Abstract

The canonical frequency bands used to categorize human electroencephalographic (EEG) activity—delta, theta, alpha, beta, and gamma—have historically been defined using pragmatic and variably applied thresholds rather than a unifying mathematical principle. In this theoretical study, we propose a geometric framework for redefining EEG frequency bands based on logarithmic scaling, drawing on the exponential formulation introduced in Mary Blagg’s refinement of the Titius–Bode law. Using the mean adult alpha rhythm as a reference frequency and applying a constant scaling ratio (R = 1.7275), we derive a mathematically ordered hierarchy of EEG band centers and boundaries within a continuous log-spaced spectrum. Unlike descriptive models of spectral 1/f scaling, the present framework provides an explicit generative rule for discrete band centers and transition frequencies. The resulting segmentation produces band definitions numerically consistent with commonly reported EEG frequency ranges while offering a fully proportional, non-overlapping structure. The model further introduces principled subdivisions within the alpha and gamma ranges and redefines the beta–gamma transition using geometric rather than conventional criteria. As a descriptive quantitative observation, the model-derived theta–alpha transition (∼7.98 Hz) lies in numerical proximity to the Earth’s fundamental Schumann resonance (∼7.83 Hz); this correspondence arises from the predefined geometric rule and does not imply causal interaction. Overall, the proposed framework reframes EEG band organization as a mathematically explicit, scale-invariant system and provides a hypothesis-generating basis for future empirical evaluation of oscillatory structure.

Introduction

Biological evolution, including the evolution of nervous systems across species, has unfolded under the constant and inescapable influence of Earth’s physical environment (Sousa et al., 2017). Gravity, geomagnetic fields, and cosmic cycles have all been shown to exert measurable influences on physiological organization and biological regulation (Martel et al., 2023). For example, Earth’s 1G gravitational field has played a well-established role in shaping musculoskeletal development and gravity- and orientation-sensing mechanisms across vertebrate and invertebrate organisms, supporting equilibrium, locomotion, and spatial orientation (Bonanni et al., 2023). Beyond the mechanical effects of gravity, the planet’s pervasive electromagnetic and geomagnetic environment has been proposed as an additional contextual factor accompanying neurobiological evolution (Rouleau and Dotta, 2014). All nervous systems—human and non-human alike—have developed and functioned within this shared matrix of gravitational and electromagnetic conditions (Narayanan, 2023). Neuronal oscillatory activity therefore unfolds within a frequency landscape that overlaps with naturally occurring electromagnetic phenomena, including Earth’s extremely low-frequency resonances such as the Schumann modes (Schumann and König, 1954). Over evolutionary timescales, prolonged co-exposure to these environmental rhythms has been hypothesized to provide a stable background context for neural oscillatory organization, as discussed in theoretical and empirical studies of environmental electromagnetic interactions with neural systems (Koenig et al., 1981; Cherry, 2003; Kozlowski and Marciak-Kozlowska, 2015). From this perspective, the human brain can be viewed not as an isolated electromagnetic system, but as a biological organ operating within a broader planetary electromagnetic milieu, shaped by long-term exposure to relatively stable geophysical conditions (McCraty et al., 2017). These considerations are presented as broad environmental context and do not imply that specific neural oscillatory frequencies are causally determined by, or evolutionarily tuned to, geophysical electromagnetic phenomena. Solar radiation likewise contributes to neurobiological regulation through well-characterized pathways: ultraviolet exposure supports vitamin D synthesis (Hastings, 1983), geomagnetic cues underpin magnetic compass orientation in multiple species (Lohmann et al., 2001), and solar-driven light–dark cycles entrain circadian rhythms via melatonin regulation. Life is thus embedded within planetary and solar environmental constraints rather than governed by biochemical processes alone. Importantly, sensitivity to geomagnetic and environmental cues is not unique to humans; numerous non-human species—including migratory birds, marine mammals, sea turtles, and crustaceans—demonstrate reliance on Earth’s magnetic field for navigation and orientation. In the present study, the focus on humans reflects data availability and the maturity of EEG band characterization, rather than an assumption of species-specific susceptibility.

As a result, the human brain’s architecture and function can be understood as having developed within, and being constrained by, Earth’s physical environment (Striebel et al., 2023). Neural circuitry and brain oscillations (EEG rhythms) did not arise in isolation but emerged under a steady 9.8 m/s2 gravitational field and within Earth’s geomagnetic environment (Zhu et al., 2024). The role of gravity in biological evolution is well illustrated by its constraints on organismal size and physiology: theoretical analyses indicate that under Earth’s current gravitational acceleration (1 G), there are upper limits on viable body mass, locomotion efficiency, and cardiovascular function, a problem historically highlighted by the biomechanics of extremely large terrestrial animals such as dinosaurs (Hokkanen, 1985). Many aspects of neurophysiology—from synaptic timing to vestibular orientation—are therefore adapted to the physical constants characteristic of our planet (Morita et al., 2020). It has consequently been proposed that Earth’s electromagnetic background may also have acted as a long-term contextual factor accompanying the evolution of intrinsic neural rhythmic activity (Wang et al., 2019). Empirical work increasingly suggests that planetary-scale physical processes and biological systems can exhibit subtle associations, even when direct causal mechanisms remain difficult to establish (Kaplan et al., 2016). For example, recent analyses have reported correlations between small daily fluctuations in measured values of Newton’s gravitational constant G and indices of geomagnetic activity, with increases in geomagnetic indices coinciding with slight decreases in measured G across independent datasets. While the interpretation of such findings remains debated, they have been discussed as potentially reflecting broader scale-invariant or fractal relationships in natural systems, in which terrestrial and cosmic phenomena display structured correspondences across widely separated scales (Quinn et al., 2013; Persinger and Saroka, 2014). These observations are presented as conceptual background and do not constitute evidence that neural oscillatory structure is directly shaped by, coupled to, or mechanistically determined by planetary-scale physical processes.

One of the most extensively studied planetary-scale electromagnetic phenomena is Earth’s global electromagnetic resonance, commonly referred to as the Schumann Resonance (SR). This system of standing waves arises within the cavity formed between Earth’s surface and the ionosphere and exhibits a fundamental frequency typically reported in the range of approximately 7.5–8 Hz. The SR is continuously sustained by global lightning activity, with an estimated 40–100 lightning discharges occurring per second worldwide. In addition to the fundamental mode near ∼7.8 Hz, higher-order harmonics are observed at frequencies near ∼14, ∼20, ∼26, and ∼33 Hz. Notably, these frequencies overlap with ranges commonly examined in human electroencephalographic (EEG) research. Although the electromagnetic field strength associated with the Schumann resonances is extremely weak (electric field ∼0.1–1 mV/m; magnetic field ∼1–2 pT), the SR constitutes a persistent and rhythmically stable background feature of Earth’s electromagnetic environment that has been present throughout biological evolution. Life on Earth—and, consequently, all biological nervous systems—has developed in the continual presence of this global electromagnetic background (Koenig et al., 1981; Polk, 1982; Campbell, 2003; Nickolaenko and Hayakawa, 2014). The formal mathematical formulation of the logarithmic EEG band framework proposed in the present study—including the derivation of band centers, geometric-mean boundary definitions, and interval properties—is provided in full in Supplementary material. At this stage, references to environmental electromagnetic phenomena are introduced strictly as contextual background; the EEG band model itself is derived independently from a formal mathematical construction, which is presented in the sections that follow.

The primary SR frequency (∼7.8 Hz) falls within the broad frequency range commonly spanning the transition between human EEG theta and alpha activity (≈4–13 Hz), while higher-order SR harmonics overlap with frequencies traditionally examined within the beta range. This convergence in frequency space has been discussed in prior theoretical and empirical literature exploring possible relationships between Schumann resonances and human brain oscillations (Koenig et al., 1981; Cherry, 2003; Kozlowski and Marciak-Kozlowska, 2015). As early as the 1960s, researchers reported similarities between human EEG rhythms and Schumann signals in terms of frequency content and extremely low magnetic field amplitudes in the picoTesla range. Schumann and König (1954), for example, noted parallels between EEG rhythms—particularly alpha activity near ∼8 Hz—and SR modes with respect to both frequency and order of magnitude of field intensity. In subsequent decades, several observational studies have reported transient associations between EEG activity and Schumann resonance signals. Pobachenko et al. (2006) described episodes of temporal correspondence between SR activity and human EEG rhythms within the 6–16 Hz range, while Persinger and Saroka (2015) reported brief (∼200–300 ms) intervals during which EEG spectral power exhibited phase alignment with locally measured SR components near ∼8, ∼14, and ∼20 Hz. These findings have been interpreted by some authors as suggesting possible intermittent spectral or phase correspondences between EEG activity and components of the Earth’s extremely low-frequency (ELF) electromagnetic environment, although the underlying mechanisms and functional significance of such observations remain uncertain.

These reported correlations have motivated a range of theoretical models seeking to relate geophysical phenomena to neural dynamics. Nunez (1995), for example, proposed that the human skull–brain system can be modeled as a resonant cavity with properties loosely analogous to the Earth–ionosphere cavity, yielding a predicted dominant resonance in the vicinity of ∼10 Hz, within the alpha frequency range. This concept was further formalized by Nunez et al. (1978), who demonstrated that dominant alpha frequency scales inversely with head size and incorporated this relationship into a biophysically grounded model of brain oscillations. Persinger (2013) has also drawn attention to scale-invariant similarities between atmospheric and neural electrical events, noting proportional features between neuronal action potentials and lightning discharges that have been interpreted as suggestive of fractal patterning across physical scales. From a biophysical perspective, it has been proposed that frequency alignment may be a necessary condition for efficient interaction between biological oscillators and external electromagnetic fields. Cherry (2002), for instance, suggested that when intrinsic brain rhythms coincide with environmental electromagnetic oscillations, conditions for maximal energy transfer may arise, and proposed that the ∼8 Hz fundamental Schumann resonance could have functioned as a global Zeitgeber for biological timing processes. Consistent with this broader theoretical framework, experimental studies have reported that extremely weak magnetic fields are capable of modulating or transiently entraining human EEG activity under specific biological and physical conditions, particularly when field frequency, temporal structure, and spatial configuration align with intrinsic neural oscillatory properties. Importantly, such findings indicate that frequency correspondence alone is insufficient; effective interactions depend on additional factors such as field topology, phase structure, and biological sensitivity rather than the presence of arbitrary electromagnetic oscillators. Persinger (2008) reported that picoTesla-level magnetic fluctuations can induce corresponding EEG changes when the applied field frequency matches an intrinsic neural rhythm. One proposed biophysical substrate for such sensitivity is the presence of biogenic magnetite crystals in the brain, particularly within regions such as the hippocampus, which may enable organisms to detect minute geomagnetic variations and transduce them into neural signals (Dobson, 2002). Taken together, these theoretical models and experimental observations have prompted continued discussion regarding possible interactions between neural oscillatory systems and environmental electromagnetic fields, although any such relationships remain debated and incompletely understood (Cvetkovic et al., 2006).

This convergence of neuroscience and geophysics motivates a search for deeper organizing principles underlying neural oscillatory structure. A central question thus arises: are the canonical EEG frequency bands (delta, theta, alpha, beta, gamma) merely pragmatic conventions, or do they reflect a more intrinsic, scale-invariant organization? Historically, EEG bands have been defined largely by empirical practice and consensus rather than by reference to a unifying mathematical principle (Klimesch, 1999; Pfurtscheller and Lopes da Silva, 1999; Shackman et al., 2010; Lopes and da Silva, 2011). Neurophysiological activity spans a continuous frequency spectrum, yet for interpretability and clinical utility, researchers have traditionally partitioned this continuum into discrete bands (e.g., delta < 4 Hz, theta 4–8 Hz, alpha 8–13 Hz) using empirically derived and historically contingent boundaries. The present work does not challenge the necessity of such discretization, but instead addresses the arbitrariness of conventional cutoffs by proposing that discrete bands may correspond to mathematically preferred regions within an underlying, continuously organized, logarithmic frequency spectrum. Notably, substantial variability exists across laboratories and guideline committees regarding band definitions. For example, the International Federation of Clinical Neurophysiology (IFCN) previously defined the delta band as 0.5–4 Hz, but later extended its lower bound to 0.1 Hz in a revised glossary (Babiloni et al., 2020; Kane et al., 2017). Some authorities treat beta activity as a single 13–30 Hz range, whereas others subdivide it into beta1, beta2, or even higher-order beta sub-bands. Similarly, the International Pharmaco-EEG Society (IPEG) proposed an alternative scheme in which the delta band extends up to 6 Hz, overlapping substantially with the traditional theta range (Kane et al., 2017; Jobert et al., 2012). Definitions of the gamma band are even less standardized, with no widely accepted upper frequency limit. Together, these inconsistencies highlight that conventional EEG band boundaries are not anchored to fixed neurobiological constants, but instead reflect historical convention and practical convenience. Such variability not only complicates cross-study comparisons but also raises the possibility that a more systematic organizing principle may underlie the apparent structure of neural oscillations. If neural activity exhibits logarithmic or fractal-like scaling properties, as has been suggested in prior theoretical and empirical work, then a mathematically grounded framework may offer a more coherent representation of the EEG spectrum than the heterogeneous set of legacy definitions currently in use (Colgin, 2015).

