sethoflagos

Your point was well taken and agreed with. Your 'harmonic excitation' = My 'brassy distortion'. Different words, same tune.

As you can probably guess from my avatar (from a 50 year-old newspaper article) my musical activities were (until quite recently) mainly high brass. I've a number of Bb trumpets ranging in nominal bore from an 11mm small bore German rotary to a 12.2 mm wide bore Wild Thing; an 11.9mm wide bore C trumpet, and a 12.5mm bass trumpet. The lowest available musical note available in each of them, for any performer, is set absolutely by the 2nd harmonic of the tubing length. The highest pitch available comfortably attainable by any reasonable player is set by the maximum stiffness of their lip. For me, pretty well throughout my playing career, this was a concert high D. This was the case for every instrument I've played over 50+ years (a lot!). Better players than me could usually get a bit higher, some considerably more so. But most serious professionals I've discussed this with (again, a lot) report the same personal experience: they top out at the same pitch on any trumpet irrespective of bore size. So much for personal testimony. Technical literature on the influence of bore size on pitch is hard to locate. If there were such then one would expect papers such as ... An Exploration of Extreme High Notes in Brass Playing (Proceedings of the International Symposium on Music Acoustics (Associated Meeting of the International Congress on Acoustics) 25-31 August 2010, Sydney and Katoomba, Australia (2010) (J. Chick, S. Logie, J. Kemp, M. Campbell, R. Smith)} & Its all in the bore! (Journal of the International Trumpet Guild (USA), 42-45 (May 1988) (R. Smith)) ... both available at https://smithwatkins.com/library/technical-papers.html to at least refer to the phenomenon. Bore size certainly changes the balance between oral cavity pressure and air volumetric flow for a given pitch and intensity - wide bore instruments trade a higher flowrate for a lower pressure compared to smaller bore instruments. This lower pressure amplitude directly results in lower characteristically 'brassy' distortion in passages that are high and/or loud. Hence they sound relatively 'mellower' which was the thrust of your OP. On that point at least we are agreed.

To vibrate a reed (or lip) we need to create a periodic displacement - (generally) to permit the flow of air through it. From Hooke's Law, we know that displacement = force x length / (elasticity x x-section) Only somewhat loosely, we can equate force / x-section to characterise the gauge pressure required to open the air pathway into the instrument. Also, we note that force/displacement is the 'stiffness' term in the analysis of simple harmonic motion, which yields the result that natural undamped frequency is proportional to the square root of the stiffness. Therefore by adjusting the stiffness of our reeds (or lips), we increase both the pressure required to maintain vibration and the natural frequency of that vibration. In short, there is a strong correlation between the pressure amplitude and frequency of the air column. And with a higher pressure amplitude the required x-section area is reduced in proportion to achieve any given acoustic intensity. Hence high-pitched wind instruments tend to feature narrower bores than their lower-pitched relatives. I've skated over (ignored) a huge amount of fine detail here, but the above line of reasoning summarises the underlying physics as I understand it. Hope this helps.

Oil and gas reserves can only accumulate in strata that are at least somewhat permeable. These fluids are free are to migrate towards a low pressure zone within the formation (they are not enclosed in a solid crystalline matrix) and therefore the issue is not accessibility but whether or not the permeability is high enough to support an economic extraction rate. Hydraulic fracturing widens a proportion of pre-existing pores, and wedges them open with appropriately sized 'proppants' suspended in the fracking fluid. It's the geological equivalent of coronary bypass surgery. These 'hard rock deposits' have essentially zero permeabilty. The gold is enclosed in a welded crystalline matrix and therefore the issue in this case is accessibility. In order to get access to the gold particles for any form of liquid extraction process, the rock matrix has to be ground into fine particles otherwise significant contact between liquid and gold particles simply won't happen. Hydraulic fracturing cannot achieve this. Without the initial permeability, you cannot even get the fracking fluids into the formation in the first place other than possibly along pre-existing fault planes. Trying to widen these is probably not a great idea.

+1 While there are undeniably many cultural differences in musical traditions around the world, I've yet to hear, or hear of, any that are not firmly rooted in the natural harmonic series. For neolithic references, Chinese forms are best attested - https://en.wikipedia.org/wiki/Chinese_musicology This clearly predates known weatern forms by many millenia. My favorite example of playing around with acoustic resonances is

Do you play the violin? If you do then you know that you can play chords by multiple stopping. Then you can play out the notes of that chord sequentially as an arpeggio - you still 'sense' the full chord don't you? Then intersperse a few passing notes between the intervals to make the line less 'gappy' and you have a melodic line based on that chord. I think there's less of a difference than you imagine.

