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The hands down best way to learn about synthesis is Syntorial http://syntorial.com. It's paid but it's well worth it.
Syntorial looks like it will teach you how to use existing synthesis software, but won't be that useful if you want to write your own?
How do you want to write your own without understanding how synthesis works?
I hope he tries his hand at building an AM analogue synth.
I started on that journey about 18 months ago. It took me five months to have a useful software instrument, and I've spent the remaining time learning how to fab a suitable hardware controller.

Within the software, the hardest parts were understanding and controlling aliasing and getting a virtual ladder filter to perform well.

I already knew how to play piano, had a little music theory, and had spent a few hours playing with other people's analog synths.

tl;dr: these form two parallel tracks in an imaginary self-study course: Practice and Performance of Synths, and Implementation of Synthesized Sound.

Even if you're trying hard to be avant garde and break new sonic ground or whatever, it's hard to imagine that you won't have some significant overlap with the problems addressed by synth builders to date. That background will also be useful when trying to understand everything from published papers to existing synths built in pretty much any existing way, from analogue modular, to "rack-style" digital like Logic and Reason, to programmatic work in Max, Pure Data, SuperCollider, or even direct-coded in a traditional programming language, etc.

Interesting - reminds me of these excellent books:

http://synthesizer-cookbook.com

I have the books as well as syntorial and syntorial blows the books out of water.
Thx - good to know I have the first book and it's an interesting read but I found it difficult to put to practice.
the hands down best way is to buy a synthesizer
and if you're going to do that Korg's new minilogue has a great omled screen that shows the waveform of whatever sound you are playing.
thanks - this looks really good (as a producer - www.soundcloud.com/decklyn )

Bookmarked - will read!

If you like this I can recommend you to look into the raw audio format [1]. It is a very simple format but it helped me a lot to learn about digital audio.

Just take your fav programming language (I took PHP) and write some bits to a file:

  $file = fopen('output.s16', 'w');
  $amp = returnANiceSynthValue();
  fwrite($file, pack('s', $amp));
[1] http://www.fmtz.com/misc/raw-audio-file-formats
I dunno, the classic Sound On Sound "synth secrets" covers all this and more, in a lot more detail IMO

http://www.soundonsound.com/sos/allsynthsecrets.htm

Yeah, a lot of this information is pretty low-value relative to the Sound On Sound stuff. What good is it to know the difference in how a square wave and triangle wave look, if you don't understand how they sound different and why you might prefer one to the other? The "synth secrets" series is much more technical and showing its age, but it's comprehensive.
It also helps to have a real or soft synth in front of you so you can tweak knobs and play around as you go If you don't have anything, my recommendation is ZynAddSubFX. It is an amazing free synth with enough knobs to do almost anything
This is THE STANDARD

Almost 6 years old and still referenced everyday. Amazing documents. I would love to see more added to it.

Nit: The first articles in the series came out in 1999, so more like 17 years old :)
This is THE STANDARD

Almost 6 years old and still referenced everyday. Amazing documents. I would love to see more added to it.

Thank you for the link! :) I've recently gotten a MIDI controller to finally take some first steps into electronic music (I'm used to playing "regualr" keyboard) and this is what I was looking for!
A fascinating read that I rarely see mentioned in these kind of threads is Rick Snoman's Dance Music Manual - for anyone interested in synths and their role in electronic music, it provides tons of information on the when/where/why/how various synth techniques became popular in different styles of dance music and gives good introductory information on how to take the raw synth components you can create in software or hardware (waveforms, filters, envelopes, etc) and turn them into the kinds of sounds common in various dance music genres.

The book is written with a slant towards the aspiring dance music producer, but, it should be plenty interesting for anyone who wants to have a bit more musical interest in synthesis beyond the purely mathematical angle.

I love that book, covers the whole chain of tools including samplers, sequencers, arpeggiators, DAWs etc. Also Shepard "Refining Sound" steps people thru oscillators, LFO's, envelope, filter/resonance and after effects. Great books.

Some other classic books are Welsh's Cookbooks (i think there are 2), and those by Miller Puckette (the guy who wrote Max), Martin Russ, Perry Cook, Allen Strange, Mark Jenkins, Mark Vail. I'm forgetting a few.

Or you could just buy a mopho/tetra, microbrute, bass station2, MS20 mini, ms2000, minilogue, sh201, jp8000 etc and start watching youtube tuts.

Yeah actually the csound manual not only has all this and more but csound is open source and free all the examples run right there in the browser thanks to asm.js and can be run locally, but this seems like a nice intro for synth beginners who may only have a hardware synth for reference.
i would be a great moment to post a link to it.