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An expression of Pythagoras' 'First Law' 'Well-Tempered' Tuning and UNFCCC-compliance with Contraction & Convergence [C&C].

For being In-Time and In-Tune to be in Harmony with Mother Earth and Nature all musicians really do, especially string players, is to 'count' so as to be 'in-tune' and 'in-time'. It can be said that, 'All life aspires to the condition of music'.

Some physicists even argue that, "Music is the very essence of the Universe" Michio Kaku
- More about this here

Compare Richard Dawkins and Stephen Hawking.

Where Dawkins denouces Plato [read Pythagoras] in his recent book "The Greatest Show on Earth", in his recent book 'The Grand Design', Hawking cites Pythagoras' discovery of the empirical basis of this counting in-tune and in-time: -

'According to legend, the first mathematical formulation of what we might today call a law of nature dates back to an Ionian named Pythagoras [who] is said to have discovered the numerical relationship between the length of the strings used in musical instruments and the harmonic combinations of the sounds. In today's language we would describe that relationship by saying that the frequency - the number of vibrations per second or 'Herz' value - of a string vibrating under fixed tension is inversely proportional to the length of the string. One could call that simple mathematical formula the first instance of what we now know as theoretical physics.'

Here, 'harmonic structure' is discovered as a 'universal law' governing the sub-divisions of a length of vibrating string as a measured and universal 'constant' that is 'in-tune' and 'in-time'. The first two instances of this law give the cadence basis of 'diatonic harmony' and are 'perfect': -

  • 'Halve-the-string-length' and you 'double-the-frequency' - perfect octave [tonic] [l/2=Hz2]
  • 'Third-the-string-length' and you 'treble-the-frequency' - perfect fifth [dominant] [l/3=Hz3] and so on . . . .

In combination, the unified 'hemiola-structure' - 1:2:3 - of this 'overtone' series shall be called 'stringularity'. It emerges like this: -

Together these fractions constitute the 'Hemiola'. This is a 'unity-divinity-trinity' - or a 'monad-dyad-triad' - whose internal
'pulse' is exactly where 'Two Groups of Three' and 'Three Groups of Two' take precisely the same length of time.

This proportionate truth or 'stringularity' is true when there are hundreds of vibrations per second or Herz [Hz] as pitch [see below] where for e.g. orchestras tune to A 440 Hz. It is also true at one, two and three Hz per second as rhythm or metre. Think of West Side Story: -

'I-like-to, Be-in-a, Me-Ri-Ca', or if you prefer, 'Wel-come-to, Pre-si-dent, Gor-Ba-Chev'.

Stringularity is true at any frequency and it is the basis of C&C, the 'Well Tempered Climate Accord'

Zoom on the example of Contraction and Convergence [C&C] rates imaged below for quantitative detail. On the 'y' axis it is 'in-tune' and on the 'x' axis 'in-time' i.e. it is not short-term versus long-term, it is full-term or 'in-time' and shorter than realized: -

Pythagoras' law of stringularity as a universal constant, is easily demonstrated audio-visually here
Click nodes - 'blue' for chromatic-scale and 'gold' for harmonic series - 'on' and 'off' in 'series/parallel' to make 'melody/'harmony'.

Pythagoras' 'law' is revealed in a string at constant tension and length, in the cello G-String shown for example. The image has Hz values based on G'o'=100 Hz, which is equivalent to a Cello G-String. Here, Herz values have been slowed visually by a factor of 100, so the Herz doubling for the octave and trebling for the fifth [etc] can more easily be 'seen' as well as 'heard' [as 'doublings', treblings' etc].

The audio-visual file for this is here Mouse-touch [one click 'on', second click 'off'] the gold-coloured buttons in the image to both hear and see how the 'harmonics' of the whole string or the 'overtone series' emerges 'harmoniously' from the 'fundamental' G. The blue-coloured buttons for 'chromatic' notes are in between but not 'harmonics'.

Like this over future 'decades' as well as 'seconds', C&C is a function of this 'long/short-wave-structure'. It yields internally consistent events per unit time that add up precisely to the stated outcome - which is UNFCCC-compliance. Being 'in-tune' and 'in-time' with C&C are a function of this structure at any rates: a concept animation of this is below and here or here: -

As the UNFCCC Executive said in 2004, UNFCCC-compliance inevitably requires C&C and C&C is a 'Well-Tempered Climate Accord'
The 'Perfect Fifth Cycle' [PFC] in columns 1 and 2, compared with The 'Well-Tempered Fifth Cycle' [WTFC] in columns 3 and 4.