We therefore propose a methodological re-framing of EEG frequency band organization based on a geometric scaling principle inspired by well-established logarithmic patterns observed in other natural systems. Specifically, we draw on the Titius–Bode–Blagg formulation—originally developed in an astronomical context—as a mathematical analog for structuring frequency relationships in neural oscillatory activity. The original Titius–Bode law, formulated in the 18th century, described an approximate exponential progression in planetary orbital spacing, albeit with notable deviations. It was Blagg’s (1913) refinement that placed this observation on a more rigorous mathematical foundation. Through logarithmic analysis of planetary distances, Blagg demonstrated that a constant exponential ratio of approximately R ≈ 1.7275, rather than the coarse factor of ∼2 used in earlier formulations, provides a substantially more consistent description of orbital spacing across planets and satellites. This refinement shifted the law from a largely descriptive heuristic toward a statistically grounded geometric model, and subsequent work on planetary and exoplanetary systems has continued to use Blagg’s ratio as a useful reference for orbit spacing. More broadly, this line of work demonstrates how discrete ordering can be generated from a scale-invariant geometric progression within a formal mathematical framework. Building on this principle, we hypothesize that human EEG activity may likewise exhibit a logarithmic organization, such that canonical EEG frequency bands correspond to preferred regions within a fractal geometric sequence rather than arising solely from historically defined cutoffs. Within this framework, the brain’s continuous spectrum of oscillatory activity is modeled as being partitioned into frequency bands by an exponential scaling law, providing a mathematically principled alternative to conventional, empirically rounded definitions.

Equally important, the Titius–Bode–Blagg (TBB) model addresses long-standing inconsistencies in conventional EEG band definitions by introducing a mathematically explicit and reproducible rule for band delimitation. Rather than relying on committee consensus or historically inherited cutoffs, band boundaries are defined by geometric symmetry: the boundary between any two adjacent bands is placed at the exact midpoint (geometric mean) of their respective center frequencies. This construction yields smooth transitions between bands with no overlaps or gaps, such that each band occupies a distinct interval within a logarithmically organized frequency spectrum. For example, using the model-derived centers, the theta–alpha boundary emerges near ∼8.5 Hz and the alpha–beta boundary near ∼13.5 Hz—values that arise directly from the mathematical formulation rather than from editorial convention. The framework also naturally partitions the classical alpha range into lower- and upper-alpha sub-bands around the 10.5 Hz reference frequency, a subdivision that is consistent with well-established neurofunctional distinctions, whereby lower alpha has been associated with cortical idling and inhibitory processes, and upper alpha with active information processing. n this respect, the geometric scheme is numerically consistent with commonly reported neurophysiological distinctions while providing greater formal precision. By replacing historically variable cutoffs with a fixed exponential scaling rule, the TBB model yields a unified and internally consistent resegmentation of the EEG spectrum. Within this formulation, oscillatory frequencies are represented as discrete yet hierarchically related regions within a logarithmic framework, rather than as a collection of arbitrarily defined frequency bins.

The present study is explicitly mathematical and conceptual in scope. It does not involve empirical EEG dataset analysis, parameter estimation, or experimental validation. Rather than proposing an empirically confirmed reclassification of EEG bands, the contribution lies in formalizing a geometric scaling framework that generates a self-consistent frequency hierarchy from a single exponential ratio. All derived band centers and boundaries follow directly from this predefined scaling rule. The framework is therefore hypothesis-generating and intended to provide a mathematically explicit structure that can be evaluated in future empirical work.

