A colleague of mine recently informed me of his theory about the acoustics of tone produc- tion on the euphonium. It embodied a concept that I had never considered, even after a lifetime interest in instrumental acoustics and authoring a Ph.D. dissertation on Intonation Deficiencies of Wind Instruments. Arthur Lehman, a well-known former euphonium soloist in the United States Marine Band, has a theory that the fullest quality and resonance of a tone is the result of the presence of a wide range of harmonics, both above and below the core tone.
Arthur Lehman, Euphonium, U.S. Marine Band (1959)
Below? How could that be? We usually think of them as being higher harmonics or upper partials emanating from a low fundamental vibrator. Most musicians understand the role of harmonics in a musical tone: how a vibrating string can be induced into “higher” harmonics by causing it to vibrate in equal segments, and how the single tube of the bugle can produce the tones of its harmonic series by forcing more complex wave patterns, and that the quality or timbre of the basic tone is determined by the presence and relative strength of the various higher partials. So how can this be? Aren’t harmonics always above the fundamental tone? Rather than conclude that he might not understand the matter, I decided to give the matter more thought on the possibility that he might have something there. I studied the idea and came up with a plausible explanation of Mr. Lehman’s theory involving harmonics below the core tone. Before explaining this, it is necessary to review the dynamicsof the harmonic series, especially regarding difference tones.
Chief Warrant Officer Donald W Stauffer in 1959 pictured with his Ophicleide in C, which was featured in the 1900 Paris International Exhibition. Stauffer purchased it from William Bell and had it fully restored by Navy repairmen. It is presently in the collection of Dr. Charles Conrad, Director of the Indiana Wind Orchestra.
The chord of nature is a family of tones made up of frequencies that are divisions of the frequency of a fundamental. The basic pulse vibrates with a complex pattern that includes the various component frequencies simultaneously. The number of and patterns formed by these are primary determiners of the tone quality or timbre. Observe Example 1:
Example 1. The Harmonic Series.
The frequency assigned to the fundamental above is arbitrary, chosen for simplicity in demonstrating the fractional relation- ships involved. Note that the higher repetitions of octaves always have the numbers doubled. The difference in the frequencies of any two adjacent tones is always the same (100 DV), which is the frequency of the fundamental. This forms the difference tones, subtle “pulsings” in the background that are rarely perceived as tones but are contributors to the quality of the fundamental. Difference tones are similar to beats, slower pulses that can be counted. Both are caused by two tones that are close in frequency. Thus if two tones are 250 DV and 252 DV respectively, the beats will be 2 per second. Twice every second the waves will be in oppositephase and cancel each other out to create minimum intensity, and twice they will be in full synch- ronization to create a power peak. This is the beat. These slow pulsations are the ones the piano tuners seek to eliminate to create perfect unisons in tones produced by more than one string. What is important to our discussionis the fact that as the tones become farther apart, the beating accelerates to the point that it is perceived as a difference tone.
An example is the foghorn of a ship, not produced by a large pipe but by two very powerful high whistles that are mistuned appropriately to create the foghorn frequency. The 32 ft. rank of sub-bass tones are faked on some smaller organs by sounding together two pipes tuned a perfect fifth apart in the 16 ft. octave above. Another curious example occurs when two people whistle a duet in a small room. One can hear strange sounds dancing around which are the difference tones of the intervals that occur in the duet.
How does this relate to Mr. Lehman’s theory of harmonics below the core tone? Observe Example 2 below. The E-flat is produced on the euphonium with the first valve tubing whose fundamental is low A-flat. The E-flat is sounded by energizing the third harmonic of the A-flat series by a proper setting of the embouchure. It is not itself a fundamental. It cannot function as a true fundamental because the proper pipe length is not present to support all of the natural family of frequencies in the chord of nature on E-flat.
Example 2. First Valve Series.
However, when the E-flat is played, it will excite the higher harmonics of the A-flat series, as dictated by the tube length. These will produce several difference tones that will converge to form a fundamental A-flat, which then can assume its role to generate much weaker upper partials. This is viewed as a mutual aid process of supporting and influencing the core tone. As a matter of further note, none of the tones in the regular register of the euphonium are true fundamentals. There are only seven pipe lengths provided in the typical three-valve brass instrument, with only seven fundamentals available in the pedal register. All of the tones in the regular register are upper har- monics that are singled out by the action of the player. The full quality and resonance is the result of a complex interplay of the harmonic elements. One cannot normally hear the “parasitic” harmonics that accompany and are built into the core tone. This is not entirely because they are weaker, but, rather, this mutual interplay directs the energy back to the fundamental tone, thus augmenting and enriching it. A balanced distribution is desirable to enable the lows to provide full body, and the highs to give life and brilliance. Intonation can also be a factor. A preponderance of strong highs can sound sharp as well as shrill. An absence of highs can cause the tone to seem dull and flat.
The presence of harmonics necessary for the best tone quality is dependent partly on the proper bore of the instrument. A “pea- shooter” small bore will tend to weaken or eliminate the lower frequencies. The more significant determiner is the player him- self, with his skillful manipulation of the oral cavity in his embouchure formation. A sufficient air cavity must be allowed behind the vibrat- ing lips to allow their full, free oscillation of the back half of the wave to reciprocate fully with its counterpart in the instrument. Observe the Example 3 sketch: This is a sketch of the author’s concep- tion of a fundamental standing wave in an air column. Dand Fare nodal points where the molecules bunch up to a stop and reflect back while pushing on a new pulse. A is the node positioned for the maximum cavity, most suitable for the low fundamental. Bis the vibrator power source (lips) that excites the air in the tube. B and Eare antinodes where the moleculear density and pressure are least and the back and forth pulsating activity is greatest.
Example 3. The Oral Cavity.
The above drawing shows only the fundamental standing wave and its need for a sufficient cavity to accommodate the matching inner portion of the fundamental. It is the longest and slowest wave possible. When the third harmonic is singled out as a core tone as in Example 1, there would be three inner nodes (letter D) and the waves would be one third as long. This would require less oral cavity for proper backup support. Too large a cavity would dissipate the energy by not reflecting the pulse in synchronization with the standing wave in the tube.
It is shown in the well known “aw” and “ee,” a teaching device for the production of low and high register tones, and to assist in wide slurs between the two. The reality of this can be shown by a reversal, as in the following experiment. Forming an “ee” will cause a low tone to be thin and nasal and, an “aw” will fail to support the high tones, which are usually flat and dull if produced at all. Matching the volume of the oral cavity to the wavelength of the core tone is necessary for full and free resonance, which is the result of a balance of brilliance and depth. The goal is a musical tone that is usually achieved by careful practicing and critical listening. It is caused by a complex standing wave that incorporates a spread of higher harmonics that generate lower sub-harmonics as difference tones in complex inter-relationships.
Master Gunnery Sargeant Arthur W. Lehman was a leading euphonium soloist with the United States Marine Band during the late 1940s and into the 50s, with hundreds of performances in the Nation’s Capitol and throughout the United States. He was born in Doylestown, Pennsylvania and began playing the baritone at the age of 10. He played in many bands in the Keystone State and a short enlistment in the Army before join- ing the Marine Band in 1947. He cites among his teachers Simone Mantia and Harold Brasch. After retirement in 1971, Art performed several years in the National Concert Band and taught private lessons. His advice and counsel is highly respected, and he has been sought out frequently by young players who are aspiring to get into one of the major service bands.
Dr. Donald W Stauffer, Commander, U.S. Navy (Retired) was born in Canton, Ohio in 1919. He holds B.M. and M.M. degrees from the Eastman School of Music of the University of Rochester, and a Ph.D. from the Catholic University of America. He played tuba and double bass in the Rochester Philharmonic and the United States Navy Band before conducting with the Navy Bands in New York and Norfolk. He was the head of the Academic Training Department of the Navy School of Music before being named Leader of the United States Navy Band in 1969. Upon retirement he was appointed Associate Professor at Birmingham Southern College and later at Samford University. He founded the Birmingham Community Concert Band in 1980 and served as conductor until 2003.
He has composed and arranged over 40 published numbers and authored five books. His best known composition is Fugue ÔN Swing,and his most recognized books are Intonation Deficiencies of Wind Instrumentsand A Treatise on the Tuba.He is a recognized researcher in the Acoustics of Musical Instruments and has published studies in the Journal of the Acoustical Society of Americaand the New York Academy of Science. His articles on musical topics have appeared in the Journal of the 1962 All- Eastern Band, Bandworld,and the Instrumentalist.