I'd like to offer some comments on just one song mentioned in the article, "Dreamland." Monk may or may not have written it. Recordings exist from 1958, 1969 (available as part of a Mosaic Records DVD box set), and 1971. Monk never copyrighted it.
Iverson posts two charts for Monk's "Dreamland," one that was done by Paul Motian (without barlines), and a version re-charted by Bill Frisell (with barlines). You can find the charts towards the end of Iverson's article, along with a discussion of the tune.
Here's the 1958 version. Monk did not approve this track for release, and it was not included in the original "Thelonious in Action" album pictured below; Orrin Keepnews eventually released the track on an album called "Blues Five Spot" (1984), after Monk passed away in 1982. On the 1984 album it was titled "unidentified piano solo."
Here is Monk's only studio recording of "Dreamland," done in London for Black Lion Records in 1971:
When the 1971 track was released, the record company titled it "Meet Me Tonight in Dreamland." Most people who are familiar with "Meet Me Tonight in Dreamland" would call this a mistake, as Monk's tune seems to have nothing to do with the old 1909 waltz that has been recorded countless times over the years.
Although Monk's tune is definitely not "Meet Me Tonight in Dreamland," I do hear some similarity to the original verse (introductory lead-in) of the 1909 tune. Here are Elizabeth Wheeler and Harry Anthony singing the tune, starting with the verse, recorded in 1909:
Here are the first two pages of the original sheet music, showing the verse (click to enlarge):
Here are the opening bars to each song, with "Meet Me..." transposed to Eb, for easier comparison. In "Meet Me...," if you remove the third note in bars 1, 3, 9, and 11, you will see a melodic curve very much like "Dreamland."
It seems at least possible that Monk took the melodic shape of the verse to "Meet Me" as a starting point, then wrote his own tune. In his recordings, Monk plays "Dreamland" straight through as an arrangement, with no real improvised solo, much as he did in his performances of "Crepuscule With Nellie."
Note that the melody of "Dreamland" falls on the #11 of a dominant chord in bars 4 and 8, and on the b9 in bar 16. Also note the pickup/triplet shape in bars 3 and 7. These are characteristic bop features, although it's always possible that they were added as part of a Monk interpretation.
There's a notation at the top of the first page of the 1909 sheet music that cautions us:
PLEASE NOTE:--Owing to the phenomenal and unprecedented success and sale of this beautiful song, there have been placed on the market, imitation "Dreamland" songs with very similar titles.
This song written and composed by LEO FRIEDMAN and BETH SLATER WHITSON is THE ORIGINAL song of this title and WE CAN PROVE IT.
If Iverson is right that Monk's "Dreamland" is an old parlor tune, one might guess that it could be one of those "Meet Me Tonight" knockoffs that the sheet music warns us about - but the style of both melody and changes is all wrong for 1909.
In his definitive biography Thelonious Monk: The Life and Times of an American Original, Robin D. G. Kelley writes,
"Dreamland" has been mislabeled and misrepresented many times...I have reviewed both songs ["Meet Me..." and a song titled "Dreamland" by Goetz and North], along with dozens of other songs with the title "Dreamland" (Harry L. Newman's "Take Me Back to Dreamland," Harold Arlen's "Hit the Road to Dreamland," Francis Paul, "Dreamland," ad nauseum). None of these songs bear any resemblance to what Monk played on those two occasions. After ten years of searching, querying, and digging, I have come to the conclusion that...it is a Monk original. Perhaps it is a sketch of a song never quite finished.
My own opinion is that there's a pretty good case that Monk composed the tune, taking off from a paraphrase of the verse to "Meet Me Tonight in Dreamland."
Finally, here's one last video, a 1950s Les Elgart big-band version of "Meet Me Tonight in Dreamland." Sitting in the sax section of local big bands, I played this arrangement countless times, but it took a few gigs before I recognized the original melody buried in the arrangement: