Why would a scale that starts on Eb, sounds bright and major, and feels at home on Eb, carry four flats in its key signature instead of the three that Eb major uses? The answer is one note, Db, and once you understand where it comes from, the whole Eb Mixolydian mode falls into place. A mode, by the way, is simply a scale built by starting a familiar set of notes from a different home note. Mixolydian is the major-sounding mode with a twist, and in Eb that twist has a warm, brassy, slightly bluesy color.
Here are the notes of Eb Mixolydian: Eb, F, G, Ab, Bb, C, Db. Seven notes, then Eb again at the top.
The first rule of spelling any scale like this is that each letter name, A through G, appears exactly once. That's why you'll never see both C and C# in it, and why the seventh note is Db rather than C#. We've already used C for the sixth note, so the next step up has to carry the letter D, and it's lowered with a flat.
Walking up from Eb, the pattern of steps is W - W - H - W - W - H - W. A half step (H) is the smallest step in Western music, like moving from one piano key to the very next one, black or white. A whole step (W) is two half steps.
Notice where the two half steps fall: between the 3rd and 4th notes (G to Ab), and between the 6th and 7th (C to Db). In an ordinary major scale, that second half step sits right at the top, between the 7th and the root, which creates the strong upward pull you hear at the end of "do re mi." Mixolydian moves it down one spot. The result is a whole step from Db up to Eb, so the top of the scale feels relaxed instead of straining toward home.
A key signature is the set of sharps or flats written at the start of a piece so you don't have to mark them on every note. Eb Mixolydian borrows its signature from Ab major (Ab, Bb, C, Db, Eb, F, G), which has four flats: Bb, Eb, Ab and Db.
Same seven notes, different home. Play Ab major on a piano, then play those same keys starting and ending on Eb, and you've got Eb Mixolydian. What changes is the center of gravity: your ear treats Eb as the resting place.
Compare that to Eb major, which has three flats and uses D instead of Db. That single flattened note, the flat 7th, is the whole difference. Formula-wise, the mode reads 1, 2, 3, 4, 5, 6, b7: a major 3rd for brightness, a perfect 4th and 5th for stability, a major 6th, and that minor 7th for grit.
You might see the same sounds written as D# Mixolydian. It's exactly the same pitches on the keyboard, but spelled with eight sharps, meaning some notes need double sharps. Nobody wants to read that. Eb is the sensible spelling, and it's the one you'll meet in real charts, so don't worry if the alternative looks strange.
Stack every other note of the scale and you get these chords:
Two chords carry the Mixolydian flavor. The home chord becomes Eb7, a dominant seventh that sounds settled here rather than restless. And Db, the bVII, is the classic rock-and-roll move: a major chord a whole step below home. Even the v chord turns minor (Bbm), because its middle note is Db.
Try this progression, two beats or a bar each: Eb - Db - Ab - Eb. That descending slide through bVII and IV back to I is all over classic rock, gospel-tinged soul and horn-driven funk. Eb itself is a natural home for horn sections, so a riff in this mode sits comfortably for saxes and trumpets.
Sing or play a tune in Eb Mixolydian and you'll feel two pulls. Ab, sitting a half step above G, likes to lean back down onto G, the major 3rd. Db leans down to C, or can step up a whole step to Eb with a shrug rather than a push.
Melodies here often circle Eb, G and Bb for brightness, then dip to Db for attitude. Landing a phrase on Db over the Eb chord sounds bluesy and unfinished; following it with C or Eb settles things. Songwriters lean on that tension-without-urgency for choruses that feel big but laid-back, and fiddle players in Celtic dance tunes use the same mode for its lilting, open feel.
Find the Eb root on the low E string at fret 11 (or 23 on guitars with that many frets), or on the A string at fret 6 (and 18). From the A-string root at fret 6, play the scale up across the strings and listen for the Db; on the G string it falls at fret 6, right under your finger. Strum an Eb7 shape, then a Db, and you've got the mode's whole personality in two chords.
On any keyboard, hold an Eb in the bass. With your right hand, play the scale slowly from Eb up to Eb, then play it again replacing Db with D. Go back and forth three times. Then hum a short phrase of your own that ends on Db, and another that ends on Eb. Notice which one makes you want to keep going.
For the bigger picture, read The Mixolydian Mode.