In years past I have made several musical instruments using rotating disks or cylinders, in which the rotation enables some kind of recurring disturbance in the audio frequency range. These instruments have been of two types: musical sirens and Savart’s wheels. Just recently I’ve done new and more refined versions of each of these types benefitting from CNC (computer numeric control) fabrication methods. This was done with the help of friends who have the excellent CNC chops that I lack. I’ve written about both of these instruments elsewhere but in this article I’ll provide brief updates on the new versions, followed by a longer discussion of one particular area of focus — namely, the question how to manage the layout of the disk or drum to get the tunings you want. That topic may sound kind of obscure, but for anyone interested in simple scale theory in a real-world application, it makes for an interesting dive.

To start, here are brief descriptions of the two types.

**Savart’s Wheel**: I’ve made Savart’s wheels in two forms: a cylinder and a set of graduated disks. This description is of the cylinder version; for a description of the graduated disk version (and a lot of additional information) see this article. The idea in its most basic form involves a scraper-like situation, in which a stick or plectrum of some sort scrapes over a ridged surface, but with this crucial feature: the instrument is designed so that the number of ridge-bumps per second is stable, not varying as with other scrapers. That ridge-bumping frequency stands out as the audible pitch. Savart’s Wheel brings together many tracks of carefully spaced ridges to make a scale’s worth of scraping frequencies available. The instrument consists of a cylinder or drum, about 8” in diameter and 15” long, mounted horizontally on a speed-controlled motor-driven spindle. The surface of the drum is engraved with closely spaced valleys, creating ridges between them. The player plays by holding a special plectrum against the ridges as they rotate past. This agitates the plectrum at some number of ridge-hits per second, and that frequency corresponds to the pitch that you hear. There are several tracks of ridges alongside one another on the cylinder, each circumferencing it, each about 3/8” wide. Within each track the ridges are spaced differently to provide different notes. For instance, the left-most track has more widely spaced ridges, resulting in fewer plectrum-hits per second and a correspondingly lower frequency, while the next track has slightly more closely spaced ridges, yielding a higher note. By spacing the ridges right, you can produce scales; the latest version can produce a chromatic scale over two octaves and a fifth. With the relative pitches thus established, you can control the actual resulting pitches — how high or low the notes of the scale are — by controlling the disk rotation speed. I use a special plectrum, which is actually more than just a pick: in addition to having a plucking end of suitable shape and rigidity for dragging over the ridges, it has a lot of extra surface area in the form of a glued-on styrofoam cup, allowing it to function as a soundboard in projecting the vibration out into the air.

**Musical Siren**: I have written about musical sirens in the past, most notably here and here. Those articles include fuller details on how sirens function, as well as a lot of colorful history. You can also see and hear a rudimentary musical siren here, and you’ll find a photo and audio clip of my new siren later in this article. In its most basic form, a musical siren is a rotating disk with several concentric rings of holes in it. A blow tube with nozzle positioned very close to the spinning disk directs a stream of air at one or another of the rings. Whenever a hole passes in front of the nozzle a puff of air is allowed through. The listener hears a tone at a frequency corresponding to the number of puffs allowed through per second. Thus if the ring of holes the nozzle is pointed at has 30 holes in it and the disk is turning at a rate of five rotations per second, there will be an audible tone at 150Hz. By aiming the nozzle at different rings with different numbers of holes, you can get different pitches. With enough rings you can have a complete scale. I should note that what I’ve just described is an ideal; in reality what you hear is not just a series of discrete puffs of air, but rather some mix of puffs allowed through, reflected air flow in those instants when no hole is present and the air stream is nominally blocked, turbulence arising around the hole edges, and so forth – but taken together the effect is clearly periodic, and the ear hears the intended pitch.

As you can see, Savart’s wheel and musical sirens are closely related conceptually, despite their sounds being very different. Both involve the use of a rotating element creating an “events-per-second” machine. Both give the designer of the instrument the job of figuring out how to space the essential elements (holes or ridges) in the right way to get some desired scale. Such systems also have in common the fact that the spacing of those elements determines a set of relative pitches (the scale) while the speed of rotation determines the actual frequencies.

To make the CNC version of Savart’s Wheel, I turned to Ian Saxton as collaborator to manage the math of the ridge spacing and the CNC programming. To complete this project (and some related projects of his own), Ian became a member of a local maker space which had the necessary machines. He took the classes necessary to learn their use, and used them to fabricate of the ridged cylinder. Our approach, arrived at after considering a lot of options, was to start with a section of heavy 8” PVC drainpipe and, using a CNC router with the capacity for rotational control, carve the valleys which between them formed the ridges. One of the greatest challenges was finding a suitable motor: one which was sufficiently strong, reasonably quiet, and speed controllable with a system sophisticated enough to maintain steady speed even with varying amounts of drag from the plectrum.

