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The major scale of C is built of 2 equal tetrachords (= four step ladders) of 2 whole steps and a half step): W W H 8notice that between the 2 tetrachords is another whole step.

C D EF and G A BC = W W H (W) W W H

(the half steps are made visible by the closer distance)

the (WH) is referring to the whole step between F and G

The relative names of a major scale are Do Re MiFa So La TiDo (mifa and tido are the halftone steps, Ti is called leading tone to Do, Fa leading tone to Mi)

You can now develop all the scales of the sharps by using the second tetrachord of the C (so la tido becomes the do re mifa of a new scale with the root note G. We get G A BC and D EF G.

The first tetrachord G A BC is now built identical and can be replaced as 1 2 34 ... but the half step of the second EF is between the 6th and 7th degree 5 67 8 and we have no leading tone from 7 to 8. By augmenting the 7 degree with a sharp # we construct the leading tone and solve the problem of the half step between 6 and 7. (5 6 78) We have a new upper tetrachord with a halftone between 78 with 1 # (F#). Continuing ths operation the upper tetrachord will become the lower one of the next key in the direction of the circle of 5ths clockwise.

CB A G ... FE D C (ToTi La So FaMi Re Do)

FE D C build a new scale with the root tone F. The upper tetrachord of F major is useful but the lower has a problem: CB A G F is not congruent with So FaMi Re Do. We need to flatten the B to Bb (a half tone lower) and get a whole tone between the upper and the lower tetrachord: FE A G (W) BbA G F and we have also construct the new leading tone Fa to get a half tone to Mi (or Me) BbA. That's the explaining why the scale of F major has 1 b-flat. Following this principal the lower tetrachord of F (BbA G F) will become the upper of the new scale Bb-major with two flats (Bb and Eb) etc.

The major scale of C is built of 2 equal tetrachords (= four step ladders) of 2 whole steps and a half step: W W H => C D EF and G A BC - the half steps are made visible by the closer distance.

notice that between the 2 tetrachords is another whole step (W). Thus the major scale has the pattern: W W H (W) W W H whereby (W) is referring to the whole step between F and G, the whole step between the two tetrachords.

The relative names of a major scale are Do Re MiFa So La TiDo (MiFa and TiDo are the halftone steps, Ti is called leading tone to Do, Fa leading tone to Mi)

You can now develop all the scales of the sharps by using first the second tetrachord of the C scale:so la tido becomes the do re mifa of a new scale with the root note G. We get G A BC and D EF G.

The first tetrachord G A BC is built like the first tetrachord of the C scale and can be replaced as 1 2 34 ... but the half step of the second tetrachord EF is between the 6th and 7th degree 5 67 8 and we have no leading tone from 7 to 8. By augmenting the 7 degree with a sharp # we construct the leading tone and solve the problem of the half step between 6 and 7. (5 6 78) We have a new upper tetrachord with a halftone between 78 with 1 # (F#). Continuing ths operation the upper tetrachord will become the lower one of the next key in the direction of the circle of 5ths clockwise.

CB A G ... FE D C (DoTi La So FaMi Re Do)

FE D C and build a new scale with the root tone F. The upper tetrachord of F major is useful but the lower has a problem: CB A G F is not congruent with So FaMi Re Do. We need to flatten the B to Bb (a half tone lower) and get a whole tone between the upper and the lower tetrachord: FE A G (W) BbA G F and we have also construct the new leading tone Fa to get a half tone to Mi (or Me) BbA. That's the explaining why the scale of F major has 1 b-flat. Following this principal the lower tetrachord of F (BbA G F) will become the upper of the new scale Bb-major with two flats (Bb and Eb) etc.

Why does this pattern span two keys in the sharp side, but only 1 in the flat side? For ex: D# is the 4th sharp note added, but key of D has 2 sharps. The difference is 4-2=2 sharps. In contrast, take Ab -- it is the third flat note added, but it is the name of the key of 4 flats. 4-3=1.

The fundamental difference between the right side and the left side of the circle of 5ths is that the additional sharp is the new 7th degree and the additional flat is the new 4th degree (so there is no congruence between the Fa and the Ti, but you will find it if you compare the old Ti with the new Ti (right side comparing the sharps and left side the naturals (♮) ... the difference will be analog in both sides.

(we used in the 8th class in secondary school in singing lessons to develope up to over 40 sharps counting in a septimal number system: the key of C = 0 sharp, C# = 7 sharps but continuing this idea might be only confusing you...)