The simplest meaning of transposition is performing or writing a musical passage in different keys.
Let's imagine for a second the two keys with only white keys, that's C major and A minor. Play the scales of these keys up and down.
Since you know the 2 keys which do not have any accidentals (sharps or flats), it is only normal to assume that there are keys with 1 sharp or 1 flat, with 2 sharps or 2 flats, with 3 sharps or 3 flats etc. until all the 7 notes of the scale are sharp or flat.
In order to transpose to different keys, first you have to know how to construct a scale with 1 accidental, 2 accidentals etc.
This is simple. The term that we need here is the Circle of Fifths - a system of organizing the keys in ascending or descending number of accidentals.
If you have C major scale and want to find a scale (or a key) which has only 1 sharp, you have to count 5 scale degrees up the current scale. For example, since the 5th scale degree in the C major scale is G, the key with 1 sharp will be constructed from the note G, that's G major.
Now go ahead and play the G major scale ascending and descending. Note that this scale also has to have same system of whole and half steps, as C major: G, A, B, C, D, E, F# G. Yes, F# is needed because between the 7 and 1 scale degrees you should have a half step.
Play this G major scale on your keyboard upward and downward. By the way, what you did just now, you transposed the C major scale to another key, to G major.
OK, now in order to find a minor key with 1 sharp, we just go down the G major scale 3 steps and find our minor key with 1 sharp - that's E minor key.
Go ahead and play this scale on your keyboard up and down. Don't forget to add F#. Notice how you just transposed the A minor scale to another key - E minor.
Now you just only need to find another pair of major and minor keys which have 1 flat. In order to do this, take the 1st scale degree of our C major scale but instead of going upward the scale, now you go downward 5 degrees - C, B, A, G, F. Yes, that's the starting note of F major scale.
Play the F major scale up and down but now add B flat (or Bb) to the scale. We need this, because, remember, whole and half steps have to be arranged in the same order as in C major.
In order to find a relative minor from F major, just go down 3 scale degrees from F up to D. That's D minor.
Now play the D minor scale on your keyboard (with Bb).
In order to fully take advantage of transposition exercises, find an easy piece of organ music in C major or A minor and try to transpose to the keys with ascending number of accidental.
If you are new to transposition exercises, then try transposing one hand at a time first and later put them together.
To make things easier, I also recommend you write in scale degree numbers with pencil above each note at first. If you find a sharp next to the note, write a + sign, or if you find a flat next to the note, write - - sign. These signs simply mean that these notes have raised or lowered scale degrees.
If you pencil in every scale degree in your organ piece, then you will be able to play the scale degrees in any key you want. Just play the scale of that key ascending and descending first.
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