Gamelan Gong Building and Construction

Table of contents :
ALUMINUM DISK GONGS (Article #1) | Bart Hopkin......Page 1
ALUMINUM DISK GONGS (Article #2) | Bart Hopkin......Page 13
FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin......Page 39
SCALE AND TIMBRE | Bart Hopkin......Page 53
THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin......Page 62

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #1) Easy to make, wonderful sound.  If you decide to make gongs in this style, you can choose whether to keep it simple or take a more sophisticated approach.   Article 1: A Simple Approach I have recently made a number of simple flat gongs out of aluminum disks.  If you can get your hands on some aluminum disks of suitable size, these gongs are wonderfully quick and simple to make, and they really sound lovely.  While you can make excellent-sounding gongs with a minimum of specialized skill and knowledge, it’s also possible, and quite rewarding, to take a more sophisticated approach in search of a more refined tone quality.  In this first of a pair of articles to be posted here I’ll describe the simple approach.  In a following article I’ll get into more sophisticated stuff.  As a teaser for the second article, I’ll tell you now that the more sophisticated approach has to do with listening for the pitches of the different modes of vibration in hopes of getting the overtones within the tone to line up in coherent relationships. The gong set I made in keeping with the principles set out in that second article Share This

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

Important disclaimer: These notes are not about the making of traditional Asian gongs. That time-honored discipline is well beyond the reach of this writing. Rather, these notes concern the making of very simple disk gongs with minimal hammering required.   THE SIMPLE APPROACH Here’s the basic idea: a flat aluminum disk of suitable thickness and diameter, hung form the right suspension points and struck at the center with the right sort of mallet, makes a lovely tone. It has many modes of vibration and thus many different frequencies within the tone, but when played as just described one of these modes stands out as the defining tone, giving it a clear sense of pitch, while the other modes blend in to help create the overall tone quality. If you’re happy with the pitch the gong is making, then no more need be done.  If you want to tune the gong to another pitch, you can do so by hammering a small nipple or boss at the center to raise the pitch, while leaving the rest of the gong flat. All this will be described as we go along. Where to get aluminum disks: My original supply came from a metal scrap yard.  I don’t know if I was lucky to find them there, or if they turn up at such places quite regularly. If you don’t find them there, aluminum disks are available from metals suppliers online (at high prices). You can also purchase sheet aluminum from metals suppliers

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

and cut the disks yourself using one of several power tools that are up to the task, or have them custom-cut. (Metalworking tools are best, but because aluminum is the softest of commonly used metals, woodworking tools such as a woodworking bandsaw with the right blade can manage the work.) Suitable aluminum alloys, suitable sizes: The most common and affordable aluminum alloy is referred to as 6061; it works fine for this purpose.  Thicknesses in the range of 3/32” to 1/8” are most suitable for flat gongs in smallto-medium sizes.  Suitable diameters range between about 6” to about 20”. The 1/8” thickness is best for gongs over about 11” inches in diameter, and 3/32” for smaller. These sizes are suitable for gongs for pitches from about C3 (C below middle C) to G5.  It’s possible to go lower with thicker material and larger diameters, but this takes us beyond the “simple gong making” we’re talking about in this article. In short, we’re focusing on this range because disks of this size and material are typically not too difficult to find or fabricate, they’re easy to work with, they’re neither so thin as to be easily damaged with heavy use nor so thick as to require too much effort in hammering, and they sound very well. Where to drill the suspension holes: Typically these gongs will be suspended on cords, two cords per gong.  The cords pass through small holes drilled in the gong (about 1/8” diameter) at points roughly corresponding to 10 o’clock and 2 o’clock on an imaginary clock face,

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

each located about 1/3 of the radius from the edge. To minimize fraying of the cords, slightly bevel the edges of the holes after drilling. A countersink bit works well for this. The reasoning behind these hole locations: the gong has several modes of vibration producing different pitches. One of these is most important for us in that it’s the one we’d like to dominate the sound and define the perceived pitch.  Accordingly, we want to suspend the gong in a way that imposes minimal damping on this particular mode. It happens that this mode has node – an area of minimal vibration — in the shape of a ring at a distance of about 1/3 of the diameter in from the edge, so we place the holes within that ring for minimal damping. Mallets:  The mallets you use will make a huge difference in the sound of the gong. The preferred sound comes from a moderately soft mallet head, and it turns out that there’s a commonplace item that is perfect for these gongs: superballs.  The 1¼” size, or anything close to that, will work well for smaller gongs, and the 1¾” size is good for larger ones. Select a wooden dowel or similar material for the handle; drill a hole very slightly smaller than the diameter of the dowel through the superball, and press-fit the ball onto the mallet. If needed, add a drop of glue such as superglue (cyanoacrylate) to keep it in place. Tuning (hammering the nipple): Your original flat gong, without any

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

hammering, may produce a pitch and tone quality that you are happy with. If that’s the case, no hammering is needed and you can skip to the mounting section below.  In my experience, however, a bit of center-hammering improves the tone quality, so even if you’re not concerned about the pitch of the gong, I recommend hammering a small nipple. And of course if you are concerned about pitch, then hammering is the process by which you tune the gong to the intended pitch.  You can only tune upward, so the original unhammered gong must be below the desired pitch, not above it.  If you want a lower pitch, you’ll need to start with a larger disk. By hammering, you can raise the gong pitch as much as a major third; more is possible but that tends to compromise tone quality and also requires more work.  The upshot is: the ideal situation is to start with a flat, untuned gong which is a little below your intended pitch. Best is if it’s about a major second or minor third below, but in any case not more than about a major third below. Here now is a description of the hammering process.  Remember once again that the procedure described here is not a traditional gong-making process at all; it’s just an approach that allows making these very simple aluminum disk gongs without too much work. Before commencing hammering,it’s useful to create a temporary way of holding the gong for pitch-testing during the tuning process. For this purpose, tie loops of very fine string or strong thread

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

through the holes. The fine string is needed because thick strings would get in the way and create unevenness during hammering for tuning. After tuning, the very fine strings can be replaced with something stronger.  To hammer the nipple, you need a flat surface that you can rest the disk on, which has a hollow that you can hammer the nipple into. Since the rest of the gong surface will remain flat, the hole diameter can be much smaller than the gong diameter – typically between about 15% and 18% of gong diameter.   For the surface you can use a piece of plywood, minimum 1/2” thick and about 18” square or larger,  with a circular hole cut out.  You can use a circle cutter for this, and because you might at times wish to hammer nipples of different sizes, you may choose to cut not one but several holes of different sizes in your plywood sheet. The sizes of these holes can range from a little over an inch for very small gongs up to about three inches in diameter for large ones.  Before making each hole cut, use a compass or pencil and string to draw a series of concentric rings around the hole-to-be, spaced not more than an inch apart, up to the diameter of the largest gong you expect to make. These will allow easy centering disks of different sizes over the hole when hammering time comes. You’ll also need to locate the center of the flat disk, if it’s not already somehow marked. There are various ways to do this. One simple method is to take half the diameter and measure that distance

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

from several points along the side to locate the point in the center at which the measurements agree. Rather than hammering directly on the aluminum, you may find it preferable to use a thick hardwood dowel of about 8” long with one end rounded. (A section of broomstick works well.) Place the rounded end of the dowel at the disk’s center point and hammer the end of the dowel.  For smaller gongs calling for a smaller nipple, you can substitute something like a 1/2” or 3/8” steel rod with rounded end. Hearing protection in the form of ear plugs or ear muffs is recommended during the hammering process. With the flat disk centered over the hole and the dowel positioned at the center point, begin hammering.  With 3/32” aluminum, make your strokes moderately firm but not hard.  With thicker material, you can strike a bit harder. After each stroke or two, lift the gong by its suspension cords and strike at the center with a suitable mallet to test the pitch of the gong. The pitch rises quite fast with the first two or three strokes, then slower as you continue. Repeat hammering and testing until the gong has risen to the desired pitch. Try not to overshoot and bring the gong up too high in pitch: while it’s possible to turn the gong over and carefully hammer back the other way to bring the pitch back down, too much of this will tend to create irregularities in the nipple shape which lead to beating effects (a slight waffling in pitch or volume) in the finished gong.

