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The table below shows constructions for squares with areas 1 through 64 using the following four methods and their combinations:

a) Squares with whole number roots (as 1, 4, 9).

b) Squares that are half (or double) the areas of other squares (as 2, 8).

c) Squares whose root is the diagonal of a rectangle (as 5, 10).

d) Squares whose root is the height of a triangle (as 3).

Squares with the area 0f 6 and 7 are combinations of methods b) and d):

6 is half of a square with the area of 12, whose root is the height of a triangle with base = 2, hypotenuse = 4.

7 is (intended to be) quarter of a square with the area of 28, whose root is the height of a triangle with base = 6, hypotenuse = 8

Hexagon with a radius of 1

in relation to squares with areas of 1, 2, 3 & 4.

Feeling a bit triumphant about constructing an enneagon from a hexagon, it seemed like an easy next step to construct pentagons from squares!

A pretty fundamental mistake was assuming that equal divisions of a line were also equal divisions of an angle. They are not! Everything (including enneagons) falls apart!

When AB = x, OC = √[x²-(x/2)²] = √[x²-(x²/4)] = √(3/4x²) = x(√3)/2

When AB = x, OD = √[x²-(x/2)²+(x/6)²] = √[x²-(x²/4)+(x²/36)] = √[(3/4x²)+(1/36x²)] = √[(27/36x²)+(1/36x²)] = √(28/36x²) = √(7/9x²) = x(√7)/3

Now I'm off to attempt pentagons by dividing squares into fifths!

90/5=18

18*4=72

18*6=108

What conditions are required for any line to become either perpendicular or parallel to the horizon?

Which repeats what was found with the Harmonic Proportion

Especially interesting to me is the presence of 8:13 - the very ratio from which, according to Ruland, the Ancient Persian scale was derived!

Thanks Grey!

The interval given by Ruland for a pentatonic based on the "Ancient Indian Cycle of Sevenths" is not so precise in practice. I was pretty crestfallen when I got to hear my first attempt (based on a 4/7 interval). I knew that I wanted the equivalent of 2.4 modern semitones but I wasn't sure what this would sound like or how to get there in a more precise way. However today I figured that the fifth root of 0.5 (=0.870551) would give me a precise ratio to work with. Actually the 4/7 was not so far off in the second octave, but precision really helps in approaching unusual tones. 

Thing still sounds weird, but I am confident in its weirdness!

Thankfully, after my students worked so hard on all that math, the ratio given for Persia of 8/13 is much more precise. This gives ten equal divisions from which seven are selected to form the scale.

Above shows:

Blue: Mathematical Scale: 1/7 1/5 2/7 1/3 3/7 1/2 4/7 2/3 5/7 4/5 6/7

which is a variation of the GiQuan based around the interval of 1/7 and leaving enough room between frets for football team stickers! 

Green: Persian Scale = Equal intervals based on Cycle of Sixths = 8/13

The above would be a nice project - four fretboards based on different mathematical intervals (with possibly and "even-tempered" one alongside for comparison).

But they also show the following as easily achievable using "natural harmonics":

Diatonic:

Do = open

Re = 8/9

Mi = 4/5

Fa = 3/4

So = 2/3

La = 3/5

Ta = 5/9

Do = 1/2

Mixolidyan but with two octaves gives all modes as previously:

. . . . also "chromatic", with only the flat ninth and major seventh elusive

(what do you mean "Quite right too!"?)

Do = open

Ra = ?

RE = 8/9

Ma = 5/6

Mi = 4/5

Fa = 3/4

Fi/Sa = 7/10

So = 2/3

Le = 5/8

La = 3/5

Ta = 5/9

Ti = ?

Do = 1/2

Here's a chromatic fretboard without those pesky semitones "Ra" (flat nine) and "Ti" (Major seventh) - all you need are two strings tuned a fifth apart.

This is really quite lovely - a fretboard with all the chromatic options (bar changing key!) and no fractional intervals in the double digits . . . . if it works!