Calculating knob distance for musical intervals on 3d-printed rubber band harp

[Gnome Alone]

Dear Nerds,

I’m trying to design a rubber band harp on a 3D printing website. I work at a library and I think it’d be a great thing to show people, demonstrating some part of the breadth of useful and fun things that can be printed with the contraptions (doesn’t always have to be bloody Pokéballs)…if only I could get the dang intervals right!

I printed someone else’s design of the same concept. It was a hollow square, with a second piece with soundhole as a lid, and then little knobs to tie the rubber bands to on either side, after wrapping the bands around (underneath) the whole thing. Despite the author’s assertion that the harp was “even tunable,” it plainly was not – each rubber band, when plucked, produced a note, but completely out of order…unless it’s some avant-garde Harry Partch micro-tonal thing.

My own first design incorporated what I thought would be a bloody obvious improvement, which is slanting the knobs so that each rubber band will produce a distinct note. This sort of worked, but certain bands were simply too close together to produce a different tone – one particular section was like four C#s in a row or something like that. And it still suffered (although less so) from the bonkers Martian scale problem, notes all out of order. I’m willing to accept that such may be the result of trying to make a musical instrument out of, y’know, rubber bands…but just in case there’s a way to do this properly, here is my question:

Is there a set formula one can use, or some way of calculating the math correctly, for how far apart to set the knobs for rubber bands? All I can think of is to make as many as I can that can possibly fit and then recording which ones actually worked; a trial-and-error test run. I’m doing them 1mm apart in height, with a row of them 5mm from each other in length. But for all I know it’s actually a multiple of 4mm that would eventually produce a whole note. Or, 3.76mm or something. Have I made my problem adequately clear? It has so far frustrated my attempts to Google it, despite how this is something that luthiers have figured out, like, thousands of years ago. Any assistance you nerds can render will by highly appreciated.

P.S. I mean, I’m a nerd too, but I can’t math; I am the softest of scientists.
P.P.S. The forum robot knows neither the word "tunable" nor "luthier," in a fascinating bit of trivia gleaned from watching the spellcheck underline them.
P.P.P.S Here is my working model of what in the nine hells I am talking about:

[halfeye]

Dear Nerds,

I’m trying to design a rubber band harp on a 3D printing website. I work at a library and I think it’d be a great thing to show people, demonstrating some part of the breadth of useful and fun things that can be printed with the contraptions (doesn’t always have to be bloody Pokéballs)…if only I could get the dang intervals right!

I printed someone else’s design of the same concept. It was a hollow square, with a second piece with soundhole as a lid, and then little knobs to tie the rubber bands to on either side, after wrapping the bands around (underneath) the whole thing. Despite the author’s assertion that the harp was “even tunable,” it plainly was not – each rubber band, when plucked, produced a note, but completely out of order…unless it’s some avant-garde Harry Partch micro-tonal thing.

My own first design incorporated what I thought would be a bloody obvious improvement, which is slanting the knobs so that each rubber band will produce a distinct note. This sort of worked, but certain bands were simply too close together to produce a different tone – one particular section was like four C#s in a row or something like that. And it still suffered (although less so) from the bonkers Martian scale problem, notes all out of order. I’m willing to accept that such may be the result of trying to make a musical instrument out of, y’know, rubber bands…but just in case there’s a way to do this properly, here is my question:

Is there a set formula one can use, or some way of calculating the math correctly, for how far apart to set the knobs for rubber bands? All I can think of is to make as many as I can that can possibly fit and then recording which ones actually worked; a trial-and-error test run. I’m doing them 1mm apart in height, with a row of them 5mm from each other in length. But for all I know it’s actually a multiple of 4mm that would eventually produce a whole note. Or, 3.76mm or something. Have I made my problem adequately clear? It has so far frustrated my attempts to Google it, despite how this is something that luthiers have figured out, like, thousands of years ago. Any assistance you nerds can render will by highly appreciated.

P.S. I mean, I’m a nerd too, but I can’t math; I am the softest of scientists.
P.P.S. The forum robot knows neither the word "tunable" nor "luthier," in a fascinating bit of trivia gleaned from watching the spellcheck underline them.
P.P.P.S Here is my working model of what in the nine hells I am talking about:

I can't maths either, so I'm completely in sympathy with that.

Don't dis the Partch, Breed was a friend (contact lost, not any falling out).

The thing is that lacking knowledgeable current friends, the best I can tell you is that maths is unfortunately necessary.

It's a matter of waves, and wave optics, and that's not simple.

The tension matters, so the overall length of the bands stretched and unstretched will matter, and maybe exactly what the bands are made of will matter too.

[LordEntrails]

Hmm, I don't know the answer. But I do know a few things that may point you in the right direction. Frequency and wavelength are inversely proportional. In order for the string to resonate properly, you will want the wavelength to be a regular fraction or integer of the length of the string.

I don't know how tension is going to affect it, but it will definitely affect the resonating/natural frequency.

Two links that might help;
- http://www.physicsclassroom.com/clas...-and-Harmonics
- https://www.omnicalculator.com/physi...-wave-equation

So, you've got two things to play with, string length and tension. If you are using the same rubber bands on each, then the tension and natural freq of each string is going to change. Not sure how you want to handle that.

