Perfect balance: Difference between revisions

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It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition.

Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. Briefly, this is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the ~~squared balance~~ <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the ~~squared~~ balance is ~~minimized~~. It's not clear from sources whether the minimum (~~guaranteed to be~~ global due to convexity) ~~is perfectly balanced~~, but it seems to be in practice.

Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. Briefly, this is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved. It's not clear from sources whether the minimum always exists and has a cost of 0 (although it is always a global minimum due to convexity), but it seems to be in practice.

For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file:

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 It is easy to construct perfectly balanced scales that are not a subset of any EDO, by superimposing two scales within EDOs transposed by an irrational amount. There also exists a continuum of perfectly balanced scales that have no such decomposition.
-Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. Briefly, this is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the squared balance <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the squared balance is minimized. It's not clear from sources whether the minimum (guaranteed to be global due to convexity) is perfectly balanced, but it seems to be in practice.
+Milne at al. showed that an efficient convex optimization procedure exists that, given an arbitrary scale, computes the closest perfectly balanced scale according to a simple squared-difference metric. Briefly, this is accomplished by the following steps: place the scale on a circle in 2D space about the origin, translate the points by a vector (''u'', ''v''), project the points back onto the original circle by dividing by the norm, then compute the cost function <math>\left(\sum \mathbf{x}\right)^2 + \left(\sum \mathbf{y}\right)^2</math> where <math>\mathbf{x}</math> and <math>\mathbf{y}</math> are vectors of the ''x''- and ''y''-coordinates. Use any standard unconstrained optimization procedure to find ''u'' and ''v'' so that the cost function is minimized. It can be seen that the cost is 0 iff perfect balance is achieved. It's not clear from sources whether the minimum always exists and has a cost of 0 (although it is always a global minimum due to convexity), but it seems to be in practice.
 For example, a perfectly balanced approximation to Ptolemy's diatonic scale [1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8] displaces by the following cent values: [0, +8.61, +20.00, +23.60, +20.03, +9.97, +0.72]. The resulting scale is given by the following [[Scala]] file: