5}\right) & \operatorname{W}\left(\mathbf{2}, \mathbf{7}\right) & \operatorname{W}\left(\mathbf{3}, \mathbf{5}\right) & \operatorname{W}\left(\mathbf{3}, \mathbf{7}\right) & \operatorname{W}\left(\mathbf{5}, \mathbf{7}\right)}.</math> | ||
More generally, if one takes ''r'' independent [[vals]] V<sub>1</sub>, …, V<sub>''r''</sub> in a rank-''n'' [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub>, then the wedgie for the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub> is defined by: | More generally, if one takes ''r'' independent [[vals]] V<sub>1</sub>, …, V<sub>''r''</sub> in a rank-''n'' [[JI subgroup]] ''q''<sub>1</sub>.[…].''q''<sub>''n''</sub>, then the wedgie for the rank-''r'' temperament V<sub>1</sub>&…&V<sub>''r''</sub> is defined by: | ||
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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 [[Wikipedia: Group homomorphism|group homomorphism]] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [[Wikipedia: 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 {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|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 [[Wikipedia: Group homomorphism|group homomorphism]] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [[Wikipedia: 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 {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|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 <math>\ | 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 <math>\tval{31 & 49 & 72 & 87 & 107}</math>. 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 {{nowrap|"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 {{nowrap|meantone(2, 3) {{=}} 1}}, {{nowrap|meantone(2, 5) {{=}} 4}}, {{nowrap|meantone(2, 7) {{=}} 10}}. With 3 as a period and 3/2 as a generator, we get {{nowrap|meantone(3, 5) {{=}} 4}} and {{nowrap|meantone(3, 7) {{=}} 13}}. Finally, with if we take 5 as a period we find that four 3/2s give 5, so 5<sup>{{frac|1|4}}</sup> (or equivalently, {{frac|3|2}}) is the basic period. Using {{frac|3|2}} as a period and {{frac|9|8}} as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5<sup>{{frac|1|4}}</sup> gives us {{nowrap|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. | ||
Given an ''n''-map ''f'' and an ''m''-map ''g'' we may define a new ({{nowrap|''n'' + ''m''}})-map, the [[Wikipedia: Exterior algebra|wedge product]] of ''f'' and ''g'', written {{nowrap|''f'' ∧ ''g''}}, as follows: | Given an ''n''-map ''f'' and an ''m''-map ''g'' we may define a new ({{nowrap|''n'' + ''m''}})-map, the [[Wikipedia: Exterior algebra|wedge product]] of ''f'' and ''g'', written {{nowrap|''f'' ∧ ''g''}}, as follows: | ||
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<math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math> | <math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 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 <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 <math>\twedgie{1 & 4 & 10 & 4 & 13 & 12}</math>. 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]]s such as [[meantone]], trivals to [[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, {{nowrap|E<sub>19</sub> ∧ E<sub>31</sub>}} is the same object we were calling {{nowrap|"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 [[Wikipedia: 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 temperament]]s. | This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[Wikipedia: 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 temperament]]s. | ||
These reduced ''n''-vals, and particularly reduced bivals, are called ''wedgies'', and the fact that they are reduced both makes the name unique and tells us that wedgies are [[Wikipedia: Projective space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, <math>E_{24} = \ | These reduced ''n''-vals, and particularly reduced bivals, are called ''wedgies'', and the fact that they are reduced both makes the name unique and tells us that wedgies are [[Wikipedia: Projective space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, <math>E_{24} = \tval{24 & 38 & 56}</math> is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called ''contorted''. Wedgies do not name or signify contorted temperaments. | ||
===== Computing the previous example in Maple ===== | ===== Computing the previous example in Maple ===== | ||
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== Conditions on being a wedgie == | == Conditions on being a wedgie == | ||
If we take any three integers <math>\ | If we take any three integers <math>\twedgie{a & b & c}</math> such that {{nowrap|GCD(''a'', ''b'', ''c'') {{=}} 1}} and {{nowrap|''a'' ≥ 1}} the result is always a wedgie, the wedgie tempering out the [[The_dual|dual]] [[monzos|monzo]] <math>\tmonzo{c & -b & a}</math>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined. | ||
