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What is this mode called?

? Never heard this definition. AFAIK the term dominant refers to the chord function.

The major 3rd of the dominant chord is the leading tone, which creates a tension to resolve back to the tonic. This seems to emphasize the importance of the major 3rd.
Even more so when combined with the tritone (created between 3 and 7 when 7 is included) which when resolved inwards (3rd up to tonic and 7th down a half-step to 3rd of I) is compelling in its power.

On the other hand I get the concept of "devaluing the influence of the 3rd". A major triad in root position is highly stable, restful and motionless thanks to the highly consonant intervals. Heard in isolation and without other harmonic or melodic context it could be I, IV or V, but will almost certainly be perceived as I. Putting in the minor 7th upsets that stability, much of which was created by the 3rd.
 
C D E F G Ab Bb
It's like "major aeolian."
I heard a recording of a Puerto Rican band playing a tune which used it.

Like others have said: Mixolydian flat 6th.

Why that and not Aeolian major? First of all, if you’re going to see it as a variant of Aeolian, you should call it Aeolian dominant, because ‘major’ also implies a major 7th. That’s why C Db E F G Ab Bb is called Phrygian dominant, not Phrygian major.

But even then, it makes more sense to see it as a variation of Mixolydian instead. The reason is that it contains both the major 3rd and the minor 7th. That interval is unique to Mixolydian and gives it its characteristic sound.

For Aeolian, the minor 3rd is a very characteristic note, so once you change that, there’s nothing Aeolian about it anymore. It would be like calling a scale without a flat 2nd a variant of Phrygian.
 
Thanks, everyone. Mixolydian b6 makes sense to me.
Sorry, but I can't provide a recording. A friend showed me a cellphone video she shot of a group of singers, instrumentalists and dancers performing at a restaurant in Puerto Rico. It's a custom which was previously unknown to me-- these amateur groups are sort of the descendants of troubadours. They show up at bars or restaurants, a space is cleared, and they perform (ritually, at least) for food. To honor the "sing for my dinner" tradition, their outfits are adorned with colorful spoons and forks. The ones with the most utensils have the highest status.
The song that I heard moved from the Mixolydian b6 mode to the Phyrigian mode.
 
Way I learned it
A 13 implies the presence of natural 9 and 11
YMMV
A 13 implies natural 13, which implies a natural 9; flat 13 does not. Not to me. But, I think we are getting into academic territory, which I loathe despite my background, So I will present it differently.

If someone handed you a piano chart with a C7b13 and you played a D on it, everyone would cringe. If you played a Db, birds would sing to it.

b13 + b9 = very common.

b13 + #9 occasionally in jazz.

b13 + natural 9 = unicorn.

I actually cannot name one song that has a b13 natural 9 as an important part of its structure.

Can anyone name any popular song with a 7b13 that has a natural 9 in it?
 
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Slight change of subject: Just for fun and just now, I paged through about 50 pages of the Real Vocal Book, in the middle somewhere (I think it was page 90 to page 140 or so) and couldn't find any 13th chords at all. My general impression of what I see in a Real Book that 13th are more often melody notes that aren't specified in the chord names.

The one example I remembered was in All The Things You Are, about 7 bars before the end, where the chord symbol says "Gb7 (13)". (Key is Fm/Ab - 4 flats.) I assume that to mean Gb-Bb-Db-Fb plus an Eb, which is the melody note. I add in a M9, a Ab, which sounds good to me so I'm playing Gb-Bb-Fb-Ab-Eb - no 5th. The 5th in that chord ruins the impression of the 13th as an appoggiatura, a leap up to the note over the Db which then resolves to the Db in the melody. The way I learned chord notation what I'm playing I'd called Gb9 add 13, meaning to leave out the 11th but keep the 3rd.

Where is @Don Kasper when you need him? :)

-S-
 
Before I knew better, I thought of Phrygian Dominant as MixoPhrygian

With apologies to the poster who requested no Dad Jokes: "MixoPhrygian" sounds like the brand name of a does-everything kitchen appliance!

The way I learned chord notation what I'm playing I'd called Gb9 add 13, meaning to leave out the 11th but keep the 3rd.

The way I orginally learned chord notation mirrors ^^^that experience...but at some point in my ~50 years of practical experience playing and writing/arranging for working ensembles I came to believe that every musician worth their salt should know that the 11th (altered or not) is a special situation requiring some spur-of-the-moment editorial decisions by the chord instrument players: They should know whether including that tension will work regardless of what the chord symbol on the page says.
 
...
My general impression of what I see in a Real Book that 13th are more often melody notes that aren't specified in the chord names.
...

This is a very true and astute observation. I think it comes down to an almost obsessive desire in some quarters to fully qualify/specify every chord at every turn, even when it makes no difference to the harmonic function of the moment. A V chord in a V-I cadence still functions as dominant no matter how many extensions you pile on top of it. I suppose eventually it will start to sound more and more like a vii7°, but vii7° still functions as dominant in the key.
 
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C D E F G Ab Bb
or
F G Ab Bb C D E
??

I also would really love to hear the recording. Or at least the chord progression for context.

Regardless of nomenclature, in context it may possibly be chord V of an F harmonic minor. It’s also fine if that’s the “home chord” aka “key” of the song.

I like my music like my chicken wings…. extra spicy.