Transposition

    The smallest musical interval we recognize in the normal music of Western culture is called a half step.  Our tonal melodic music is usually composed using scales or modes. These scales are built with half steps, whole steps, minor thirds and occasionally major thirds. If you build a scale of any kind, even one of your own invention, you must put together a sequence of these intervals either going up or going down, using the letters of the alphabet. We usually describe a scale by the octave. An octave being eight notes from the Greek “octo,” like an octopus.    
    An octave is an interval which refers to the distance between the lowest pitch and the highest pitch. The lowest pitch will vibrate exactly half as fast as the high pitch, so if you have a note vibrating at 440 cps. (cycles per second) an octave above that would be 880 cps. and an octave below it would be 220 cps. Thirteen (13) half steps  also make an octave, so if you start with C the next note would be C-sharp followed by D followed by D Sharp followed by E followed by F followed by A sharp, followed by G followed by G sharp followed by a, followed by A sharp followed by B followed by C.     The distance between each note is referred to as an interval so that the distance between C and the G above it would be called  some kind of a fifth.
There are diminished, double diminished, major, minor and perfect intervals. The perfect intervals are called that because they are the same in both major and minor scales and most “modes.” In today’s music, a mode may be thought of as a scale. The term mode is still used today, but you can use the terms interchangeably.
    The C major scale is usually considered to be the simplest scale because it has no sharps or flats in it. The sign for a sharp is what we call today #. The sign for a flat when using text would be analogous to a small b. although at this level you won't find it too many of them you will also find double sharps and double flats. The # in front of the note indicates that note is one half step higher.
A double sharp indicates two half steps higher. On a piano the note C double sharp would be the same note as D, but it's not called it D. it's called C double sharp.
The order for the scale C Major is as follows:
A B C D E F G A
Rule number one
The interval between E and F is always a half step. Always!
The interval between B and C is always a half step. Always!
In a major scale there are only two half steps: they occur between the third and the fourth step and between the seventh and eighth step. This is the natural order for a major scale.
A major scale also shares a key signature with a minor scale three half steps lower.
So C major would share a key signature with a minor. (it is usual to use small letters with minor scales or chords and caps with the major scales and chords.
As a consequence the half steps in a minor scale are not between three and four and seven at eight,
but between steps two and a three and the steps five and six.
Now
If you write the alphabet out     a b c d e f g a, you can see where the half steps are in the minor scale. If you remember that between b and c and between e and f our always half steps, you can build your own minor scale starting on any letter by simply writing an alphabet out four yourself. Here's an example
B C D E F G A B
Let's make it a minor scale first. Once again remember the half step is between B and C, but we don't want to halfstep there: we need a whole step. What do we do?
Since we're starting on B we cannot alter that note by adding a sharp or a flat because that's the root note, meaning that it's some kind of B scale.
We need to alter the C, making it one half step higher so we added a sharp. Now we have B to C-sharp and that is a whole step. Whoops! Since we’ve raised the the note C to C#, we’re left with a half step between C# and D. What to do?…….. raise the D to D#….is that right? Whoops….no it’s not!! That’s a MAJOR tetrachord!
Why?
Well, we are trying to build a minor scale. Where are the half steps in a minor scale?
Between the second and third notes. Oh! …. so we didn’t need to make the D a D#.    OK!
Now we have the notes B, C# and D - so, probably, the note E comes next (yes it does). We now have the first tetrachord in a B minor scale (or any minor scale or mode!) The second tetrachord of the scale starts one whole step above the fourth note of the first tetrachord. But you need a half step between five and six; so this is not the same tetrachord as the first one. This tetrachord is called a Phrygian tetrachord. It begins with a half step followed by two whole steps. We have now created a minor scale beginning on the note B. This scale is known as the “natural” minor scale.
If you are a pianist, the half steps occur in the same order as it were from A to the A above it.
Using some of what we’ve just learned, let's build a major scale starting on the note B.
We're going to build a scale with two tetrachords. The first tetrachord is a major one. So that would be a tetrachord with a single halfstep between three and four. So let's lay down the alphabet starting on B. B, C, D, E. That's it that's the tetrachord. Is it correct?
No it's not! The interval between D and E is a whole step. This tetrachord has a half step between B and C -that’s the Phrygian tetrachord! Whoops! We need a major tetrachord. Whole step, whole step, half step. Look back above where we added the D sharp in the minor tetrachord. That was not right, was it? But it is right for a major tetrachord. So now we have a major tetrachord. B, C# D# E. (sounds like the theme from The Addams Family)…… guess what?
The next tetrachord is the same as the first. That would be F# G# A# B. Whole step, whole step half step. Yup! The major scale contains two major tetrachords: easy PZ!
 You now also know the major, minor and Phrygian tetrachords! There is only one other tetrachord needed for you to build almost any musical scale in Western music, and that is the Havah Nagila tetrachord (can be called the Middle East tetrachord or the Ahavah rabbah mode) it begins with a half step followed by an augmented second and another halfstep. Starting on C it would go like this C Dflat E F. We need this tetrachord to build the harmonic minor scale.
Now…. using some manuscript paper (paper with music staves) try building some major and minor scales for yourself. Remember! Start with an alphabet. If you are beginning a scale on G, then write out G,A,B,C,D,E,F,G. If you wish to separate the scale into two tetrachords G,A,B,C,  D,E,F,G, - find the half steps, change them if they are in the wrong place by altering the alphabet note with sharps or flats, according to the scale’s need.