TuneFast Harp: 3D Printed Single-string Instrument

Note: If you just want to make a harp, skip ahead to the build instructions.

Tuning musical instruments can be a pain. This is especially true with instruments like pianos or harps, which can have dozens of strings (all of which slightly deform the instrument as they are tightened, affecting the other strings around them making tuning a long and tedious process).

Which raises the question: can you use a single string that zig-zags back and forth to drastically reduce the number of tuners necessary?

Design Constraints

A harp consists of an array of tuned strings, often in a scale that makes it easier to play. We’re choosing the diatonic scale (the white keys on a piano) since this is commonly used with harps and makes it possible to play a wide range of simple tunes (like campfire songs often in the key of C Major).

Since the string will be under constant tension, we can simply adjust the relative position of each endpoint to select which notes are present. Allowing for 8 notes gives us a full octave (all the notes of a diatonic scale, plus one additional note), while keeping it small enough to 3D print with ease.

The instrument should be as rigid as possible, since the string will be under significant tension and we want the harp to bend as little as possible.

We’ll need a single tuning mechanism (inexpensive but reliable versions cost about a dollar or two each and still seem to hold tension well) and a string (guitar strings are cheap and easy to obtain). We’ll also need something to act as low-friction endpoints for the string as it weaves back and forth (grooved bearings are a natural fit).

And finally, we’ll need the printed part itself and some basic nuts, bolts, and washers to hold it all together.

Quick Background Math

But back to the big question: will this zig-zagging string idea even work? Let's employ some basic math to find out.

Consider the string vibration frequency equation (which looks intimidating at first, but is actually quite simple):

√T f = ---------- 2 × L × √μ

This equation shows that the frequency (f) of vibration is increased by the square root of the string’s tension (T) and decreased by the string’s length (L) and the square root of the string’s linear density (μ, mass per unit length).

For our purposes, the big takeaway here is that if the string has uniform density (most strings do) and the entire string is under uniform tension (i.e. all parts of the string are pulled upon equally as it zig-zags around), then we can change its frequency linearly by just varying the length between endpoints.

The other equation we'll need explains how notes on musical instruments are related by the equal tempered scale (this is how most western instruments are tuned):

l_low × 2^-(s/12) = l_s

There are 12 semitones (s) per octave. Since we know from the previous equation that the frequency will change linearly with length, we can pick the length for the lowest note (l_low) to be anything we want and derive all of the other lengths (l_s) from that.

So for example, if the lowest note had a length of 100mm, and the next note in the diatonic scale is 2 semitones higher (going from C to D on the piano), the length for that note on the harp would be:

100mm × 2^-(2/12) = 89.09mm

The note after that is 4 semitones higher which would be:

100mm × 2^-(4/12) = 79.37mm

And so on up the scale.

And with that, the engineering logic checks out and we're ready to make some harps!

Experiments Along the Way

The end result we'll come to shortly is rather simple, but it took a lot of iterations to figure out what worked and what didn’t. We tried various numbers of notes, strings, string types, bearing types, spacing, overall instrument shape, and tuner placement.

We even tried adding sliding mechanisms to pluck the strings as chords with little delrin pics (known as plectrum/plectra) and magnets to keep them from resting in the middle against the strings, but the added complexity and fussy nature of tight tolerances on plastic under tension made it hard to reliably control (and overly tricky to assemble and play).

All the while, we kept tweaking the design, getting the placement just right so that the notes would all be in tune (accounting for the instrument’s deformation under stress so that everything is correctly placed when tuned).

On to the latest version!

The TuneFast Harp

Here’s the final result of our single-string harp experimentation, the TuneFast Harp on Thingiverse.

The total cost with all hardware is under $7 per harp and only takes around 3 hours to print.

We designed the harp to be easy to assemble and fairly quick to print so that even young makers with shared resources could try it out as makerspace/classroom projects.

Parts List

You’ll need a handful of parts to complete this project. Everything is relatively inexpensive, but many of the parts come in large packs, so you may want to plan to make a few and scale quantities accordingly.

