Theories about tuba pedal notes

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imperialbari
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Theories about tuba pedal notes

Post by imperialbari »

We have discussed pedal notes some years ago, where I by pedal notes mean the notes that we usually would call the first partial of a tuba with its various fingering combinations. Not false notes, not second partials with a whole lot of valves pressed.

My terminology isn’t strong enough to make a good archive search on this topic.

I am mostly interested in getting a memory from the old discussion confirmed, or if wrongly remembered having it rejected.

As I remember old discussions, research had it that at least some tubas didn’t really sound proper pedal notes with any significant strength. What we hear as an often strong pedal note then really is an accumulation of the difference notes between adjacent partials. Because the difference note between adjacent partials always is the first partial. But as these difference notes are imaginary insofar that they only exist in the human perception of musical sounds, then they cannot be measured electronically.

Is this memory reasonably correct?

Can somebody please provide links or references to available, preferably online, research reports?

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Re: Theories about tuba pedal notes

Post by swillafew »

I can keep a better grip by using the terms "overtones" and "harmonic series". I believe overtone is interchangeable with "partial" but sometimes I have my doubts when others use them.

I am always more confused after reading others' posts here.
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Re: Theories about tuba pedal notes

Post by Donn »

I think the most recent and extensive discussion here is this one: Pedal Tones (Physics Perspective)???. I'm not sure I can easily make out what came of it, but you may be able to follow it better.

On the specific question of whether pedal tones are real at all and therefore measurable, or purely figments of our imagination - I fired up my cellular phone and tried the tuner applications, and I couldn't make it recognize a tuba pedal -- but could make it recognize a bass trombone pedal. Down to F; Eb was too sloppy. That happened to be the pedal note I was trying for on (Eb) tuba, so I reckon it could very well be me that can't ring the bell on the tuba, not an inherent property of the instrument so much. This may not prove anything - the pitch recognition could very well depend partly on other partials, couldn't it? and thus discover a pitch with no 1st partial present.

My recollection of the conventional wisdom, in any case, is that the pedal 1st partial is present but weak.
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Re: Theories about tuba pedal notes

Post by imperialbari »

swillafew wrote:I can keep a better grip by using the terms "overtones" and "harmonic series". I believe overtone is interchangeable with "partial" but sometimes I have my doubts when others use them.
Overtones are derivates of the fundamental.

Partials include the fundamental, which makes more sense, as fundamental and overtones are thereby connected in the same simple Pythagorean matematical rows, where

F, 2F, 3F, 4F, 5F, ------, nF describes the relationship between frequencies of the partials

and

L/1, L/2, L/3, L/4, L/5, --------, L/n describes the relationship between the lengths of oscillation of the partials.

You may set up mathematical rows for overtones also, but they would be less simple, because your formula would have to compensate for the natural starting point having been removed from the row.

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Re: Theories about tuba pedal notes

Post by imperialbari »

Donn wrote:I think the most recent and extensive discussion here is this one: Pedal Tones (Physics Perspective)???. I'm not sure I can easily make out what came of it, but you may be able to follow it better.

On the specific question of whether pedal tones are real at all and therefore measurable, or purely figments of our imagination - I fired up my cellular phone and tried the tuner applications, and I couldn't make it recognize a tuba pedal -- but could make it recognize a bass trombone pedal. Down to F; Eb was too sloppy. That happened to be the pedal note I was trying for on (Eb) tuba, so I reckon it could very well be me that can't ring the bell on the tuba, not an inherent property of the instrument so much. This may not prove anything - the pitch recognition could very well depend partly on other partials, couldn't it? and thus discover a pitch with no 1st partial present.

My recollection of the conventional wisdom, in any case, is that the pedal 1st partial is present but weak.
Thanks for the link, which I will look at, when I am a bit more awake later in the day.

Your memory of the pedal note’s measurable strength is that it is weak. Mine is that it is without significant strength. I would say that our memories are pretty much in agreement on this matter.

I use a strobotuner app on phone and on iPad. The weak link is about the microphones. While much better than phone microphones of traditional phones of my youth and of most of my adult life, they still are small. And their main purpose still is to catch the human speaking voice, which does not operate in the frequency ranges of the pedals of low brass instruments.

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Re: Theories about tuba pedal notes

Post by Donn »

This page - Brass Instrument Acoustics at Georgia State University Physics & Astronomy, is mentioned late in the above discussion. Its propositions are
  • brass instruments are closed pipes
  • for musical purposes the full harmonic series of an open pipe is more convenient
  • therefore the bell causes those odd closed pipe harmonics to be compressed to form a full harmonic series
  • except for the first, which doesn't work out and becomes an "unused lower resonance" below the artificial series, and the pedal octave is purely the combination of upper partials.
I'm no physicist, but ... I don't find that very convincing at all. In the reeds, it's clear that despite the physical similarity at the top of the clarinet and saxophone, one is "closed" and the other is "open" for acoustical purposes. Or if you like, the saxophone is a "cone" as portrayed at the GSU site, with the same full series properties as an open pipe. They don't need to invent a story about how the saxophone got its full series by artful transformation of the odd series, and of course they have a rather clear and evident first partial there, as that's the main woodwind register. Brass instruments are for sure less solid in the first partial, but it's a matter of a design that revolves around the upper partials - and subject to some variation, taking the bass trombone as an example.
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Re: Theories about tuba pedal notes

Post by imperialbari »

No, a compressed odd series that ends up very close to the full Pythagorean series doesn't sound likely.