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

Mounting: At this point you may need to replace the fine string used for suspension during hammering with something more durable.  Make a loop of cord through each hole large enough to accommodate whatever you’ll be suspending from.  You can suspend the gongs from whatever you find suitable for the purpose: hang them from a convenient tree branch, use microphone boom stands, or design a nice frame. But here’s an important consideration: when the gong is suspended, the suspension cords should not lie flat against the surface of the gong. If they do, they will rattle when the gong is played.  To avoid this, the cords should angle upward and out from the suspension hole.  Typically this means that whatever they’re hanging from – whatever the loop is looped over – should not be too skinny; it should be wide enough to spread the top of the looped cord out a couple of inches. That’s it.  Once mounted, the gong is done. And you’ll find that while a single gong is nice, a set of gongs tuned to several different pitches is that much more delightful still. The follow-up to this essay has been posted here. It describes a more sophisticated approach to the tuning of disk gongs, taking into account the relationships between the overtones which affect the tone quality in subtle but important ways.  

ALTERNATIVES FOR PITCH CONTROL IN

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

WIND INSTRUMENTS → ← ALUMINUM DISK GONGS (Article #2)

MORE WRITINGS

ADDITIONAL NOTES PERTAINING TO HOMEMADE WOODWIND MAKING MORE ‘MOE FORCED VIBRATION NAMES AND APPEARANCES SORRY-ASS ORGAN THE WINDOW OF AUDIBILITY IMPEDANCE INSTRUMENTARIUM HOPKINIS SAMPLE LIBRARIES PALINDROMES EVERYONE WANTS BEACH-FRONT REAL ESTATE ELASTIC STRINGS AGITATION PIPES MUSICAL INSTRUMENT CATEGORIZATION SYSTEMS ADVENTURES IN FRICTION FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES ALUMINUM DISK GONGS (Article #2) ALUMINUM DISK GONGS (Article #1) ALTERNATIVES FOR PITCH CONTROL IN WIND INSTRUMENTS PLAY HERE SOUNDS AND SILENCE ENGRAVINGS OF EARLY ACOUSTICAL APPARATUS NORTH-SOUTH/EAST-WEST CHIMES MASTERY VS. GO-FOR-A-RIDE OVER-UNDER SCALES

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

THERE IS NOTHING NEW UNDER THE SUN OVERTONES HARMONIC AND INHARMONIC THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE ORIENTATION OF THE OSCILLATION MAN, WHAT A WEIRD WORLD THAT WOULD BE (Where would we be without Hooke’s Law?) SYNTHESIS vs SAMPLING SLOPPINESS SCALE AND TIMBRE

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #1) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #2) Article 2: A More Sophisticated Approach In an article posted prior to this one, I presented the basics for making simple aluminum disk gongs. This follow-up article assumes that you’re familiar with the approach described there, so if you haven’t read that article already, have a look at it before reading this one.  That article discussed things like materials and where to obtain them, and described the simple method for hammering the nipple in the center, by which the gong is to be tuned. This article picks up from there, describing an approach to tuning that is more sophisticated than the rudimentary approach described in the first article.  This article also touches on the underlying acoustics more than the previous one. For photos and audio of the gong set in keeping with the ideas set out in this article, see here.  To reiterate an important Important disclaimer pertaining to both articles: These notes are not about the making of traditional Asian gongs. That timehonored discipline is well beyond the reach of this writing. Rather, these notes concern the making of simple disk hot forging required.

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

One of the key points of discussion in this second article is the musical relationships between the overtones within the gong tone, and I’ll start with an overview of this topic. Gong sounds, like the sounds of most musical instruments, contain not just a single frequency, but many frequencies sounding simultaneously. In their blend, these frequencies help create the characteristic tone quality of the gong. With most musical instruments, our ears effortlessly recognize one of the many frequencies present as the fundamental. We hear that frequency as the defining one which gives the tone its recognizable pitch. Normally this is the lowest of the frequencies present.  Other frequencies above this are thought of as overtones. They contribute to the overall tone quality but are less essential to the sense of pitch. For many instruments, such as most string and wind instruments, the overtones are harmonic. This means that the overtone frequencies appear in a series reflecting a simple mathematical relationship between the frequencies. The timbre of sounds with harmonic overtones typically seem coherent and integrated, and they have well defined pitch. Gongs, however, are different in this regard. Not only are the many pitches present in the gong tone not harmonic in their relationships, but even the idea of a single  linear series of frequencies, with fundamental as the lowest and overtones arrayed in a single orderly sequence above, is a bit shaky. 

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

(This is a characteristic that gongs share with drums and bells.) To get a picture of how this works, we need to look at the physical basis of the frequencies that are sounding. In gongs, as in strings, winds, and other instruments, each of the frequencies within the composite tone arises from one or another pattern of vibration which the physical sounding body engages in, commonly called a mode of vibration. If you’ll excuse a momentary digression (one that I hope will be an instructive), I’ll discuss the examples of strings and some other instruments before returning to gongs. With strings there is a certain recognizable pattern of oscillatory movement that can be referred to as the first mode of vibration. This is the motion that produces the frequency that we hear as the fundamental and that gives the string its recognizable pitch. Simultaneously, as part of the overall string motion, there is a second mode, a third mode, and so forth, each producing its own frequency. In strings, this series of frequencies happens to fit the mathematical definition of a harmonic series. In this case the overtones, with their harmonic relationship to the fundamental, tend to blend very closely with the fundamental in our perception; the ear scarcely recognizes them as separate tones. And, lining up mathematically with the fundamental as they do, they tend to reinforce the pitchsense of the fundamental in our ears, creating a coherent and clearly pitched composite tone. In some other vibrating bodies, such as

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

marimba bars and kalimba tines, a similar series of tones is in play. Once again the lowest frequency present is usually heard as the fundamental, and there is a series of overtones arrayed above. But in these instruments, the overtones typically are not harmonics of the fundamental; they appear at inharmonic frequencies. In such instruments we still usually hear the lowest tone as the fundamental providing a sense of recognizable pitch, but the effect is different from that of harmonic tone qualities. The inharmonic overtones may stand out rather than blending in, and the pitch-sense of the tone may be less clear.  Sometimes, in such instruments, the bars or tines are reshaped to bring one or more of the more prominent overtones into a harmonic relationship with the fundamental. This helps clarify the pitch sense, and creates the impression of a more integrated tone quality. And now back to gongs. Gongs are more complex in that there’s not a single linear series of overtones. With their large two-dimensional surfaces, gongs are capable of many different modes organized along two main types of vibratory patterns. One type of pattern is characterized by circular nodal rings, with movement in which adjacent ringshaped concentric regions flex in opposite directions. The other involves nodes along diameter lines, with sideby-side areas moving in opposite directions.  (These things are difficult to describe in words; have a look at the diagram for a clearer picture. These

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

diagrams are based upon illustrations in Thomas Rossing’s Science of Percussion Instruments, and the thinking underlying this essay owes much to Rossing’s discussion of vibrating disks on pages ­79-80.) Some gong modes are characterized by one or the other type of motion; other modes involve both.  The upshot is that many different modes of vibration are available in a single gong, and when you strike a gong, many modes are excited, producing a blend of many pitches. The frequencies of these modes don’t follow any familiar or musically meaningful pattern; they’re pretty much all over the place. Which of the many available modes are most prominent in the resulting tone depends on several factors including striking location, hardness of the beater and the size of the beater surface making contact. So how does the ear make sense of the gong tone?  Does it recognize one of the many frequencies present as the defining tone, giving the sound a recognizable sense of pitch?  Does it hear it as a sonic wash without meaningful pitch? Does it hear a chordal blend of pitches, some of them recognizable but none serving as a defining pitch?  The answer is that any of these responses may arise with different gong sounds. As one example, http://barthopkin.com/1329-2/[02.07.2019 23:23:34]

ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

in many of the beautiful large Asian gongs you’ve probably heard, there’s a rich sense of tonality, but in the lush complexity of frequencies present, the pitch sense may be ambiguous. And this brings us back to the simple disk gongs that are the subject of this writing. They differ from very large gongs in that with these the tone is not quite so complex, making it easier for the ear distinguish particular frequencies within the tone. Let’s start by talking about the simplest possible gong: a flat circular disk with no boss or nipple at the center and no curvature to the surface. To be more specific, in keeping with the suggestions in the previous article, we’ll take as our model a disk of aluminum, between 3/32” and 5/32” thick, and between about 6” and 20” in diameter. If you strike a disk gong within this size range somewhere off-center with a hard mallet, you’ll get a clangy tone with a mess of different frequencies. None of the modes will be clearly dominant, and as a result you’d be hard put to identify a single defining pitch in the resulting sound.  On the other hand, if you strike in the center with a fairly heavy, soft mallet, the picture clarifies considerably.  There is one frequency that stands out strongly enough that most people would not have difficulty hearing that tone as the defining pitch. In other words, even though there are many frequencies present, the gong produces what the ear can interpret as a recognizable note.  My experiments suggest that this note corresponds to the mode of vibration which has no diameter line nodes, and

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

one concentric nodal ring – that is, the mode identified as {0,1} in the diagram above. Center-striking brings it out strongly, because this mode’s pattern of vibration is most active at the center point. In the sounds of most other musical instruments, the pitch-defining mode, called the fundamental, has the lowest frequency of all the modes present. With these gongs, however, this pitchdefining {0,1} mode is not the lowest frequency present.  That honor goes to the mode identified in the diagram as {2,0}, with two  diameter lines node and no nodal rings.  In a perfectly flat and uniform gong, this tone typically sounds at a musical interval a little more than a major 6th below the {0,1} tone.  But this lower mode is much quieter.  Indeed, when the gong is center-struck, it is barely audible. This is partly because for {2,0} the center point is at the intersection of the two nodal diameter lines, meaning that this mode is not active at the center and the center-strike thus scarcely excites it at all. In practice, a center-strike or near-center strike actually will usually activate this low mode to some degree, but not enough to carry the day in the ear’s unconscious analysis: the much louder {0,1} mode dominates the tone enough to establish itself as the defining pitch. Borrowing from terminology used for carillon bells, I’ll refer to the dominant mode {0,1} as the prime tone, and I will refer to the quieter, lower mode {1,0} as the hum tone. With careful listening it becomes

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

possible to identify several more frequencies in the flat gong sound, all of them higher than the prime tone. Depending on your choice of beater and on how perfectly centered your centerstrike is, different ones among these will stand out more or less. But if you’ve developed your ear to where you can distinguish the more prominent frequencies in a complex tone such as this, a couple of them can be heard standing out fairly strongly. In a perfectly flat gong, one of them appears just over a major ninth above the prime.  Analysis shows that this tone is associated with the mode {1,1} with one diameter-line node and one concentric nodal ring. Another can be heard about two octaves and a minor second above the prime. This is the mode {0, 2}, with no diameter line nodes and two concentric nodal rings. As discussed in the previous article, the flat gong can be tuned by hammering a small rise in the center, called the nipple or the boss. Doing this has the effect of stiffening the disk, causing the pitch to rise. In an ideal situation, the prime tone of the flat disk is below the pitch you intend to tune the gong to, and you can bring it up to its intended pitch by hammering, being careful not to hammer too much and push it sharp. At this point you might ask the question, will hammering the center raise all the modes simultaneously, and by the same amount?  The answer is that it will tend to raise all modes, but not by the same amount. And this suggests an interesting possibility. In the flat gong,

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

the various audible modes produce a set of seemingly random, musically unrelated pitches. If the different modes can be affected differently by reshaping the gong, would it be possible to reshape in such a way that the several audible modes realigned themselves in a more coherent set of pitch relationships? To clarify this idea and provide background information, let me make another digression.  As discussed above, in most of the most familiar musical instruments, including most winds and strings, the overtones are aligned harmonically over the fundamental. With this relationship in place the overtones blend closely with the fundamental to produce a tone of clearly recognizable pitch. In other instruments, such as marimbas and kalimbas, although there may be a clear fundamental with overtones arrayed above it, the frequency relationships are not harmonic.  Listeners can usually still clearly recognize the fundamental and identify the pitch, but the tone quality is more complex and the pitch sense is not as clear; in some contexts there may even be musical confusion created by the prominent and harmonically unrelated overtone pitches present in the tone. Some makers of these instruments – most notably marimba makers – carefully reshape the bars to alter the patterns of movement for certain modes of vibration in a way that shifts the frequency relationships of the fundamental and the most prominent overtones. The intent is to place the overtones frequencies in harmonic http://barthopkin.com/1329-2/[02.07.2019 23:23:34]

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relationships with the fundamental. This has the effect of clarifying the tone and making the pitch sense clearer. Some might say this makes the tone less colorful and more boring, but there can be no doubt that it has the effect of reducing ambiguity and making the tone seem more “in the pocket.” The most obvious frequency relationships to strive for in doing this would position one or more of the overtones in octave relationships to the fundamental. If one or two of the overtones stand at one or two or three octaves above the fundamental, that goes a very long way toward clarifying the tone and disambiguating the pitch sense. That done, there will still be other modes producing overtones at various inharmonic pitches. These may add spice and color, but with those octaves in place the pitch sense still locks in clearly. The techniques for reshaping bars or tines to align the overtones are known and understood, and not dauntingly difficult to apply.  But how might this apply to gongs?  It turns out that, as a practical matter, it is a lot easier to apply these reshaping rules to linear sounding bodies like bars and tines than it is to two dimensional shapes like flat gongs.  In theory it could be done; in fact it could be a very enjoyable project for someone who is both mathematically inclined and skilled in computer-aided manufacturing techniques. But for someone operating in a home workshop, it would be quite a challenge.  For myself, I haven’t so far come up with a viable approach.

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

… But wait!  In an unlikely bit of luck, it turns out that in hammering the small nipple at the center of a flat going, it can fortuitously happen that three of the prominent flat gong modes described above come reasonably close to octave alignment.  As shown in the diagram, the alignment is between the prime mode {0,1}, mode {1,1} at an octave above the prime, and mode {0,2}  at two octaves above the prime. The result is the creation of a gong tone that is more coherent and clear in pitch than a flat gong with the usual inharmonic relationships in the dominant overtones. Yet plenty of inharmonics remain present in other modes, creating a tone that is clear and full, yet still distinctive, interesting, and complex. I happen to love this tone, and that’s why I’m writing this article with a lot of enthusiasm.  Notice that this situation is not so much a matter of deliberate analysis leading to controlled overtone tuning; it is mostly just a matter of luck and happenstance in the way the physics of flat gong modes works out. That said, there is room for skill on the part of the gong maker here, because getting the three tones into alignment is a matter of hammering the nipple just right.  Make the nipple a little different – a little too big or too small, or a little too narrow or a little too spread out — and you’ll miss the alignment. So, starting with a flat gong of reasonable diameter and thickness, it should be possible to create a finished gong with suitably sized nipple in which the overtones thus align, at least approximately.  But now we need to ask, http://barthopkin.com/1329-2/[02.07.2019 23:23:34]

ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

at what pitch?  If you want to tune your gong to any particular pitch, or if you want to make a tuned set, you will need for two things to line up correctly: The point at which you’ve done just enough hammering of the nipple to bring the overtones into alignment must also happen to be the point at which the gong has arrived at its target pitch. Is there a way to plan for that? To answer this question, we can start by asking, “In the process of raising the original pitch of the flat gong by hammering, how much does the pitch of the prime tone need to go up before the overtones come into octave alignment?” If the answer to that is a consistent value, then it should be possible to start with a flat gong whose prime is that much lower than the target pitch, and hammer it up to pitch knowing that, if all goes well, the octaves will align when the target pitch is achieved. In my experience, the needed pitch-rise is about 220 cents.  (A cent is one onehundredth of a semitone. 220 cents is two semitones plus 20/100 of a semitone. Most electronic tuners are calibrated in cents.)  Thus, if you start with flat gong whose prime is 220 cents below the target pitch, and you hammer the nipple just right, then with luck you’ll come reasonably close to octave alignment between the three prominent modes at the point where the prime reaches the intended pitch. But it may be simpler to think in terms of frequencies than cents, so here’s the equivalent information stated that way:

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

the interval of 220 cents corresponds to a frequency ratio which, written as a decimal, comes out to .880.  Thus, you want the frequency of the prime prior to hammering the nipple to be .880 of the frequency of the intended pitch.  We can also translate this into diameters: for two gongs of the same thickness and material, for gong A to be 220 cents higher than gong B, the diameter of gong A should be .938 that of gong B; or (same thing) the diameter of gong B should be 1.066 that of gong A. From here on out, this article gets more technical and possibly boring.  Feel free to bail.  To start, let’s get more explicit about the underlying rules of flat gong tuning: Frequency is inversely proportional to the square of the diameter Frequency is directly proportional to thickness To put this information to use, you need to somehow determine the diameter for a flat gong tuned 220 cents below your target pitch – or, stated the other way, whose frequency is .880 of the target frequency.  There are various ways to do this.  Here’s one: Start with a sample gong of any diameter within the general range of diameters you expect to be working with.  Make note of the frequency and pitch of the gong’s prime note. Hammer the nipple at the center of the gong to raise that to the point where the overtones align reasonably well.  (This hammering/tuning procedure will be discussed in shortly.) If what I’ve said here is true, the desired alignment will

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

happen when the gong is about 220 cents above its original pitch. Make note of the new pitch and frequency. Now you can (assuming some math chops) calculate diameters to yield desired frequencies in other gongs of the same thickness and material based upon this sample gong. The formula is: where f(sample) is the frequency of the sample gong, f(new) is desired frequency for the new gong to be made,  d(sample) is the diameter of the sample gong, and d(new) is the suggested diameter for the new gong to be made, the formula is: d(new) = [square root of f(sample)/ f(new)] times d(sample). In theory, the flat gong you get by this calculation should be the famous 220 cents below the intended pitch, and will arrive at the intended pitch when you tune it up. If you’re tuning to pitches within the standard 12-equal tuning system, then once you’ve got one gong tuned to a pitch within that system, you can use standard factors to determine diameters for gongs to be tuned to other pitches within the system.  I won’t give the full list here, but here’s a start: to make a gong a semitone higher than a known gong, make the diameter 0.9715 as large.  For a semitone lower, make it 1.0293.  These semitone-up and semitone-down factors can be applied repeatedly to determine other intervals within the 12-equal system. All this assumes that you’ll be able to cut gong disks to very precise diameters. 

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

This is a challenge in a home workshop, but not impossible. For my own gong set, I arranged to have the disks cut on a laser cutter. My specs were input and the cutting done by Ian Saxton, with flawless results. Thanks, Ian.  This also assumes you’ve got a good tuner on hand. The simplest approach these days is to use one of the smart phone or computer tuning apps: you play the tone; the app tells you the frequency and the pitch with cents deviation from the nearest 12-equal A440 pitch.  I often use an app called Plusadd Tuner; it’s just of many available. Note that this type of tuner app will decide for itself which of the many frequencies present to regard as the prime or fundamental, indicting the frequency for that tone.  If you strike the gong at the center with the right sort of beater, they app will usually hear your prime correctly.  (For a more sophisticated picture, try using a spectrum analysis app — there are many of these available too — which will create a graph showing all of the frequencies present, allowing you to zero in on whichever of them you’re interested in to get its frequency.) Here are a couple of important caveats.  I have found the inverse-squared relationship for diameter to be admirably accurate for flat gongs as long as the material is consistent and the thickness tolerance is very tight. But if there has been any bending of the supposedly flat gong, even seemingly minuscule and scarcely observable, all bets are off. As for the direct proportion rule for

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

thickness: I have found this to be much less dependable.  For that reason, I recommend that you work from a sample gong for each thickness you intend to use, making calculations from there based on diameter, and avoid using the thickness rule to calculate expected frequencies from one thickness to another. You could ask: are there ways to calculate flat gong frequencies from scratch without having to work from samples? The answer is yes, but to do so you need to know the properties of the material you’ll be working with. The math is much more complex and in any case these properties may vary for materials from different sources, and it’s probably both simpler and more dependable to work empirically from the samples. And now, more details about the actual gong cutting and hammering processes. As discussed in the earlier article, you can sometimes find aluminum disks ready cut at metal scrap yards.  If you aren’t tuning to a particular scale, you can make very nice gongs at low cost from this material. For tuning to particular pitches and scales, however, you’ll need disks of certain precise sizes, and so will have to cut them yourself from larger sheets or have them cut for you. The hammering process is described in the previous article. Please refer back to that for the basic step-by-step.  But where that article speaks of tuning the prime tone only, we’re now going to talk

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

about tuning two or three modes. To start with, you’ll be listening for the pitch of the prime mode {0,1}.  To bring out this mode when you check for pitch, strike at the center with a fairly large, soft, heavy beater such as a large superball on the end of a stick.  If you use an electronic tuner, the tuner will probably be able to read this mode without too much difficulty.  You can also tune this mode by ear by comparing the gong pitch to the desired pitch sounded on an electronic keyboard or some other tone source. As you continue the process, you’ll also be listening for the two overtones. For these it will be more difficult to recognize and identify the pitch. Under normal circumstances the tuner won’t be able to recognize these pitches, and without a lot of practice most people find it very difficult to hear and them clearly with the naked ear. (This is where spectrumanalysis software may be useful, if you want to get into it on that level.)  To help with this, here are some tricks for bringing out the sound of these modes more prominently. With these methods the overtone pitches will be easier to hear by ear, and you may even be able to isolate them enough to allow the tuner to pick up on them. For the prime mode {0,1}, as mentioned above, you can do quite well by center-striking with the heavy, soft beater. Suspending from points somewhere in the nodal ring at about 1/3 of the disk’s radius from the outer edge, as recommended above, also helps bring out this http://barthopkin.com/1329-2/[02.07.2019 23:23:34]

ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

mode. For the mode {1,1} which we’re hoping to align at an octave above, strike not at the center, but in that 1/3 r ring, using a slightly smaller, harder mallet (but not overly hard, which will produce too much clang sound).  This trick of striking in the nodal area of the prime will tend to excite the prime less, making it easier to hear the {1,1} mode instead. You can further improve the result by resting a finger gently at the center of the gong while striking, which will also tend to damp the prime along with other modes that are active at the center. To bring out the {0,2} mode that we’re hoping to align at two octaves above, strike the gong gently but very accurately at the center with a small, hard beater such as the end of a ¼” steel rod.  This may be good enough to let you hear {0,2} clearly, but you can also try a gentle finger at a point a little more than half the radius in from the edge. This may help to quiet the prime, allowing you to hear {0,2} a bit better. The idea is that as you bring the prime up to its intended pitch, the other overtones will align at something close to the hoped-for octave relationship as well. I confess that I’m still depending a bit on luck to get those overtones to align, and I’m not always successful.  But my luck has been a little better when I make the nipple fairly narrow.  This can be done by hammering it into a relatively

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

small hole in the board you use for backing when hammering – close to one inch for smaller gongs; a little bigger for larger gongs. In this article I’ve been talking about a particular set of overtone relationships involving octave alignments. This is by no means the only possible set of relationships to strive for or to enjoy in the gong sound.  Approaching the hammering a little differently will yield different relationships. For instance, hammering over a much larger hole will result in a larger nipple and a generally more curved gong surface, resulting in other overtone configurations.  Similarly, doing more hammering to raise the prime more than I’ve suggested will likewise alter the outcome. These things can be fun to explore.  Effective tunings can be found in which a prominent overtone is tuned an octave and a fifth, or perhaps two octaves and a fifth, above the prime. And please remember that you are in no way required to pursue the idea of overtone tuning if you don’t want to. You may prefer to ignore the overtone question entirely!  If you do that (as described in the first of this pair of articles), you’ll get gong tones with whatever relationships between the different modes you happen to end up with. The results may sometimes seem odd or exotic, but more often than not the resulting gong sound will be pleasing to the ear.

