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Since many on the forum asked about music software for their beginner production, I decided to write this guide on which free or cheap music software are best for your choices. Like other ppl, I also came from the same way that you're going through, and I have many experience with these software.
Freeware vs shareware: Most of the stuff posted here are freeware that have not restriction. Shareware are useless and doesn't work most of the time.
Linux vs other platform: I would expand all the tools to Linux, and at least to Wine-based. Because there are many great tools that only work well on Linux, and I wish their developers would do it for other platforms. Beside, Linux is 100% free, as in both free beer and freedom to tinker, so you could pick up a distribution and use it solely for music.
To put it simple, as "why Linux?". You save money for an operating system, while you could instead spend it on the necessary software you prefered. Since Linux, and many DAWs are free, there is no reason to spend on extra. Unless you do the illegal way.
Digital Audio Workstation (DAW)
Best free and open source multiplatform DAW ever. Work across any platform, especially Linux. Literally has everything you needed, from piano roll to audio editor.
One of the best free DAW you could get around and use without problem. It is the best alternative to LMMS.
Once you get pass the learning curve, OpenMPT is a very powerful tool to use. In contradicted to what people believe, you could track a whole track in less than an hour, in compared to a couple hours on other DAW. This was tracked in less than 2 hour, including sample editing.
Linux has its own gem, like many other free DAW, qtractor is amazing on its own.
Free alternative to Finale and Sibelius. Support VST via JACK.
Here is MuseScore in action with Guitar Rig on Kubuntu (KDE version of Ubuntu).
Tracktion 5 Win/OSX/Linux
Rising star of alternative to professional paid software.
VST and Library
Kontakt Player Win/Mac/Linux(Wine 1.8.x)
Best free sampler plus the Factory Selection. And despise rumor, Kontakt does work on Linux (Ubuntu/Mint/Debian). Here is Kontakt in action on LMMS with Linux Mint.
Reaktor Player Win/OSX/Linux(Wine)
Modular programming software by Native Instruments.