Pythagoras

"The School of Athens" is one of my favorite paintings & is one of the most famous frescoes by the Italian Renaissance artist Raphael, painted between 1509 and 1511 as a part of Raphael's commission to decorate the rooms now known as the Stanze di Raffaello, in the Apostolic Palace in the Vatican. 

Let's check out some of the main characters  -->

#5: Averroes      #6: Pythagoras      #14: Plato       #16: Diogenes      #18: Euclid 

Closeup of what Pythagoras is looking at

Notice the chalkboard at his feet --> The Epogdoon

The Epogdoon is composed of Perfect 4ths, Perfect 5ths & The Octave

... it also contains the Tetractys symbol, another way of looking at the same concept ...

2:1 = The Octave

3:2 = A Perfect Fifth

4:3 = A Perfect Fourth

The Hammers  -->

According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously. 

According to Nicomachus in his 2nd century Enchiridion Harmonices, Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be "singing" the same note. Pythagoras rushed into the blacksmith shop to discover why, and found that the explanation was in the weight ratios.

The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were (12:9 = 4:3 = perfect fourth) and (12:8 = 3:2 = perfect fifth). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval.

This type of system, attributed to Pythagoras, is the same type of system as the Sumerian Cyclical Tuning System, which takes into account the purity of the octave --> Only then do we fill in the rest of the locations.

This hammer and weights system based on the idea of 6 & 12 is NOT the same as the spiral tuning they have attributed to Pythagoras {Pythagorean Tuning}.  This spiral tuning does NOT take into account the pure octave locations.  

The dissonance created by this pythagorean spiral tuning system is know as The Pythagorean Comma.

More about this later ...

Here is video #1 in the educational series --> 

1∆ The Octave, Tuning, Intervals, Circle of Fifths, Proportion...

Lesson 1: We explore the history, composition and subdivision of the octave... musical intervals ...both "equal" and "perfect" temperaments... the "Circle Of Fifths"... proportion.. and more !

...Observe how each of the intervals in "equal temperament" has a specific geometric shape, spiraling either clockwise or counterclockwise...based on the reference point of the CIRCLE . . .

...Based on the stated notion of movement by a perfect 5th as being represented as movement around the perimeter of the CIRCLE, rotating clockwise, and the reciprocal, the perfect 4th, being represented as counterclockwise movement around the perimeter of the circle.  The diminished 5th can be seen as a mirror to the note opposite from it, 180* directly across the circle...

...The remaining intervals create paired shapes - Triangle (M3,m6), Square (M6, m3), Hexagon (M2, m7) and 12-pointed star (M7, m2)....

-The process to create the geometry: Lock the "circle of fifths" in place, and observe intervals only, from one point to the next, one point to the next...watch them cycle in shapes and direction !  It is the individual interval that is of importance -- it does NOT matter what "note" you choose to start on .... All "keys" show the exact same geometry !  12√2 

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