However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that {{nowrap|W ∧ W {{=}} 0}}, where the "0" means the multival of rank 2''r'' obtained by wedging W with W. For prime limits 7 and 11 this condition suffices for rank two, but in general we need to check, for every prime {{nowrap|''q'' ≤ ''p''}} and every basis val ''v'' sending ''q'' to 1 and everything else to 0, that {{nowrap|(W ∨ ''q'') ∧ W {{=}} 0}} and {{nowrap|(W ∧ ''v'')º ∧ Wº {{=}} 0}}, where "∨" denotes the [[interior product]]. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the [[Wikipedia:Plücker embedding|Plücker relations]]. Note that the Plücker relations must be satisfied, since for a rank-''r'' multival, {{nowrap|W ∨ ''q''}} is a rank-({{nowrap|''r'' − 1}}) multival corresponding to tempering out all the commas of W, as well as ''q''. | However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that {{nowrap|W ∧ W {{=}} 0}}, where the "0" means the multival of rank 2''r'' obtained by wedging W with W. For prime limits 7 and 11 this condition suffices for rank two, but in general we need to check, for every prime {{nowrap|''q'' ≤ ''p''}} and every basis val ''v'' sending ''q'' to 1 and everything else to 0, that {{nowrap|(W ∨ ''q'') ∧ W {{=}} 0}} and {{nowrap|(W ∧ ''v'')º ∧ Wº {{=}} 0}}, where "∨" denotes the [[interior product]]. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the [[Wikipedia:Plücker embedding|Plücker relations]]. Note that the Plücker relations must be satisfied, since for a rank-''r'' multival, {{nowrap|W ∨ ''q''}} is a rank-({{nowrap|''r'' − 1}}) multival corresponding to tempering out all the commas of W, as well as ''q''. | ||
In the 7-limit case, if we wedge a prospective rank two multival <math>W = \ | In the 7-limit case, if we wedge a prospective rank two multival <math>W = \twedgie{a & b & c & d & e & f}</math> with itself, we obtain <math>W \wedge W = 2\left(af - be + cd\right)</math>. The quantity {{nowrap|''af'' − ''be'' + ''cd''}} is the [[Wikipedia:Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space '''P⁵''' in which wedgies live, the wedgie lies on a (four-dimensional) [[Wikipedia:Hypersurface|hypersurface]], known as the [[Mathematical theory of regular temperaments#Geometry_of_regular_temperaments|Grassmannian]] {{nowrap|'''Gr'''(2, 4)}}. For an 11-limit rank-two wedgie <math>W = \twedgie{w_1 & w_2 & w_3 & w_4 & w_5 & w_6 & w_7 & w_8 & w_9 & w_{10}}</math> we have that <math>W \wedge W = 2\quadtval{w_1 w_8 - w_2 w_6 + w_3 w_5 & w_1 w_9 - w_2 w_7 + w_4 w_5 & w_1 w_{10} - w_3 w_7 + w_4 w_6 & w_2 w_{10} - w_3 w_9 + w_4 w_8 & w_5 w_{10} - w_6 w_9 + w_7 w_8}</math> is zero. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that '''Gr'''(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[Wikipedia: Algebraic variety|algebraic variety]] in nine-dimensional projective space '''P⁹'''. Wedgies correspond to rational points on this variety. For 11-limit rank three temperaments, we have {{nowrap|''w''<sub>6</sub>''w''<sub>1</sub> − ''w''<sub>5</sub>''w''<sub>2</sub> + ''w''<sub>4</sub>''w''<sub>3</sub>}} = {{nowrap|''w''<sub>9</sub>''w''<sub>1</sub> − ''w''<sub>8</sub>''w''<sub>2</sub> + ''w''<sub>7</sub>''w''<sub>3</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>1</sub> − ''w''<sub>8</sub>''w''<sub>4</sub> + ''w''<sub>7</sub>''w''<sub>5</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>2</sub> − ''w''<sub>9</sub>''w''<sub>4</sub> + ''w''<sub>7</sub>''w''<sub>6</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>3</sub> − ''w''<sub>9</sub>''w''<sub>5</sub> + ''w''<sub>8</sub>''w''<sub>6</sub> {{=}} 0}}; again, this leads to a six-dimensional variety, this time {{nowrap|'''Gr'''(3, 5)}}. | ||
== Constrained wedgies == | == Constrained wedgies == | ||
Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank-w wedgie <math>W = \ | Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank-w wedgie <math>W = \bitval{a & b & c & d & e & f}</math>. | ||
By definition, <math>E = \left\|J \wedge Z\right\|</math>, where Z is the weighted version of W; if ''q''<sub>3</sub>, ''q''<sub>5</sub>, and ''q''<sub>7</sub> are the logarithms base two of 3, 5, and 7, then <math>Z = \ | By definition, <math>E = \left\|J \wedge Z\right\|</math>, where Z is the weighted version of W; if ''q''<sub>3</sub>, ''q''<sub>5</sub>, and ''q''<sub>7</sub> are the logarithms base two of 3, 5, and 7, then <math>Z = \bival{\frac{a}{q_3} & \frac{b}{q_5} & \frac{c}{q_7} & \frac{d}{q_3 q5} & \frac{e}{q_3 q7} & \frac{f}{q_5 q7}}</math>. We now have | ||
<math>\left(\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3}\right)^2+\left(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3}\right)^2+\left(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5}\right)^2+\left(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5}\right)^2 = 4 E^2</math> | <math>\left(\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3}\right)^2+\left(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3}\right)^2+\left(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5}\right)^2+\left(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5}\right)^2 = 4 E^2</math> | ||
