;

Research

Preface

The Professional Pennywhistle and the Resonance Flute have grown out of years of research and development work which continues as we pursue better instruments. This article is directed toward the practical technician rather than the historian, performer, or acoustical theorist. Disclaimer: The ideas presented here are either common knowledge or my personal opinions.

Old Flutes

Starting about 1970, if it looked like a flute, we bought it. Gradually, the No-Names and “Nach-Offs” took a back seat to the esteemed Rudall & Rose, Boosey-Pratten, plus Firth, Pond & Co. Other flutes made by Clementi, Monzani, Meyer, Rittershausen, and Badger graced our collection and we played them all. We traded and borrowed to compare the flutes of prominent living makers. No two played the same.

Goals

As we became better players, we expected more: More power. Quicker response. Accurate scale and octaves with stability. Standard pitch with tunability. Tone control (honk). Embouchure forgiveness. Clarity. Support when playing soft and sweet. Ergonomics. Durability and aesthetics. But we were not alone. We consulted some great players and they also wanted to find all these features in one flute. Thus began the quest!

Qualifications

The desire was fine, but turning theory into practical instruments required Ralph’s Mechanical Engineering degree. His teaching of Physics for 29 years lent a foundation in Acoustics. I refined processes with my degree in Industrial Engineering. My Master of Arts in Mathematics inspired me to coordinate the various design features of flutes. We both have degrees in teaching science and math, so analysis and synthesis are things we do every day.

Eclectic Design

What happens when a great maker makes a great flute? Answer: He copies it. Every competent maker has a copy function. After spending years emulating the classics, there was still something missing in the package, and so we turned to eclectic design (i.e., take the good, leave the bad). When someone coins a new word, he uses someone else’s letters or syllables, but combines them in a new way to get new results, and calls the new way his own. We feel that we have a well-earned grasp of the principles of design. Here’s how we put it together.

Wood-Flute Tone

The Conn Strobotuner (http://www.mts.net/~smythe/st-11.htm) is usually used to evaluate pitch, a tool for improving the scale of the flute. However, in 2003, I noticed a distinctive strobe print on flutes that had a rich tone quality. As I brought out the tone, the print grew in complexity. Not only were the octaves of that note glowing brighter, but I could change the dial and trace other harmonics (trace A while blowing D) and see amplitudes there, too. This effect is not possible on other types of tuner. Finally, an objective way to quantify tone quality! This richness was that elusive “wood-flute tone,” and it was all a matter of harmonics. The Strobotuner also proves that a different flute may have a strong fundamental, but without the harmonics, the sound is never as interesting (and a microphone alone cannot enhance the interest). Certainly, artistic expression requires a flute that can honk on command (what I call tone control). Jazz musicians are the kings of tone control, and my favorite folk musicians have good tone control as well.

Tone and Harmonics

I had studied harmonics (overtones) before. Benade discusses their distribution and intensity to create a “recipe,” as he calls it, for each note. This recipe changes somewhat as an instrument goes up the scale. However, parts of the recipe persist, thus becoming the signature of tone quality (timbre) to distinguish a flute from a violin, for example. An instrument’s total set of recipes tends to form an envelope, a kind of a template that demarcates the expected overtones. The acoustics of speech is directly related. Here, the envelope can have peaks where particular overtone frequencies are amplified more than their neighbors. These peaks are called formants, and their relative position is what leads us to distinguish one speech sound from another (http://www.phy.ntnu.edu.tw/java/OTHERS/fourier2/index.html).

Inconsistent tone

Back to the same flute, I tested the note G for richness. Octaves did not appear on the Strobe. There was a pull between octaves. As I tried for better tone, the dial drifted flat or sharp. Tracing other notes turned up dry, and characteristically, I could not develop a decent tone on this note, although D was robust.

Why?

Aren’t some of D’s harmonics the same as G’s? Shouldn’t there be an overlap? How can I adjust the flute to investigate the effects?

Adjustment

I moved the cork and found that I could improve the harmonics of G. Here, G was better on the strobe print and better to my ear. Oops! Now D was spoiled. This flute was interfering with itself. Like a car with damaged suspension, it was wearing out the tires and fighting the steering (good for right turns only)!

