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The Ionian mode is identical to the major scale. Both have the following pattern of half and whole steps: WWH W WWH
If you look at the dulcimer, this pattern makes sense. Remember that the only half steps on the dulcimer are between a marked course and the one below. So from C to D to E is two whole steps, then a half step to F. A whole step takes us over to the other side, where we repeat the WWH pattern. Since people use sharps and flats nowadays, we can easily apply this pattern to any other key note:
F Ionian: F G A Bb C D E F -- The B is flat to ensure a half step from A and a whole step to C.
G Ionian: G A B C D E F# G -- The F is sharp in order to ensure a whole step from E and a half step to G.
D Ionian: D E F# G A B C# D -- F is still sharp for the same reason. C is sharp for a whole step from B and half to D.
A Ionian: A B C# D E F# G# A -- F and C are still sharp. Now we add G sharp for a whole from F# and half to A.
To summarize:
Shape: Box
Starts on: Marked course
Steps: WWH W WWH
Examples: Liberty, Soldier’s Joy, Southwind, Simple Gifts
G Mixolydian and G Ionian share the same key note, G, but G Ionian has one sharp, and G Mixolydian has none. Mixolydian's pattern of whole and half steps is WWH WWH W. Compared to Ionian, then, the Mixolydian mode has a flatted seventh note.
C Mix: C D E F G A Bb C -- In C Ionian B is natural.
D Mix: D E F# G A B C D -- In D Ionian, C is sharp.
A Mix: A B C# D E F# G A -- In A Ionian, G is sharp.
E Mix: E F# G# A B C# D E -- In E Ionian, D is sharp.
G Mixolydian and C Ionian share the same key signature, no sharps or flats. When two modes share a key signature, we say they are related. (You may have heard of relative minors; well, relative modes is the same idea.) For any key signature, the Mixolydian scale begins on the fifth note, e.g. G, of the relative major scale, e.g. C.
Shape: Vertical
Starts on: Marked course
Steps: WWH WWH W
Key Note: Fifth note of the related major scale
Examples: Over the Waterfall (A part), June Apple, Red-Haired Boy, Old Joe Clark, March of the King of Laois
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