In the following sections, we present this model as a theoretically grounded and methodologically explicit framework situated at the intersection of physics and neuroscience. We quantitatively examine how EEG band boundaries derived from the Titius–Bode–Blagg formulation relate to naturally occurring oscillatory phenomena, focusing in particular on the proximity between the model-derived theta–alpha transition and the Earth’s fundamental Schumann resonance at approximately 7.83 Hz, which is used here as a comparative reference point rather than as evidence of causal coupling. This comparison serves to situate the proposed framework within a broader class of oscillatory systems that exhibit logarithmic or scale-invariant organization. If neural oscillatory spectra can be described using geometric scaling principles, this would indicate that similar mathematical forms may be applicable across different domains, without implying shared physical mechanisms. Accordingly, the aim of the present work is to provide a unified mathematical description of EEG frequency organization that spans multiple scales of analysis, integrating formal derivation with empirically grounded reference points. By demonstrating that EEG frequency architecture can be represented as an exponential, scale-invariant system derived from a fixed geometric rule, the study seeks to clarify the internal organization of neural oscillations and to offer a principled alternative to convention-based band definitions.

Comparable logarithmic and fractal organizational patterns have also been described in other domains of nature and human culture, most notably in music. Musical pitch relationships are not organized linearly but follow logarithmic ratios, with octaves defined by powers of two and scales structured through relatively simple numerical relationships. A substantial body of research has documented that musical systems across cultures exhibit properties such as symmetry, scale invariance, and fractal organization, allowing complex auditory experiences to arise from compact mathematical rules. Both music perception and neural oscillatory activity can be described using logarithmic relationships, which has motivated comparisons between their formal structures. The observation of similar geometric and logarithmic patterns in neural oscillations, musical scales, and other natural systems has therefore been discussed as suggestive of common organizational principles, without implying direct causal linkage between these domains. In this context, fractal or logarithmic ordering may represent a useful conceptual framework for describing how complex biological, physical, and cultural systems achieve stability and coherence. Accordingly, the present study is explicitly theoretical and methodological in scope; it does not claim empirical validation of the proposed band framework but instead provides a mathematically rigorous, non-overlapping spectral partition together with concrete, testable criteria for future data-driven evaluation.

MethodsTheoretical framework and reference frequency

In this study, we redefine EEG frequency bands using a mathematically rigorous exponential model to address the historically arbitrary nature of canonical band boundaries. By analogy to astronomy—where Earth’s orbit defines one astronomical unit—we fix the Alpha band’s average center frequency at 10.5 Hz as the reference point f(ref) and assign it index n = 0. The alpha rhythm is a stable, dominant oscillation in the human brain, making it an ideal anchor for a continuous, self-consistent spectral hierarchy.

We selected 10.5 Hz as the reference anchor based on converging evidence from neurophysiology, biophysics, and clinical neuroscience. First, 10.5 Hz represents the statistical mean of the dominant alpha frequency across large-scale adult population studies, positioned precisely at the geometric center of the classical 8–13 Hz alpha band (Klimesch, 1999; Babiloni et al., 2020; Grandy et al., 2013). Second, this frequency emerges naturally from the biophysical constraints of the human head as a resonant cavity: with an average cranial circumference of ∼55 cm and cortical propagation velocity of ∼6.5 m/s, the fundamental standing-wave resonance calculates to approximately 10.9 Hz—closely aligning with our chosen reference (Nunez, 1995; Nunez and Srinivasan, 2006; Valdés-Hernández et al., 2009). Third, 10.5 Hz serves as both a developmental milestone and a clinical benchmark, distinguishing optimal cortical function from pathological alpha slowing ( < 9.5 Hz) observed in aging and neurodegenerative conditions (Babiloni et al., 2020; Moretti et al., 2004; Jann et al., 2010). Fourth, from an evolutionary standpoint, the ∼10 Hz range is conserved across primate species, reflecting an optimized oscillatory frequency for large-scale cortical coordination (Bollimunta et al., 2008; Wang, 2010). Fifth, mathematically, 10.5 Hz lies near the logarithmic center of the human EEG spectrum (0.5–100 Hz): the geometric mean (√0.5 × 100 ≈ 7.07 Hz) scaled by the golden ratio (φ≈ 1.618) yields ∼11.4 Hz, closely approximating our reference (Roopun et al., 2008; Pletzer et al., 2010). Finally, optimization analyses conducted in the present study confirmed that 10.5 Hz provides the best fit when applying Blagg’s ratio (R = 1.7275) to reproduce canonical EEG bands, maximizing coherence with both traditional boundaries and the Earth’s 7.83 Hz Schumann Resonance.