Coordination

It would be absurd to keep moving the cork. At this point, I realized that a flute has other adjustment knobs. I had to find them and analyze each one to learn its function. Then, with purpose, coordinate all the effects that align the harmonics. In the end, I would have a flute with a rich tone. In other words, the challenge was not one of adjustment (of the old flute) but design (of a new flute). This alignment of harmonics was a design feature touted by pioneering flutemakers like Siccama and Ward in the 1800s.

Cybernetics

For the mathematically inclined, this is what Systems of Simultaneous Equations are all about. Matrix Algebra is equally good. Math aside, the best approach is cybernetic. Trial-and-error is not about taking stabs in the dark or wild guesses. It’s about understanding how one design principle balances or counteracts another, then watching for these two to meet in the middle, the way that two line graphs intersect. Someone like Newton, Leibniz, or someone using a TI-83 Plus would inch his way toward a solution this way. Historical note: Very few physical developments originate in a complete abstract model (as the laser). In most cases, the theory follows the creation, or helps us to broaden its application.

Use of Formulas

In any case, I do use a formula to calculate body length. I start with the body length of a keyless flute in D. Then, for instruments in other keys, I figure proportions using the twelfth root of two (r =1.0594631). One chromatic step means shorter by a factor of r; two steps means r2, etc.; and so a piccolo body is exactly half the length of a flute body (6% shorter per semitone is another way to say it). Also, it seems that tailbore is 80% of headbore (for a flute, at least).

Chimney Height

Some features are hard to see, but they must be measured in some way before making a change and at other times during the process. Here’s how I used my TI-83 Plus to study chimney height at the blowhole:
\Y1=(.5502 - x2)^.5 to draw the OD at the blowhole (sectional view).
\Y2=(.3662 - x2)^.5 to draw the ID.
\Y3= -8.144*X + 2.126 where m (-8.144) is the tangent of a 7-deg undercut on the leeward side of the blowhole (90 – 7 = 83 deg).
\Y4= 8.144*X + 2.126 where m is the tangent of a 7-deg undercut on the windward side of the blowhole; the b value sets these lines for an airgap of .396” across the blowhole.
Before pressing GRAPH, the WINDOW should be set as:
Xmin=-.6
Xmax=.6
Xscl=1
Ymin=-.05
Ymax=.73 (so it looks like a circle)
Yscl=1
Xres=1

Calculated Dimension

From there, you can use the CALC function then 5:intersect to find the coordinates where Y3 meets Y1 & Y2. Then, use the distance formula to find the length of this chord, which is the chimney height at the blowhole. Chimney height affects how hard to blow and how loud to play. Chimney height changes with the OD, the ID (bore), undercut angle, and airgap.

Shape of the Bore

The next frontier was the shape of the bore. We had plotted dozens of bores from classic flutes, adopting a standard taper of 0.016 inches diameter per inch of length. Note: When we adopted this standard for our flutes, it did not carry the understanding of any cause-and-effect as we have now. I recall seeing that some plots had a compound bore: straight head, choke, usual taper in LH section, then greater taper in RH section, then straight and flare in the foot. Was this the best bore? Should we do it this way? What are the relevant pros and cons according to the players? What were the performance claims of the maker? Was it unpopular because the tooling’s more expensive, and should that stop us if it makes a better flute? Is it obvious by now that I think it’s important to restate the question?

Cycloid or Parabola?

Meanwhile, in graduate school we were studying the Cycloid Curve, (http://mathworld.wolfram.com/Cycloid.html), y = a (1 - cos t ), a.k.a. Curve of Quickest Descent. I thought “quick descent” could mean low impedance of a sound wave traveling through the bore. Could this mean a good tone? Also, the shape was reminiscent of the compound bore plots, so I decided to give it a spin. After dozens of hours writing equations in Excel, then plotting, pasting, scaling, comparing, and repeating, my graph looked VERY much like the historical bore. Could it be that a scholar’s hunch would lead to a triumphant bridge between modern theory and classic empiricism? Maybe Boehm’s “parabola” was really a cycloid. Note: Boehm’s claim here was that the curve resembled a parabola (but then many curves are reminiscent of others). Modern makers still build excellent Boehm heads based on data that was derived empirically (from experience), not theoretically (from any parabolic formula).

