The Circle of Fifths

~~Lord Larehip~~ · 12-14-2013, 10:49 AM

Are the pentatonic (5-note) and diatonic (7-note) scales related? Yes, intimately.

Now, if you go to the Circle of Fifths and connect every 5th or 7th mark with a line you will get the identical geometrical figure:

We could also make a 12-sided polygon if we connect every mark in sequence or by counting by 11s. If we count by any other intervals, however, we will not hit all 12 marks on the circle. They will form only squares, triangles or hexagons. Why is that? Because a 4th is 5 half-steps, a 5th is 7 half-steps, a minor second is 1 half-step and a major 7th is 11 half-steps, and the octave is 12 half-steps. 1, 5, 7 and 11 are what we say in mathematical jargon co-prime in modulo 12. That is, they are not divisible with any number other than 1 and themselves and also not factors of 12. All the other intervals are not prime or are factors of 12 or both and so end up creating circles that are factors of 12 but never all 12.

Geometrically, a Circle of 5ths should be a large 12-pointed star with a smaller 7-pointed star at each of the 12 points.

12-14-2013, 10:49 AM	#20 (permalink)
~~Lord Larehip~~ Account Disabled Join Date: Jun 2013 Posts: 899	Are the pentatonic (5-note) and diatonic (7-note) scales related? Yes, intimately. Now, if you go to the Circle of Fifths and connect every 5th or 7th mark with a line you will get the identical geometrical figure: We could also make a 12-sided polygon if we connect every mark in sequence or by counting by 11s. If we count by any other intervals, however, we will not hit all 12 marks on the circle. They will form only squares, triangles or hexagons. Why is that? Because a 4th is 5 half-steps, a 5th is 7 half-steps, a minor second is 1 half-step and a major 7th is 11 half-steps, and the octave is 12 half-steps. 1, 5, 7 and 11 are what we say in mathematical jargon co-prime in modulo 12. That is, they are not divisible with any number other than 1 and themselves and also not factors of 12. All the other intervals are not prime or are factors of 12 or both and so end up creating circles that are factors of 12 but never all 12. Geometrically, a Circle of 5ths should be a large 12-pointed star with a smaller 7-pointed star at each of the 12 points.