Two items to ponder: 1) Notes don't just stop: they bounce around the room as echoes and gradually fade. 2) We have pitch memory. Even when a sound fades into imperceptibility, we can still hold it in memory almost indefinitely. How long do you have to be parted from someone before you forget what their voice sounded like?

We don't need the pitches to sound together to sense the harmonic relationship between them. Also, I don't know about you, but I would find music built entirely out of consonant intervals unstimulating to say the least. Good music tells good stories and good stories need some level of conflict. You can't have a Beowulf without Grendel and his mum tagging along in the background.

Consider this: Per joigus' post, any given structure tends to resonate with a series of acoustic waves that are an integer multiple of some fundamental frequency. That fundamental frequency along with the acoustic intensity gives us an idea of the physical size of the structure, and its proximity. New point: these resonances are related not only harmonically, but also in phase. Therefore if our ears detect a number of simultaneous frequencies that have a simple harmonic relationship and are in phase with each other, then we can reasonably deduce that they came from a single source - perhaps prey, perhaps predator. We could learn to match these complex sounds to precise sources critical to our survival. If we sense either phase shifts or non-harmonic tones within the sound, this indicates that there is more than one source object - useful to know if you are up against a single wolf or a pack. This suggests to me that our distant ancestors may well have learnt to associate simple harmonic waveforms as 'safe' and complex non-harmonic, out-of-phase sounds as 'dangerous'. Not much established science to back up this hypothesis. But it seems a reasonable one to explain why we find frequency ratios of 2, 3 and 5 'pleasant'. And since all twelve notes of the chromatic scale (at least in western music) are constructed from these three ratios (at least approximately), the roots of both harmony and melody seem to follow with some logical consistency.

Because it is a simple case. And because of its simplicity, it's also fairly uninformative. You're not creating any complexity here - just warm water.

Makes sense. "Are you talking about the mover, or that which is moved?"

Sorry. Senior moment. Please substitute 'negative' for 'positive' in the above. (assuming HIP/WIN is the standard)

It isn't a great example to work with. Better would be the case of tree growth in an oxygen rich environment. The occasional forest fire reminds us that the delta G of tree + oxygen to hot fog and ashes is very positive. And yet trees thrive, even sometimes utilising forest fires to suit their own purpose. High positive free energies provide the possibility of multiple simultaneous processes and vastly more chemical diversity than near equilibrium systems. In fact, so much diversity is possible in certain favourable conditions, that long term predictions rapidly become approximate in general trends and utterly speculative in detail. Hence the evolution of life as we know it.

No problem. And be sure to keep us informed of your progress. Goodnight, John.

Well I wish you luck with your endeavours.

John, Just how easy do you think it is to realise that ambition? The heart of your OP, the nature of far-from-equilibrium thermodynamics, wasn't even a recognised field of study until the the work of the Russian chemical engineer Ilya Prigogine was brought to general attention with his award of the Nobel prize in chemistry in 1977 - that's half a century after the first firm principles of quantum mechanics and general relativity were established! It isn't an easy subject. Do you really expect to become fully conversant with it with a few brief exchanges on a general science internet forum? I was a first year chemical engineering student when Prigogine received his recognition, and it was something of an inspiration to us at the time. But mastering just standard thermodynamics is a major undertaking, and after a 40-odd year career development mainly focused on its application in the energy sector, I still often feel that I've only scratched the surface. Can you see how the assumption of being able to pick it up in five minutes might rub some people up the wrong way? Never mind ignoring any content that didn't quite fit in with your preconceived ideas. That level of understanding takes work, A great deal of work. And if you request help, as we all should do when we can't see the wood for the trees, then you really need to be switched to 'receive' rather than 'transmit'.

The strong force at distances oto 0.7 fm? Proton-proton electrostatic repulsion? Others .....

