Análisis e interpretación de resultados: Encuesta a docentes

ccording to the Limited Growth Model, the shape of a simple closed form can be explained as resulting from a combination of simultaneously unfolding tendencies: tendencies to rise, fall, expand, and contract. The apparent parts of the shape can be understood, not as successions of unrelated entities, but as consequences of dynamic properties that are set at the beginning and remain relatively invariant throughout. An arch, for example, is not merely a rising motion followed by a closing motion. An Open-Low-High-Close form is not merely a falling motion followed by a rising motion followed by another rising motion, or, alternatively, a trough followed by an arch. The Limited Growth Model interprets an arch as what results when an expansive tendency to rise is countered by a contrary goal-directed tendency to contract back toward the starting point. An Open-Low-High-Close form results when the goal-directed tendency to contract is imposed on an opposition between simultaneously rising and falling tendencies to expand. The Limited Growth Model is abstracted from theories of melodic shape by Kurth, Meyer, Narmour, Gjerdingen, Adams, and Huron. In the stochastic form of the model, the expansive tendencies are represented by parameterized quasi-random walks. Cyclical motion, also generated stochastically, can be added to an Open-Low-High-Close form to create a more complex shape.

In realistic stochastic models, state variables would be drawn from tables containing empirically observed values. For purposes of illustration, however, the demonstration models will use computed variables.

Fixed parameters used by the Limited Growth generator are shown in Tables C1.1 and C1.2. All of the pre-set parameters are arbitrary. Two groups of parameters are needed, one for each of the components of the Limited Growth Model.

Parameter Value Range

Table C1.1 Fixed parameters for the Limited Growth generator, Primary Curve.

A

Parameter Value Range Contraction Exponent 0.200 > 0

Reversal Probability 0.100 0 to 1 Initial Direction Up Up or Down

Bias 1.700 > 0

Initial Walk 0.000 Fixed

Table C1.2 Fixed parameters for the Limited Growth generator, Contrary Curve.

The generator for the Limited Growth Model creates tables for both the Primary and Contrary curves, and each row of each table contains the following variables:

 Item Number

 Time

 Contraction Factor

 Normal Variates

 Absolute Value of the Normal Variate

 Reversal Variable

 Direction

 Step

 Walk

The calculations are performed as follows:

 The Contraction Factor is calculated by counting down the time backward from 1.000 to 0.000. Experiments were made using sequences of thirty-five points in time, a number chosen because this study supplements an analysis of Debussy’s composition for solo flute, Syrinx, which happens to be thirty-five measures long. In these experiments, the tendency to contract imposed by a linear countdown was too strong, so the countdown was raised to a fractional power, called the Contraction Exponent.

 The individual steps are simulated using a modified normal random variable.

To approximate a normally distributed random variable, one takes the sum of twelve uniformly distributed random variables, ranging from zero to one. In the long run, the standard deviation of the sum will tend toward one. After subtracting six from the sum, the mean will tend toward zero.

 We are going to skew the random variable that has just been generated; so, we take its absolute value. What we do to the absolute value depends on whether we want the current Step to ascend or descend.

 We arbitrarily decide that the direction of the first Step will be up. The direction of succeeding Steps depends on another uniformly distributed random variable, the Reversal Variable. The interpretation of the Reversal Variable depends on a fixed parameter, called the Reversal Probability. If the Reversal Variable is less than the Reversal Probability, the polarity of the direction is changed to the opposite of its last state; otherwise, the direction stays the same.

 Furthermore, we scale the size of the Step, referring to one more fixed parameter, called the Bias. If the direction is up, we multiply the absolute value of the normal variate by the Bias; otherwise, we multiply it by minus one and divide by the Bias. A Bias of one is neutral. Values of the Bias greater than one will tend to make the Walk ascend, and vice versa.

 The first value of the curve (the Walk) is arbitrarily given a value of zero.

Subsequent values are calculated by adding the most recently calculated Step to the last previously calculated value of the Walk.

 This sum is multiplied by the Contraction Factor. The last value of the Contraction Factor will be zero; so, the last value of the Walk will also be zero, returning to its initial value.

The Resultant Curve is calculated by subtracting values of the Contrary Curve from corresponding values of the Primary Curve.

More research is needed to evaluate the full range of effects of the parameters. All of the parameters are assumed to be positive. Values of the Bias greater than one tend to produce arches;

lower values tend to produce troughs. The smaller the Contraction Exponent, the later the peak of an arch (or the bottom of a trough). All other things being equal, the lower the Reversal Probability, the larger the range.

The accompanying illustrations (Fig. C1.1-C1.5) show the variety of shapes that can be generated using a fixed set of parameters. The parameters are chosen to generate late-peaking S-curves from combinations of late-peaking primary arches and early-peaking contrary arches. In every case, the Initial Direction is up; and the Bias Coefficient is 1.70. The primary curves have a small Contraction Exponent of 0.060 and a relatively large Reversal Probability of 0.500. The

contrary curves have a somewhat larger Contraction Exponent of 0.200 and a small Reversal Probability of 0.100. Each chart shows the first, second, and third quartile of the generated values, measure-by-measure, from a sample of twelve generated curves of each type.

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The following is, loosely speaking, a stochastic interpretation of Simple Harmonic Motion. In this model, random motion is tailored to resemble the action of a body in motion under the constraint of a restoring force that tends to bring the body back toward a central value. The motion cycles about the central value indefinitely.

The Perpetual Cycle generator uses its own collection of fixed parameters, shown in Table C2. Again, all of the pre-set parameters are arbitrary.

Parameter Value Range

Table C2. Fixed parameters for the Perpetual Cycle generator.

Like the Limited Growth Model, the stochastic cycle generator creates a table, but the table is somewhat more complex. Each row of the table contains the following variables:

 Item Number

 Standard Deviation

 Normal Variate

 Absolute Value of the Normal Variate

 Reversal Criterion

 Reversal Variable

 Direction

 Bias Exponent

 Bias

 Step

 Relative Amplitude

 Interpolation Factor

 Walk

We begin, as before, by generating random variables that approximate a skewed normal distribution. As an approximation to the behavior of velocity in Simple Harmonic Motion, however, we want to make the Walk move more quickly through the middle of a cycle than it does near the reversal points. To do this, we will control the Standard Deviation of the Normal Variate by obtaining feedback from the generated Walk. We will also control the probability that the Walk will reverse direction, in a similar manner. This refinement helps to maintain persistent motion through the middle of a cycle.

 The rows of the table are numbered consecutively. Each row represents a Step of the Walk. Each Step depends on all of the variables discussed below.

 The Walk is initialized to zero.

 The Relative Amplitude is the value of the Walk, divided by a pre-determined Amplitude. In the present examples, we will use an Amplitude of 6.00.

 The Interpolation Factor is the absolute value of the Relative Amplitude, truncated to a maximum value of one.

 The Standard Deviation is a linear interpolation between pre-set minimum and maximum values. To calculate the interpolation, we first take the product of the

Interpolation Factor and the difference between the minimum and maximum allowed values for the Standard Deviation. We then subtract that product from the maximum. When the Walk is at zero, nothing is subtracted from the maximum value of the Standard Deviation. The wider the deviation of the Walk from zero, the larger the Relative Amplitude (up to a maximum of one). The larger the Relative Amplitude, the smaller the Standard Deviation. When the Relative Amplitude is at a maximum of one, nothing is left of the Standard Deviation but the minimum allowed value. The interpolated Standard Deviation, therefore, will be high near the middle of a cycle and low near the extremes. In our examples, we will use a minimum Standard Deviation of zero and a maximum of one.

 Calculation of the Normal Variate begins with the second row, since it depends on a previously calculated value of the Standard Deviation. Values of the Normal Variate approximate a normal distribution. Each estimate is the sum of twelve uniformly distributed random variables in the range from zero to one.

This sum will have a standard deviation of one. We subtract six from the total to center the range and multiply the difference by the Standard Deviation calculated for the previous row.

 The Absolute Value of the Normal Variate will be needed later. We will divide the Normal Variate into two parts, which can be treated differently. The Absolute Value will serve for the calculation of either part.

 The Reversal Criterion governs the probability that the Walk will reverse direction. The Reversal Criterion is intended to be low near the middle of a cycle, but high near the extremes. The Reversal Criterion is a linear interpolation between a minimum and maximum value of the Reversal Probability, which we will set to zero and one, respectively. The calculation of the Reversal Criterion is similar to that of the Standard Deviation, except that they vary in opposite directions. The product of the Interpolation Factor times the difference between the maximum and minimum probabilities is added to the minimum probability.

 The Reversal Variable, as before in the Limited Growth Model, is a uniformly distributed random number between zero and one. This variable is first needed in the second row.

 The Direction is calculated from the Reversal Variable and the previous value of the Direction, as before. A value of one signifies upward motion, and a value of minus one signifies downward motion. If the Reversal Variable is less than the Reversal Criterion, the Direction is changed to the opposite of its previous value, otherwise the Direction stays the same.

The restoring force of Simple Harmonic Motion is simulated by the Bias. This is one of the most important features of the model. It is used to skew the steps of the Walk, to bring the Walk back toward the zero line if it strays too far afield. We will calculate the Bias from a Bias Base and a Bias Exponent. In the present examples, we will use a Bias Base of 16.00.

 The Bias Exponent is simply the negative of the Relative Amplitude.

 The Bias, after an initial value of 1.00, is equal to the Bias Base raised to the power of the previous Bias Exponent. The Bias is used to calculate the Step.

 The Step is initialized to zero. From that point on, if the Direction is up, the Step is equal to the Absolute Value of the Normal Variate multiplied by the Bias. If the Direction is down, the Step is equal to the negative of the Absolute Value of the Normal Variate divided by the Bias. Both upward and downward motion can take place with any value of the Bias. If the Bias is equal to one, upward and downward motion will balance each other, on average. However, if the Bias is greater than one, upward motion will be augmented and downward motion will be diminished. If the Bias is less than one, the opposite will occur.

Limits are imposed on the Step size. It is truncated to fall between plus or minus half the Amplitude.

 The Walk is simply the sum of the Step and the previous value of the Walk, as it was in the Limited Growth Model.

The next set of examples (Fig. C2.1-C2.5) shows stochastic cycles added to curves resulting from the stochastic Limited Growth Model. The cycles are arbitrarily multiplied by three before they are added. The charts are stacked area graphs, in which cycles are built on top of Limited Growth curves.

Fig. C2.1

Fig. C2.2

Fig. C2.3

Fig. C2.4

Fig. C2.5

The results shown here are not necessarily typical. As often happens with stochastic processes, there is considerable variation between examples.