For the musical siren, I turned to Aaron Soloway for the math and CNC chops. Aaron did the math and programming for hole spacing within the disk, and arranged with a fabricator to make the disk out of 1/8” MDF fiber board.

For the remainder of this article, I’ll be talking about the thinking that goes into decisions about hole or ridge spacing in these instruments. I’ll speak primarily about the process as it pertains to the musical siren, but the underlying thinking for Savart’s Wheel is pretty much identical. If the readers’ interest is not in making a siren but rather in Savart’s wheel, it should be easy to see how the thinking would apply.

The first thing to make note of is a practical matter: if you want a siren disk with a large enough selection of notes to have a complete scale, you have to drill an awful lot of holes. For things to work well they should be positioned accurately. This can be a lot of work. I know this, because for my earlier sirens I drilled the holes by and one at a time on a drill press. On those early sirens, to keep the job manageable I contented myself with rather limited scales. For a siren with more notes and a greater range, the numbers are correspondingly larger. That’s why my thoughts turned to CNC fabrication, where you can program the machine to cut a circular disk of suitable size and drill a very large number of holes in precisely specified locations. For the new siren, Aaron submitted the necessary data in the required format to the fabrication company and in a short time we had a perfectly made siren disk with close to 2000 holes in it. Close to 2000!? Yes, that’s what the chosen scale and range called for. What we were aiming for with this instrument was a chromatic range of two octaves. For that we needed 25 rings of holes (twelve per octave plus one more for the octave-repeat note at the top). And how did we determine how many holes should be in each ring? This question calls for a bit more explanation.

As intonation theorists and probably many readers of this essay know, musical intervals correspond to the ratios between the frequencies of sounding pitches. The simplest example for this is the octave, with a ratio of 2:1. When presented with two tones for which the frequency of one is twice that of the other, the ear hears the interval of an octave. For sirens, it follows that if one of the rings of holes in a siren disk has, let us say, sixteen holes, and another has 32 holes, then those two rings will sound an octave apart when the disk spins and the nozzle blows, as twice as many holes will whiz by the nozzle with each disk rotation for the 32-hole ring as did for the 16-hole ring. Each musical interval of the scale will have its own corresponding ratio, so all you need to do is make the numbers of holes in the rings correspond to the ratios of the scale you want. Except …

…Except that it’s not always possible to get the ratio you want. Example: imagine you have a ring with twenty holes in it, and as part of your intended scale you want another ring which will sound a perfect fourth above that. The interval of a fourth corresponds to a ratio of 4:3, so this new ring will need a hole count of 4/3 times 20 = 26.6666 holes. Hmmm… Trying to conceptualize a ring with twenty-six and two-thirds equally spaced holes in it is a bit of a mind-bender. Jacques Dudon, in an excellent article on the closely related subject of light sirens, discusses a couple of sophisticated ways you can approach this dilemma, but for the purposes of the current discussion we can simply say that the number of holes in each ring needs to be a whole number. And we find that in order to arrive at a set of whole numbers that will give you the ratios for the scale that you want, you need to do some careful planning concerning the number of holes in each ring. For any ratio-based scale it’s possible to find a set of numbers that work, but in many cases it turns out that the solution calls for very large numbers of holes. This may be impractical if only because it may require an awkwardly large disk to accommodate so many holes. All of which is to say that if you get into musical siren making, you can expect to have some fun and some headaches juggling numbers, as well as hole sizes and disk sizes, in search of the most practical way to get the scale you want. Alternatively, you can look for a set of ratios — that is to say, a scale — that sounds good to the ear and works well musically, but that also happens to be conveniently realizable without requiring a very large number of holes. That’s how I arrived at the scales I used in the simple musical sirens I made long ago. For one siren I settled on a scale of only four notes per octave just because its ratios happened to require conveniently low numbers. (Yet it’s surprising how much musical enjoyment those notes, replicated over two-plus octaves, have provided in the years since.)

