This illustration uses an SVG container to show three squares—X, Y, and Z—whose sizes are related by simple measurement ratios. All squares are centered in the same container to make their relative sizes easy to compare.
The SVG container represents the reference space. Within it:
Here Z, X, and Y are treated as pure values in octave space: a constant, order‑0 encoding with no explicit time or geometric extent. Each value is a single coordinate on a linear axis of doubling and halving.
The three symbols are mapped to integers that describe their octave shift:
As order‑0 constants, these values summarize a whole “complex” of possible measurements into a single scalar: the average position of all sensitive dimensions collapsed into one octave index.
The following strip shows the three octave values as rounded boxes, emphasizing their symmetric placement around the reference:
Even though we speak of a “tree”, with only three values the structure is linear: there are no branches, just a chain of positions along the octave axis. The SVG below encodes this order‑0 tree, where each node is a distinct value, similar to basis vectors like [1, 0, 0], [0, 1, 0], [0, 0, 1] in a trivial identity matrix.
Once a single octave value is understood, we can scope it by repeating the symbol. A three‑digit code like “ZZZ” means “take Z at radius R = 3”: the value is pushed to its own extreme within a three‑step window. Similarly, “XXX” and “YYY” mark the neutral and upper extremes in the same radius.
In this sense, R = 3 behaves like a small, symmetric horizon around the origin: a finite shell that is still topologically coherent with an underlying infinite, linearly ordered space. The simple codes Z, X, Y act like basis directions; their repeated forms (ZZZ, XXX, YYY) hint at how higher‑order structures can be built, much like identity matrices and Hilbert‑style constructions extend trivial coordinates into richer, potentially infinite‑dimensional spaces.
The octave values Z = −1, X = 0, and Y = 1 can be paired to form a two‑dimensional, order‑0 complex digit system. Each digit is a pair of octave positions, forming a 3×3 grid of equally scaled, equally precise values. This creates a hologram‑like pointer: a compact, two‑dimensional value whose internal phase acts as a local exponent.
Each axis uses the same symmetric set {Z, X, Y} mapped to {0, 1, 2}. Pairing them produces nine possible combinations:
These pairs behave like complex digits: two components, equal scope, equal scale, and no branching. The system remains order‑0 because each pair is still a single point, but the two‑dimensionality introduces a local exponent through the phase between the components.
Mapping Z→0, X→1, Y→2 turns each pair into a base‑3 number. Converting to decimal yields nine distinct values, assigned to digits 1–9:
All digits share identical resolution: each is a pair of octave values, each axis has three states, and the grid is uniform. This symmetry ensures equal precision, scope, and scale across the entire system.
Although the digits are two‑dimensional, they remain order‑0 because they do not branch or extend in time or space. The exponent‑like behavior arises from the relation between the two axes: a phase shift that acts locally, influencing the value without creating higher‑order growth. The number itself becomes a measure of internal scope and branching potential.
The following SVG shows the nine paired values arranged in their natural base‑3 lattice:
Multi‑digit codes such as “ZZZ”, “XXX”, or “YYY” extend the same symmetry into radius R = 3. These sequences behave like small, finite horizons that remain topologically coherent with infinite linear spaces. The base‑3 complex digits form the foundation for such higher‑order constructions, preserving identity‑matrix‑like triviality while allowing dimensional extension.
The four symbols I, O, A, E form a two‑bit system: one bit distinguishes I from O and A from E, and another bit distinguishes the two groups (IO vs AE). Adding lowercase forms i, o, a, e introduces a third bit, dividing the set into two parallel layers. This creates an order‑2 digit system whose internal structure is measured across several bit‑levels.
The first two bits define the classical four‑value system:
These two bits create a stable, symmetric 2×2 structure. Each symbol is a point in this binary square, and the system is still order‑0: no branching, no time, no spatial extension.
Introducing lowercase symbols creates a second layer:
This expands the system to eight values:
The result is a 2×2×2 cube of values: an order‑2 digit, because its internal structure is measured across three bits, not one. Yet it remains order‑0 in the sense that each symbol is still a single, indivisible point.
