Evaluating Instruments with the Tuning Table and Tuning Graph - edwardkort/WWIDesigner GitHub Wiki
Introduction
This tutorial illustrates how you can use the Calculate tuning and Graph tuning tools to evaluate the tuning of an instrument. It assumes you have followed the instructions in Working With the Whistle Study Model or Working With the Flute Study Model to measure the tuning of your whistle or flute, and calibrate the model for your instrument. As our example, we use the sample PVC high-D whistle included with the WIDesigner release: the instrument definition is at WhistleStudy\instruments\SamplePVC-Whistle.xml, the tuning is at WhistleStudy\tunings\SamplePVC-tuning.xml.
Release 1.0.7 of WIDesigner introduces an additional tool, Graph note spectrum, which you can read about in Reading the Note Spectrum Graph. The next release will include the Supplementary Information Table tool.
The tuning table and tuning graph are also available with the NAF and reed study models, although they display less information.
The Tuning Table
For the sample PVC whistle, with the Blowing Level option set to 4, the Calculate tuning tool produces the table below.
About the Tuning Table
In this table, the whistle model and flute model display three sets of information as listed below. For other study models, the table does not include predictions for minimum or maximum frequency.
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For the nominal playing frequency: the expected or target value from the tuning file, the predicted value from the model, and the difference between the two, in cents.
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For the minimum frequency: the actual value as given in the tuning file, the predicted value from the model, and the difference between the two, in cents.
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For the maximum frequency: the actual value as given in the tuning file, the predicted value from the model, and the difference between the two, in cents.
The bottom two rows in the table summarize the deviation columns, giving the average difference, and the root-mean-square difference.
What the Table Tells Us
When we have measured fmax and fmin values, the tuning table tells us how well the whistle model or flute model are predicting these numbers.
For the sample PVC whistle, we see that most of the fmax deviations are small, with the exception of the high A and D. This indicates that the Window Height in the instrument file is set appropriately, and except for those two notes, the whistle model can accurately predict fmax.
In the upper octave, the fmin deviations are generally flat at the bottom and top of the range, and sharp for the lower B, C and C#. While the average error is low, we could consider reducing beta to improve the prediction in the upper octave since fmin is not as well defined in the first octave, and second-octave notes are generally played closer to fmax than fmin, so fmin has less influence on the nominal playing frequency than fmax.
The tuning table also gives us a quantitative indication of how well the whistle plays in tune, and which notes might be troublesome.
Looking at the deviation between the target scale, in the second column, and the predicted playing frequency, in the third column, we see that the low C-natural and C# are both likely to be played sharp. Other than the middle and high D, the entire second octave is likely to be played flat. The flat second octave can be compensated by blowing harder, which we can model by increasing the blowing level. However, this isn't enough to help the high C-natural, and is likely to make the high D come out even sharper than it is at the lower blowing level.
The Tuning Graph
For the sample PVC whistle, The Graph tuning tool produces the table below.
About the Tuning Graph
The x-dimension of the tuning graph is frequency; the y-dimension is reactance, or the imaginary part of the impedance, as calculated by the whistle model or flute model. (To be precise, the y-dimension is the ratio of reactance to resistance, X/R.) The graph has one solid line for each note in the tuning file. To help you keep track of which note is which, every fourth line is coloured blue. This colouring works well with a typical whistle or keyless flute range, two diatonic octaves plus a flattened seventh (C-natural): the blue lines represent the Ds (tonic) and As (dominant) of a D whistle or flute. This line colouring scheme may not work as well for other forms of tuning.
The diamond dots at the top and bottom of each line represent fmax and fmin, as predicted by the model. (The graph does not show the measured fmax or fmin.) The predicted fmax is always at a reactance of zero.
The filled dots--green circles, yellow triangles, and red triangles--show the calculated reactance at the target playing frequency. These dots are yellow when the target is close to fmax or fmin, red when the target is beyond fmax or fmin, and green when the target is comfortably between fmax and fmin. A green dot predicts that it is possible to play the note at the target frequency, if you have good breath control. However, that doesn't mean it will be easy to play the target frequency in the middle of a tune.
The open blue circles mark the frequencies that a hypothetical player would comfortably produce at the specified blowing level: playing closer to fmax at the lowest note, and closer to fmin at the highest note, with the air speed increasing uniformly from the lowest note to the highest. If you change the blowing level, only the position of these blue circles will change, the rest of the graph remains the same. The assumption is that a whistle with a uniform blowing pattern will be easier to play in tune. Where the open blue circle is below the filled (target) dot, this blowing pattern will play the note flat. Where the open blue circle is above the filled dot, the blowing pattern will play the note sharp.
What the Graph Tells Us
The tuning graphs gives us a visual indication of how well the whistle or flute plays in tune, and where the trouble spots might be.
For the sample PVC whistle, other than C-natural and C# the first octave should be easy to play in tune. For C-natural and C#, the filled circle (target) is well below the open blue circle (prediction), suggesting they will tend to play sharp unless you drop your breath pressure when you play them. The red triangle on low D indicates that the target frequency is higher than the predicted fmax. However, the difference looks small, and is likely within the prediction error we can expect from the model. (In fact, the target frequency is less than the measured fmax for the low D on this whistle.)
At this blowing level, the second octave would tend to play flat: the filled circles are below the open blue circles. For the most part, increasing breath pressure in the second octave should be enough to compensate, but the transition from C/C# to D to E could be a challenge. High C-natural will be very flat without a dramatic increase in breath pressure. High D is likely to be quite sharp, especially relative to the other notes in the second octave. It might be possible to blow high D in tune with a significant decrease in breath pressure.