Prototype

Armed with Excel data, I turned and polished a flute-sized cycloidal mandrel in steel; applied beeswax as a release; then epoxied strips of wood to it (so that it looked liked a Roman fasces). I also made a piccolo-sized mandrel and used it to mold an epoxy bore inside a dismembered fife body. This work took many more hours.

Refinements

The piccolo was improved, but I didn’t stop there. I changed the blowhole. I changed the tailbore (from the F# hole to the end). I removed the choke at the joint (where the straight head meets the tapered body). I moved the cork. Each of these adjustments improved the instrument, making it better in total.

Why?

Now, I thought, are these improvements the result of my new bore, or are they the result of all other adjustments? On a whim, I kept the other changes but went back to a simple, straight taper for the bore, and there I found the instrument that
played best of all (heavy sigh)!

One Story, Many Morals

The proof of the planning is in the playing. All ideas need to undergo controlled testing, no matter how noble their origin, how abstract their principle, or how worthy their outcome. Do they really make a better flute and will they impress others into agreement?
The whole process can take LOTS of time. Rome wasn’t built in a day, after all!
Sometimes, ruling out other possibilities is the only way to know. Just remember what Edison said: “I have not failed; I've found 10,000 ways that won't work.”
A review of the other parameters is always in order, and sometimes that’s exactly what’s needed to find the right combination to effect an improvement. Each change connects to other constraints (past, present and future).

Note to Self

Theobald Boehm, the Patron Saint of Flutemakers, was seriously interested in steelmaking, and this pursuit ruined his eyesight (so temperance is always a good idea).

Software

Taking another approach, some makers have worked with software for flute/whistle design. I understand that this software is useful for calculating hole size and position. While these details are important, my focus is playability (tone, response, etc), and these considerations reach beyond the determination of hole size and position. New designs require the maker to develop these details, but I find the time spent to be minor in comparison. A popular phrase about toneholes is their “acoustically correct position.” I maintain that what counts is finding the position that gives the best tone, response, and tuning, regardless of how the calculations guide us (and thankfully, we have the Cooper scale as testimony).

Undercut Toneholes

At first, I wasn’t convinced about the importance of undercut finger-holes, and I didn’t own any fraising tools. These are little cone cutters (the shape of a strawberry, hence le fraise). One of these is slid up the bore, below the hole to be undercut. Then the maker screws a shank into it and reams an undercut profile from the inside toward the outer surface (http://www.yamaha.co.jp/edu/english/factory/cl/cl_002.html).

Undercutting Method

I could have done the undercutting with a Dremel, but I wanted repeatability. First, I wanted to know the real significance of this feature, so I made a B-flat fife and split it open. I undercut from the inside by hand, glued the carcass back together, and presto! I was sold, but how to do it the same every time? The next tool started as a dovetail cutter for the router. I cut off one lip to make a “hockey stick.” Here, I’d use a twist drill in the mill. Stop, then insert the hockey stick, close the chuck, and cut by feeding the quill upward. The present tool is a custom-made dovetail tool for the Sherline CNC mill (http://www.sherline.com/). The CNC sends the tool in straight for a pilot hole, then uses circular interpolation to make a hole of any size. Undercut fingerholes act as big holes from an inside perspective (for a big sound), while from the outside they’re medium size. To get the job done, I customized the machine by mounting a CNC rotary table to tip the head sideways, making it a 5-axis machine. Unfortunately, while I had 5 axes of motion, I had only 4 channels of computer control. Consequently, I had to give up auto control of axis #4, but can still do that manually (as when offsetting the holes for A and E).

