Many students fail to get the purpose of the cycle of 4ths and 5ths, so here is how and why they are constructed.....it may look long but the idea is simple and stays with you.
For any scale to be considered a 'major' scale the leading note (the 7th note of the scale) has to be a semi-tone from the 8th note (the tonic)
Scales are constructed by ordering tones and semitones, a C major scale is considered a 'perfect' scale as it does not need any alterations to it to maintain its interval structure.
That interval structure is, T - T - ST - T - T - T - ST (T = Tone, ST = Semitone)
As the diatonic notes are named after the first seven letter of the alphabet we get
A B C D E F G, these seven notes repeat over and over to give us the use of
A B C D E F G A B C D E F G A B C D E F G A B C D E F G A B C D E F G etc etc
The seven intervals follow the patten of use, they also repeat their order,
T - T - ST - T - T - T - ST - T - T - ST - T - T - T - ST - T - T - ST - T - T - T - ST etc.
So any starting point within either the order of the notes or intervals will produce new scales or construct modes.
Look at the note lists below, these are the diatonic notes set out, and this is how we would write them to construct scales, and therefore keys.
Look at the first scale C major, its diatonic notes follow each other and the intervals comply to T - T - ST - T - T - T - ST, so there is nothing to do to it, as said it is considered 'perfect'.
Look at the next scale G major, Look at where is was cut from the C major Scale....it was cut from the 5th note. Count the notes through, see that in fact the C major scale is just cut in half and we use the upper half as the new starting point for each new scale.
(When cut in this manner we have two sets of four notes, this is the basis of Greek Tetra Chord theory, Tetra the Greek word for four.)
5ths
CDEF GABC
GABC DEFG
DEFG ABCD
ABCD EFGA
EFGA BCDE
BCDE FGAB
FGAB CDEF
CDEF GABC
If we examine the G major scale we see it does not follow the order of
T - T - ST - T - T - T - ST because F - G is a tone, so the 7th note is not a semitone, and E - F is a semitone. To fix this we simply add a sharp (#) to the 7th note and this will move it a semitone away from the E and a semitone closer to the G.
By adding a # to make F# we solve the problem and the scale now complies and we have used one sharp for the key of G major
Look at the next scale in the order, D major.
First thing we do is add the F# to make the lower half
DEF#G.
If we look at the upper half we have the same issue as before with the 7th note.
Again we just add a # to the 7th note to make it C#, this will move it a semitone away from the B and a semitone closer to the D, now we have used two sharps for the key of D major.
Go through the rest of the scales in this manner and you will 'cycle through the 5ths' creating new scales and keys by adding sharps to the 7th note each time to make the intervals comply.
You will add a mew sharp each time to the maximum of seven sharps by the time you finish with the C# major scale, but that is to be expected as that scale is a semitone higher than the C major scale...so all seven notes will need to be altered.
If we look at the same Diatonic notes again and start on the 4th note rather than the 5th, we get the flats (b) to be used in creating new scales and keys.
4ths
CDEF GABC
FGAB CDEF
BCDE FGAB
EFGA BCDE
ABCD EFGA
DEFG ABCD
GABC DEFG
CDEF GABC
As we know the C scale is perfect so go to the 4th and use that to construct a new scale, FGAB CDEF now we have F major
First off we check the intervals and that 7th that gave us the problem in using 5ths to construct scales.
E - F is a semitone so that is fine, but looking at the A - B -C that is not.
In order for it to comply to T - T - ST - T - T - T - ST of a major scale, the A - B should be a ST and the B - C should be T.
By adding a flat (b) to the B we move it a semitone closer to the A and a semitone further from the C so now we have FGABb CDEF and the scale complies.
By adding one flat we create the key of F major.
Go to the 4th of F major and we get B, and of we go again.
In the same way we always altered the 7th with a sharp, we will always alter the 4th with a flat and continue till we get back to Cb major which will use seven flats as it is a semitone lower that C major.
This brings in the conclusion that B major uses sharps and Cb major used flats to alter the same pitch with different consequences to the key.
It is the key that will determine whether en-harmonic notes use sharps or flats as we do note mix sharps and flats, as a rule, with in keys.
Since each new scale/key shares notes then so it will sound pleasing to use I IV V structures in songs and harmonise the major scale.
This simple lesson i hope has explained a few of the ideas used in music, and leaving you to complete the rest of the scales and keys will help teach you better about scales and keys if you write them out and work out where they should go and in what order.
The sharps are ordered as F C G D A E B, The flats are B E A D G C F....read forward for sharp, read backward for flats........see the patterns in it all to understand and learn the construction rather than try and memerise.