Herz Perfect
Fifth Cycle

Herz Well Tempered
Fifth Cycle
Herz difference between
Perfect Fifth & Well Tempered Fifth Cycles i.e. the 'Pythagorean Comma'

Pythagorean tuning preserves the mathematically given fractions for the different 'intervals' within the octave. This makes 'modulating' away from the home key more and more out of tune. Well Tempered tuning adjusts for this by making each of the 12 'semi-tone' steps 'equal in the octave'. The PFC starts with an 'octave' based on 'G-G1' at 100-200 Hz and plots the Perfect Fifth [PF] 'D' at 150 Hz. It then takes D-D1 based on 150-300 Hz and plots PF 'A' at 225 Hz, and so on . . . The WTFC starts with an octave based on 'G-G1' at 100-200 Hz and plots the Well Tempered Fifth [WTF] 'D' at 149.831 Hz. It then takes D-D1 based on 149.831-299.6614 Hz and plots PF 'A' at 224.492 Hz, and so on . . .

When you complete the 12 note fifth-cycle at the these Hz, the Perfect Fifth Cycle continues through G-G8, going from 100 - 12974.63 Hz as an 'imperfect fit' with multiples of 100Hz. But when you complete this 12 note cycle at the these Hz, the Well Tempered Fifth Cycle completes through G-G8 as you go from G 100 to G8 12800 Hz as a 'perfect fit' with multiples of 100Hz.

The differences between the values for perfect and well-tempered fifths in the 'fifth cycle' is shown in columns 1 and 3 where the 'values' of the emerging 'Pythagorean Comma' is shown column 5 of the table.

This is a significant, audible and increasingly obvious difference of pitch [as it is a very out-of-tune]. This is why 'Well-Tempered' tuning was formalized [for example Bach's Well-Tempered Clavier - 48 Preludes and Fugues in all twelve keys both major and minor] in the 18th Century and has been widely adopted as the basis of all music-making since that time.

Pre-tuning keyboard instruments in what is also called 'Equal Temperament', makes 'modulation' between all keys possible, while remaining 'in-tune' in all keys. In other words, by slightly 'compressing' the perfect fifth so we can have twelve exactly equal semi-tone steps per perfect octave, 'well-tempered' tuning makes this twelve-note 'chromatic' division of that octave into a perfect internally consistent whole that is consistently 'in-tune' based at any Hz rates or pitch.

Now, for reasons of transparency with 'climate politics', C&C is primarily 'well-tempered' or 'equalized' rather than 'perfect' and this is the basis for saying that C&C is a 'well-tempered climate accord'. This argument for 'equality of rights per person per unit time' is politically expedient to avoid the futile further defence of various forms of global inequality. But perhaps more importantly it is technically necessary to preserve internal consistency when calculating all the different rates of C&C for UNFCCC-compliance and so to keep in-time and in-tune to be 'In Harmony with Mother Earth'.

However, whatever rates of C&C are chosen to respond to the objective of the UNFCCC and actually commit to achieving it, this means simply that the 'unit of measurement' [numeraire] is not first and foremost 'money'. It means that C&C or carbon, per person, per unit time, subject to the objective of the UNFCCC is the primary unit of measurement or the 'numeraire'. This makes it possible to quantify and share now the limited total of future emissions - the Contraction-and-Convergence-Event - that is UNFCCC-compliant. Because of urgency and equity these rates are faster than is being talked about in the 'Copenhagen Accord' for example and 'Money' - as a numeraire - is only meaningful as a function of rates of C&C that are UNFCCC-compliant.

Like this, C&C can be understood as 'A Well Tempered Climate Accord': -

  1. the octave [the whole/the unity or the 'full-term contraction-event' needed for UNFCCC-compliance] is preserved;
  2. the sub-divisions of that event [carbon, per person, per unit-time, subject to the objective of the UNFCCC] are quantized as equal or 'well-tempered' units of that whole for consistency over time.