ALUMINUM DISK GONGS (Article #1) → ← FUNDAMENTAL, HARMONICS,

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

OVERTONES, PARTIALS, MODES

MORE WRITINGS

ADDITIONAL NOTES PERTAINING TO HOMEMADE WOODWIND MAKING MORE ‘MOE FORCED VIBRATION NAMES AND APPEARANCES SORRY-ASS ORGAN THE WINDOW OF AUDIBILITY IMPEDANCE INSTRUMENTARIUM HOPKINIS SAMPLE LIBRARIES PALINDROMES EVERYONE WANTS BEACH-FRONT REAL ESTATE ELASTIC STRINGS AGITATION PIPES MUSICAL INSTRUMENT CATEGORIZATION SYSTEMS ADVENTURES IN FRICTION FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES ALUMINUM DISK GONGS (Article #2) ALUMINUM DISK GONGS (Article #1) ALTERNATIVES FOR PITCH CONTROL IN WIND INSTRUMENTS PLAY HERE SOUNDS AND SILENCE ENGRAVINGS OF EARLY ACOUSTICAL APPARATUS NORTH-SOUTH/EAST-WEST CHIMES MASTERY VS. GO-FOR-A-RIDE OVER-UNDER SCALES THERE IS NOTHING NEW UNDER THE SUN

http://barthopkin.com/1329-2/[02.07.2019 23:23:34]

ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

OVERTONES HARMONIC AND INHARMONIC THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE ORIENTATION OF THE OSCILLATION MAN, WHAT A WEIRD WORLD THAT WOULD BE (Where would we be without Hooke’s Law?) SYNTHESIS vs SAMPLING SLOPPINESS SCALE AND TIMBRE

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

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ALUMINUM DISK GONGS (Article #2) | Bart Hopkin

Copyright Bart Hopkin, 2019. All rights reserved. Designed by Blue Oak Web Design

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES As many readers will already know, most musical sounds contain multiple frequencies. These frequencies come about as the results of different modes of vibration — that is, different patterns of vibratory movement — coexisting in the physical body that is the source of the sound. Certain terms are commonly used in talking about how the frequencies blend and how the ear and brain make sense of them to arrive at a sense of pitch and timbre. These include words like fundamental, overtones, partials, and harmonics. For many musical sounds this terminology works well insomuch as it reflects the ways that people perceive and respond to multi-frequency tones. But for many other sounds, the terminology doesn’t fit quite so well, in that it doesn’t correspond very well to either the way we perceive the sounds or the physical reality that underlies them. In this essay I’ll be talking mostly about the cases where the fit isn’t so good … but first, let’s have a look at the cases in which the terminology does work well. The good-fit cases are the ones where the tone is made up of steady frequencies which can be seen as lining Share This

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from lowest to highest. For this to work, we also have to assume that the lowest tone of the series isn’t so low in frequency as to be below the bottom of the hearing range, and that higher tones in the series aren’t confusing the ear by being far louder than the lowest. The best examples of this are the sounds of strings and tubular wind instruments. Many synthesized electronic instrument sounds also fit the bill nicely, as they are actually designed to work this way. For sounds such as these, the ear typically recognizes the lowest frequency as the defining pitch, commonly called the fundamental. The overtones above blend more or less imperceptibly into the overall tone, but at the same time they contribute to the perceived timbre or tone quality. An important additional consideration in these cases has to do with the relationships between the overtone frequencies and the fundamental. If the frequencies of the overtones have a certain simple arithmetic relationship to that of the fundamental, then they qualify as harmonics. If they don’t happen to fall into that mathematically defined relationship, then they are inharmonic. Harmonic timbres are easiest for the ear to make sense of. The best examples of this again are most winds and string instruments, which typically meet fairly accurately the criteria for harmonicity. And, again, a lot of synthesized tones are deliberately built this way as well. In these harmonic tone qualities, the ear easily recognizes the pitch in an

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

instantaneous, effortless and unconscious process. By contrast, some other instruments produce inharmonic overtone series, and here the process of pitch recognition is a bit less dependable. With many inharmonic timbres, particularly those which happen to have a single series of widely spaced overtones, pitch recognition still happens pretty easily much of the time. The ear picks up on the lowest component as the fundamental which anchors the pitch-sense. However, when it comes to reinforcing this pitch sense, the inharmonic overtones don’t do as good a job as harmonics. Nor do they blend as closely: while they do contribute strongly to the characteristic tone color, they may seem to stand out more as separate tones. Examples for these sorts of inharmonic tone qualities are kalimba tines and simple rectangular marimba bars (although, as we’ll see in a minute, things can get more complicated with marimba bars.)   DIFFERENT SORTS OF VIBRATION PATTERNS AND THEIR PERCEIVED EFFECTS RIPTION

EXAMPLES

ndamental

Most wind

onic

and string

PERCEIVED EFFECT

Clear, integrated tone; unambiguous pitch sense above

instruments

Overall pitch sense usually ndamental

kalimbas,

rmonic

marimbas,

recognizable but sometimes confusing; overtones may stand above

some drums out as separate pitches 

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin armonicly

Some sense of pitch, but it may Many bells,

equencies,

be fickle or confusing; Some of many gongs,

dest but

the frequencies may tend to some drums

e lowest

stand out rather than blending in

armonically

No identifiable pitch. Individual Many gongs,

equencies,

frequencies may or may not most cymbals,

them

seem to blend into a single some drums, 

s

sound. 

shifting ble

Wind, traffic,

No identifiable pitch, more like

es crowded

ocean

noise than tone. 

In these cases where there’s a coherent overtone series but it’s inharmonic, if the inharmonic overtones are too strong and the fundamental is insufficiently prominent the sound, the ear may fail to recognize the fundamental. Then, the sound may come across as a blend of pitches with none standing out as the defining pitch. Or it may happen that the ear may focuses on some other particularly loud overtone to provide pitch-sense. To repeat, the cases I’ve just been describing are those in which there’s a single overtone series in which the lowest tone of the series stands out clearly as a fundamental. These are the cases where it usually makes sense to think in terms of a pitch-defining fundamental joined by overtones which may be harmonic or inharmonic. (I say it makes sense usually rather than always because of the exception noted at the end of the last paragraph.) Let’s