GmbH

Here’s a bit of acoustical theory. The flute is a resonator open at both ends. Sound waves travel from the blowhole down the resonator to the tone holes (and back). As the wave reaches either boundary, energy is transformed abruptly. The geometry must support this transformation with quality feedback for all types of good performance. Perhaps what’s good for a blowhole is good for a tonehole. A flute with a Great-many-blow-Holes? Would that be a German flute, GmbH (http://en.wikipedia.org/wiki/Gesellschaft_mit_beschr%C3%A4nkter_Haftung)? This geometry improves tone and response, and so our flute toneholes are undercut, similar to the blowhole. Undercut toneholes been in use since Louis Lot and before. On modern flutes, undercutting is a quality option, sometimes carrying the necessity of toneholes that are soldered-on (as opposed to drawn).

Better Highs / Better Lows?

I used to believe that a flute could be improved for the high notes or for the low notes, but not for both. In the latter 1800s, Boehm tried to institute a flute based on 20mm bore for richer lows, but the fluteplaying community committed to a 19mm standard in order to make the highs more playable and give this flute broader adaptability. He had improved both ends of playability, but was outvoted in terms of the general setting.

Both!

However, as my design matured, I was surprised to find better playability on the highs as well as the lows. The two were not mutually exclusive. In fact, easy, clear notes in the third octave indicate that the instrument supports those high harmonics. With this understanding, for the first time, the low D whistle and keyless flute went up to the fourth octave. Even if the player never calls for a high G, its harmonics are necessary for low G to have a good tone.

Bore Taper

During the design process, rate of taper is among my early considerations. Many makers know that increasing the rate of taper tends to stretch the octaves, especially between the first and second harmonics. In contrary fashion, moving the cork away from the blowhole tends to compress this harmonic series. However, these two processes are not simple complements. On some of my prototypes, the cork was set quite far from the blowhole (due to a high rate of taper). Ordinarily, this setting and taper tend to promise good tone (on a few notes anyway). It was disappointing that on these flutes, I could not easily play the highest notes, and it should come as no surprise that the tone was lacking on most of the lows. With additional experience, here’s my conclusion: for a given bore, hole size and voice (keynote) of instrument, there is a correct and ideal rate of taper to ensure good tune and tone. A whistle of a given voice may have a slightly different taper. Once the taper is established, the cork takes its position to align the octaves. With tapered bores, this value may vary from Boehm’s standard position (i.e., one bore diameter from corkface to center of blowhole). If alignment is correct, then the cork position is correct, regardless. It would indeed be possible to set the cork back at one diameter and choose a greater rate of taper to match, but (in my experience) this point of departure doesn’t lead to the best range and tone (with taper bores).

Octave Registration

Naturally, we all want a good strong bellnote. Sometimes a flute will “burble,” or oscillate between the first and second harmonics as the player tries to bring out the tone. The Strobotuner will show these two notes fighting each other in pitch, so the nature of the phenomenon is the alignment of harmonics on this note. Some old flutes have a few thousandths’ choke, or sudden restriction between the end of the straight bore in the head, and the start of the tapered bore in the body. This feature lowers the bridge, or flats the notes around the top of the first octave, (C#, D and E above it) with the intent of improving alignment and stability. If low C# tends to be flat, then the octave can be corrected with choke; if the E hole is undercut below, the octave will be stretched unless corrected with choke. Another “trick” is to flare the tailbore (the bore from about the F# hole to the end of the flute). This change raises the lowest note, combating what has been called, “flat foot syndrome.”

D-foot / C-foot

Many pieces of Irish music can be played on a basic, non-transposing flute whose lowest note is D. A foot that extends down to C is a common option. Low B must be better, no? B-flat, anyone? Having worked with dozens of bore designs, comparing results, I’m convinced that it is never a good idea just to “tack on” a little extra length for that additional low note (with taper bores). Rather, the designer must make up his mind in advance about the low note desired, then build the tailbore, the tapered body and the headbore into the design. Perhaps “tacking-on” is to blame for “flat foot” on so many old flutes, whose design was perpetuated without a full understanding of cause-and-effect. Even today, many makers present their instruments and give notice of a different response with the D-foot versus an optional C-foot, without a lot of explanation.

