Usually an interval is named after counting the keys along the scale (c−g is called fifth because c d e f g are five keys). This is impractical as first the concerning scale has to be determined (to know the fifth of e with this method one has to count along: e f# g# a b). Moreover it is inaccurate to count the keys in stead of the intervals between them (the interval fifth is the total of 4 intervals on the scale). (Remark for mathematicians: an interval is actually a quotient). And you count one more than necessary.
The way of counting I recommend is along both the black and white keys. E.g. from c to its neighbor d I consider as 2 key steps: from c to c# and from c# to d. So, to make a fifth you must count 7 key steps to the right or faster 5 to the left.
In lesson 03 of 2003/2004 we used the counting of key steps to make chords, e.g. a major triad was made by 4 plus 3 key steps. For who hasn't used it (often) here is an example: to make the chord F#m7 you start with the root note f# and then step +4, +3 (a, c#) and for the seventh note −2 from the root note (f# minus 2 steps gives e). In less than 5 seconds you have found the chord notes (f# a c# e).
Transposing a chord scheme can efficiently be done by counting
of key steps. Suppose you have to transpose chord scheme F D7 Gm
C7 from F to A. Then you must shift every chord 4 steps to the right. In
case you have no keyboard in front of you, you may write a keyboard
(of say 2 octaves) on paper. You will only have to write the white keys,
because the black ones may be indicated by a just a little stripe:
Now it is easy to see you will have to count 4 steps to the right in order to go from f to a, in short: +4. The other chords of the scheme are: d+4 gives f#, g+4 gives b and c+4 gives e. So the new scheme will be: A F#7 Bm E7. The additions (in this case 7 and m) stay the same, making the transpose easier than it seems at first.
Even transpose during playing is possible with counting of key
steps. It only requires some preparation. Suppose you know a song with
the scheme F D7 Gm C7 and you will have to accompany someone
who is going to sing it, but you don't know in which key he will do that.
Then you may determine the intervals between the chords beforehand.
There are two ways for it: every chord relative to the root or every chord
relative to its predecessor. We will choose the first method. The steps
relative to the root (f) are respectively −3, +2, −5 (from f to d is −3, from f
to g is +2 and from f to c is −5). So at home you may write the chord
scheme as follows: −3, +2, −5 or more detailed as follows:
(X) (X−3)7 (X+2)m (X−5)7. Here (X) is the initial key the singer wants. For example, if he wants A, then you must count quickly: A−3=F#, A+2=B en A−5=E. So the transposed chord scheme is: A F#7 Bm E7. The additions 7 (seventh) and m (minor) will give no difficulties, as you are familiar with the song already, though in another key. This method works fast for who knows the new chords already, otherwise he will have to make those too by counting of key steps, making the performance slow.
More applications are: from root to dominant = −5 (reverse +5), to subdominant = +5 (reverse −5), to relative key = −3 (reverse +3), from subdom. to dom. = +2 (reverse −2), major scale upward (stepping to the neighboring key) = +2+2+1+2+2+2+1, harmonic minor scale = +2+1+2+2+1+3+1 (this stepping to a neighboring key may be used not only for melodies but also for chord schemes), etc.