I actually didn't realize that the notes of the major scale and it's modes were contiguous on the circle of fifths until seeing guitardashboard.com last night. That's what the down one, up five is saying.
A fifth is seven semi-tones above the root. It has a 3:2 frequency ratio to the initial note. It so happens that if you jump up by seven semi-tones five times, you've got most of the major scale, albeit strewn over 3 doublings of the initial frequency. Usually, the major scale is thought of as occurring within a single doubling so in the key of C you'd have to divide the frequencies D and A by 2; E and B by 4 to get back into the original n<->2n range. Any time you double or halve the frequency you get a note with the same mood/purpose/name but one octave higher or lower. The frequency ratios end up something like this:
C = 1
G = 3/2
D = (3/2)(3/2) (/2 to get back to the initial range)
A = (3/2)(3/2)(3/2) /2
E = (3/2)(3/2)(3/2)(3/2) /2/2
B = (3/2)^5 /2/2
There's one note missing, though--the one with a 4:3 frequency ratio: F. To get that one, we invert the 3/2 relationship and take the frequency that's 2/3 of our initial frequency. That's in the halved range, n/2<->n, though so we've gotta double to get back into our starting range. This is the one time we go down seven semi-tones instead of the five times that we go up.
I'm not sure how useful it is to think in these terms but it does show that you can derive the major scale by using simple ratios which, psycho-acoustically speaking, are generally considered more pleasant than complex ratios when played at the same time. The relationship between B and C, (3/2)^5 == 243/32 is already pretty tense. E.g., you could alternate between two notes with that ratio to make it sound like Jaws is lurking somewhere nearby:
A fifth is seven semi-tones above the root. It has a 3:2 frequency ratio to the initial note. It so happens that if you jump up by seven semi-tones five times, you've got most of the major scale, albeit strewn over 3 doublings of the initial frequency. Usually, the major scale is thought of as occurring within a single doubling so in the key of C you'd have to divide the frequencies D and A by 2; E and B by 4 to get back into the original n<->2n range. Any time you double or halve the frequency you get a note with the same mood/purpose/name but one octave higher or lower. The frequency ratios end up something like this:
C = 1
G = 3/2
D = (3/2)(3/2) (/2 to get back to the initial range)
A = (3/2)(3/2)(3/2) /2
E = (3/2)(3/2)(3/2)(3/2) /2/2
B = (3/2)^5 /2/2
There's one note missing, though--the one with a 4:3 frequency ratio: F. To get that one, we invert the 3/2 relationship and take the frequency that's 2/3 of our initial frequency. That's in the halved range, n/2<->n, though so we've gotta double to get back into our starting range. This is the one time we go down seven semi-tones instead of the five times that we go up.
I'm not sure how useful it is to think in these terms but it does show that you can derive the major scale by using simple ratios which, psycho-acoustically speaking, are generally considered more pleasant than complex ratios when played at the same time. The relationship between B and C, (3/2)^5 == 243/32 is already pretty tense. E.g., you could alternate between two notes with that ratio to make it sound like Jaws is lurking somewhere nearby:
https://youtu.be/BX3bN5YeiQs
Sticking with the simpler 3/2 makes it sound like Superman is here to save you so you can relax:
https://youtu.be/e9vrfEoc8_g