From Mathematics to DSP I'm currently working on multiple mini pet-projects in my spare time, grounding myself in the basics of audio signal processing, and trying to develop algorithms for polyphonic pitch detection. As I'm having a lot of joy (even in the failure) in doing so, I am considering moving into a career in signal processing (I'm aware this is a vast area!) My background is in mathematics (I have a  bachelors degree in applied mathematics, so the theory and practice excites me) and I have no grounding whatsoever in electrical or electronic engineering.

I'm therefore wondering the following:


Assuming I can develop a project portfolio demonstrating basic competence, what specific careers would be open to someone of my background in something highly geared to signal processing? I would be particularly interested in sound engineering, but would also be interested in compression algorithms and coding theory.
Is a supplementary MSc. necessary to get my foot in the door to job positions? I am open to both industry and academic positions.
Would an MSc. in something involving machine learning and neural networks be a.) relevant, and b.) worth the dough? I am considering getting out of industry for a year to work more on my projects in the context of such a degree programme. I am getting the impression that combined with a knowledge of signal processing, some knowledge of A.I. and self-learning systems would go a long way.


Thanks in advance for your answers.
In general, Maths to audio signal processing is a very natural progression. I seem to recall my Master's and PhD supervisors both did their undergraduate degrees in Maths. 

To your questions:


This is the hardest question to answer. Depends what's in your "portfolio of basic competence". In addition to the understanding of signal processing theory that you're developing you need to show you're able to implement these algorithms probably either in hardware or software. Given little EE grounding, I'd suggest software. Look at developing your software engineering skills. They will serve you well in either academia or industry.
MSc will definitely help but is not absolutely necessary. The types of algorithms you are describing could conceivably be the subject of an MSc level project. I work at a company that makes hardware for audio signal processing. The statistic they gave me is that around 60% of staff have a Master's or higher degree.
Yes, combining signal processing and machine learning is a very wise idea if you want to work towards the research end of the field. Several problems in audio signal processing - e.g. audio source separation and speech recognition- are increasingly finding their state-of-the-art solutions come from machine learning.