As I understand it, kumoi is a mode of the hirajoshi scale:
http://en.wikipedia.org/wiki/Hirajoshi_scale
Wiki is not always 100% reliable of course -

- but that seems a knowledgeable page.
However, it states (and take this as you like): "The pentatonic formula [1-3-4-5-7 of a scale] can also be used over ... the locrian mode to produce the kumoi scale".
That gives - from A - A C D Eb Bb. This doesn't correspond with any other "kumoi" formula I've found. There's also no way locrian mode can produce any of the likely candidates for kumoi mentioned below.
So - black mark for wiki there!
I have another extensive scale resource which has no specific "kumoi" scale, but gives the following (again all measured from A for comparison):
Kata-kumoi (aka Hira-joshi): A B C E F
Hon-kumoi-joshi: A Bb D E F (5th mode of the above)
Han-kumoi: A B D E F
Other (perhaps less reliable) sources for plain "kumoi":
A B C E F# - 1 2 b3 5 6 (This is a mode of the "han-kumoi" above, and jazz musicians know it as the "b3 pent" or dorian pent.)
http://www.freakguitar.com/kumoi.html
http://www.all-guitar-chords.com/gui...t&t=0&choice=1
http://www.playbassonline.com/kumoi-scale.html
A Bb D E F - 1 b2 4 5 b6 (as "hon-kumoi-joshi" above, 5th mode of hirajoshi)
http://www.ehow.com/how_4457596_play-kumoi-scale-
http://hubpages.com/hub/EXOTIC_SCALES
http://www.ultimate-guitar.com/colum...ony_style.html
http://www.shredaholic.com/user48.html
This last person seems to know what they're talking about, and refers to it as "Hon Kumoi Shiouzhi". I guess "Shiouzhi" is the same word as "joshi", just phoneticized differently; and seems to meaning "tuning", so would be a dispensable part of the scale name.
IOW, "hira", "hon-kumoi" and "han-kumoi" are all types of "joshi" or "shiouzhi". (? - perhaps a Japanese speaker could correct me here! *)
This site has the "kata-kumoi" (1 2 b3 5 b6):
http://pianoencyclopedia.com/scales/...ata-kumoi.html
This piece:
http://www.youtube.com/watch?v=XCbRFZqEI_s
says it's using the kumoi scale, and the notes are G A Bb D Eb, with the apparent keynote as G (most commonly played bass or drone note). Transposed to A, that's A B C E F, and is what seems to be usually known as Hirajoshi (or kata-kumoi).
The piece does occasionally rest on the 5th of the scale (D) - and the 5th mode is "hon-kumoi-joshi", the 1 b2 4 5 b6 scale some of the above websites favour.
However, that doesn't sound like a keynote to me. But then my ears are not Japanese!
This lesson goes with A Bb D E F (hon-kumoi):
http://www.associatedcontent.com/vid...le.html?cat=33
* EDIT: here's some online translations of "joshi":
http://www.freedict.com/onldict/onldict.php
- in a musical context, I imagine it could easily mean "tuning" or something like "system" or "string arrangement".
Another translation site has 35 meanings (mostly referrring to women), so it's clearly a non-specific word with lots of uses: no musical ones listed, but it does include "arranging seats by seniority".
"Hira", meanwhile, means "the broad; the flat; palm".
"Kumoi" can mean "sky; cloud; distant place; high place; imperial court"