The organising principle is true whatever C&C rates are computed. It is essential that the different rates of applying this structure are openly considered in the assessment of our 'global' problem and the negotiation of its 'solution' in advance of all the agreements that are made to implement the solution: - see the user interactive animation for many examples of C&C at different rates of both contraction and convergence below and animated user inter-actively here and here and here: -

In the animation of the strings, the 'functionally harmonic' or 'cadential' structure of music can be seen and heard, as here for example: -

  • The 'Dominant 7th' chord [touch D, A, D, F#, A, C, D, F#] 'resolving' to
  • The Tonic chord [touch G, G, D, G, B, D, G, B, D, G]

This same 'Amen Cadence' structure is functional in C&C. Sub-dividing the whole, it is 'constitutional' and stable, where the 'numeraire' is 'carbon per person per unit time subject to the UNFCCC objective. Thus UNFCCC-compliance requires that: -

  • in the finite period of time left for UNFCCC-compliance [achieving safe and stable concentrations];
  • negotiated in a rational, non-random, transparent and proportionate global structure of harmonic feedback or 'cognitive resonance';
  • a finite weight of carbon [the full-term, global carbon contraction budget];
  • is consumed equally and simultaneously by a finite number of people ['cadentially resolving' through convergence to equal per capita rights globally by an agree date].

This structure 'knows-where-it-is' because it is rooted in this constitutional 'source-code' or fundamental structure. Like an 'Amen Cadence', the functionality of the C&C structure means it 'knows-where-it-is-going' - in other words to closure from Dominant to Tonic - and that for reasons of UNFCCC-compliance, it 'needs to know' this.

We don't simply make-all-this-up-as we-go-along. C&C knows by how much and by when, because we tell it where to go by how much and by when, or at what 'rates'. Thus, the C&C framework is 'radar' [where are we?], 'rudder' [where are we going?], 'throttle' [by when do we want to get there?] and 'ration' [and against how much of the resource is left, on what principle of pre-distribution do we agree to share this?]. All this information is processed in the model from the outset and all rates of C&C generically show that in authentic climate-strategy, the 'guesswork' of 'evolution' is subordinate to this 'framework' of 'constitution'. From this the UNFCCC Executive could say about the UNFCCC objective: 'Stabilization inevitably requires Contraction and Convergence.' COP-9 Milan 2004 J Waller Hunter, UNFCCC Executive Secretary

This means that 'UNFCCC-Compliance' is dependent on a C&C framework. It doesn't mean that a C&C Framework is dependent on 'UNFCCC-Compliance'. In a phrase, C&C is input before it is output and this is simply understanding and accepting this global sequence - like 'cap-and-trade' - of cause and effect. Deceptively simple, it is the same as seeing that the overtone series is a function of the fundamentals on which it rests and not vice versa. Evolution is a function of constitution - like learning to 'stop-and-play' the violin, you learn 'it', because it doesn't learn you. Climate-institutions that are not a function of this constitution will not succeed in achieving UNFCCC-compliance.

Thus, C&C projects a 'Well Tempered Climate Accord' - a 'constitution' for 'Equity and Survival'. C&C's Pythagorean structure creates a musical framework of proportionality and feedback, within which we can yet globally orient, measure and integrate with resource limits by which we are governed and negotiate how to be 'in-tune' with each other [equity] and crucially 'in-time' together to avoid dangerous rates of global climate change [survival].

However, the 'economists' menu-heading on the GCI web-site refers to how a Woody Allen one-liner became the 'design-free' answer some economists tended to give to that question, 'how long is piece of string?

Ask a string player 'how long is a piece of string?' and the answer will always be 'exactly twice half its length' [l/2=Hz2].
However, economists assume questioning 'string-length' is the 'growth issue' and evade it with the 'touch-woody joke' where, 'eternity is a very long time, especially towards the end' - in other words, make-it-up-as-you-go-along.

From that perspective, economists now argue that 'biological diversity' is a function of 'portfolio diversity' in a market where the numeraire is the direction-less dollar [monetary unit] and that, assuming it happens at all, C&C will be a function of that.