From this we can conclude that ''d'', ''e'', and ''f'' satisfy <math>\left|d - q_3b + q_5a\right| \leqslant 2Eq_3q_5</math>, <math>\left|e - q_3c + q_7a\right| \leqslant 2Eq_3q_7</math>, and <math>\left|f - q_5c + q_7d\right| \leqslant 2Eq_5q_7</math>. This has an interesting interpretation: since <math>\ | From this we can conclude that ''d'', ''e'', and ''f'' satisfy <math>\left|d - q_3b + q_5a\right| \leqslant 2Eq_3q_5</math>, <math>\left|e - q_3c + q_7a\right| \leqslant 2Eq_3q_7</math>, and <math>\left|f - q_5c + q_7d\right| \leqslant 2Eq_5q_7</math>. This has an interesting interpretation: since <math>\tval{1 & q_3 & q_5 & q_7} \wedge \tval{0 & a & b & c}</math> <math>= \bitval{a & b & c & q_3b - q_5a & q_3c - q_7a & q_5c - q_7b}</math>, if {{nowrap|E ≤ {{frac|1|4''q''<sub>5</sub>''q''<sub>7</sub>}}}}, then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <math>\tval{1 & q_3 & q_5 & q_7}</math> and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples <math>\bitval{a & b & c & \ldots}</math> (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <math>\tmonzo{1 & q_3 & q_5 & q_7}</math>, and rounding, then checking if the GCD is one and the Pfaffian is zero (i.e. {{nowrap|''af'' − ''be'' + ''cd'' {{=}} 0}}). Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound. | ||
If {{nowrap|C {{=}} {{!!}}W{{!!}}}} is the TE complexity, then the formula for the [[Tenney-Euclidean_metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: {{nowrap|B {{=}} CE}}. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since {{nowrap|E ≤ {{frac|1|4''q''<sub>5</sub>''q''<sub>7</sub>}}}}, {{nowrap|B ≤ {{frac|20|4''q''<sub>5</sub>''q''<sub>7</sub>}} {{=}} 0.767}}. This badness figure is easily met. While simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06. | If {{nowrap|C {{=}} {{!!}}W{{!!}}}} is the TE complexity, then the formula for the [[Tenney-Euclidean_metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: {{nowrap|B {{=}} CE}}. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since {{nowrap|E ≤ {{frac|1|4''q''<sub>5</sub>''q''<sub>7</sub>}}}}, {{nowrap|B ≤ {{frac|20|4''q''<sub>5</sub>''q''<sub>7</sub>}} {{=}} 0.767}}. This badness figure is easily met. While simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06. | ||
== Reconstituting wedgies in general == | == Reconstituting wedgies in general == | ||
Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if <math>E \leqslant \frac{1}{\binom{n}{3}\log_2\left(q\right)\log_2\left(p\right)}</math> then wedging <math>K = \ | Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if <math>E \leqslant \frac{1}{\binom{n}{3}\log_2\left(q\right)\log_2\left(p\right)}</math> then wedging <math>K = \tval{1 & \log_2\left(3\right) & \log_2\left(5\right) & \ldots & \log_2\left(p\right)}</math> with the val consisting of 0 followed by the first {{nowrap|''n'' − 1}} coefficients of the wedgie and rounding will give the wedgie, where ''p'' and ''q'' are the largest and second largest primes in the prime limit. | ||
More generally, we can reconstitute W by rounding {{nowrap|Y {{=}} (W ∨ 2) ∧ K}} to the nearest integer coefficients, where K is the JI point <math>\tval{1 & log_2\left(3\right) & \log_2\left(5\right) & \ldots & \log_2\left(p\right)}</math> in unweighted coordinates. Then we have <math>\left\|\left(W - Y\right) + Y\right\| \leqslant \left\|W-Y\right\| + \left\|Y\right\|</math> by the triangle inequality, and since {{nowrap|{{!!}}W − Y{{!!}}}} is bounded by the fact that W has been obtained by rounding, complexity, which is {{nowrap|{{!!}}(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 {{nowrap|{{!!}}((W − Y) + Y) ∧ K{{!!}} {{=}} {{!!}}(W − Y) ∧ K{{!!}}}}, hence relative error is also bounded by the fact that {{nowrap|{{!!}}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]]. | More generally, we can reconstitute W by rounding {{nowrap|Y {{=}} (W ∨ 2) ∧ K}} to the nearest integer coefficients, where K is the JI point <math>\tval{1 & log_2\left(3\right) & \log_2\left(5\right) & \ldots & \log_2\left(p\right)}</math> in unweighted coordinates. Then we have <math>\left\|\left(W - Y\right) + Y\right\| \leqslant \left\|W-Y\right\| + \left\|Y\right\|</math> by the triangle inequality, and since {{nowrap|{{!!}}W − Y{{!!}}}} is bounded by the fact that W has been obtained by rounding, complexity, which is {{nowrap|{{!!}}(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 {{nowrap|{{!!}}((W − Y) + Y) ∧ K{{!!}} {{=}} {{!!}}(W − Y) ∧ K{{!!}}}}, hence relative error is also bounded by the fact that {{nowrap|{{!!}}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]]. | ||