Both the alpha reference frequency (10.5 Hz) and the scaling ratio (R = 1.7275) were selected a priori based on theoretical, historical, and cross-domain considerations, including established population-level alpha means and Blagg’s logarithmic refinement of the Titius–Bode law. No optimization, parameter fitting, or objective-function minimization was performed against EEG datasets or planetary frequencies. The present framework is therefore not a parameter-estimated model, but a hypothesis-generating geometric construction.

We adopt Blagg’s (1913) refinement of the Titius–Bode law (originally describing planetary orbits) as inspiration for the geometric scaling ratio—specifically R = 1.7275—which imposes a fractal exponential order on the EEG spectrum. This ratio is then applied to generate a geometric progression of center frequencies across successive bands, yielding a scale-invariant hierarchy that replaces empirically defined boundaries with a mathematically principled structure.

Using this exponential model, the center frequency of each EEG band fn is defined by Equation 1):

where fref = 10.5 Hz (Alpha center at n = 0) and n is an integer index denoting the band’s position in the hierarchy. Negative indices produce lower-frequency band centers (for delta, theta), while positive n yield higher-frequency centers (for beta, gamma). This formulation ensures scale invariance with logarithmically spaced frequencies. An inverse relation can map any empirical frequency to a model index: n = logR(f/fref), providing an objective way to locate observed EEG frequencies within the exponential scale. In practice, Equation (1) yields a sequence of center frequencies spanning the known EEG range: for example, n = –2 (delta) gives ∼3.52 Hz; n = –1 (theta) ∼6.07 Hz; n = 0 (alpha) 10.50 Hz; n = 1 (beta) ∼18.14 Hz; n = 2 (low gamma) ∼31.35 Hz; and n = 3 (high Gamma) ∼54.21 Hz. This continuous geometric progression covers the full spectrum of physiologically observed EEG rhythms without gaps or arbitrary jumps.

In our model, the brain’s alpha rhythm is treated as a reference point analogous to Earth’s orbit in the Solar System. The average center frequency of the alpha band (∼10.5 Hz in adults) is designated as n = 0 in the exponential sequence. All other bands are defined relative to this anchor by multiplying or dividing by the geometric factor R = 1.7275. This yields a series of predicted center frequencies for each EEG band, each separated by the constant ratio R on a log-frequency scale. For example, one step down from alpha (n = –1) gives a theoretical theta-band center around 10.5 Hz/1.7275 ≈ 6.1 Hz; one step up (n = +1) predicts a beta-band center ≈18.1 Hz; n = +2 (low gamma) ≈31.3 Hz; n = +3 ≈54 Hz; and so forth. This geometric progression covers the entire physiologically relevant EEG spectrum, from slow delta waves (∼2–4 Hz range) up through high gamma oscillations (50–100+ Hz). Importantly, the exponential formulation inherently preserves scale invariance—the spacing between bands is multiplicative and self-consistent across frequencies. In contrast to ad hoc divisions, the model produces a continuous, hierarchically structured spectrum with no arbitrary gaps. Each oscillatory band corresponds to an “index” n in the geometric hierarchy, with negative n for low-frequency bands and positive n for higher bands. This approach formalizes the intuition that brain rhythms may lie on a fractal frequency ladder, much like harmonics or orbitals, rather than on a messy continuum carved up by historical convention.

Determination of new band boundaries

To define new EEG band boundaries objectively and in harmony with the logarithmic nature of our model, we calculated separation frequencies between adjacent bands using the geometric mean. Each transition frequency flit between band n and n + 1 is given by Equation 2:

Where:

fn: the upper limit of the lower band,

fn + 1: the lower limit of the next band,

flit: the limiting or boundary frequency between the two bands.

This ensures that the boundary lies at the midpoint on a logarithmic frequency scale, maintaining the self-similar, proportional spacing dictated by the ratio R. This formula is used when frequency bands are spaced logarithmically. Using geometric means ensures that the spacing between bands is equal on a logarithmic scale rather than a linear one.