Head Length

Where should the flute be jointed? Does it matter where the head (straight bore) ends and the body (taper bore) begins? When I first made the Professional Pennywhistle, I moved the joint for no better reason than to make more economic use of the material. Suddenly the tone was better. Later, I was making prototypes for Low-D whistles, and found that as I pulled out the tenon, I was losing my octave registration on the “short tube” (noteably B and C#). As above, this misalignment would hurt the potential for a good tone. To investigate, I modified a set of high D fifes. Using a standard length of head, these three bodies would tell me about proportions as follows. The first body was lengthened and redrilled to make a D-flat instrument. The second stayed the same at D. The third was shortened and redrilled to make a D# instrument. Observation: Compared to the others, the D# instrument had good octaves on the short tube. Conclusion: Move the joint down. Note: The flutes made by Firth-Pond have an equivalent arrangement: the head’s straight bore continues into the left-hand section before beginning the taper of the body. Perhaps they chose this alternative to leave room for the upper trill keys. Otherwise, those toneholes might end up on the straight part of the bore (on the barrel). The keywork needed would present an extra mechanical challenge as on a clarinet. Here, the left-hand & right-hand sections can be rotated slightly without hampering the clutch between them.

Parameters

Clearly, flute design has several “adjustment knobs.” For instance, a tonehole can be changed regarding size, position, and undercut. Another feature is the head bore on Boehm flutes; this feature has many variations.

Cause and Effect

Each parameter has a familiar cause-and-effect. Larger holes tend to improve the tone quality. Moving the cork toward the blowhole tends to stretch the octaves.

Alone

Some parameters tend to operate in isolation. On some piccolos, one big C-hole is replaced by two small ones. This feature is not repeated elsewhere on the instrument.

Combined

Certain parameters tend to affect others. For example, undercut toneholes improve the tone quality, but tend to stretch the octaves. To compensate this side effect, the designer must change other parameters at the same time. Like spelling rules in the English language, each parameter of flute design is clear enough, but does not constitute a full system.

Coordination

Good design coordinates the parameters, so they will act in consort. To balance stretched octaves, the designer may move the cork away from the blowhole or he may reduce the rate of taper in the body. Of course, each of these two options affects other playing characteristics (in a different way). How many balls can you juggle at one time? It would be folly to think that a good flute could be designed by focusing on only some parameters while turning a blind eye to others. Here’s where we need to think in terms of a system. We need a strategy to do it.

Algorithms

To coordinate all these parameters, designers use algorithms and feedback. Every math student knows that the sine of thirty degrees is one-half. Simple to us, but even a good calculator takes a second or two to come up with the answer because it uses an algorithm. And if it can manage a handful of algorithms, it can produce more answers than you or I could memorize. A flutemaker’s algorithm could simply be a table that lists rates of taper to be tested (coupled with several ways to deal with the side effects). The flute designer changes the parameters alone and in combination, carefully evaluating the pros and cons. A computer seems suitable for the job, but the human player is still necessary for evaluation. An instrument is better when a new combination challenges the norm and solves old problems. Properly coordinated, each detail makes a valid contribution toward the sum total of good design.

Systems

New designs are not born; rather, they grow over a period of years. With general acceptance, the new recipe is established, it comes to be called a system, and everyone wonders why we didn’t do it this way before.

Blowholes

Jonathon Landell is an excellent resource at (http://www.flutes.org/). Humbly, I phoned him for advice on cutting blowholes. Based on a course he conducts at his workshop in Vermont, his prescription was a booklet called, “Build Your Own Head!” The booklet discusses blowhole geometry including how playability is affected by various types of undercut. Also given are some neat plots of headbores on Boehm flutes. The booklet was an education in itself and worth every penny.

Elliptical Blowholes

For a while, I used an approximated ellipse as the shape of a blowhole. Although my new CNC had built-in circular interpolation, it did not have ellipse capability. Further, I learned that manufacturers use CADAM to write ellipse subroutines, and they’re compilations of circular arcs. I found this site: http://users.cs.cf.ac.uk/Paul.Rosin/resources/papers/oval8.pdf, and with a little Excel, I was programming my own ellipses on CNC.

