Haven't bumped into it specifically yet, but the location of the various harmonics in cents is easily available online info, and "overtones" are a resident property of a vibrating string.
Probably better off just using the guitar for those explorations.
If you absolutely, positively, had to do it with an app, there must be some synth apps with a pair of oscillators you could sweep until smooth. That'd give you the sine wave stuff at least. .
Not entirely sure what you're hoping to accomplish but odds are you could do it with existing musical instruments to your satisfaction.
One interesting app is Scalegen, it allows you to generate a WIDE variety of scales, including ones made of the overtone series.. an MUCH, much more.
Then you can use the app to play the scales, if you have a recorder app that can record MIDI you can also record the output as Midi (or sound) thus enabling you to create tonal clusters (chords) of infinite complexity..
Basically any DAW with some VERY simple synth can be used.. the best thing would be some sort of additive synth which allows you to chose fundamental and have controls for all harmonics, like Verbos Harmonic Oscillator..
I haven't used it in a couple years but you could also look at using "Sonic Visualizer". It's not a mobile app( that I know of) but a free software package for frequency analysis. You could record your instrument, use the spectral analysis and then alter the overtones with the software. This might give you the tools that ( I think) you are looking for to link pitch and timbre. Sort of the opposite of the additive synth approach and using Audacity. ( Actually, there might be a plug in for Audacity to do the same thing. I haven't used some of this software in years.)
A multi channel tone generator that is not bound to 12 tet can be sort of hard to find, perhaps something called "Harmonic series" would to it http://appcrawlr.com/ios/harmonic-series
Looks like it produces the overtone series on a fundamental and that you can control amount of each overtone.. That being said the overtone series is just a small part of tone quality.. the string on a guitar does not produce a pure sine wave, amp feedback gets you closer.. the amount of overtones from a guitar is also different over the envelope, more high pitch overtones when you first hit the string, and less of those as the tone sustains.. sort of a Low Pass Gate behavior if you are into Buchla Synths.. or a envelope controlled low pass filter.. the the guitar also filters and resonates frequencies, much like a synth filter, and not just tones in the over tone series.. a guitar has some resonance frequencies that will act much like a resonance control of a synth filter.. then the amp and room acoustics does similar things, the amp ant guitars controls ARE high and lowpass filters.. You also have sympathetic resonance from other strings that depends on how you damp, or don't damp string you don't actively play.. You will probably also add Low Frequency oscillations by adding vibrato.. and so on.. so learning a bit a about synth helps to understand the relation between pitch and timbre, which are two sides of the same coin..
I just bought a book that discusses the relation between "tuning, timbre, spectrum, and scale" (title of the book) basically what makes sounds dissonant or consonant..
Basically these are old ideas, that were articulated by Stockhausen..
It is a interesting subject.. using a guitar and some tone and spectral analysis tool could be one way of exploring the subject, listening to your playing with short recording in conjunction should be a good learning tool, understanding additive and subtractive synthesis should also give some insights..
Ok i read more in the EXCELLENT book "Tuning, Timbre, Spectrum, Scale: Edition 2" by William A. Sethares (https://play.google.com/store/books...pt_read&pcampaignid=books_booksearch_viewport)
And of course you can use the guitar to understand the overtone of a vibrating string, using the harmonics.. and the other "revelation" is that the overtones of a string (which just 12 tone and 12 tet is built on) are NOT overtones of other sound sources, such as metal or wooden bars. So a scale based on the overtones of a metal bar would be different that that of a string..
There's got to be a tone generator app out there. I wonder there is a pink noise analyzer for the phone. Although, the mic and speakers on a phone might not be worth a crap for playing or hearing such details for them to be useful. I know I have a DB meter app that's not very sensitive at all. It's kind of like it hears things as on or off but not much change in between.