Harmonic series (Maths and Music) I am a high school student trying to apply calculus to music (harmonic series). I am just wondering, how can I collect data from any online music app (with music tones that form harmonic series - which I do not know yet) and analyze it using calculus (series – specifically, harmonic series?
Any ideas/resources/links will be appreciated.
 A: The free online book of Music: a Mathematical Offering by Dave Benson is comprehensive, and may be a good start. Chapter $1$: "Waves and harmonics" talks about frequency analysis, and how to listen for/understand sound. Chapter $7$ talks about Digital Music, and how to understand and apply various methods to change/analyze music. To be completely transparent, I'd be incredibly surprised if there does not exist a class somewhere that was started because of this book, as it's incredibly comprehensive.
In addition, there is the free book The Development of Musical Tuning Systems by Peter A. Frazer that discusses the physical aspect of sound using tuning systems, and how and why they were made. Harmonic analysis and similar are mentioned throughout the book, and there exists sound examples in most chapters.
Here are some introductory YouTube videos on the topic that describe the Harmonic Series in way that'll be easier to understand, and more comprehensive, than a text forum like this one:

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*Calculus $10.17$ Harmonic Series Math and Music

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*The video discusses the harmonic sequence and series in mathematics, and they look into how it connects to the harmonic series in music.



*The Harmonic Series and The Harmonic Series

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*Discusses why the harmonic series is called the harmonic series, and gives proofs on how it works.



Most importantly are the types of sound analyzer apps you'll want to be using, as for the most part, you'll want something that provides the most amount of information possible (it's easier to work with excess data than missing data) and is easily exportable. There are only a few apps I have personally used that fit this criteria, but the most notable one is Spectroid as it's incredibly transparent and comprehensive data-wise (and most important: free). If you're using a device that is not Android, then you're going to have to browse around for similar apps, but remember this: look for an app that provides the most data possible and is easy to export said data. In addition, look deeply into the reviews of any paid apps before you get them, as most apps are not worth the payment.
A: Maybe my calculator could be of some use...Either ignoring the idiosyncratic music notation (or choosing to learn it), you can calculate frequencies, cent deviations, harmonic space coordinates (prime factorisation as a matrix), etc. for any integer ratio $\frac{a}{b}$ forming an harmonic series aggregate, i.e. a just intonation interval (http://www.plainsound.de/HEJI/). We also have a number of other texts and resources freely available on the https://www.plainsound.org/ page for people interested in this topic.