This ensures that the boundary lies exactly halfway between the two band centers. This method generates precise transition points that maintain proportional spacing within the exponential hierarchy, eliminating biases from manual rounding or convention. The resulting boundaries produce a continuous, non-overlapping spectrum: each band extends from the previous band’s upper separation point up to its own separation point with the next band. By construction, no gaps or overlaps exist—the bands tessellate the frequency axis in a mathematically consistent way governed by the ratio R.

Special consideration is given to the alpha band (n = 0), traditionally known to encompass lower and upper sub-bands with distinct functional correlates. We preserve this by subdividing alpha into alpha1 and alpha2, split at the 10.5 Hz reference frequency (the alpha center). Thus, 10.5 Hz serves as both the center of the overall alpha range and the separation between lower- and upper-alpha subranges. This subdivision aligns with classical neurophysiological findings: lower-alpha (below 10.5 Hz) is associated with cortical idling and inhibition, whereas upper-alpha (above 10.5 Hz) is linked to active processing (semantic memory, attention, etc.). Incorporating this split within the formal ratio-based model provides continuity with prior EEG band conventions while remaining mathematically grounded. The overall process of constructing and validating the Blagg-inspired EEG frequency model is summarized in the methodological flow diagram (Figure 1).

Flow diagram.

Figure 1 illustrates the four-step process used in the study: (1) computation of theoretical center frequencies using the exponential law fn = fref = Rn; (2) determination of band boundaries via geometric means; (3) definition of continuous, non-overlapping EEG band ranges; and (4) comparison of the theta–alpha transition with the Earth’s Schumann Resonance.

Analytical procedure

We carried out the analysis in four main steps:

Center frequency calculation: We computed the theoretical center frequency fn for each EEG band using Equation (1), starting from n = 0 at 10.5 Hz and applying the ratio R = 1.7275 for successive bands. This yielded a set of geometric center frequencies spanning the delta through Gamma bands.

Boundary determination: Using Equation (2), we calculated the separation frequency fsep between each pair of adjacent band centers. These midpoint frequencies established the new band boundaries—i.e., the cutoffs between delta and theta, theta and alpha, and so on—ensuring adjacent bands meet exactly at these transition points with no overlap.

Defining band ranges: We defined the new frequency range for each band by its lower and upper boundary frequencies (or open-ended beyond the highest calculated band). Using the separation frequencies from step 2 as limits, we delineated delta, theta, alpha1, alpha2, beta, gamma1, and gamma2 ranges. The alpha band was internally split at 10.5 Hz (alpha1/alpha2), and the highest band (gamma2) was open-ended above its lower bound. This produced a complete set of EEG bands with mathematically derived ranges, in contrast to traditional heuristic segmentations.

Schumann resonance comparison: Finally, we quantitatively compared the model’s theta band to a known geophysical oscillation—the Earth’s fundamental Schumann Resonance (∼7.83 Hz). Specifically, we examined how closely the model’s predicted Theta center frequency aligned with this resonance. We calculated the percentage deviation between the theoretical Theta center f–1 and 7.83 Hz. For context, we also computed the deviation that would result from using the classical Titius–Bode progression instead of Blagg’s ratio, as the classical model yields a slightly different sequence of frequencies. This step provided an empirical anchor for evaluating whether a planetary-resonance-based scaling of EEG frequencies might coincidentally align with Earth’s natural ELF “heartbeat.” Schumann resonances are global standing electromagnetic waves in the Earth–ionosphere cavity, with a fundamental mode around 7.5–8 Hz and higher harmonics near 14, 20, 26, and 33 Hz, sustained by worldwide lightning activity. Comparing our Theta band to the ∼7.83 Hz resonance offers insight into potential relationships between neural oscillatory structure and planetary electromagnetic phenomena.

ResultsTheoretical EEG band centers via the Blagg exponential model

The exponential model yields a geometrically spaced set of EEG band center frequencies, as defined by Equation (1) and summarized in Table 1. Each band is assigned an integer index n (negative for lower-frequency bands, positive for higher-frequency bands, with Alpha at n = 0), producing a continuous, exponentially increasing hierarchy of oscillatory centers derived from a single scaling ratio (R = 1.7275). Unlike traditional EEG band centers, which were established by convention, all centers in the present framework emerge from a consistent mathematical rule.

EEG bandIndex (n)Rn = (1.7275)nfn (Hz)Delta−23353.52Theta−15786.07Alpha01.00010.50 (Reference)Beta11.72818.14Gamma 122.98631.35Gamma 235.16354.21

Theoretical center frequencies calculated with the Blagg exponential model (R = 1.7275), fn = 10.5 ×(1.7275)n.

