Edit: I will be offering a retraction of certain statements made in this post as of May 2 or so.
The actual money supply according to the market is #_transactions * average_transaction_size. If April saw 100 transactions with average price $100, then the supply of money in April was $10,000. Goods demanded $10,000 dollars, and $10,000 dollars demanded goods. The actual amount of coinage in circulation is irrelevant except that it has to be at least $100 so that this set of transactions is logically possible.
If you increase transactions, you increase the money supply, lowering the price of money. Inflation.
However, to increase transactions, you also have to increase the amount of goods the market sees, causing deflation.
As it happens, these effects exactly cancel out.
Put another way, if additional good production is zero (e.g. it's all arbitrage) then the velocity of goods rises in lock-step with the velocity of money. If it's not all arbitrage you get production increases of equivalent goodness.
Put a third way, velocity is measuring the price of money in money. If you increase transactions you increase the money supply, but if you increase transactions you also increase the demand for money. Price of money in money is always going to be 1.
Which is why, for example, the price of a bushel of wheat is 11 pence (or local equivalent; dram, drachma, denarius, &c). Any time production should cause deflation, it also increases velocity, cancelling the deflation with inflation, and wheat prices remain the same. Along with the price of everything else. Indeed it is likely that the very slight price increase that wheat did see (actually there was a low of 8 pence) was due to ongoing silver mining.
Absent central banks, long-run prices are absurdly stable.
Population growth: more people demand the same amount of money, causing deflation, but also demand goods, causing inflation. The additional food purchases raise the price of food, and then the additional velocity raises the price of money. Meanwhile, of course they are also adding labour, thus adding to goods, thus causing price to fall. Blah blah etc.
Check: the long-run price of oil (as in, neglecting the many price shocks) in grams of gold is absurdly stable.
Is productivity the same?
If you raise productivity you definitely decrease the price of the good. However, you then also free up some labour to go make something else....which seems like it shouldn't have a net effect. There might be something I'm missing, but it makes sense that productivity changes should properly affect good prices. It changes their relative cost. Indeed it's productivity which determines prices in the first place. In a sense, the labour theory of value is correct, at economic equilibrium: the only limiting input for any good is human labour, because labour can be transmuted into more supply of any good that can be supplied. Just don't forget labour is not created equal.
When you print or otherwise debase money, you don't increase the number of goods, so you don't change the number of transactions, you just increase the average transaction size near the money printer. Higher supply of money, lower price of money.
P.S. Population growth is equivalent to immigration from a market perspective. More demand for jobs + more demand for output of jobs => more supply of jobs etc etc. If the immigrants can take all your jobs, then your children can take all your jobs too. Eventually population grows so much that all the jobs are taken long before anyone can find one...that's how it works, right?