Reading the Note Spectrum Graph - edwardkort/WWIDesigner GitHub Wiki

Introduction

Release 1.0.7 of WIDesigner introduces a new tool for evaluating instrument performance: Graph note spectrum. This tool plots impedance characteristics for a specific fingering over a range of frequencies around a specific note. You can use this tool with any study model.

As our example, we use the "middle D" fingering, OXXXXX, on 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.

The Note Spectrum

To display a note spectrum:

• Select an instrument in the Study panel.
• In the editor pane, display the editor tab of a corresponding tuning file.
• Select a specific note in the fingering list on the tuning file dialog.
• Select Tool-->Graph note tuning.

For the middle D of the sample PVC whistle, the Graph note spectrum tool produces the graph below.

About the Note Spectrum

The note spectrum graph displays two quantities as a function of frequency:

1. The calculated impedance is plotted as a black line, more specifically ratio of the imaginary part to the real part: reactance divided by resistance. It is impractical to graph reactance alone, because between resonances, the reactance tends to infinity. The imaginary/real ratio, however, remains finite, so it is a much more practical quantity to graph. Flute and whistle resonances occur where the reactance is zero; at these points the imaginary/real ratio is zero and increasing. For frequencies at which the imaginary part is infinite, the imaginary/real ratio is zero and decreasing.

2. The estimated loop gain for oscillation is plotted as a series of circles: green circles if the gain is 1 or more, red circles if the gain is less than 1. (Loop gain is available only for the whistle and flute study model. For other study models, the loop gain always appears as 1 on the graph.)

When the tuning file gives actual playing frequencies, the graph also marks these frequencies with grey diamonds. Harmonics of the note are marked with small grey triangles.

In general, resonant frequencies - frequencies at which the instrument will play a note - occur where the reactance is zero. These will be places where the Impedance Ratio graph goes through zero with a positive slope. For the NAF and reed study models, these are the playing frequencies of the note. The whistle study model predicts the instrument will play at a range of frequencies where the reactance is less than or equal to zero, and the loop gain is greater than 1. These are necessary conditions, but they aren't always sufficient.

What the Graph Tells Us

The sample graph above shows several potential resonances, where the reactance is zero and the gain is greater than 1, the first four around 900 Hz, 1200 Hz, 1950 Hz, and 2450 Hz. Our target for this note is D6, 1174.7 Hz; the graph shows a robust resonance around this frequency, and the instrument actually can play a good D6.

However, the resonance around 900 Hz suggests that if we don't blow strongly enough, the whistle will play a lower, un-intended note around this frequency. That is what actually happens: if I blow gently enough, the whistle plays a note around 870 to 900 Hz. All the whistles I've tried exhibit this behaviour when blown gently with OXXXXX fingering: a pitch lower than the expected note.

In contrast, the predicted resonance at 1950 Hz cannot be played in practice. This whistle jumps directly from around 1200 Hz to around 2400 Hz. A spectrum analyzer does show a hint of the 1950 Hz resonance, but I can't actually play the whistle at that frequency.

The whistle does play at the fourth resonance. However, the playing range at this resonance is a bit too high for D7. The sample tuning file uses the fingering OXXOOO for D7, which may still come out a bit sharp, but not as sharp as OXXXXX.

Tonehole Lattice Cutoff Frequency

In Fundamentals of Musical Acoustics (1976), Arthur Benade defines the concept of the tonehole lattice cutoff frequency, the frequency above which acoustic waves propagate significantly past the first open tone hole, and gives an estimation formula for it. Some sources suggest the cutoff frequency has a significant effect on the timbre of an instrument. However, formulas for the cutoff frequency generally assume that the line of toneholes is infinitely long; it is not easy to identify a precise cutoff frequency for a finite instrument. Fortunately, the note spectrum graph can give a general idea of where the cutoff falls.

The graph below shows the spectrum for a fingering with few open toneholes, OXXXXX. This is same graph as shown above, extended to higher frequencies. You can generate it by changing the Max Note Spectrum frequency on the WIDesigner Options dialogue to 6.5 from its default of 3.17.

In this spectrum, the extremes in the impedance ratio and gain, particularly the negative extremes of the impedance ratio, get progressively smaller, in a gradual and regular fashion. There is no evidence of a dramatic cutoff.

In contrast, consider the spectrum for a fingering with many open toneholes, XOOOOO, as shown below. There is a sudden drop in the magnitude of the extremes, particularly the negative extremes of the impedance ratio, between 2000 Hz and 2600 Hz. From this spectrum, we infer that the cutoff frequency is somewhere in the neighbourhood of 2600 Hz.