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

continue to look a little more closely at these examples, starting with strings. One of the reasons things turn out so neatly with strings is that musical strings represent a uniquely simple physical vibrating system. To wit: In the ideal, a musical string is in effect a onedimensional body, its mass uniformly distributed over its length, with no inherent stiffness, held under high tension at perfectly immobile anchor points. Of course strings in the real world don’t live up to the ideal, but they come close enough to give us something very close to the desired result: the lowest of the frequencies present (the fundamental), is usually loudest, and assuming that the string is well made and uniform in shape, the overtones generally come out quite close to the harmonic ideal. As a result, in our perception the pitch sense is clear and unambiguous, and the harmonic overtones blend into the whole so closely that the tone seems integrated and coherent. With a few caveats, the situation is similar for well made tubular wind instruments: although the physical basis is different, a relatively simple and orderly, more or less one-dimensional physical situation gives rise to a coherent sound with a clear fundamental and an orderly harmonic overtone series. But as soon as you start to get into other sound sources, things get more complex. Consider, for instance, a flat disk gong. Like the string, this is in many ways a very simple form — in this case effectively two-dimensional, defined by a single radius value (which is to say, http://barthopkin.com/fundamental-harmonics-overtones-partials-modes/[02.07.2019 23:24:22]

FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

circular), uniform in thickness and rigidity, and so forth. But the flat gong turns out to be considerably more complex in its resulting frequency blend. The first thing to note is that the relationships between the frequencies produced by simple disk gongs are not harmonic; they’re pretty much all over the place and any seemingly musical frequency relationships you might be able to pick out are purely coincidental. Furthermore, being two dimensional, gongs don’t have a single series or progression of overtones going from low to high. A plausible analysis would be that they have two series, one corresponding to modes of vibration featuring concentric nodal rings, and one corresponding to modes featuring nodal diameter lines … but then, it’s more complicated than that, because there are also modes that combine the two. The ear knows none of this, of course; it just hears a whole lot of inharmonically related frequency components combining to create a gong-like tone quality. For some gongstrokes the ear may perceive one of the many frequencies present as the defining pitch, and it would be tempting to call that one the fundamental. But it won’t necessarily be the lowest frequency present, nor necessarily even the lowest of either of the two main types of vibration pattern. And in fact, two listeners hearing the same gong tone might perceive different frequencies as the fundamental; even the same listener might hear different frequencies as the defining pitch with different strokes. This, then, is a case

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

where thinking in terms of a single pitchdefining fundamental, plus a supporting cast of higher overtones adding color to the tone, doesn’t work so well. Consider another example. Rectangular bars, such as xylophone bars, are mostly played in a way that brings out one particular set of modes which can plausibly be seen as having a lowest “fundamental” tone followed by a series of (generally inharmonic) overtones. But those same bars have many other physical vibrating patterns as well, including separate series for transverse modes in the side-to-side direction, torsional modes (vibratory patterns based in twisting motions), and longitudinal modes (vibratory patterns involving pressure wave fronts running back and forth along the length of the bar.) All may affect the tone. The result is decidedly complex and irregular. For this reason, bar-percussion instrument makers and players do many things to tame the complexity and present something more coherent to the ear — things such as choosing the right mallet, and striking only at the center of the top surface of the bar (not the sides or ends). Most importantly, bar makers may reshape the bar in ways that alter the relationships between the frequencies of the different modes, bringing some of the most prominent ones into harmonic relationships. Here’s another very interesting case study: carillon bells. In some ways bells are similar to the gongs discussed earlier: they tend to produce many inharmonic modes involving

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

combinations of nodal lines and nodal rings. Makers of large bells go to great ends to manage the relationships between these modes, with the idea of tuning the most prominent among them to relationships that will make musical sense to the ear and help the listener to focus on the frequency of one mode in particular as the defining pitch. Nonetheless, you may have had the experience of listening to a melody played on a set of carillon bells, and at some point realizing that your ears were tracking some unintended mode in the bell sound as the defining pitch. Interestingly, the mode that is intended and usually heard as the defining pitch is not the lowest of the frequencies present. There is quieter but still audible lower frequency within the carillon bell tone known as the hum tone. The mode that gives the bell its defining pitch, meanwhile, is variously known as the strike tone, the prime, or sometimes, confusingly, as the fundamental. I have found this bell terminology to be useful in other contexts. Recently I made a set of disk gongs in which I managed to adjust the inharmonic relationships between three of the most prominent modes to where they aligned in much more coherent relationships. (Specifically, I got them pretty close to octave relationships. I did this through a combination of careful sizing of the gongs and careful hammering of the nipple at the center. You can read about it here.) Like the big bells, these tuned gongs have a clear defining pitch. Borrowing the carillon terminology, this pitch can be called the strike tone or http://barthopkin.com/fundamental-harmonics-overtones-partials-modes/[02.07.2019 23:24:22]

FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

prime. Then there’s a quieter lower pitch, which can be thought of as the hum tone. And of course there are a lot of other frequencies present. The tone perceived as the prime is one of the three prominent modes tuned to octave relationships; there’s another an octave above and another two octaves above. Together these reinforce very nicely the sense of pitch, while additional quieter inharmonics add a bit of color. I could continue with further examples of increasingly complex or irregular shapes, pointing out their their resulting complex and irregular frequency blends  … considering that, indeed, most objects in the universe are complex and irregularly shaped … but I’ll refrain and just jump to the main point of this article, which is: Aside from specialized cases such as strings, carefully designed wind instruments, and the artificial timbres of many electronic instruments, most physical vibrating bodies are not well described by saying they have a single fundamental followed by a series of harmonics. Even when we open the door to inharmonics, the notion of a fundamental plus a single orderly series of overtones doesn’t fit all or even most sounds in the real world.  I’m happy to use the terminology of fundamentals and overtones in those cases where it aptly describes what’s happening both physically and in our perceptions. In some other cases, as just discussed, it seems useful to borrow terminology from the carillon bells.

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

Those are cases where we can speak of a prime tone as the defining pitch, without implying that that defining pitch is the lowest tone present, nor that it is the fundamental tone of a single overtone series. And then there are the other cases, most common in nature even if they aren’t the most favored in musical contexts. These are the cases in which many modes of vibration are present with all of their corresponding audible frequencies, and the effect on the ear tends to be ambiguous as to which of them, if any, is most naturally heard as a defining pitch. In these cases (which for me are often most interesting) I’ve come to feel that the best way to think and talk and describe the situation is to reduce it to basics without preconceptions about the functions of the different frequencies present. One can simply think of the vibrating body as having multiple modes of vibration producing different frequencies. Then try to be realistic and open-minded which modes are most audible, what the relationships between them happen to be, and how the ear makes sense of the blend. Words like fundamental, overtone, harmonic and inharmonic become less important in such cases. Modes becomes the key word and the essential concept. For another essay exploring ideas related to this one, see here.    

ALUMINUM DISK GONGS (Article #2) →

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

← ADVENTURES IN FRICTION

MORE WRITINGS

ADDITIONAL NOTES PERTAINING TO HOMEMADE WOODWIND MAKING MORE ‘MOE FORCED VIBRATION NAMES AND APPEARANCES SORRY-ASS ORGAN THE WINDOW OF AUDIBILITY IMPEDANCE INSTRUMENTARIUM HOPKINIS SAMPLE LIBRARIES PALINDROMES EVERYONE WANTS BEACH-FRONT REAL ESTATE ELASTIC STRINGS AGITATION PIPES MUSICAL INSTRUMENT CATEGORIZATION SYSTEMS ADVENTURES IN FRICTION FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES ALUMINUM DISK GONGS (Article #2) ALUMINUM DISK GONGS (Article #1) ALTERNATIVES FOR PITCH CONTROL IN WIND INSTRUMENTS PLAY HERE SOUNDS AND SILENCE ENGRAVINGS OF EARLY ACOUSTICAL APPARATUS NORTH-SOUTH/EAST-WEST CHIMES MASTERY VS. GO-FOR-A-RIDE OVER-UNDER SCALES THERE IS NOTHING NEW UNDER THE SUN

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FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES | Bart Hopkin

OVERTONES HARMONIC AND INHARMONIC THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE ORIENTATION OF THE OSCILLATION MAN, WHAT A WEIRD WORLD THAT WOULD BE (Where would we be without Hooke’s Law?) SYNTHESIS vs SAMPLING SLOPPINESS SCALE AND TIMBRE

