Farey Sequences, Noble Gas Configurations, Stable Isotopes, Extended Tertian Harmony, and Cities

Suppose someone told you that they took a survey that showed that 57.14% of people preferred apple juice, 28.57% of people preferred ruby red grapefruit juice, 14.29% of people preferred cranberry pomegranate juice, and 0.00% preferred any other types of juice. What would you conclude from this?

You would be right if you decided to be suspicious of the sample size involved in this survey and call out misuse of excessive significant digits from the insufficient sample size, because what happened really looks like someone used a sample size of 7, from which four people said they preferred apple, two liked ruby red, and one liked cranberry pomegranate. Okay, maybe it might have been 14, or 21, but it would be very unlikely if the interviewer actually interviewed, say, 63 people, and the number that chose each category somehow turned out to be a multiple of 9.

Now suppose they instead reported 57.13% of people preferred apple juice, 28.56% preferred ruby red grapefruit juice, 14.31% of people preferred cranberry pomegranate juice, and 0.00% preferred any other types. Now, you might still be suspicious that they really interviewed 7 people but now didn’t even bother to publish truthful answers, but given that they are honest results, you should be intrigued. You might even be interested in researching a bit to find a psychological reason why so close to twice as many people prefer ruby red to cranberry pomegranate if that was the case, as the sample size needed to obtain such results, if decimal representations were rounded to the nearest hundredth of a percent as shown, would be at least in the thousands, and such a ratio would seem like a thing that isn’t coincidence.

But my point here is how a small disturbance from a well-established low-denominator fraction leads to an area of exclusively high denominators in fractional equivalents. One can see how this is with other low denominators: 0.5 can be a fraction with denominator 2, but 0.49 and 0.51 are both fractions with denominator at least 35 (considering the possibility of rounding). Often, in math problems, we gain more assurance from the answer we get being “nice,” because it means there probably is an underlying reason for the occurrence. Within the rational numbers of denominator less than or equal to any number, the nicest numbers (those with lower denominators in reduced form), exclusively require the traversing of not-as-nice numbers before getting to another nice number, traveling via the metric of value. Nice numbers, as they are, make a thorough job of spreading out, and the closer one gets to a nice number without being specifically at the nice number, the uglier numbers get.

In fact in any Farey sequence, like the fifth Farey sequence {0 , 1/5 , 1/4 , 1/3 , 2/5 , 1/2 , 3/5 , 2/3 , 3/4 , 4/5 , 1}, one can see that the largest gaps in numbers “as nice” as 1/5 occurs next to the numbers with denominator 1. In the above, those are 1/5 gaps of exclusively numbers with larger denominators than 1/5, followed by next-in-size gaps of 1/10 around the number with denominator 2.

The property of optimally desired states being surrounded directly by locally optimally undesired states extend beyond the theoretical world of mathematics, though, into chemistry. Take the stablest elements there are. That would be the noble gases. The next stablest? Those in the middle of the transition series, far from the noble gases. The least stable? The halogens and alkali metals, right next to the noble gases. In fact, one could characterize the reactivity of halogens and alkali metals as a “want” for the pre-selected “happy” noble gas configurations. The further one is from the noble gases, the more electron shedding or receiving required, and the less irresistible taking the path to noble gas configuration becomes. Rather, an atom is more likely to ionize to a local optimum, that often makes a subshell happy.

This can be observed in trends in isotope stability as well. Technetium, an element with no stable isotopes, is next to molybdenum and ruthenium, both elements with many table isotopes. There is a large general tendency of Technetium isotopes to desire to decay to bordering strongholds of stability. Around the “magic numbers,” for which numbers of protons and neutrons can hold stable with a generally wider array of numbers for the other, isotope stability drops, although this example is a case of more “spreading out” and less nearby “well depth” than in the previously mentioned two cases.

This effect can also be seen when analyzing music harmonically. From an Ionian perspective, the twelve scale degrees can be viewed as 1, flat-2, 2, flat-3, 3, 4, flat-5 (or sharp-4), 5, flat-6, 6, flat-7, and 7, which when combined with the root at scale degree 1 produces the intervals of unison, minor 2nd, major 2nd, minor 3rd, major 3rd, perfect 4th, augumented 4th/diminished 5th/tritone, perfect 5th, minor 6th, major 6th, minor 7th, and major 7th, before returning to a perfect octave, made of the same notes as the unison but with a different separation in registers. Interestingly, here the most dissonant interval is the one right in the middle, the tritone, but the next most dissonant are indeed the minor 2nd and the corresponding minor 7th, right next to the perfect same-note intervals. The perfect intervals are indeed so named because they were thought to be the best musical consonances there were, at least until 3rds and 6ths became popular and perfect 4ths started seeming like dissonances (well, okay, in musical context, the implication of the unstable 6|4-position chord is probably the best explanation for why perfect 4ths are considered dissonances in baroque and early classical counterpoint). But the same pattern of optimal intervals surrounded by more notable non-optimal intervals still persists, with major 2nds and minor 7ths being next most dissonant. When writing four-part counterpoint, in a major key the scale degrees to resolve to are the 3rd, 5th, and 1st or 8th. It is very important when resolving a dominant chord to a tonic that the two greatest resolving tendencies, the 4th (7th of a dominant seventh) to the 3rd, and the 7th (leading tone, or 3rd of a dominant or dominant seventh) to the 8th, are observed, whereas the tendencies of the 6th to resolve to the 5th and the 2nd to resolve to either the 3rd or the 1st in this chord and other chords are more circumventable, as resolutions involving non-half-step stepwise movement. As we extend dominant chords to consider the dominant 9th, dominant 11th, and dominant 13th, we notice the same patterns, but with the notable extension of the 13th (scale degree 10 or 3) falling to scale degree 8 or 1 in preferred resolution. This expansion of scope of preferred resolution with increasing complexity of implicit harmony is also interestingly analogous to the increasing complexity of available oxidation states with greater number of electron shells. The effects of considered-nice intervals in music are actually even reducible to mathematical patterns, since notes are logarithmically calibrated and in a non-well-tuned instrument, ratios of small numbers are purposefully used to produce the nice intervals.

Abstracting even further, to human civilization, one can see where this effect and spreading out at peaks intersect by observing the arrangements of cities. One principle in the study of human geography is that before the effect of terrain, cities tend to arrange in hexagons, and then cities and towns of the next smaller general magnitude in hexagons around these. The hexagon usually ends up being the shape of choice because in the transferring of goods, it provides the minimal amount of collections of settling for suiting a contiguous area, since the regular hexagon is the regular polygon with the most number of sides while still tessellating the plane. By the same influence of the mathematical pattern, cities of next smaller size tend to appear more towards the midpoint between cities of larger size, but in this case, size itself is the magnitude in question, invoking the concept of suburbs, a counter-pattern to this occurrence.

Thus, just additionally pouring out that last bit of juice after pouring whole cups from a container represents not just a want for completion, it reflects a theme seen over and over again of the greater pull to reach more notable values, practically everywhere.

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