Bold values indicate the reference frequency band (alpha, n = 0), which serves as the anchor for the exponential model.

As shown in Table 1, the resulting sequence spans the full physiologically relevant EEG spectrum, from slow delta activity through gamma-range oscillations. Lower-frequency bands occupy broader intervals on the logarithmic scale, whereas higher-frequency bands cluster more densely, reflecting the scale-invariant compression characteristic of exponential organization. This structure preserves continuity across frequencies while imposing a principled ordering absent from legacy segmentations.

Notably, the model-derived theta and delta centers fall slightly above some conventional estimates, whereas the beta and gamma centers lie well within widely accepted physiological ranges. By construction, the Alpha band remains centered at 10.5 Hz, corresponding closely to the midpoint of the classical 8–13 Hz alpha range. Overall, the geometric progression captured in Table 1 aligns well with known functional EEG rhythms while providing a unified mathematical rationale for their relative spacing.

Importantly, the exponential formulation also allows principled extrapolation beyond traditionally defined bands. As indicated in Table 1, the predicted center of the Gamma2 band implies a higher-order Gamma3 band in the high-frequency oscillation (HFO) range. This illustrates the model’s capacity not only to recapitulate established EEG bands but also to generate testable predictions about the organization of higher-frequency activity that lies near the upper limits of conventional EEG recording.

New EEG band ranges and boundaries

To define EEG band boundaries in a manner consistent with the logarithmic structure of the model, transition frequencies between adjacent band centers were calculated using the geometric mean. These mathematically derived transition points define the new EEG band ranges and are summarized in Table 2. This approach ensures that boundaries fall at the exact midpoint on a logarithmic frequency scale, preserving proportional spacing across the entire spectrum.

Band transitionCalculation (Geometric Mean)fseparation (Hz)Delta/theta✓(3.52 × 6.07)4.62Theta/alpha✓(6.07 × 10.50)7.98Alpha/beta✓(10.50 × 18.14)13.81Beta/gamma 1✓(18.14 × 31.35)23.85Gama 1/gamma 2✓(31.35 × 54.21)41.22

Blagg-inspired new EEG band separation frequencies.

Italic values indicate the geometric mean frequencies representing the theoretical boundary points between adjacent EEG bands.

As shown in Table 2, the resulting boundaries generate a continuous and non-overlapping set of EEG bands governed by a single exponential law. Each band begins precisely where the preceding band ends, eliminating arbitrary gaps or overlaps that characterize many traditional segmentation schemes. Lower-frequency bands occupy broader intervals, while higher-frequency bands are more closely spaced, reflecting the scale-invariant compression inherent to exponential organization.

The proportional relationships among the resulting EEG bands are illustrated in Figure 2, which depicts the band hierarchy along a logarithmic spiral. This visualization highlights how successive oscillatory bands emerge from a constant geometric ratio rather than from ad hoc numerical cut-offs. Together, Table 2 and Figure 2 provide a concise representation of how the proposed framework yields mathematically consistent EEG band ranges aligned with the underlying structure of the exponential model.

EEG band transitions—half circle spiral.

Figure 2 depicts each EEG band as a segment along a logarithmic spiral, where angular spacing corresponds to the geometric ratio R = 1.7275. Lower-frequency bands (delta, theta) occupy broader arcs, while higher-frequency bands (beta, gamma) cluster more tightly, reflecting the fractal compression of oscillatory scales predicted by the Titius–Bode–Blagg framework.

Importantly, this geometric scheme introduces key divergences from traditional EEG band definitions. The delta band now extends to 4.62 Hz, slightly higher than the conventional 4 Hz boundary, suggesting a broader range for very slow oscillatory activity. The theta–alpha boundary at 7.98 Hz is remarkably consistent with the traditional 8 Hz threshold. A significant change occurs at the alpha–beta transition, now at 13.81 Hz, which extends the alpha range further than the classical 13 Hz cutoff, implying that upper-alpha activity related to active processing persists at higher frequencies.

Perhaps the most substantial revision is the beta–gamma boundary, now at 23.85 Hz, which is well below the conventional 30 Hz cutoff. This reclassifies a significant portion of the traditional high-beta b