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SCALE AND TIMBRE | Bart Hopkin

a



SCALE AND TIMBRE Since his death the American composer Harry Partch has had a profound impact on contemporary music, at least in academic and intellectual circles. His main contribution has been that he got people seriously thinking about tuning systems. He wasn’t alone in this, but he was arguably the strongest voice at a time when such a voice was much needed. There’s no need for me to say much about Partch, as he’s much written-about and celebrated elsewhere. But I will comment on one incongruity in his work. Partch’s concern about tuning systems was grounded in the idea that just intonation systems most accurately reflect the ways our ears and brains respond to music. Just tunings are those that, unlike the equal-temperament system in wide use today, are built around the ratios between the fundamental frequencies of the notes in play. This reflects the fact that our ears do naturally seem to interpret musical intervals on the basis of the frequency ratios involved. Partch was quite rigorous in his approach to these things – rigorous in his math, and rigorous in his insistence on accurate tuning. But here’s the odd thing: Partch was also instruments, and many of his http://barthopkin.com/175-2/[02.07.2019 23:26:14]

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SCALE AND TIMBRE | Bart Hopkin

instruments were percussion instruments which produce prominent inharmonic overtones. That means that, for all his concern with getting the pitches of his music just right, the music was filled with overtone pitches that were, as often as not, way out in left field someplace. Our ears are amazingly adept at recognizing a single pitch, usually the fundamental, as the single defining pitch within a musical tone, even when that tone may have a lot of other frequencies present in the form of overtones. The process occurs effortlessly and unconsciously. In the end result the listener is much aware of the fundamental pitch and its musical meaning, but almost entirely unaware of the overtone frequencies present except insofar as they contribute to the listener’s sense of the tone quality or timbre. Thus it’s easy to see how someone thinking about just intonation could obsess over getting the fundamental pitches just right, all the while remaining oblivious to the other pitches present in the form of overtones. Harmonic overtones are overtones whose frequencies line up with the frequency of the fundamental in specific simple arithmetic relationships. Inharmonic overtones are overtones that don’t line up that way. When the overtones are harmonic, the most prominent overtone frequencies typically are, for the most part, not in serious conflict with the scale in which the music plays. For the majority of music and the majority of scales, most of the prominent

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SCALE AND TIMBRE | Bart Hopkin

harmonic overtones align themselves nicely with the intended pitches and harmonies. (The alignment isn’t perfect for the equal tempered scale; this is a commonly-cited argument against that scale and in favor of just scales.) But when the overtones are inharmonic, those overtone pitches represent a lot of pitch information that, in all likelihood, is unrelated to the intended scale. Does this matter? After a lot of listening and thinking about this, my sense is that those unaligned overtones do affect the listener’s sense of tonality. At the same time, I think that most listeners are pretty good at unconsciously masking this irrelevant harmonic information so as to be able to focus on the intended fundamental pitch information. I think we can say that the tonal meaning of any given piece of music is likely to be clearer when the overtones align with the intended pitch information of the fundamental tones. Stated differently, when there’s a lot of unintended and unacknowledged pitch information flying around in the form of prominent inharmonic overtones, the overall effect of the music is less clear. I hasten to add: this need not be seen as a bad thing. The added complexity may just as well enrich the effect as detract from it. The most interesting work in this area has been done by William Sethares. In various articles and in his book Tuning Timbre Spectrum Scale, he discusses the idea that if you think of timbre as the tone quality of a musical sound as manifest in its particular overtone recipe, then scale and timbre are intimately

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SCALE AND TIMBRE | Bart Hopkin

related. If the frequencies of the overtones align with the frequencies of the scale, then the resulting music has a quality of integrity and clarity. If they do not, then the overtones introduce a lot of irrelevant pitch information into the mix, and integrity and clarity are compromised. Working with computers and synthesized tone qualities, Bill did some really fascinating investigations based on this idea. He constructed odd scales and odd timbres to match, such that the overtones within the timbres bore coherent relationships to the intervals of the scales. Music played in these matched scale-and-timbre systems indeed sounded remarkably coherent, even when they might seem on paper to be very peculiar scales and timbres indeed. As one example, he created stretched scale-and-timbre combinations. For example, he made scales constructed much like a just major scale, but with all the intervals proportionally enlarged so that the octave was substantially larger than the 2:1 octave that seems natural to our ears. To play these scales, he created timbres analogous to harmonic timbres but correspondingly stretched. Such a stretched scale, played with conventionally harmonic timbres, sounds bizarre and discordant. Likewise, the timbre with stretched harmonics, when used with a standard just major scale, sounds definitely off. But as a matched scale-and-timbre system, the two together sound unusual yes, but remarkably natural and coherent. A while back I built a big, multifaceted http://barthopkin.com/175-2/[02.07.2019 23:26:14]

SCALE AND TIMBRE | Bart Hopkin

instrument that I call bands and bars. One thing about it is relevant to the current discussion. The instrument includes several sounding elements in the form of metal bars long enough to make their fundamental frequencies subsonic – well below the hearing range, thus inaudible. That leaves many of the overtones in the heart of the hearing range, and the relationships among those overtones are thoroughly inharmonic. The instrument, multifaceted as it is, also includes other sounding elements in which the fundamentals sound quite clearly and for which the ear easily recognizes the fundamental as the defining pitch. So when it came time to decide on the tuning of these various elements, I found myself thinking about the Bill Sethares question: could the peculiar relationships of the audible overtones in the long bars inform my thinking about what sorts of scale to tune the other elements to? My approach was entirely non-mathematical. It was influenced by that fact that the instrument was already odd and esoteric in both construction and sound. To counterbalance this, in the tuning I wanted to create something that listeners could effortlessly relate to. So I engaged in a kind of meditation: I played the long bars, listened to their peculiar inherent tonal blends, and let them suggest to me what the tuning for the remainder of the instrument might be. I allowed those bars to create a tonality in my mind. I helped the process by sort of humming along. From what might have seemed like a wilderness of arbitrary tones I looked for relationships

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SCALE AND TIMBRE | Bart Hopkin

that seemed to have musical meaning. Somehow out of this I arrived at a feeling for a particular tonality, and I built the tuning for the instrument as a whole around that. ← SLOPPINESS

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ADDITIONAL NOTES PERTAINING TO HOMEMADE WOODWIND MAKING MORE ‘MOE FORCED VIBRATION NAMES AND APPEARANCES SORRY-ASS ORGAN THE WINDOW OF AUDIBILITY IMPEDANCE INSTRUMENTARIUM HOPKINIS SAMPLE LIBRARIES PALINDROMES EVERYONE WANTS BEACH-FRONT REAL ESTATE ELASTIC STRINGS AGITATION PIPES MUSICAL INSTRUMENT CATEGORIZATION SYSTEMS ADVENTURES IN FRICTION FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES ALUMINUM DISK GONGS (Article #2) ALUMINUM DISK GONGS (Article #1) ALTERNATIVES FOR PITCH CONTROL IN WIND INSTRUMENTS PLAY HERE SOUNDS AND SILENCE

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SCALE AND TIMBRE | Bart Hopkin

ENGRAVINGS OF EARLY ACOUSTICAL APPARATUS NORTH-SOUTH/EAST-WEST CHIMES MASTERY VS. GO-FOR-A-RIDE OVER-UNDER SCALES THERE IS NOTHING NEW UNDER THE SUN OVERTONES HARMONIC AND INHARMONIC THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE ORIENTATION OF THE OSCILLATION MAN, WHAT A WEIRD WORLD THAT WOULD BE (Where would we be without Hooke’s Law?) SYNTHESIS vs SAMPLING SLOPPINESS SCALE AND TIMBRE

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SCALE AND TIMBRE | Bart Hopkin

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SCALE AND TIMBRE | Bart Hopkin