Here’s what you’ll need for each harp:

• 1 x Right-handed Guitar Tuner ($1.25/each, typically comes in a pack of 6)
• 9 x M3x20mm bolts/nuts (~$0.10/each for large quantities; or consider a M3 bolts/nuts multipack)
• 9 x M3 Belleville Washer ($0.07/each, sold in packs of 100; or just print your own washers)
• 8 x 3124mm V-Groove Bearings ($0.33/each, comes in pack of 20 but account for the possibility of a couple bad bearings
• 1 x M3 6x4 Spacer ($0.35/each, or use a nylon spacer if you prefer)
• 1 x Nylon guitar string ($0.50/each, or use 25-50 lb test fishing line)
• ~100g of PLA (or similarly rigid filament to print the part)

Print It Out

Our print time came out to around 3 hours total on a TAZ6 printing PLA with high speed settings and 30% infill. That left us with just enough spare time to warm up a cinnamon roll on the buildplate (not directly, of course!).

Generally, the harp works well printed with fairly large layer heights (i.e. coarse/fast settings), but make sure to use at least 30% infill (to increase strength) and print with stiff plastic (PLA works well; ABS is strong, but it’s significantly less stiff than PLA, and rigidity is what we’re looking for here).

Assemble the Harp

After the print is complete, remove it from the print bed and clean up any defects from the print process (we didn’t really need to do anything, but your mileage may vary depending on which printer/filament you use). In particular, make sure there aren’t any extra strands or printer blobs that would interfere with the bearings.

Add the 8 bearings using an M3x20mm bolt, nut, and washer (we prefer metal belleville washers, but you can even print your own M3 washers). The washer should go between the bearing and the printed part to keep the bearing from touching and interfering with its ability to smoothly rotate. Tighten the bolt firmly, but not so tight that it causes the bearing to rub.

Next, add the spacer for the shortest/highest note (we don't use a bearing here because of space constraints and the minimal displacement under tension so close to the end of the string).

Now, double check that each of the bearings can rotate smoothly. If there’s additional friction in any of the bearings, the string won’t be under uniform tension and thus alter the relative tuning of some of the notes. If any bearing is rubbing, just loosen it, inspect/clean any surfaces, and slowly re-tighten. If the bearing is a total dud (as can happen with some of the cheaper components), replace it.

Then install the guitar tuner in the remaining hole. Hand tighten in place with the knob facing outward. Screw it in place from the bottom and fully tighten the nut.

Finally, string it up. If using a ball end string (common for electric guitar strings), just feed it through the small hole by the spacer. If using a nylon string, tie a knot in it first near the end, then feed it through.

Now weave the string back and forth, holding it firmly enough that it doesn’t slide off of the bearings as you go.

After the last bearing, thread the string through the hole in the metal shaft of the tuner and turn the knob to tighten it in place. Bring the string just up to tension so it holds in place for tuning. Trim off any excess string that sticks out past the tuning peg so it doesn’t get in the way or poke you later.

Tune It Up

Since all of the notes are produced from a single string under constant tension, all of the notes should be relatively in tune as soon as the string is somewhat tight. By further tightening the string, all of the notes will increase in frequency, allowing it to be tuned to any desired key.

Two things to keep in mind as you’re tuning:

1. You may need to sequentially push on each of the strings after adjusting the tuning to make sure the tension is even across all the notes. This is particularly important if any of the bearings are rubbing (i.e. have significantly higher friction).
2. You can only tighten the string so much before something bends/breaks, so adjust it slowly at first to understand its range for a given string size.

Note that the harp also works well with steel strings, but they require about twice as much tension which can deform the printed part a bit more and make it trickier to play in-tune (especially when using thicker low strings). The other downside of using a metal string is that it can be pokey/dangerous should things break or just flop around as you string the instrument. Work carefully and be mindful of others around you.

Play It!

We hope you enjoy playing this experimental single-string diatonic harp. When properly built, it produces a pleasant sound and holds its tuning quite well.

The TuneFast Harp is intended to be a hands-on tool for learning about how stringed instruments work, as well as a springboard for designing and printing your own custom stringed objects.

A final note for the musically inclined that feel limited by only having 8 notes to choose from: keep in mind that you can use chord inversions to play chords that go beyond the harp's limited range. For example: a G Major chord (G-B-D) can be played in the 1st inversion (B-D-G) or 2nd inversion (D-G-B) to fit within the harp.

Feel free to share your feedback about the harp on Thingiverse if you have questions or suggestions.

Makefast Workshop
Delaware, Ohio
Est. 2016