The Doubling Constant, d, and the Resonance of Triogenesis and Musical Scales

The Doubling Constant dand the Resonance of Triogenesis and Musical Scales Shu Jian Chen∗1 1School of Civil Engineering, University of Queensland, St Lucia, 4072, Australia Abstract This paper introduces the doubling constant d, a structurally emergent exponent that reconciles a 12-step recursive musical scale—built from alternating ±1 3frequency ratios—with the octave doubling observed in physical and musical systems. Derived from the Triogenesis framework, d is shown to correct the slight deviation between the recursive scale’s terminal ratio and the canonical 2:1 octave, through the structural growth function g(u). Remarkably, the symbolic formulation of d converges to a precision exceeding most known physical constants, without requiring an infinite tail. This finding challenges the presumed primacy of the number 2 in nature, suggesting it may be a resonance threshold arising from deeper layered recursions. ∗Corresponding author: shujian.c[email protected] (S.J. Chen) 1 The ±1 3Scale and Frequency Ratios We construct a 12-step pure scale using alternating multiplication by 4 3and 2 3, corresponding to structural +1/3 and −1/3 resonance steps. Starting from a base frequency f0, the sequence {si}is defined recursively as: s1=f0, si=     si−1·4 3,if iis even, si−1·2 3,if iis odd, for i= 2,3,...,13. This alternating structure produces a natural downward spiral in frequency across 12 steps. Reversing the ratio from s13 back to s1, we obtain a total scale ratio of: ρ−1 1/3=s1 s13 = 2.027286529541015625. The Doubling Constant d This value reflects the natural resonance doubling embedded in the ±1/3 alternation, which slightly exceeds the defined 2:1 ratio of the equal-tempered octave. The discrepancy is corrected structurally by the recurrence asymmetry growth exponent dvia: ρ1/3 g(u)d= 2 g(u)dcan be numerically evaluated as g(u)d≈1.0136432647705078125. This exponent is not arbitrary but emerges from a convergent symbolic structure that encodes recurrence alignment, asymmetry saturation, and observer delay coupling. The full symbolic expression of the doubling constant is: d= 180 + 15 + 1 3−1 187 + 15 + 15 + 3 + 1 14+ 2 3+1 173−15−15−15−2 3−1 180+δtail . Here, δtail denotes the unresolved recurrence structure beyond this symbolic depth — analogous to the infinite decimal tail of irrational constants like πor e. The convergence of dto this value ensures that the 12-step resonance-based pure scale is structurally corrected into the defined octave by the recurrence growth function g(u). Precision of Structural Doubling (Without Tail) Let the exponent dbe defined symbolically by: d= 180 + 15 + 1 3−1 187 + 15 + 15 + 3 + 1 14+ 2 3+1 173−15−15−15−2 3−1 180 2 excluding the final unresolved term δtail. Then the structural doubling expression: ρ1/3 2·g(u)d≈1.0000000000000000001631803595036782 differs from unity by only: ε= 1.63 ×10−19. This demonstrates that the symbolic recurrence structure alone — without infinite expansion — yields a convergence more precise than the known uncertainty in most fundamental physical constants. The constant d, introduced in this work, emerges as a resonance point across layered structural patterns previously derived from the Triogenesis framework—including the CMB dipole asymmetry, Earth–Sun orbital coupling, and hydrogen molecular bonding (1; 2; 3; 4; 5; 6; 7; 8). Although the structure of Triogenesis does not fundamentally assume the recurrence of a doubling—or even the number 2—the layered alignment of these structural patterns naturally converges toward a doubling ratio. In this sense, ddoes not impose doubling, but reveals where the recursive asymmetries across domains align most coherently. Doubling Constant and Foundational Mathematical Questions The discovery of the doubling constant, denoted as d, raises profound questions about the foundational role of the number 2 in mathematics and physics. Traditionally, doubling—such as frequency ratios in octaves, binary systems in computation, or geometric scaling in physical laws—is treated as a natural and self-evident operation. However, within the Triogenesis framework, the appearance of 2 may not be a primitive axiom but rather an emergent property arising from a deeper recursive structure. Specifically, the derived constant dsatisfies g(u)d=s1 s13 /2, where s1to s13 define a 12-step scale built from recursive ±1 3frequency ratios, and g(u) encodes a layered structural growth pattern defined by the Triogenesis recurrence law. The precision match between this construct and the physical doubling observed in octaves suggests that what we perceive as ’doubling’ may in fact be the outcome of a convergent process involving nested structures, rather than a fundamental axiom. This insight places dalongside other historically pivotal constants such as πand e, which also arise from geometric or exponential convergence rather than arbitrary definition. Unlike π—which measures circular closure—or e—which emerges from compounding change—d captures the recursive alignment required for structural self-similarity across scales. It governs not just a frequency relationship, but a resonance condition that appears in domains as diverse as hydrogen bonding, orbital periods, and cosmological dipoles. In this sense, the doubling constant dmay represent a new kind of mathematical entity: not a numerical shortcut, but a resonance threshold—a structural attractor at which recursion stabilizes into the recognizable ’2’ of our physical and musical experience. Its existence 3 challenges the assumption that binary division or doubling is primitive. Instead, it invites comparison with deep unsolved problems and constants in mathematics, such as the Riemann Hypothesis (concerning zeros of the zeta function) or the fine structure