Whistlesynthesis

Three-chime whistle chords

Within the Yahoo Steam Whistle Group, on 2007-04-21 Peter Ommundsen published a list of chords for 36 samples of 3-chime whistles blown on steam. Each whistle was denoted by its maker's name and (inch) nominal diameter, and the chord notes were identified on a chromatic scale. Here these data are rendered a few different ways.

Survey of chords

In the first place the listed note symbols were converted into corresponding frequencies, and the frequency ratios for the second and third notes relative to the first note was computed. The list was then sorted in ascending order of these ratios. That way items with the same kind of chord come next to each other. Within each chord the list was further sub-sorted for descending absolute pitch of the lowest note. Results are shown in the table below. In some cases two or three whistles have identical data, these are clustered on single lines in the table.

The chords are illustrated with sets of little squares below a symbolic octave keyboard at top, black blue and red for the three notes. This would suggest how to play each chord in C. But apart from the structure of their chords, the whistles differ widely in pitch depending on size. The pitch of the lowest note for each pipe is shown at right by vertical bars on a logarithmic scale in Hertz. Note that the symbolic keyboard does not match with this pitch scale even if the width of a semitone step is the same.

Synthetic sound samples

Clicking on the icons at right will invoke synthetic renditions that in some way approximate the respective pipe sounds. Each such sound is fabricated using a specially written program from three pure sinusoids, with frequencies according to the listed notes. Sample durations are 2 seconds. To coarsely mimick pipe behavior with steam blowing the following control pattern was applied.

There is a gradual turn on until time 0.15 sec and turn off at 1.0 sec. After that a level tail is to simulate ambient reverberation. During the first 0.5 sec pitch sweeps over three semitones. This simulates what happens with steam blowing when the initial air within a whistle is replaced by steam having a higher speed of sound.

These control features alone were not enough for a satisfactory result. In an initial synthesis experiment the partial tones were set precisely to equal temper, where one semitone step corresponds to a frequency ratio of 1.05946, about 6%. Then in many cases the results had 'beats', slow periodic level fluctuations, something that will not happen with a real chime whistle. In such the different component whistles will influence each other, making their frequencies synchronize to take ratios of relatively small integer numbers. One whistle will pull the frequency of its neighbor, 'Mitnahme' in German organ terminology.

Frequency ratios of the tone intervals within an octave.

At left the number of semitones in the intervals and corresponding frequency ratios on the theoretical equal temperament scale.
At right close approximations to these in form of integer ratios Num/Den, most as used with Just Temperament. The MIDI column shows this converted back into equal temperament semitones, where the decimals indicate the difference in cents.
E.g. a perfect fifth is 3:2=1.5, corresponding to 7 semitones plus 2 cents on the equal tempered scale. Similarly a 5:4 major third is 14 cents low, compared to equal temper.

In the present synthetic samples the frequency ratios have been set to integer ratios as shown to the right in this table. The following sample picture exemplifies a difference in waveforms between using equal temperament or just temperament for the frequency intervals.

Johan Liljencrants 2007-05-05