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Sound money is a phrase technically synonymous with a currency backed by or pegged to some good or commodity. It can also mean a currency that is not rapidly inflating or deflating in value. As even the technical definition is rather fuzzy, in reality, it's often synonymous with whatever economic woo its promoter is hyping. Sound money is often a cover for gold buggery, and lately bitcoin fanaticism, though it can mean whatever the hell the bullshit purveyor wants it to mean.
It's an American thing
In US history, the term itself dates back to the 1800s and the Free Silver movement, where conservative politicians advocated "sound money", or keeping the gold standard, while populists advocated the coinage of silver. Free Silver is often written off as economic woo today, but it wasn't entirely stupid. What was wrong with the Silverites was not that they advocated the coinage of silver (which some states actually allowed at certain times to fight off the deflationary nature of gold), it was that they demanded free and unlimited coinage of silver. This would be like you going up to the Federal Reserve today and asking them to just print you up a few thousand dollars, and then doing it again five minutes later, with the total amount of raw silver in existence being the only limit to inflating the currency.
Father Coughlin pushed a lot of "sound money" crankery in his day, but he meant something else entirely. Coughlin was actually something of a Silverite and his sound money plan basically involved nationalizing everything (to break up the Jewish banking conspiracy, of course) and other ridiculous plans frighteningly close to fascism.
Sound money is making a resurgence among the more populist Teabaggers, financial bloggers, and even "serious" Republicans. This sound money advocacy is only slightly less insane than re-installing the gold standard or pegging the currency to one commodity. For example, Paul Ryan is a sound money advocate who wants to tie the dollar to a "commodities basket". This makes a little more sense, as having multiple commodities reduces the chance that volatility in the commodity you're pegged to will completely screw over the entire currency. Of course, if you have a particularly volatile commodity, just one in the basket could presumably do that, although its effect would be muted by the presence of the other commodities in the basket. There's also the chance of foreign financiers targeting whatever commodities are in the basket to screw with the value of the currency to influence global trade.
Sound money advocates mistrust fiat currency because its quantity can be very elastic. A fiat currency can be manipulated to shrink or grow with (or inverse to) economic output. Having a pegged currency means, if you peg it to, say, wheat, that your currency (and hence economy) is going to be partially dependent on what some wheat farmer in China or Russia is doing. This is one argument that can be levied in favor of first-world countries using free-floating fiat money. Pegged currencies or hard-money currencies are sometimes used in the Third World, developing economies, or countries with an otherwise unstable government. They have also been some of the worst abusers of fiat currency. (See Zimbabwe.)
Sound money advocates (as well as gold bugs) also tend to populate the conspiracy theory and survivalist circuits. The reason usually has to do with some mixture of the following: fractional-reserve banking conspiracies (likely perpetrated by Jews), Federal Reserve conspiracies (also likely perpetrated by Jews), Jews, World War III, End Times, the destruction of the US economy, and did we mention Jews? (Though of course Rothbard, one of the most influential and vocal critics of fiat currency and fractional reserve banking, was himself Jewish, and presumably not anti-Semitic.) In any of these events, paper money will become worthless, so these people tend to just throw sound money crankery into the conspiracy stew as well.
Libertarians also love sound money. This tends to come from the influence of the Austrian school, who are major gold bugs. Some libertarians who advocate for commodities backed by something other than gold usually use the phrase "sound money" to distance themselves from the gold bugs. The general logic, though, is that fiat currency is controlled by the government ("government fiat"). If government is bad, then fiat currency must be bad. Result: Sound money cranks.
Bitcoin fans claim their digital pogs are strictly limited in quantity, therefore bitcoin is the perfect sound money! More conventional gold bugs point out that anyone can copy'n'paste the code and make endless other "strictly limited" digital coins, and they do. Bitcoiners usually retreat to claiming Bitcoin is special though, because reasons.
The bottom line is that the idea of sound money ignores the true purpose of money — to create an easily portable standard to make trade easier. Because of this, the standard for money only has value if the society says it does — regardless of it being paper, gold, silver, or something else. In a time of economic collapse, this standard tends to fail and society returns to a largely barter system.
This fact was hammered home April 14, 2013 when gold lost nearly 9.35% of its value in one day. Furthermore, it was pointed that from its high on August 22, 2011, gold has lost 28% of its value (but this wasn't as bad as the period between January 2, 1996 to August 25, 1999, where gold lost 39% of its value).