[Gnome Alone]

I can't maths either, so I'm completely in sympathy with that.

Don't dis the Partch, Breed was a friend (contact lost, not any falling out).

The thing is that lacking knowledgeable current friends, the best I can tell you is that maths is unfortunately necessary.

It's a matter of waves, and wave optics, and that's not simple.

The tension matters, so the overall length of the bands stretched and unstretched will matter, and maybe exactly what the bands are made of will matter too.

I love Harry Partch's music, I was just making a joke about the other harp-maker's intent; doubtless he was just similarly befuddled by all this...wave optics stuff? Oy gevalt and faith 'n begorrah.

Hmm, I don't know the answer. But I do know a few things that may point you in the right direction. Frequency and wavelength are inversely proportional. In order for the string to resonate properly, you will want the wavelength to be a regular fraction or integer of the length of the string.

I don't know how tension is going to affect it, but it will definitely affect the resonating/natural frequency.

Two links that might help;
- http://www.physicsclassroom.com/clas...-and-Harmonics
- https://www.omnicalculator.com/physi...-wave-equation

So, you've got two things to play with, string length and tension. If you are using the same rubber bands on each, then the tension and natural freq of each string is going to change. Not sure how you want to handle that.

That does indeed look like exactly the kind of stuff I've got to miraculously figure out if I'm to do this as well as I want.

Really, the best option would probably be to devise a tuning mechanism so that I don't have to do the Flintstones-style stretching out of the strings to produce different notes. That at least would give me an insurmountable problem that I can wrap my head around - "how do I make tuner" is a bit easier to troubleshoot than "how do I acquire a sufficient knowledge of wave otptics and harmonic frequency."

[Algeh]

Really, the best option would probably be to devise a tuning mechanism so that I don't have to do the Flintstones-style stretching out of the strings to produce different notes. That at least would give me an insurmountable problem that I can wrap my head around - "how do I make tuner" is a bit easier to troubleshoot than "how do I acquire a sufficient knowledge of wave otptics and harmonic frequency."

This is almost certainly the way to go. Every actual stringed instrument I've ever played has had some sort of way to adjust the tension on each string in order to tune it. Otherwise, you'd need to have a set of perfectly identical rubber bands (that did not distort differently over time) and exactly the right distances. I suspect the first problem would be harder to solve than the second in the end.

I don't know how best to make 3d printed tuning pegs for your particular application (the basic problem is that you want something that you can gradually adjust the tension on and then keep locked in place once it's in tune), but that sounds like a much more solvable problem.

[gomipile]

Once you have the ability to tune each string separately, you can assume that all the strings are under the same tension. That assumption allows you to build the harp.

You'll still need a baseline to start at. Take, say, two nails in a block of wood that you can easily measure the distance between. Stretch a rubber band to a tension that feels good, and pluck it. Use a spectrum analyzer app or a guitar tuning app, or something like that to record the sound. That tells you what fundamental frequency(note) you can get with that length. Then use v=f*λ to calculate the speed of sound in the string(v) for that string length(λ) at the frequency(f) you determined with the app.

Then apply v=f*λ to calculate the string lengths for all the frequencies you want to produce with your harp.

The value of the assumption of equal tension at the top of my post is that with tunable strings, you will be able to tune all the strings to get the notes you calculated lengths for in the above steps.

[jayem]

This is almost certainly the way to go. Every actual stringed instrument I've ever played has had some sort of way to adjust the tension on each string in order to tune it. Otherwise, you'd need to have a set of perfectly identical rubber bands (that did not distort differently over time) and exactly the right distances. I suspect the first problem would be harder to solve than the second in the end.

I don't know how best to make 3d printed tuning pegs for your particular application (the basic problem is that you want something that you can gradually adjust the tension on and then keep locked in place once it's in tune), but that sounds like a much more solvable problem.

For ideal strings, the note an octave higher should be half the length, and the others spaced inbetween. E.g for isotonic? tuning you'd end up with

Spoiler

Show

C 10.000
B 10.595
A# 11.225
A 11.892
G# 12.599
G 13.348
F# 14.142
F 14.983
E 15.874
D# 16.818
D 17.818
C# 18.878
C 20.000

I think that will still more or less be the case for bands of equal tension, if you can adjust and clamp an appropriate length.

Spoiler

Show

For stretched bands, a very rough bank of the envelope calculation (for some magical region where the elastic is under fairly high tension but still obeys hooks law)

Then I think, something like the following lengths, where the unstretched elastic is 1 unit long and doubling it's length gives a C then the following should give an Octave.
C 2.00
C# 2.06
D 2.13
D# 2.20
E 2.28
F 2.37
F# 2.47
G 2.58
G# 2.70
A 2.84
A# 3.01
B 3.20
C 3.41

I wouldn't like to bet on it, as I've probably made a mistake, and the assumptions almost contradict each other.

[DavidSh]

The other solution to adjustable tuning, besides using tuning pegs that you somehow keep from untuning themselves, is to have movable bridges. This is common with East Asian zithers, such as the Japanese Koto.

[halfeye]

You could vary the tension in the bands by winding them, which would also give you one (knobbly) strand instead of two..