Geomusic

DECONSTRUCTING EQUAL TEMPERAMENT  -->

Now that I knew the sumerian cyclical tuning system was based on the DORIAN mode, I wondered, WHY ?  What is so special about DORIAN to make it the center piece of the framework used in tuning the lyre ?

I decided to look at the modes as scale formulas.  Not much new to see there.  

But then I asked myself, "what if you looked at each mode in the SAME key? How would they look different then?" I had the revelation that what I needed to do was to LOCK the 'circle of fifths' in place, and look at all 7 modes, but only look at ONE KEY ! 
{or you could see it as ANY key, or EVERY key}

This would show me the weights of the modes, as it were.  I was astounded to see DORIAN as the most evenly weighed.

You can spin the outer circle of KEYS to whatever KEY you choose and the WEIGHT of the individual MODE stays the same !


There was the answer I was looking for !!  

The reason why the sumerian cyclical tuning method used DORIAN as the starting point was because it is the most evenly balanced of the 7 modes, manifesting 3 on either side of itself, each set being reciprocals as well.

The next question I asked myself was, do the musical intervals themselves also have a specific geometry? 

I figured why not LOCK the 'circle of fifths' in place again & only look at one 'key', but this time look only at the individual intervals.  

again, {you could see it as applicable to ANY key, or EVERY key}

I chose to put the key of 'A' at the top of my diagrams, as it is the first letter in the alphabet, and the root 'note' I am using to assign to 432. You could choose to put ANY KEY at the top - however, the intervals, geometry, proportions & relationships will remain exactly the same.


Reciprocals intervals share identical geometry
but they alternate clockwise or counterclockwise in flow.  

The whole system can be seen to flow in a figure-8/infinity/analemma
increasing or decreasing in frequency.

You are surfing frequency, from octave to octave, in any chosen KEY, in ANY chosen frequency.

Notice the reciprocals will also {digit sum} to 9 - 
P4 + P5
M3 + m6
m3 +M6
M2 + m7
m2 + M7

Notice when reciprocal intervals are displayed vertically, the outside proportion stays the same, while the inside proportion is doubling/halving -
3(2:4)
5(4:8)
5(3:6)
9(8:16)
15(8:16)

Or, if we look at them in a horizontal arrangement, we see the octave doubling/halving in an alternating sequence - inside, outside, inside, outside, etc...


Here we can see the colors of the rainbow corresponding to the reciprocal intervals -
octave 1 + octave 2   =   {Red}
m2 + M7   =   {Orange}
M2 + m7   =   {Yellow}
m3 + M6   =   {Green}
M3 + m6   =   {Blue}
P4 + P5   =   {Indigo}
d5   =   {Violet}

Looking to see what other constants were floating around, I found that √3 is nestled right in between the Major 6th & Minor 7th. Interesting that the shapes picked out for these 2 intervals are also the ACTUAL SHAPES that contain this geometry - a square/cube & hexagon.


square root jamboree key of C


square root hoedown to nowhere


square dance in C


square root shindig limbo in limbo


√2

dissonance   or   harmony


Taking the √2 proportion to the next conceptual level, 
I realized that it would be true of any isosceles right triangle, that ->

If, the sides are the root frequency {1:2:1}
Then, the hypotenuse is the tritone frequency {√2}

because = circle the square


well would you look at that... 

The isosceles triangle in the proportion of 108 on the sides gives a hypotenuse of the TRITONE location of 153.

Throw a √ on that value {√153} and you get the number of full moon's per year (12.369).

You also can see this proportion in the construction of Stonehenge, and the proportions of the Earth & Moon.


Third's & Sixth's  -->

Notice when using 6:5 {1.2} as a Minor Third, the result is a larger number than the one we see in the cyclical system -> 
C=518, instead of C=512   (six off)

Notice when using 5:3 {1.666} as a Major Sixth, the result is a smaller number than the one we see in the cyclical system ->
F#=720, instead of F#=729   (nine off)

small things add up - but also need to be evenly distributed across the whole ...


Take a look at the geometry of water ripples ->