DECODING GANN's HARMONIC WHEEL
AFTER MAKING THE CONNECTION
GANN ANGLES ARE CYCLICAL OCTAVES

continued....

The Extended Pythagorean Chromatic Scale

The use of "accidental" signs allows the power of 3 scale above to be extended. This scale was actually the 17 tones later determined by Prosdocimus:

But it is again necessary to sort the powers of 3 to define an Octave Range:

The Problem with the Pythagorean Scale Appears:

You end at a note that is higher than C in the 7th Octave. There is an interesting difficulty in Pythagorean tuning. There are 12 perfect fifths within a total of 7 Octaves. But 12 fifths (3/2) to the 12th power is 129.7463379... yet mathematically is 128.

The amount by which 12 fifths overshoots 7 octaves is called the Pythagorean comma. Instruments of ancient Greece only had one Octave...it was not a problem, but the mathematical inconsistency took centuries to resolve and was the source of many arguments in history. One argument over tempered scales that lasted a life time... Galileo Galilei's father... Vincenzo Galilei.

2b. Construction of the Just Intonation

By the sixteenth century an effort was made to correct the error in the Pythagorean Scale. A major third is introduced for the first time. (5/4) (C to E).

But... The perfect 5th between notes 2 and 6, (10/6)/(9/8) = 40/27 = 1.481 did not have the desired ratio 3/2 = 1.5.

CONVERTING THE LINEAR MUSICAL SCALE INTO A MATHEMATICAL MATRIX To CREATE GANN'S HEXAGON WHEEL:

• Calculating Intervals - Vector Addition of Exponents

  To add or subtract intervals by vector addition of exponents, simply add or subtract by vector addition the set of exponents for each tone or interval, using 0 where necessary as a placeholder:

  two negatives still make a positive:

A MATHEMATICAL MATRIX AND CREATING GANN's HEXAGON WHEEL:

Gann analysts will quickly recognize the numbers in these string lengths!

GANN'S HEXAGON WHEEL...a Big Breakthrough but not the Square of Nine: