"later codifications of bebop harmony emerged, notably in the teachings of pianist/educator barry harris, who encouraged players to learn "bebop scales" for improvising such as the bebop dominant 7th scale (1 2 3 4 5 6 b7 7) and the bebop major scale (1 2 3 4 5 #5 6 7). a feature of these scales is that when they are played in 8ths, up or down, players automatically play a tone featured in the corresponding chord on every 4/4 beat. these scales are often disguised by playing them through segments of an octave, changing direction on chord tones, or enclosing chord tones with a chromatic tone above and below the chord tone. both of these techniques allow the improviser to embellish the bebop scale without sacrificing the effect of chord tones on every 4/4 beat."

https://en.wikipedia.org/wiki/bebop
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here is a small excerpt from jazz pianist hal galper's article on "understanding forward motion" which touches on bebop scale theory.

"during this period i bought the album "everybody digs bill evans." the title came from the fact that the album cover was filled with testimonials from great jazz musicians about bill's playing. cannonball's testimonial hit a nerve. he said: bill's melodic lines sound like the best possible lines that could have been played at the moment." wow! if what cannon said was true, this meant that there were reasons why some lines sound better than others. wanting to play the best possible melodies lines myself, i began my research by trying to analyze and understand how to use "one" of the bar. i started with a series of questions and answers.

what defines a 'best possible line?'

answer: a line that is strong.

what defines a strong line?

answer: a line that spells out the chord changes, either basic or superimposed.

how are chord changes spelled out by melodic lines?

using the system of tension and release analysis the obvious became clear:  

answer: by synchronizing the strong beats of the bar with the strong tones of a chord scale.

the release beats of a bar ("one" & "three" and the "on" beats of every quarter note) are the strong beats for the bar. the tension beats of the bar ( "two" & "four" and the "ands" of each quarter note) are the weak beats of the bar.

the release tones of a chord scale are the root, third, fifth & seventh. they are the strong tones of the chord scale.the non-chord tones are the weak tones.

note: when analyzing alterations of a chord scale: 9, b9, #9, 11, #11, b13, etc. the notes that fall on the release beats will be found to be either the root, third & fifth of a superimposed triad. for example: the c# on a c7 can be called a b9. in fm, it could be called the third of an a triad.

because they are the stronger beats and tones of the bar, the "on" beats of the bar and the chord tones have a natural emphasis within them that are enhanced when synchronized. they then spell out changes. when chord tones (basic or superimposed) are synchronized with the "on" beats of the bar, the chord changes are being "spelled out" by the melodic line. the melodies become so strong that even without a chord being played behind them, you can hear the movement of the chords as they progress through a tune.

for example: every musician knows that the f major scale is common to the ii-v-i of the key of f. many beginning improvisers use this understanding to improvise scale wise. they realize that as long as the piano or guitar player is playing the g-7- c7-fmaj., they can just run the scale and sound more or less like they are improvising in the key. however as soon as the accompanying chords are removed from the background, the melodies sound weak because they are un-synchronized. "

"article" continues at:  
https://www.halgalper.com/
excerpt from "the jazz theory book" by mark levine:

"the bebop scales are traditional scales (the ionian, dorian, and mixolydian modes of the major scale, and the melodic minor scale) with an added chromatic passing note. play a traditional descending c7 mixolydian scale over a c7 chord. rhytmically, this sounds rather clunky, because the chord tones (root, 3rd, 5th, and 7th) are played in awkward places in the bar. the root is played on the first beat, but the bb(the 7th) is played on the "and" of the first beat, g(the 5th) is played on the "and" of the second beat, and e(the 3rd) is played on the "and" of the third beat.  

now play a descending c bebop dominant scale over a c7 chord (c b bb a g f e d c). hear the difference? the c bebop dominant scale sounds rhytmically much smoother than the c mixolydian mode. the reason is very simple. the chord tones of the c bebop domninant scale are played on the beat. c(the root), e(the 3rd), g(the 5th), and bb(the 7th) are all played on the beats of the bar. the non-chord tones d(the 9th), f(11th), and a(13th) are all played off the beat. even though the context is a melodic line, playing chord tones on the beat accentuates the harmony of the c7 chord.

the bebop scales were an evolutionary step forward from traditional seven-note scales such as ionian, dorian, mixolydian and melodic minor scales. louis armstrong was playing bebop dominant scales as early as 1927 and can be heard playing a bb7 bebop dominant scale during his solo on "hotter than that." bebop scales were occasionally played by jazz musicians in the 1930s, but they didn't become an every day part of the jazz language until the 1940's. all bebop scales have an added chromatic passing note, transforming them from their seven-note origin into eight-note scales.  

in david baker's words, adding chromatic passing notes to traditional scales make the scales rhythmically "come out right."

you can add chromatic passing notes to any scale or mode, but the most commonly played bebop scales are the bebop dominant, the bebop major, the bebop dorian, and the bebop melodic minor."

__________________________________

there are countless examples of bebop scale licks on recordings, here are just a few:

parker used a descending ab major bebop scale over a i vi in the first 2 meausres of "donna lee".  

john coltrane played a descending bb7 bebop scale over a ii v in measure 8 of the first and fourth choruses of his giant steps solo.

miles played descending eb major bebop in bar 31 of his solo on four.

sonny rollins played descending eb7 dominant bebop scale over a ii v in bars 17 and 18 of his solo on airegin.

gene ammons played f dorian bebop scale over f-7 in bar 9 on miss lucy.

joe henderson played a f bebop dominant over f7 on lee morgan's "totem pole".

freddie hubbard played ab bebop dominant over ab7 in "your my everything".

freddie hubbard played f bebop dominant over f7 on his tune "for spee's sake".

coltrane playe numerous descending bebop dominant scales on his "lazy bird".

sonny stitt played f bebop dominant over a ii v on "eternal triangle".

coleman hawkins played f melodic minor bebop over f minor in his tune "night hawk".
no responses? interesting.
forgot to say thanks?

ok, thanks.

you rock!!!

seriously
ok, i try to be careful about joining this kind of discussion because while i can talk about some of this theory, i can't, for the most part, play it.  still, i want to throw out something that is germane from my teacher.  it's in the blues and he calls it the soloist's award.  it's the ability to understand when the 6th chord is coming and to play, for instance, the f# in the f blues to announce that change.  the ability to outline the chord, presumably without just playing chords, helps us feel the tune.  

and, by the way, i find the bebop scales much more fun to run than the majors.  something about the symmetry of the 8-note scales just is pleasing to the hand.

absolutely agree with 7 on this one too!
anybody else?
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