At my high school we can take a class where we basically learn about a subject on our own for a semester. I was thinking that I want to learn about "sound programming," but I realized that I have no idea what that entails. I'm interested in learning about, for example, how a synthesizer works and how sound works in computer science. I really want to focus on the low-level code part, not so much the composition part. Is this a feasible subject? Are there any good tutorials out there for somebody completely new to this? I know C++ and am using Windows. The first answer in this is something that interests me (although it's over my head).
"Sound programming" is a very broad field. First of all, it is definitely a feasible subject, but since you need to cram stuff into a single semester you will need to limit your scope. I can see that you're looking for a place to start, so here are some ideas to get you thinking.
Since you have mentioned both "how sound works in computer science" and "synthesizers", it's worth pointing out the difference between analogue sound, sampled sound and synthesized sound, as they are different concepts. I'll explain them briefly here.
Analogue sound is sound as we humans typically interpret it -- vibrations of air sensed by the human ear. You can think of sound as a one-dimensional signal, where the independent variable is time and the dependent variable is amplitude of vibration. Analogue sound is continuous both in the time and amplitude domain. Older sound recording methods (e.g. magnetic tape) used an analogue sound representation. Analogue sound is not frequently used with computers (computers aren't good with storing continuous-domain data), but understanding analogue signals is important nevertheless. Expect to see plenty of math (e.g. complex numbers, Fourier transforms) if you go down this path.
Sampled sound is the sound representation that lends itself well to processing with a computer. People are most familiar with sampled sound through CDs and other musical recordings. An analogue signal is sampled at some frequency (e.g. 44.1KHz for CD recording). So a sampled sound signal is discrete in the time domain. If the signal is quantized then it will be discrete in the amplitude domain as well. Formats like MP3 are sampled formats. There's lots of things to study in this field if you're interested, such as restoration (removing static, etc) and compression (again, codecs MP3, Ogg Vorbis). It's a lot of fun because there's lots to experiment with and code.
Both analogue and sampled sound dig deeply into a field called Digital Signal Processing. Google around for that to get a feel of what it's like. It's often taught as a course at universities, so if you're really keen you can have a look at some lecture slides or even try some of the earlier, simpler projects.
Synthesized sound is a representation that is suited for reproduction of a music track, where the instruments playing the track are known beforehand. Think of it as sheet music for the computer. Somebody has to write the sheet music -- you can't just record it like analogue or sampled sound. This makes synthesized sound a completely different representation to analogue sound and sampled sound Also, the computer needs to know what the instruments are (e.g. piano) so that it can play (synthesize) the track. If it doesn't know the instrument, it either gives up or picks a close match (e.g. replaces the piano with electric keyboard). I have never worked with synthesizers before so I can't comment on the learning curve for them.
So, based on what I wrote -- pick a direction that interests you more, Google around and then refine your question.
A good book to read is this. You can probably look around related titles in Amazon and find something newer, but it's been a while since I did my audio processing shopping.
And if you have half an hour to spare, then have a look at this video tutorial. It covers sound, image and video processing -- they're actually closely related fields.
Consider working through the book "Who Is Fourier?: A Mathematical Adventure". You could adapt the examples to make small programming assignments that demonstrate the basic concepts. After you're done you should be able to use the fft to make a spectrogram of your voice as you pronounce the vowels a,e,i,o,u -- identifying the fundamental frequency and the formants of each vowel.
I recommend learning Python and the modules NumPy, SciPy, and matplotlib (there's a ton there, so beyond the basic tutorials, just learn as you go). The iPython shell has the option "-pylab -p scipy" to automatically import the most common tools into your namespace. You can record and play audio using PyAudio. There's also Pygame, which expands on SDL (Simple DirectMedia layer), and pyglet, which uses OpenAL (the OpenGL of audio; it does 3D audio and effects).
As to C/C++, there's IT++, SPUC, and FFTW for signal processing, and SDL/SDL_mixer and OpenAL/ALmixer for interfacing with hardware and audio files.
Does anyone have a favorite "go-to" paper, web site, etc., for explaining the basics behind Fourier theory / discrete Fourier transforms?
I am not overly inclined mathematically, and while I know this particular domain requires some math skills I'm hoping for documentation that eases me into it so I have some understanding of intent by the time I have to understand the equations. I have been hunting around on Google for some time without success.
(I am aware of this question but want to understand the theory/math and not just use an implementation.)
If you want a book that is both very accessible to non-mathematicians and provides good coverage of larger Fourier theory, I'd recommend "Who is Fourier?"
It's a math book like few others -- it blurs the boundaries between narrative and expository formats, and there's a conscious choice to avoid most of the density involved in conventional math texts. Lots of illustrations.
Some technical people feel pandered or talked-down to, though -- you can find a negative review on Amazon where the reviewer recommended a Schaum's Outline instead. :)