How Pythagoras' simple string experiment, conducted in Crotone around 530 BC,

embeds a per-unit-time-based-derivation of the beautiful and irrational 'Phi'.

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The value of ‘Phi’ (.618…) is irrational.
It is commonly derived from a ‘quadratic equation’

((the square root of 5 + 1)/2) or the Fibonacci Series (as in the Pentagon:Pentangle).

However, the value of ‘Phi’ can be quantized per-unit-time (Hz) from measuring and back-feeding

the differences in the phenomenon known as the ‘Pythagorean Comma’. Derived from ‘Stringularity’,

‘Phi’ is a point on a curve at the ‘Phi-point’, between steps 12 and 13 of the Pythagorean Comma.

It is constant, anywhere from nano-to-macroscale. So, using the derivation of ‘Phi’ as a function of

steps per-unit-time,
can be a relevant step in changing the cognitive paradigm where particles in

wave-fields are only *probability perceptions* still regarded as the ‘quantum measurement problem’.

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The ‘Stringularity’ Hypothesis; the ‘path-integral’ of Pythagoras’ String Experiment

The now famous string experiment of Pythagoras (~ 530 BCE) demonstrates that: -

- When we take a string of constant length and constant tension tuned at 100 Hz
- And halve the length of the string, the frequency (Hz) doubles from 100 Hz to 200 Hz
- And that if we third the length of the string the frequency (Hz) trebles to 300 Hz etc . . . .

As the path-integral basis of the universally constant ‘harmonic series’, these steps start with: -

- the ‘Fundamental’
- the ‘Perfect Octave’
- the ‘Perfect Octave plus the Perfect Fifth’ etc . . .

sub-division by* any* rational number will produce a 'harmonic' in this series
. . . .

The Pythagorean Comma (see chart above)

Proceeding from above, the Pythagorean Comma emerges as a growing ‘gap’: -

- when we sequence seven ‘Perfect Octaves’ in Hertz (Hz), in the example shown we go through

seven doublings 1 (each enlarging by 200%), from 100 Hz to 12,800.00 Hz,

- when we sequence twelve ‘Perfect Fifths’ we go in the example shown we go through 12 steps

(each enlarging by 150%) from 100 Hz to 12,974.63 Hz,

- it becomes clear that seven Perfect Octaves (12,800.00 Hz) do not ‘commute’ exactly with

twelve ‘Perfect Fifths’ 12,974.63 Hz.
At this ‘twelfth-step’, the gap is 174.63 Hz (1.345...%)

Well-Tempered Tuning (see chart above)

- In order to get the twelve Perfect Fifths to commute with the seven Perfect Octaves,

the Perfect Fifths (enlarging at 150%) are ‘Well Tempered’, enlarging at 149.83070768…%)

- The result of adopting this procedure is that twelve Well-Tempered Fifths commute exactly

with seven Perfect Octaves (at 12,800 Hz), giving rise to what is called 'Well Tempered Tuning' (3, 4,)

The Hz Differences between Perfect and Well-Tempered Fifths lead to ‘Phi’

- When step-by-step, we subtract 12 Well Tempered Fifths from 12 Perfect Fifths,

a sequence of differences between the two paths is the result (see table above),

- When step-by-step, we back feed these differences into the sequence, so the
largest

difference is divided into the smaller difference that precedes it etc . . . (see table above),

- A twelve step sequence with negative curvature (concavity) is revealed, decreasing from

0.615731…. just below the value of ‘Phi’, to 0.000 (see table above),

- When we go one step further and start at the thirteenth step of the Pythagorean Comma,

the difference is 283.62 Hz, yielding a value 0.6199396…. just above the value of ‘Phi.

‘Phi’ is exactly at the Phi-Point between per-unit-time steps 12 & 13 of the Pythagorean Comma

- The value of ‘Phi’ is on the overall curve at exactly the Phi-point between steps 12 and 13

of the Pythagorean Comma http://www.gci.org.uk/movies/PC_12_13.mp4

- Exponential growth curves at 200% 150% and 149.83070768…% etc are positively governed

by
acceleration (convexity) and unfold without limit towards infinity.

- Path-integral to that, the growth curve to ‘Phi’, is governed by deceleration within the

feedback
limits or concavity that define it, countervailing the convexity of acceleration.

- Deriving the value of ‘Phi’ from measuring the path-integral per-unit-time/space of

‘Stringularity’ is distinct from the time-free quadratic equation and Fibonacci Series.

- This path-integral procedure gives rise to an array of time-based features which suggest corroboration.

The beauty and the deceptive simplicity of ‘Phi’ underlies the structure, the sequencing,

the symmetry and the curvature of natural phenomena, from the nanoscale to the macroscale.

Could this 'musical-derivation' of ‘Phi’ be a simple yet beautiful idea that helps to answer Erwin Schrodinger’s

famous question, *‘What is Life’?* Might it underlie a ‘cognitive framework’ that would help better to understand

the wave:particle perceptions-dichotomy, the path-integral complexity, the moment:momentum continuity and

life:death
challenges
that will always face us?

**Concomitant corroboration . . . . ?**

- Phi squares the circle
- Khufu Squares the Circle
- The equivalence of Phi and Pi
- Phyllotaxis - the spirallic, nested, emergent, folded, ubiquitous, awesome Phi
- The 'symmetry binding' properties of Hz-derived Phi integration
- the relevance of this to the Standard Model when introducing the Higgs Boson
- Phi-Spiral (vortex) 'black-holes' from excitons to galaxies

Footnotes

2. This ‘non-commutation’ phenomenon has been

recognized for Millennia.

Unaffected by issues of scale and whatever base Hz value is set for ‘doublings,

if 12 Well Tempered Fifths are 100%, 12 Perfect Fifths are always 101.36433…%.

3. In musical narrative, the purpose of adopting ‘Well Tempering’ was to enable music

to be written and played in all the keys
(of all the 12 semitone steps in the octave)

so they are all equally ‘in-tune’ and where modulating between these keys these is

smoothed by the equality of the twelve semitone steps in the Perfect Octaves arising.

4. The classical example of this is the Well-Tempered Clavier of JS Bach (1721).