And then there’s the fact that not all scales are ratio-based. Although it’s true that the ear interprets musical intervals in the “rational” way just described, it’s also true that the mathematics underlying some scales employ irrational numbers. As difficult as it is to imagine a ring with 20 ⅔ equally spaced holes, it’s even weirder trying to picture a ring with an irrational number of equally spaced holes. And the most widely used scale in the world — the standard twelve-tone equal temperament to which your piano and guitar are probably tuned — falls in this category. Given those irrational numbers, it follows that it’s not possible to make a perfectly tuned scale of this sort on a siren, even with an insanely huge number of holes. However, the more holes you’re willing to make, the closer you can come to the ideal. So if you’re willing to have enough holes, and are willing to compromise by having a slightly imperfect tuning, then you should be able to come up with something acceptably close. Remember, most instruments in the world are to some degree imperfect in their tuning, and that does not prevent us from enjoying their music.

Scales that are ratio-based are called just scales. The most common scales that are not ratio-based are equal-tempered scales, with 12-tone equal temperament as the most prominent example. Many people feel that just scales are preferable because they reflect the ear’s ratio-based ways of hearing musical intervals as well as the physics of many vibrating systems, while tempered scales like 12-equal are seen as inherently compromised. For that reason, many would look at the situation with siren scales, built as they necessarily are on ratios of whole numbers, as a fine opportunity to do the right thing and dive into the happy realm of just tunings. Indeed, it would be would seem perverse to do otherwise! And that is my recommendation for makers of either Savart’s Wheels or musical sirens: go just. (But! The other option is also discussed later in this article, so do read on.)

And how do you go about figuring out how many equally spaced holes should be in each siren ring to get the scale you want? For just scales the process is pretty straightforward. It is typical to represent the degrees of a just scale as a series of ratios, and for the current discussion we’ll start from there. Scale making for the siren disk is then a matter of doing the arithmetic to recreate those scale ratios in the numbers of holes in each ring. Here’s the arithmetic: For any given set of ratios, the number of holes in the ring with the least holes needs to be the least common denominator for all the ratios. After figuring out that LCD, you can find the hole numbers for the subsequent rings by multiplying that number by each scale ratio in turn. To cite what may be the most useful example: The most common version of a just major scale is comprised of these ratios: 1:1, 9:8, 5:4, 4:3, 3:2,5:3, 15:8, 2:1. The least common denominator for this set is 24, so that will be the number of holes in the smallest ring. For hole numbers for all the rings, multiply 24 by each of the scale ratios in turn and you get: 24, 27, 30, 32, 36, 40, 45, 48. If you want to continue up through another octave, multiply all of these numbers by the octave-up factor of 2.

Note that in this discussion I have spoken only of the relative pitches comprising a scale; I haven’t said anything about specific notes or frequencies. The actual notes, of course, depend on the disk’s speed of rotation. Increase that speed of rotation and you raise all the resulting scale pitches in parallel, preserving the ratios and intervals. By controlling rotational speed, you can tune the whole gamut up or down, but you can’t change the intervals within the scale.

Despite what I said a moment ago about how just scales are the natural way to go for siren-like instruments, for my recently fabricated siren as well as my recent Savart’s Wheel I did not go just. I wanted something that would be able to play with other instruments in the standard scale. So in both cases I opted for a large number of holes or ridges which would allow me to create an approximation to a 12-equal scale, necessarily imperfect but, I hoped, close enough to be acceptable.

The process of figuring out hole numbers for non-just scales such as 12-tone equal temperament is messier than the process described earlier for just scales. You’ll start with a set of decimal approximations of the scale’s pitch relationships. For example, for a minor 2nd in 12-equal, the frequency of the upper pitch should be 1.05946… times that of the lower pitch. Comparable multipliers pertain for each degree of the scale. (The complete sequence can be found online.) The trick is to find a suitable number of holes to put in your smallest ring, such that it’s possible to come up with hole numbers for the subsequent rings that come as close as possible to the desired set of relationships when multiplied out. So how to come up with that suitable number for the first ring? This question was made simple for me by my siren and Savart making co-conspirators, Aaron and Ian. (Working separately from one another, they arrived independently at similar results.) To explain what they did, let me introduce a useful term: for the purposes of siren-scale making, we can use the term *ordinal number* for the number of holes in that first ring with the smallest number of holes. This number then serves as the starting point for calculating the number of holes to make in the subsequent rings. In creating a scale like 12-equal, for smaller ordinal numbers it may be possible to produce only badly detuned scales, whereas with larger ordinals it’s typically possible to do better. But it’s not a uniform progression: even among smallish ordinals, some do better than others. Naturally we prefer the smallest feasible ordinal number, because with larger ordinals you have to have larger numbers of holes, commensurately larger disks, and so forth. So the question is, what’s the smallest ordinal number that will allow a set of hole numbers that together produce an acceptable tuning? To address this, Aaron wrote a program which allowed the computer to generate a chart of ordinal numbers. The chart includes, for each ordinal, the degree of detuning for the worst intervals in the best realization of that ordinal’s scale, as well as and the average detuning for the scale as a whole. You can look at this chart and say things like “Well, ordinal number X looks pretty good; lots of very accurate notes in it, except it has these two problem notes that are unacceptably far out of tune, so I won’t use that ordinal number. Ah, but with ordinal Y, most of the notes are reasonably close to the ideal, and none are drastically out of tune, so maybe I’ll use that one.” Hopefully, in reviewing the chart one can find an ordinal number which is not so large as to be impractical, yet which produces acceptable tuning of the notes throughout the resulting scale.

Using this program, Aaron produced a chart for twelve-tone equal temperament showing best achievable results for ordinal numbers from 1 to 500. At the low end, having one hole in the first ring is a kind of reducto ad absurdum, there just for completeness and interest; at the high end, 500 was an arbitrary cut-off reflecting the fact that as a practical matter much larger numbers become increasingly unmanageable when it comes to fabricating the disk. In between were the potentially useful numbers. Looking at the lower end of the chart I found a sweet spot at 32. At 32 there is a dip in the detuning curve — no notes drastically out of tune, and most notes respectably close, making 32 considerably better than any smaller ordinal. Going up from there, there isn’t anything significantly better until 46.

So it was decided: with 32 holes in my innermost ring I would be able to produce what I hoped would be an acceptably close approximation to the scale intervals of 12-tone equal temperament. The next step was to determine the number of holes to go in the subsequent rings. This was done by multiplying the ordinal number 32 by each of the multipliers for the 12-equal scale and, in each case, choosing the whole number closest to that result. Those whole numbers would be the number of holes in the rings. As mentioned earlier, I had decided on a disk with a range of two chromatic octaves, which amounts to 25 rings of holes. The hole diameters were set at 1/4” and the ring widths at 3/8”, and the disk diameter came in at about 23”. I sent the specs off to Aaron, who created the CNC file for those specs and handed it off to the fabricator to cut the disk. A while later the disk came back looking great.

Aaron had also researched and selected a suitable quiet motor and speed controller. With disk and motor on hand I put together a simple-as-can-be support. After miming the playing motions, I had decided that the most ergonomic positioning for the disk would be not horizontal (as with my earlier sirens), but a little off of vertical, with the top leaning back slightly away from the player. My mounting frame allows this lean angle to be adjusted. Another valuable feature: it helps greatly to have something to brace the nozzle-holding hand against, steadying the hand while holding the air tube nozzle as close as possible to the disk without touching. For this purpose my frame has a vertical bar running in front of the disk.

Also needed was a blow tube. This I made from 30” of flexible tube, ½” inner diameter, with a nozzle at the end. Coming up with the right sort of nozzle called for a bit of experimentation. I tried various sorts of nozzles, some that I found readymade from various sources, and others that I fabricated from scratch. What works best, I found, differs depending on air pressure. A long, very narrow nozzle creating a very focused air stream is effective at very high pressures – higher, that is, than lungs can provide. At human lung pressure levels something with a little less resistance works better. I settled on relatively short a nozzle with an aperture of about 3/32” diameter over a length of about 1/2”. I attached a tiny electret microphone alongside the nozzle to provide the option of amplification.

With blow tube in hand I was able to start the disk spinning, bring the nozzle close to the disk, and play, moving the nozzle up and down over the surface of the disk with the aid of the hand-brace bar to aim it at different rings of holes.

And how does it sound? Well, it sounds much like earlier sirens I had made, but with larger range and more complete scale. It’s an oddly trumpet-like sound, but with the peculiar articulation and phrase patterns that tend to arise from the playing gesture of moving the nozzle from ring to ring. It’s also far quieter than anything like a trumpet; the little built-in microphone proves its worth in all but the quietest performance contexts.

The motor that Aaron chose is wonderfully quiet. However, there’s a lot of noise from another source, namely, air-turbulence sound. This noise is of two sorts. One arises from the spinning of the disk, and can be heard even when there’s no blowing through the nozzle. The spinning disk turbulence sound is negligible as long as the disk isn’t spinning at excessively high speeds, and the instrument plays well enough at lower speeds, so this sound is not a major issue. The more substantial turbulence noise comes from the player’s blowing, as the air stream creates a lot of turbulence as it hits the edges of the holes or reflects off of the solid surface between the holes. As you can hear in the audio clip above, it’s a pretty serious noise and I haven’t yet come up with an effective way to eliminate it. So I’ve been telling myself, “Just accept it; it’s part of the character of the instrument.”