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Re: Theories about tuba pedal notes

Post by Donn »

tuben wrote:Stopped pipes propagate:
go, e1, BBb1, etc.... If you 'overblow' an stopped organ pipe, this is the pattern of notes you get.
I see you start at the 2nd or 3rd partial - no fundamental here?

My uneducated hunch is that "closed pipe" is not correct - that they've put together this elaborate story about how the closed pipe is made to sound just like it's an open pipe, but a simpler explanation is that it is acoustically an open pipe. Maybe the cylindrical tube without a bell does come out closer to a closed pipe - just as in the case of the reeds, with the clarinet vs. saxophone.

The effective difference between the two theories is, what happens with that pedal octave. In their odd-partials transformation scheme, it isn't there at all, while in my simple minded version it should be there acoustically speaking (though not necessarily useful.) [And I wouldn't have that "unused resonance" derived from the odd series fundamental.]
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Re: Theories about tuba pedal notes

Post by Donn »

bloke wrote: ... most likely, none of this is related at all…
Well, at the risk of applying logic in the absence of thorough understanding - if the GSU account of the pedal, as a fictitious pitch formed only by resonance of its harmonics, is correct, it would be interesting to hear them explain why it would have a different intonation tendency than the pitches formed directly by those harmonics.
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Re: Theories about tuba pedal notes

Post by sloan »

Short answer: the "harmonic series" that makes so much musical sense is not at all a natural effect in brass instruments. Instead, it is a highly engineered artificial effect. The mouthpiece, the bell, and the taper combine to produce a close approximation to the simple "harmonic series". As with most artificial, engineered solutions, it is possible to move the errors around, increasing the accuracy where it is needed and pushing the gross imperfections somewhere where they don't interfere with normal operations. So it is with the "false tones".
They exist in the THEORY we use to make sense of the "harmonic series", but they don't really exist in reality.

Long answer: see the original thread(s) for citations. Alas, my brass theory library fell victim to my recent downsizing. Perhaps Rick Denney can re-post the references.

Bottom line: brass instruments are engineered to provide a reasonable approximation to PART OF the "harmonic series". You are on reasonably safe grounds in applying simplified musical theory to MOST OF the range. But, there is no "fundamental" (except in your head).
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Re: Theories about tuba pedal notes

Post by pjv »

What, so whatcha all sayin' is that bass trombonists spend 1/3 of their careers playing notes that aren't even there and THEY GET PAID FOR IT?!
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Re: Theories about tuba pedal notes

Post by sloan »

Yes, but...the fundamental of a tuba is not part of the harmonic series the player has in mind.
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Re: Theories about tuba pedal notes

Post by timothy42b »

bloke wrote:I tend to suspect that the "perfect" harmonic series (numbers/products that are all simple error-free multiplication problems) only exists on paper, and that no conical nor cylindrical tube (regardless of the method of setting a vibration in motion) can be found nor designed which offers pitches which line up perfectly with calculated numbers.
I bolded a piece of your post because that's the part I would disagree with.

I agree you can't design anything that will have a simple integer multiple harmonic series. Per our last discussion, if the slot is wide enough you can still play it in tune. But if you start at the bottom of the harmonic series and play up the series, one note at a time, they're all going to vary a bit from that simple relationship.

But I think this is not true if you're playing the fundamental. Then I think that all the overtones that sound above it WILL BE simple integer ratios. Even though these are not quite where the horn would resonate if you were playing one of those overtone notes directly.
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Re: Theories about tuba pedal notes

Post by Donn »

Good to see someone trying to untangle the semantic confusion between partials/overtones/harmonics, vs. the pitches that we may play that correspond to a natural (idealized?) partial series for the horn. (Argh, not much untangling happened in that sentence. Oh, well.)

Could you elaborate on why you expect the pitches following that natural series wouldn't quite match the integer-ratios of the corresponding harmonics? If I got that right.
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Re: Theories about tuba pedal notes

Post by timothy42b »

Donn wrote:
Could you elaborate on why you expect the pitches following that natural series wouldn't quite match the integer-ratios of the corresponding harmonics? If I got that right.
When you pluck a violin string, it vibrates at its natural frequencies. If you bow it, the timbre is totally different, because when you are driving a system with an input, as opposed to exciting it and letting it respond, the driven system responds to the input frequencies.

When you drive a wind instrument with a lip buzz, it is my belief that it is forced to respond with overtones that are simple integer ratios.

However it is very clear that the natural series of the horn is different. The different pitches available for a given fingering differ from integer ratios, and in fact differ from horn to horn. A horn is not a cylinder nor a cone, but a complex combination of various size tubing, valves, bends, restrictions, etc. (To the sound wave, a bend represents a wide spot in a tube: the sharper the bend, the wider the diameter if it were straight)
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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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Re: Theories about tuba pedal notes

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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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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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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.

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