There's three problems with this...
1. 802.11 legacy/802.11b DSSS channels are recommended to be spaced at least 22MHz apart, as they have a 20MHz carrier (allowing 2MHz of guard space between users). Since this isn't possible as designed (except channel 14), 1, 6, and 11 are used as the non-overlapping channels. They're 25MHz apart, resulting in a 3MHz waste of spectrum. OFDM channels have a 16.25MHz carrier and the total space recommended is 20MHz. Channel 14 is 22MHz above 11 and is DSSS only, only ever approved in Japan, and essentially useless today even there given how slow DSSS is. But the channel plan is still designed around it.
2. Based on this, much equipment ships defaulted to 6 or 11 and many people know "1, 6, 11." It's NOT good to have 5 and 6 or 11 and 9/13 used in the same space. It's much better to just stick to 1,6,11 than let this happen.
3. As far as I'm aware, no 802.11 equipment supports channels 12 and 13 when set to a US regulatory domain, or has been approved for operation on 12 and 13 in the United States. 12 and 13 are not strictly illegal, per se, but the band directly above Wi-Fi is restricted in the United States so allowing a 10MHz guard space has been the de facto regulatory requirement. Even if you decide to break the law and "risk it" on 13, you have to deal with neighbors on 11 and incompatible client cards. Why bother?
The ONLY channels should be 1,5,9,13. But that's not reality. You have to look at the radio spectrum in your area to make sure the channel you want to use has no strong adjacent channel users.