The Z‑X‑Y scale (−1, 0, 1) is not a dimension of truth values but a frequential axis. It behaves like seasons: a three‑step cycle that repeats across local octaves. This is an order‑1 structure because it measures how a value changes across octave‑mapped symmetries.
In continuous space, this corresponds to extremely slow exponential drift: a number whose growth rate is tied to its last infinitesimal value. When slowed to the limit, this drift becomes octave‑like. Thus, the discrete Z‑X‑Y axis mirrors a smooth, continuous frequency line.
The discrete system restarts at each digit length. This allows linear approximation of higher‑order spaces: each digit is a small step, and the sequence of digits accumulates precision. The continuous system, by contrast, never restarts; it flows smoothly.
Yet both systems converge when the symbolic sequence becomes long enough. After enough steps, the discrete digit‑space can pinpoint a value “in infinity”: the limit of an infinite series of symbolic moments.
The following SVG shows the eight values arranged as a cube of bits x, y, z:
The order‑2 digit cube is discrete, but its Z‑X‑Y scaling is continuous in spirit. The discrete digits act as integral steps; the octave‑like drift acts as a differential flow. Together they form a hybrid system: symbolic precision built from small steps, and smooth frequency encoded in the transitions.
This duality allows the symbolic system to approximate continuous growth, quadratic curves, or exponential behavior, depending on how the digits accumulate. The structure remains identical at each digit length, but the magnitude changes, giving the system its power to represent both finite and infinite values.
When octave = 2 and radius R = 2, the first‑order function collapses into a binary axis: O = −1 and A = +1. This is the classical False/True polarity. The earlier Z‑X‑Y values (−1, 0, 1) now become 0‑order constants that can be encoded as three‑bit binary strings. These strings represent the same information density as the previous order‑2 system, but one degree lower.
At octave = 2, the system reduces to a pure binary axis. The two values are:
This binary axis is the first‑order projection of the richer Z‑X‑Y structure. It is the minimal “frequency” representation: a single octave split into two halves.
The 0‑order constants Z, X, and Y can be represented using three binary digits. Each digit is either A (1) or O (0). These three bits correspond to the three‑valued structure of the original Z‑X‑Y axis:
The mapping is not arbitrary: each bit corresponds to a structural component of the three‑valued system. The final bit (rightmost) determines whether the value “steps up” into the next octave. Thus OOA (Y) is one power higher than OAO (X), which is higher than A00 (Z).
A three‑valued system cannot be represented by a single binary bit. Instead, it requires three bits to encode the same amount of information. This is why the Z‑X‑Y axis, although 0‑order, is informationally denser than the binary O/A axis.
The three bits act like a compressed hologram: each bit contributes a part of the structure, and the full three‑bit string reconstructs the value’s position in the octave.
Any value expressible in information space is one integral level (one octave) more dense than the raw three‑value representation. The three‑bit encoding is therefore the “next octave up” from the Z‑X‑Y constants. But compared to the earlier eight‑value cube (I/O/A/E with lowercase), this is one degree lower: the cube was order‑2, while this is order‑1.
The system scales smoothly: each increase in order adds a new dimension of measurement, but the internal structure remains digit‑wise identical. This allows discrete systems to approximate continuous ones. When numbers restart at each digit length, the system becomes locally linear even in higher spaces.
The following SVG shows the three values mapped into their binary strings:
In continuous space, growth based on the last infinitesimal value becomes octave‑like when slowed to the limit. This creates a smooth rival to the discrete system. But because the discrete digits restart at each length, they approximate the continuous line with surprising accuracy. Each digit is a small step; the full sequence is a path through symbolic space.
After enough steps, the symbolic sequence can pinpoint a value “in infinity”: the limit of an infinite series of discrete moments. This is how discrete octaves and continuous exponentials converge.
