Theories about tuba pedal notes

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DonShirer
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Re: Theories about tuba pedal notes

Post by DonShirer »

     You may want to see a previous discussion of this topic at viewtopic.php?f=2&t=47557&start=24" target="_blank

     One of the problems is that people have different perceptions of what a "pedal tone" (a poor name in the first place) is. Brass instruments do NOT have the resonances associated with a conical or cylindrical tube with a closed end often quoted in books. The bell cuts off the higher frequencies, and it as well as the mouthpiece and valve section compresses the resonance spectrum, especially toward the high end. In a "well-tuned" instrument, useful resonances or "privileged tones" occur at nearly 2, 3, 4 and 6 etc. times a "fundamental" frequency (not present), roughly matching a harmonic scale, and the players lips bring those notes into tune. By loosening the lips, tuba players can (often) sound a note perceived as having a pitch near that of the missing fundamental, and many call that a "pedal tone".

     There seems to be a controversy as to whether the "pedal tone" contains any content at that fundamental frequency. The resonance curves of a trumpet or trombone (I once taught Musical Acoustics and found these in my library) do show a lower resonance, but it is nowhere near the "fictitious fundamental" and usually unplayable. The French horn does have a lower resonance much closer to its fundamental. I could not find quickly a resonance curve for a tuba, but it would help if someone could find such on the web to see if the lowest resonance of a tuba is anywhere near its fundamental. In that case, the waves at the "2" frequency (the 2d harmonic of a missing fundamental) might help to reinforce the players lip vibrations to produce sound at the pedal frequency.

     The perception of pitch by the ear is quite complex and besides the lowest frequency present in the spectrum, depends on the amplitude and frequencies of the higher overtones as well as other factors. Another possibility is that the predominance of harmonics of the "fictitious fundamental" might persuade us that it exists in the "pedal" tone. If someone could find a good frequency spectrum of a "pedal" note, that might confirm or deny the existence of any content at the "fictitious fundamental" frequency and help to distinguish between these two possibilities.

     For those wishing an enjoyable elementary treatment of musical acoustics, look for a copy of "Horns, Strings & Harmony" by Arthur Benade (originally published in 1960 but still available online).
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Art Hovey
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Re: Theories about tuba pedal notes

Post by Art Hovey »

Here are my 2 cents:

The “frequency” of a note is the number of repeated cycles per second. If the waveform of a sound with definite pitch has any shape other than that of a perfect sine curve, then that wave shape is the sum of many sine or cosine curves with frequencies that are integer multiples of its fundamental frequency, i.e. a “harmonic” series. If the fundamental frequency is “f”, then the harmonic series of frequencies is f, 2f, 3f, and so on. Any sound with a wave-shape that repeats periodically can be shown to be the sum of a harmonic series of sine-shaped waves. For example, if you record one clap, put that recording in to your computer, copy and paste it over and over again to create series of evenly-spaced claps with a reasonably high frequency you get a sound with definite pitch. Most of that sound will be in the higher harmonics, but the pitch that you hear will be that of the fundamental.

Brass instruments utilize the familiar “bugle-tone series”, which closely resembles the harmonic series. If “f” represents the frequency of the imaginary “pedal” note, then the useful frequencies are 2f, 3f, 4f, and so on. The sound is generated by periodic puffs of air from the lips into the mouthpiece. Each puff forms a sound pulse which travels through the instrument to the bell, where it is partially reflected back. If the returning pulse arrives at just the right time it triggers the lips to release the next pulse. In that way we build up a strong, steady air vibration in the instrument. The process resembles how we push a child on a swing, building up large swinging motion with a series of small pushes at just the right frequency.

If you repeat that pushing motion with just half of the swing frequency you can still maintain a large swinging amplitude. That series of pushes is the sum of a harmonic series of cosine curves with a fundamental frequency just half of the swing’s frequency. (The lower-frequency members of that series will have rather small amplitude.) A graph of the swing’s motion will then resemble a cosine curve with alternating cycles of larger and slightly smaller amplitude. That graph is also the sum of a harmonic series with a fundamental frequency just half of the swing’s frequency. In this case the second member of that series (corresponding to the swing frequency) will have large amplitude, but there will also be a small contribution at half the swing frequency.

That’s how “pedal notes” work on a brass instrument. The player sends puffs of air from well-trained lips at just half the frequency of the lowest “good” note in the bugle series. Since the instrument resonates at twice that frequency, return pulses arrive at twice the frequency of the puffs. Half of those return pulses arrive at the right time to trigger the next puff, and the other half don’t. The result is tone that has a small component of fundamental frequency but also many other components that are integer multiples of the fundamental frequency. We hear it as a tone at the “pedal” frequency.

If we produced the sound with a pure sine or cosine generator instead of a series of puffs the result would be quite different because the fundamental resonant frequency of the air in the bugle (as described by Benade in Horns, Strings, and Harmony) is actually much lower. But the harmonic series formed by puffs of air at the instrument’s true fundamental resonant frequency does not match up at all with the frequencies of the instrument’s higher resonant frequencies. That’s why the true fundamental is virtually unplayable.

By the wa, my little Korg tuner recognizes pedal notes on a BBb tuba quite well, although it is probably responding to the higher harmonics.
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Re: Theories about tuba pedal notes

Post by humBell »

My working theory of pedal tones as pourly thought out as it is...

Only half the waveform is guided by the tuba.

So you can depending on the geometrics of the tuba, lip from roughly Eb to an octive below it on either side of the fundamental.

My suspicion is that it is easy to hit tha Eb because it is the boarder of what resonates, and even a board the usual palyet not used to holding notes that low will slide upward to.

They are generally quieter, because the instrument doesn't do as much to support it by virtue of only guiding a part of it.

Anyway, kudos to anyone who follows my poorly worked out thought enough to make sense of it. Someday i will think more. or at least to take the time to express it more clearly...
Thanks for playing!
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Art Hovey
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Re: Theories about tuba pedal notes

Post by Art Hovey »

Further discussion can be found here:

http://www.galvanizedjazz.com/tuba/PedalNotes.html
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imperialbari
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Re: Theories about tuba pedal notes

Post by imperialbari »

I tend to disagree with those saying that the overtones making it possible to hear the imaginary pedal will line up in a perfectly in tune row of Pythagorean partials.

At least some brass instruments have very wide slots for the said imaginary pedal note. I tend to ascribe that to the availability of several sets of two adjacent partials being able of producing the imaginary pedal at various pitches according to the pitches of the involved partials themselves tending high or low.

I have played a lot of different brasses at the same time trying to minimize the number of necessary embouchures by using as few rims as possible. Before I had some trumpet mouthpieces threaded to take my preferred horn rim, I played trumpets, cornets, and flugels via my horn mouthpiece. In the trumpet range called for, 2 octaves in the mid-low range, the pitch and the intervals were fine.

On my rotary German Bb Meister K. Wolfram trumpet the open pedal also was perfectly in tune. Only it was not a Bb, but an Ab. When using trumpet cups the open pedal became a Bb again.

Klaus
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