It won't. Just as the economy is a wholly-owned subsidiary of the environment, so the numeraire for UNFCCC-compliance is 'C&C' or carbon per person, per unit time subject to the objective of the UNFCCC. For economists this means that the numeraire is not primarily the direction-less dollar [monetary unit], it means that if there is a relevant monetary unit, it is a function of C&C. All this is why GCI's contribution to Working Group Three of the IPCC 2nd Assessment Report in 1994 stated the following: -

"We believe that any proposed solutions to the problems [which both cause and proceed from global climate change] which are not equitable will not work. In a very real and fundamental way, equity is the solution - i.e., properly valuing each other and the planet. A failure to understand and apply this is a failure to appreciate the double-jeopardy in which humanity is now situated. We face the actuality of scarce resources (sink capacity etc) and the increasing potential for conflict with each other over these scarce resources. We do not imagine the solutions that emerge will be based exclusively on the principle of rights to equal carbon usage. However, the analytical tools that we are developing and making available are based on the principle of equal rights to carbon usage, and the results our that our work reveal can be used as a network of reference points. Anyone who wishes to diverge from or ignore the principle can then describe what they propose, and this can be judged against our results. It would then be for the international community, through a reformed and better advised negotiating process, to decide whether or not the degree of divergence proposed was socially and ecologically viable."

All-in-all, UNFCCC-compliance will have to be structured and organised. It will not be a 'design-free-happening' in a 'free-market-accident', following the dictum that 'eternity is a very long time, especially towards the end'. If UNFCCC-compliance happens, 'Kyoto's market-based-framework' will have transmuted to become 'C&C's framework-based-market'. Quite simply, unless we subordinate the fuzzy monetary-numeraire of economics to the fundamental numeraire of C&C and UNFCCC-compliance, we will become ever more deeply caught in non-compliance, the 'green-wash' that conceals a 'death-wish', where the economics of genocide leads to what Ban Ki Moon has now called 'a global suicide pact' - especially towards the end . . .

In his latest book, 'The Greatest Show on Earth', the 'evolutionary biologist Richard Dawkins promised this time to simply present the 'evidence' for his Darwinian thesis, rather than engage with the arguments he has said are attacking it. However, in Chapter Two he launches into a scathing attack on his 'real enemy' and it transpires that it is not the 'creationists', it is 'the dead hand of Plato'. This dead hand, he says, is the single thing that over a period of now well over 2,000 years is most responsible for why people do not see the secular reality of 'evolutionism'.

This extraordinary statement fails to realize that while institutions may be a function of evolution, evolution is a function of constitution.

Emphasizing four things that are relevant to the rationally cooperative - or 'well-tempered' framework - of C&C: -

1. Plato was a Pythagorean
2. C&C is Pythagorean
3. Chapter 11 of Dawkins original book 'The Selfish Gene', was about 'memes and it clearly made the point that if it takes cooperation to survive, we'll learn that too
4. Cooperation needs a rational framework of agreement.

Dawkins might consider that as things stand, the failure of the 'Kyoto Protocol' to the UNFCCC reveals that in the competitive struggle to survive by being the fittest, we are sadly no longer fit to survive climate change and we probably won't. Tim Smit once observed the world must get behind C&C but quipped about 'homo sapiens' and climate change as, 'the wise hominid, what a joke.'

So is it worth looking again at what Stephen Hawking recognises as the first instance of what we now know as theoretical physics:

"In today's language we would describe the number of vibrations per second of a string vibrating under fixed tension is inversely proportional to the length of the string. One could call that simple mathematical formula the first instance of what we now know as theoretical physics.'

Is this why Tim Flannery once observed about C&C: -

'When facing a grave emergency, it's best to be single-minded.'

As Patanjali said millennia ago, 'heyam dukham anagatam' or, 'pain which has not yet come is avoidable'.

In Sanskrit this 'single-mindedness' is called 'ekagrata'.

For reasons of UNFCCC-compliance, this requires us to go now beyond ideological banter to teleological purpose.

A Well-Tempered Climate Accord is prerequisite to the fulfillment of this purpose.

Stringularity is l/2=Hz2.


Wonderful David Hare Interview with ABC Rowan Williams in the Guardian Magazine 8th July, 2011

I ask if that's the criterion 'that the world should make sense'. "Make sense not in a great theoretical system, but that you can see the connections somehow and 'I tend to reach for musical analogies here' you can hear the harmonics. You may not have everything tied up in every detail, but there's enough of that harmonic available to think, 'OK, I can risk aligning myself with this.' Because you're never going to nail it to the floor and eat your heart out, Richard Dawkins!

"You know that scene in the Woody Allen film where they have an argument in a cinema queue, and Marshall McLuhan is standing behind them and able to interrupt to settle the argument? Woody Allen turns to the camera and says, 'Wouldn't it be wonderful if life were like that?' Well, no Marshall McLuhan will ever step forward in the queue and say to Richard Dawkins, 'The archbishop's right.' It's not going to happen."

