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Arguments about rational, irrational & aspirational numbers began at least when Pythagoras conducted his 'string experiment' around 6 centuries BC
The rational whole numbers were revealed as intrinsic to the overtone-series. What is less well-known is that derivation of the irrational number of 'Phi' is too.

After 2500 years the story is still told that the Pythagoreans allegedly murdered Hippasus for discovering & also revealing irrational numbers,
as these challenged the Pythagorean orthodoxy of there being only rational or whole numbers. This is deeply ironic.



This is deeply ironic.

When the Pythagorean Comma is measured, the supremely irrational number of Phi is generated very precisely at step 12.618
of the harmonic series. This is a number that is ubiquitous in Nature, in for example the numerous expressions of the Golden Spiral.

The Harmonic Series: even numbers & odd numbers, separately & together & the curve arising generating Phi

Harmonics - odd & even numbers (the rule is halve the wave-length-double the frequency; third the length-treble the frequency etc) . . .

1. Within the overall structure of the 'harmony' of the harmonic-series, the rates & ratios of each harmonic
   one-by-one has an individual wave-length (frequency) defined by each whole numer from 1 to infinity.
   
   
2. 'Opening' from the absolute zero position at the centre, individually these 'strings' oscillate equally net-positive
   to the right, in relation to oscillating net-negative to the left, giving 'net-zero' overall at the centre (amplitude).
   
   
3. Each string oscillates individually 'net-zero', at all & any [a] wave-length & at all & any (b) amplitudes
   as defined by the rates and the ratios of the harmonic series.
   
   However when observed all together. . .
   
   
4. As these 'strings' vibrate at different rates, a key structural difference between odd & even numbers emerges: -
   even-numbered harmonics (2, 4, 6, 8, 10, 12 etc) are never at the absolute zero position at the same moment,
   while they are together at the fully open position half way through & also again at the completion of a full cycle.
   
   
5. Odd-numbered harmonics (1, 3, 5, 7, 9, 11, 13 etc) on the other hand, all reach the net-zero position simultaneously,
   at exactly halfway through a full cycle of the even numbered harmonics, while all harmonics (odds & evens) are all fully
   together at the fully open net-zero position emerging at the completion of a full cycle.
   
   
6. The odd numbers generate the so-called dominant 7th chord which leads through the modulations of the '5th cycle'.
   
   
   
   
7. Only strictly within the fully defined rates & ratios of the harmonic series (where 7 perfect octaves equal 12 well-tempered fifths) can
   the void be described as 'net-zero' & this truth - "the strictly ordered, regulated, symmetrical & well tempered void" inferred by Tonelli,
   this is a structured, symmetrical & scheduled catalyst at the birth of space-time & it is true for any formulation of net-zero at the void.
   
   However when observed all together, the generation of Phi at step 12.618 is quite clear
   
   The fundamental clarity of this well tempered certainty shows that 'uncertainty' is an inadequate refuge from it.

Counts all visits to all pages of the GCI website since ~2022.