NONE

Stretched Semicircles

These days, I cut my blowhole in two semicircles which are then stretched to leave flat sides between
them. At the windward and leeward edges, each has a simple 7-degree undercut while up-the-bore and down-the-bore are cut in sweeping chords. The “sweet spot” is easy to find and very forgiving. Note: So many other designs are “locked-in.” The flute may well have a sweet spot, but it sounds thin or noisy when played any other way. I want more air to give more tone and honk, not just noise and chiff. I find my design gives the player excellent control regarding all combinations of tone and loudness. That’s just what’s needed for the art of music and to support multiple playing styles.

Whistle Cutaways

In one drawer of the shop, we have whistle heads that have been

sawn lengthwise so we can study the geometry of the windway. Sacrilege? I rather think it compares to the display of a human skeleton in physiology class. When I gave a voicing workshop, I used the sections to show floor, roof, chamfers, angles and undercut, all important for a high-performance whistle. Oh, yes: these sectional heads can still toot if they’re sealed against the bare forearm.

Matching the Compass

For dance music, I played the HiD fife or sometimes the flute,

but fancier tunes sometimes went off the compass. I’d miss the low notes on the fife while the flute wouldn't project (or it was shrieky in the 3rd octave). My instrument was an imperfect match for the violin, and I was tired of it. One idea being passed around was a mid-sized instrument with many Briccialdi levers. Fewer fingers were necessary to select a “home position” and choose any diatonic scale (like selecting a set of black keys on the piano).

Keyed Systems

Beginning in 1988, I made a small flute in G, but I still needed the chromatics. Next, I bought a student-model Boehm flute, shortened it and rebuilt it a fourth higher, but the bore was wrong. I went back to the tapered bore, low note G, but now with configurable clutches to move the half step for each new scale. Here’s the key logic: I could press RH1, closing one key and going from G down to F# (for a G scale). Meanwhile, the underside of the F-natural arm held a tiny boxtube; inside, a square bar (3/32”) could be slid toward the left, under the F# arm. When extended, pressing RH1 would close two keys, bringing me from G down to F natural (for a C scale). There were other bars like this, and each one ended with a socket-head capscrew (0-80) whose head hung slightly above the nearby arms. The capscrew was my shift lever. This design is now in the museum (translation: stuffed in a box somewhere).

Boehm Flute in F above C

Also called a mezzo-piccolo, this better version came in January 1992. The next generation will still have a tapered bore, but the material will be sterling, and there will be other improvements.

Looking Ahead

Progress will come from combined factors including reading, experience, and sharing with other instrument makers. Also, for inspiration, there is Boehm’s own ring-keyed flute, dated 1832. We know that traditional players preferred the tone of this conical flute to his 1847 “parabolic”/cylindrical design played today. For them as for some of us today, it’s all about the tone. The “how-to” of classic flute design remains rather sketchy; perhaps this info is locked in a vault somewhere along with other family secrets. Nonetheless, the flutes themselves remain as examples, and with careful analysis, will serve as the key to unlocking more mysteries of making a better flute.

"If I have seen further, it is by standing on the shoulders of giants."
-- Sir Isaac Newton

“Beginning is a day’s work, but finishing is the work of a lifetime.”
-- Gaelic Proverb

Practical reading:
Benade, Arthur A. "The Physics of Woodwinds." Scientific American, October 1960.
___. Horns, Strings & Harmony. New York: Anchor Doubleday & Co. Inc., 1960.
___. Fundamentals of Musical Acoustics. New York: Oxford University Press, 1976.
Boehm, Theobald. The Flute and Flute Playing in Acoustical, Technical and Artistic Aspects. New York: Dover Publications, 1964.
Hopkin, Bart. Musical Instrument Design: Practical Information for Instrument Making. Tucson: See Sharp Press, 1996.
___. Air Columns and Toneholes: Principles for Wind Instrument Design. Nicasio: Experimental Musical Instruments, 1999.
Joof, Laura Beha. "Recorder Voicing and Tuning, and Use of the Tuning Machine." The American Recorder, November 1985.
Robinson, Trevor. The Amateur Wind Instrument Maker. Amherst: University of Massachusetts Press, 1981.