constant (a dimensionless physical coupling). If confirmed, the structural origin of doubling could redefine our understanding of numerical foundations, suggesting that even the number 2 is not given, but grown. Perceptual Preference and Tuning Structures in Music The 12-tone equal temperament (ET) system, dominant in Western music since the 18th century, divides the octave into 12 logarithmically equal parts. This design enables modulation across keys at the cost of small deviations from the acoustically pure intervals found in just intonation or other structurally defined scales. In contrast, the ±1/3 ratio scale we introduced—alternating between ×4 3and ×2 3at each step—constructs a complete 12-note scale that converges on the octave through a cascading structural pattern. This scale maintains internally consistent frequency relationships without key modulation flexibility. While our structurally generated scale resonates more naturally with the recursive layering principles of Triogenesis, one might ask whether such tuning actually feels better to the human ear. Interestingly, empirical studies suggest that listener preference does not always align with structural or physical purity. In a controlled A/B comparison experiment, Long (9) found that adult listeners tended to prefer equal temperament (ET) over just intonation (JI), even when JI tones were acoustically purer. However, this outcome is widely interpreted not as a psychoacoustic superiority of ET, but rather as an artifact of cultural familiarity and habituation. As Huron argues in Sweet Anticipation (10), musical expectations and aesthetic judgments are largely shaped by statistical learning through repeated exposure—much like a perceptual tuning to the dominant tuning system of one’s culture. From a physical and mathematical standpoint, just intervals—whether based on ratios of small integers or recursive laws such as the ±1 3scaling used here—are measurably more consonant, as demonstrated in psychoacoustic models and resonance theory (11). Intriguingly, studies on infants who have not yet been culturally conditioned support this interpretation. Plantinga and Trainor (12) found that six-month-old infants showed stronger sensitivity to mistuned just intervals than to deviations from equal temperament, indicating a possible innate or resonance-based auditory bias toward structurally simpler frequency relationships. In this light, our 12-step pure scale generated by alternating ×4 3and ×2 3steps may not only embody recursive structure as predicted by Triogenesis but also reveal a deeper resonance with untrained perceptual systems—one that may have been gradually overridden by cultural adaptation to ET over centuries. This dichotomy mirrors the discovery we make with the doubling ratio constant d. While the number 2 appears throughout physical laws, including octave doubling in music, it emerges not as a primitive but as a resonance point of layered structural ratios, converging through deeper recursions like those found in the Triogenesis g(u)dformulation. The 12-step ±1/3 scale thus reveals itself not just as an alternative tuning system, but as a physical and perceptual bridge between structural convergence and emergent symmetry. 4 Acknowledgments The author received no external funding for this research. Portions of this manuscript, including language refinement and structural organization, were supported by the use of AI tools (OpenAI’s ChatGPT). These tools were employed solely for information distillation, clarity enhancement, and formatting assistance. All scientific concepts, formulations, and interpretations presented herein are original and authored by the corresponding researcher. Conflict of interest The authors declare no conflict of interest. CRediT authorship contribution statement S. J. C. conceived, developed, and wrote the manuscript in its entirety. Data availability The data that support the findings of this study are available from the corresponding author upon request. References [1] Shu Jian Chen. Triogenesis-driven entropy growth and unified relativity: Recurrencebased origin of blackbody radiation and the cosmic microwave background. Zenodo, 2025. [2] Shu Jian Chen. Triogenesis: Structural mechanism for beta decay, neutrino oscillation, and emergent strong–weak interactions in muon, tau, free neutron, helium-4, helium-3, and deuterium systems. Zenodo, 2025. [3] Shu Jian Chen. Triogenesis: Disherence–herence projection as the origin of wavefunctions, thermal statistics, gaussian fields, and conscious fluctuation. Zenodo, 2025. [4] Shu Jian Chen. Triogenesis: Resolving the fermi paradox and exoplanet search criteria from the perspective of the generative replica. Zenodo, 2025. [5] Shujian Chen. Triogenesis: Hydrogen molecular bonding and the projection of universal structures in and beyond spacetime. Zenodo, 2025. Preprint. [6] Shu Jian Chen. Projection of straintity as mass and energy in space-time: Emergence of e=mc2in triogenesis, 2025. [7] Chen Shu Jian. Triogenesis: A closed-form framework for the emergence of physical order and consciousness. Zenodo, 2025. https://zenodo.org/records/15116919. 5 [8] Chen Shu Jian. Closed-form derivation of the hydrogen spectrum from triogenesis. Zenodo, 2025. https://zenodo.org/records/15123220. [9] Derle Ray Long. A Study of the Perceptual Differences Between Just Intonation and Equal Temperament in Harmonic Intervals. Ph.d. dissertation, University of Kentucky, 2008. [10] David Huron. Sweet Anticipation: Music and the Psychology of Expectation. MIT Press, 2006. [11] F. Richard Moore Thomas D. Rossing and Paul A. Wheeler. The Science of Sound. Addison-Wesley, 3rd edition, 2002. [12] Judy Plantinga and Laurel J Trainor. Infants’ preference for consonant over dissonant intervals. Developmental Psychology, 45(1):1–6, 2009. 6