ABC is on record with accomplished C&C advocacy in the CoE dcoument 'Sharing God's Planet' and on page 66 of the DAVOS document Faith and the Global Agenda: Values for the Post-Crisis Economy - and Richard Dawkins will yet make his peace with Plato.


Music-making as rouser and healer

World Orchestra for Peace

East West Diwan Orchestra

The Quartet of Peace

Liberated Carbon Andy Revkin, indefatigable Environment Editor of the New York Times

Music-made - [also here linked to animations of 'Pythagorean' Earth:Venus orbits . . . . ] . . . .
[1] Uskudar Giderken - Middle Eastern Folk Tune arranged for Clarinet, Viola Cello

[2] Das Lied van de Merwe - Clarinet Viola and Piano Trio in Three Movements [1] [2] [3] - [Purcell Room]

[3] Bells, Lillies and Clouds Without Rain [Studio]
[4] Tilted Windmills [Studio]
[5] "Choros" - [Live performance SWRB - Poor Quality Recording - hand-held mike]


Pentagram Venus:Earth orbit - perfect 'golden section'

On the numbers and 'Venus', I came across these perceptions in two lovely little books:

"A Little Book of COINCIDENCE in the Solar System" John Martineau


"The Golden Section" Scott Olsen

and am still in a state of stunned silence by these 'vibrations' of what to me are expressions 'stringularity' . . . . .


"A Little Book of COINCIDENCE in the Solar System" John Martineau

THE PERFECT BEAUTY OF VENUS - the things they don't teach you at school

With the Sun in the centre, let us look at the orbits of Venus and the Earth. Every couple of days a line is drawn between the two planets (below left). Because Venus orbits faster she completes a whole circuit in the same time that the Earth completes just over a half-circuit (below centre). If we keep watching for exactly eight years the pattern opposite emerges, the sun-centred version of the five-petalled flower on the previous page. The ratio between Earth's outer orbit and Venus' inner orbit, i.e their home, is intriguingly given by a square (below right) (999%). Venus rotates extremely slowly on her own axis in the opposite direction to most rotations in the solar system. Her day is precisely two-thirds of an Earth year, a musical fifth. This closely harmonises with the dance opposite so that every time Venus and Earth kiss, Venus does so with her same face. In other words paint a spot on Venus' surface and every time she lines up with the as seen from Earth the spot will point back at Earth. Over the eight Earth years of the five kisses, Venus spins on her own axis twelve times in thirteen of her years {from Kollerstrom}. All beautiful musical numbers.


"The Golden Section" Scott Olsen


Aphrodite's golden kiss - Not only the microcosm and mesocosm display a liking for the divine proportion. Golden ratios abound in the solar system, and, strangely, seem to occur particularly frequently around Earth. For example, both the relative physical sizes and the relative mean orbits of Earth and Mercury are given by Phi squared : 1, or a pentagram to 99% accuracy (opposite top left) . Nothing, however, compares to the extraordinary relationship between Earth and our nearest planet, Venus which draws a beautiful fivefold rosette around us every eight years. Eight years on Earth is also thirteen Venusian years, the Fibonacci numbers 13: 8: 5 here appearing to connect space and time. Furthermore, Venus' perigee and apogee (her furthest and closest distances to Earth) are defined as Phi to the power 4 : 1 to an accuracy of 99.99%, shown here as two nested pentagrams (opposite, after Martineau). The two largest planets, Jupiter and Saturn, also produce a perfect golden ratio from Earth. Line them all up towards the Sun and a year later Earth is back where she started. Saturn will not have moved far and 12.85 days later Earth is again exactly between Saturn and the Sun. 20.79 days later Earth is found between the Sun and Jupiter. These synodic measures exist in space and time and relate as 1: Phi to 99.99% accuracy (after Richard Heath). Moving yet further into the macrocosm, irrespective of whether or not they become reconstrued as dark energy stars, Paul Davies discovered that rotating black holes flip from a negative to a positive specific heat when the ratio of the square of the mass to the square of the spin parameter (rotation speed) equals Phi


Is this awesome proportionality a deep simplicity that is true both sides of the Schwarzchild radius?
Is the new reality negative entropy? Do dog-days go backwards inside a Black Hole?
Did Schrodinger find the cat that couldn't miaow or the frog that couldn't croak?