The guitarist worked with producer Stephen Street to transform shelved recording sessions the band undertook in Japan in 2013 into tracks for 'The Magic Whip' and says one of the reasons he undertook the project was to ''make amends'' with the singer for the problems they have had in their friendship.
Graham said: ''Part of the reasons why I wanted to do it was to make amends.
''Damon and I have an increased respect for each other because of this record, and we're not ashamed to let each other know about that increased respect.
''But what we also have a lot of is history and our friendship - like any friendship between two people who are in a band together - has had to go through a lot.
''It's been put to the test and we've often let each other down. This record was a way of saying, 'Sorry for being such a pain in the a**e for the last 20 years.'''
Meanwhile, Damon addresses the pair's relationship in a track on the album, 'My Terracotta Heart', which surprised Graham when he heard it.
He told NME magazine: ''I knew it was going to be an incredibly sad song, which is why I put the crying guitar on there.
''What I didn't know at the time was that the lyrics would turn out to be about Damon and I, our long friendship and the ups and downs we've had.
''It's a lovely song. The four of us have kind of met in the middle with this album - we've all been off on our own individual journeys, but when we come together and something like 'My Terracotta Heart' is the result, that's a good marriage of all our different tastes and outlooks.''
Corgan took to Instagram to confirm rumours of new Pumpkins material, saying the first songs could arrive as early as May.