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE What is it about some sounds that gives them clear pitch, while other sounds come across as pitchless noise? The simple answer is that pitched sounds have a steady repetition rate, or frequency, that the ear can pickup up on and identify. Noisy sounds are more chaotic and irregular; the ear does not recognize a steady frequency. A deeper look at the question reveals a bit more complexity: some sounds – many sounds, in fact – have one or more steady frequencies, but those steady frequencies are mixed in with some amount of noisy irregularity. In fact, virtually all musical sounds aside from pure synthesizer tones have some unpitched components. For instance, the flautist blows across the hole in the mouthpiece and a pattern of air flow develops at the edge of the hole which sets up a clear tone in the resonant air column, and that’s most of what we hear. But there inevitably is also some chaotic turbulence at the blow-hole edge as the air stream rushes across. The unpitched edge-turbulence sound is typically much quieter than the clear tone of the air column resonance, and it’s usually not consciously noted, but it Share This

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can often be heard as a sort of breathy

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

overlay. Even unnoticed it contributes meaningfully to the sound quality ̶ a subtle unpitched component adding a bit of character to the otherwise clearly pitched flute tone. The most famous maker of noise instruments was the Italian Futurist Luigi Russolo, working in the 1920s. He built and composed for a series of instruments which he called Intonarumori – “noise intoners” – designed to have some degree of recognizable pitch in otherwise very noisy sounds. Many of them used strings and membranes agitated in ways that brought out strong noise components. The original instruments no longer survive, but recreations have been built, based upon written records. While I haven’t made a noise orchestra to rival Luigi’s, I have made quite a few instruments in this in-between territory myself. Several but not all have been flute-like instruments. A few examples: The breathy component of flute tone described above interests me, so I recently made a flute-like instrument specifically to bring it out. This breath flute (as I’ve been calling it) is an endblown flute set up in a way that makes it hard to get the strong air column resonance that flautists learn to produce with such clarity. Instead, it yields a much less focused tone in which the pitch of the air column resonance can be heard, but half-buried in the breathy, unpitched edge-turbulence sound. The lack of a strong resonance in the air column means that this flute is pretty

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

quiet, so a tiny electret microphone is fitted into a hole in the pipe where it can listen directly to the sound within. In addition to reinforcing the volume, the mic brings out a distinctive inner-aircolumn sound quality. When the breath flute is heard alone, the listener’s ear recognizes the pitch of the air column resonance and easily follows melodies the flute plays, despite the presence of all that unpitched information in the tone. It’s an evocative effect, the flute tone in among the all the windy breathiness, the pitched and the unpitched somehow moving together and integral to one another. But in more crowded musical contexts, as when other instruments are present, it becomes harder to sort out the flute’s sound. Then it may no longer be possible to follow the flute’s melody, and the effect in general is noisier, less musically meaningful, and in most cases less appealing. Another instrument that mixes tone and noise is my waffle-gurgle flute. It’s a flute-like pipe with toneholes, but instead of blowing across an opening to create a clear edge-tone, the player blows through something similar to the comic novelty noise maker known as a rubber razzer. “Den ve Heil! Pphhttt! Heil! Pphhttt! Right in der führer’s face!” … If you If know this World War Two song from the Spike Jones band, then you know the sound of a rubber razzer. For the waffle-gurgle flute the razzer-like component is basically a sort of droopyfloppy latex tube a couple of inches long, easily made from a balloon neck. The neck droops into what would normally be the blowing end of the flute http://barthopkin.com/249-2/[02.07.2019 23:25:15]

THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

pipe. When you blow through it, the razzer flaps around crazily trying to make the pphhttt sound within the restricted space of the tube, creating a chaotic raspy sound. From out of the resulting generalized agitation of the air, the air column within the pipe selectively reinforces its own natural resonance frequencies. As with standard flutes, the pitch of this resonance depends on the length of the tube and the configuration of open and closed tone holes. What the listener hears is the razzy sound, as crazy and chaotic as ever, but with the clearly pitched tone of the air column resonance highlighted enough to give it an identifiable pitch. By the usual flute fingering techniques you can play melodies. I’ve also made a gurgle organ, with air tubes leading into water vessels of different sizes and resonances. The player blows into selected tubes by means of a harmonica-style mouthpiece, to create semi-pitched bubbly sounds.  The bubbling agitates the air in the vessels, and the vessels bring out a sense of pitch by selectively resonating their own natural frequencies.   One more flute-like pitched-noise instrument: Scraper Flutes are tubes made of medium-hard plastic with guirolike ridges filed into the top surface. Using a short stick to scrape over the ridges, you get the guiro-like sound you’d expect, but with the tones of the pipe resonances highlighted. The scraping action agitates the walls of the pipe enough to excite an air resonance at a pitch determined by the pipe’s

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

length. I’ve made these instruments in two forms: as individual tubes with tone holes to vary the pitch, and as tuned sets of tubes producing one note each through their differing lengths. If you don’t like working with plastic, you can take a cue from the late instrument maker Darrell DeVore and make a similar instrument with bamboo. I group together all of the abovementioned flutes under the term “agitation flutes”. Here’s one more pitched-noise instrument, this one not flute-like. You know you can make a nice raspy sound by scraping a fingernail or some sort of plectrum over the teeth of a comb. Often (depending on the comb) the ear can recognize some identifiable pitch among the noisy scraping sound. This pitch, if it is present, is the pitch of the individual teeth behaving a little like tiny lamellaphone (kalimba) tines. If all the teeth are the same length and thickness, they should all produce more or less the same pitch, and this is the pitch you hear cumulatively when you scrape. But with most combs this pitched tone is to some degree masked by all the unpitched noise mixed in there. A while back I made a tuned set of about 30 combs covering over two octaves by filing the teeth of each comb to the length that gave the desired pitch. The combs were mounted on a soundboard for musical scraping. The noise component in the sound is of course very strong, and it’s fair to say that it’s a pretty irritating sound, but irritating in an interesting way.

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

All of the instruments described here – the combs especially – have the quality mentioned above, that their pitched qualities get buried and lost whenever the musical context is full or complex. Their appeal is most apparent in more spacious contexts when the listener can take in the tonal complexity without losing the thread.

ORIENTATION OF THE OSCILLATION → ← OVERTONES HARMONIC AND INHARMONIC

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ADDITIONAL NOTES PERTAINING TO HOMEMADE WOODWIND MAKING MORE ‘MOE FORCED VIBRATION NAMES AND APPEARANCES SORRY-ASS ORGAN THE WINDOW OF AUDIBILITY IMPEDANCE INSTRUMENTARIUM HOPKINIS SAMPLE LIBRARIES PALINDROMES EVERYONE WANTS BEACH-FRONT REAL ESTATE ELASTIC STRINGS AGITATION PIPES MUSICAL INSTRUMENT CATEGORIZATION SYSTEMS ADVENTURES IN FRICTION FUNDAMENTAL, HARMONICS, OVERTONES, PARTIALS, MODES ALUMINUM DISK GONGS (Article #2) http://barthopkin.com/249-2/[02.07.2019 23:25:15]

THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

ALUMINUM DISK GONGS (Article #1) ALTERNATIVES FOR PITCH CONTROL IN WIND INSTRUMENTS PLAY HERE SOUNDS AND SILENCE ENGRAVINGS OF EARLY ACOUSTICAL APPARATUS NORTH-SOUTH/EAST-WEST CHIMES MASTERY VS. GO-FOR-A-RIDE OVER-UNDER SCALES THERE IS NOTHING NEW UNDER THE SUN OVERTONES HARMONIC AND INHARMONIC THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE ORIENTATION OF THE OSCILLATION MAN, WHAT A WEIRD WORLD THAT WOULD BE (Where would we be without Hooke’s Law?) SYNTHESIS vs SAMPLING SLOPPINESS SCALE AND TIMBRE

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

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THE TERRITORY BETWEEN CLEAR PITCH AND PURE NOISE | Bart Hopkin

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