I'm no expert on cosmology, far from it, but the general picture I have is of periods of free expansion (which I presume to be near isentropic) punctuated by intermittent phase changes (eg quarks 'condensing' into hadrons, nucleosynthesis etc) as and when the temperature falls to to the point where the free energy change for that transition becomes positive. At each phase change, we see the matter components of the universe transform into a more structured lower entropy state, accompanied by a large release of energy (the associated 'latent' heat for that phase change) into the surrounding 'photon gas', the radiative component of the universe. The overall process may well approximate to a period of expansion at constant temperature with a corresponding significant overall entropy increase. As regards the influence of your four fundamental forces on entropy, I guess it depends on whether those forces are attractive or repulsive in nature. Attractive forces tend to create structure, so I suppose you could view them as battling with entropy in some sense. But the repulsive forces do quite the opposite. I'm not sure that alloting a 'purpose' to these forces is helpful to a true understanding of them.

The CMBR map tells us that the early universe was extremely close to thermal equilibrium (and in a relatively low entropy state) at least until recombination. Significant departure from equilibrium began with the ensuing localised gravitational collapses and formation of the first stars. Only then did we have the large thermal and density gradients that are necessary to drive far-from-equilibrium processes.

Ignore any references to thermodynamic work for now - it's irrelevant to what you're trying to get your head around. The stipulation "at a constant temperature" simply implies that transitions between significantly different temperatures require multiple calculations over small temperature changes (ie. integration wrt temperature) Free energy values, especially for water, can easily be obtained from thermodynamic tables or calculated from standard values. But your interest is more in changes in free energy. Consider the process of water freezing. At high temperatures the free energy of liquid water is higher than the free energy of ice. The freezing of water is associated with a negative change in free energy and is therefore disallowed whereas ice will melt spontaneously. At low temperatures the reverse is true. At around 273 K the free energies are equal and so the changes in free energy for both the freezing and melting processes are zero. This defines thermodynamic equilibrium: there is no nett tendency for more water to freeze nor more ice to melt. Having established that principle, we can now look at a slightly more complex process that is (I think!) relevant to your main interest. Let us take our liquid water at (or at least close to) freezing point and expose it to a large departure from equilibrium via a significant source of negentropy: a sealed, insulated vacuum chamber The change in free energy for the transition liquid water to water vapour is positive, and so the liquid starts to boil. But this chills the water, making the transition of water to ice positive also, so we get simultaneous freezing. Eventually the vapour pressure reaches 611 Pa and the free energy of vapourisation falls to zero. As does the free energy of condensation, the free energy of melting, the free energy of freezing and incidentally, the free energies of both sublimation (ice to vapour) and deposition (vapour to ice). Thermodynamic equilibrium is reached at the triple point. But the key point to take home is that by adding a medium entropy material (cold water) to a substantial source of negentropy (the vacuum chamber) we've managed to create a relatively low entropy state (ice) without recourse to external energy input. This simple experiment belies the idea that thermodynamics cannot create ordered systems spontaneously. Given a large enough departure from equilibrium and a good mix of building blocks to play with, structures of arbitrary levels of complexity are not only possible, but inevitable. Even brain matter!

On further consideration, by omitting the pressure terms from the Navier-Stokes, haven't you lost control of the conservation of energy? For a rotating system, I can visualise material falling into an equatorial disk from both sides, but there appears to be no mechanism in your system for it to cross the boundary. The momenta will just cancel (conserving momentum) but then so will the corresponding kinetic energy! Restoring the pressure term (which in context, is an expression of the system internal energy) will keep the 1st Law books straight and avoid some embarrassing infinite densities.

In principle, your equations look okay given your stated constraints, but I would strongly recommend converting to polar coordinates to get as much help as possible from its symmetries. Given that your gravitational field terms are directly analogous to the pressure gradient terms in Navier-Stokes applications that I'm somewhat familiar with, you may find that at least for some simple starting conditions, the method of characteristics may help convert your PDEs into ODEs which would then be amenable to numerical integration. However, you're using Gauss' Law where I would normally be inserting a (simpler) equation of state, and that may complicate matters significantly.

So in this viewpoint, there is no preferred (spatial) direction for any of the forces at play here. I was going to proceed to my follow-up question of what happens (to us) when there's no preferred direction for gravitational forces. But I'm now getting the feeling that you've already answered that. In that it's not about 'us' - it's about how mass acts on spacetime. And we're just little specks riding on that ebb and flow. Humbling thought. Thank you once again, Markus.

Yes, that's the general picture I'm asking about.

All is clear. Many thanks, Markus.