how to harmonize a melody whitout a specific scale?

Chords, scales, harmony, melody, etc.
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Thx for your nice explainations! :)
It helps alot!

But can I use also every kind of chord cadency or must I always just use chords which exactly fit with the melody?

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You should generally use chords which include at least some of the notes you're playing them under, but not always and not exactly. Look at some jazz or classical fake books. Here's a random example:

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First measure you've got an F major chord, with the notes F, G and A in the melody over it. There's no G in the chord, but that's fine. Take a look at some stuff like that, and some simpler jazz standards like "Autumn Leaves" and you'll see how theory is applied in practice.

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There's no G in the chord, but that's fine
Is this called passing notes and can such a chord on a down beat?
you'll see how theory is applied in practice.
The practice is exactly the reason which leads me to my questions...

Especially they are sound always in key but if I try to harmonize my melodies, it sounds most of the time like grap.

One the one hand I try to use cadencies like V I or IV V I with tritonus and pre quarte and sixte ajourte(I dont know how it is spelled in English) ... on the other side I try to fit it with most of the notes... But it dont work...
Especially if one step intervalls occur, which I cant avoid because there is always a note which is just one step away of a chord note... Yeah, this is my problem.

Another problem is the harmonic rhytm: when to change and how long to hold a chord a.s.o. ...

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If the G was between a A and an F, it would be a passing tone, but since it's between two As it's technically a neighbor tone.

What cadences to try using and how long to hold a chord depends on the style. Bach changes chords a lot more often than 2 Unlimited... but generally if you can make the same chord work for one or two measures that's a good rule of thumb to aim for in a lot of styles.

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