Musical and energetic states can be encoded using three binary bits based on letters O and A, representing values -1 and 1. Each bit corresponds to a distinct plane of existence:
| Bit | Plane | Interpretation |
|---|---|---|
| Y | Spiritual / Activation | Engagement of higher consciousness |
| X | Mental / Frequency | Oscillation or wave pattern in mental plane |
| Z | Physical / Octave | Octave or material manifestation |
Each bit can take the value O (-1) or A (1). Triplets define the full state:
| Sequence | Y | X | Z |
|---|---|---|---|
| OOO | -1 | -1 | -1 |
| OOA | -1 | -1 | 1 |
| OAO | -1 | 1 | -1 |
| OAA | -1 | 1 | 1 |
| AOO | 1 | -1 | -1 |
| AOA | 1 | -1 | 1 |
| AAO | 1 | 1 | -1 |
| AAA | 1 | 1 | 1 |
Each bit has a fixed weight when computing a static value:
| Bit | Weight |
|---|---|
| Y | 4 |
| X | 2 |
| Z | 1 |
Value = Y × 4 + X × 2 + Z × 1
Octaves are measured multiplicatively. Each bit corresponds to a multiplication level:
| Bit | Octave Multiplication |
|---|---|
| Y | 3rd multiplication |
| X | 2nd multiplication |
| Z | 1st multiplication |
Examples:
| Sequence | Octave Value |
|---|---|
| OOA | 2 × 2 = 4 |
| OAA | (2 × 2) × (2 × 2) = 16 |
| AAA | ((2 × 2) × (2 × 2)) × ((2 × 2) × (2 × 2)) = 256 |
Octaves can be represented in a linearized 1:2 sequence, treating O as 0 and A as 1. Each triplet maps to a distinct octave slice:
| Sequence | Linear Index | Octave Value |
|---|---|---|
| OOO | 0 | 1 |
| OOA | 1 | 2 |
| OAO | 2 | 4 |
| OAA | 3 | 8 |
| AOO | 4 | 16 |
| AOA | 5 | 32 |
| AAO | 6 | 64 |
| AAA | 7 | 128 |
This shows how the three-bit encoding generates octave values in a **binary-linear fashion**.
Binary encoding of O and A across three planes allows a formalized model of wave dynamics, octave structure, and activation within a multi-level framework. Each bit represents a plane: spiritual (Y), mental (X), and physical/octave (Z).
The three planes exhibit distinct behaviors in terms of their influence on functions, accelerations, and linearity:
f(f(n)), providing exponential modulation.Each bit can take O (-1) or A (1). Triplets define the state of the system across three planes. These states allow **wave inference and symmetry calculations**:
| Sequence | Y | X | Z |
|---|---|---|---|
| OOO | -1 | -1 | -1 |
| OOA | -1 | -1 | 1 |
| OAO | -1 | 1 | -1 |
| OAA | -1 | 1 | 1 |
| AOO | 1 | -1 | -1 |
| AOA | 1 | -1 | 1 |
| AAO | 1 | 1 | -1 |
| AAA | 1 | 1 | 1 |
Static weights can still be applied to describe **quantitative contributions of each plane**:
Octave values follow multiplicative scaling:
| Sequence | Octave Value |
|---|---|
| OOA | 2 × 2 = 4 |
| OAA | (2 × 2) × (2 × 2) = 16 |
| AAA | ((2 × 2) × (2 × 2)) × ((2 × 2) × (2 × 2)) = 256 |
Linearized octave indexing allows mapping of these triplets into **distinct, non-overlapping wave slices**.
Using the plane encoding, we can analyze **wave symmetry and bounds**:
Mathematically, a state (Y,X,Z) may induce transformations:
n → f_Y(f_X(f_Z(n)))
reflecting how planes combine while maintaining their symmetry properties.
After the symbolic constructions — octave constants, base-3 grids, binary encodings — the natural question is: where does this appear in daily life?
The Z-X-Y system can be read without mysticism. It describes three layers that operate continuously in ordinary modern activity.
| Symbol | Plane | Everyday meaning |
|---|---|---|
| Z (−1) | Material | Execution, hardware, typing, clicking, file generation |
| X (0) | Mental | Structure, models, abstraction, decision logic |
| Y (+1) | Directional | Purpose, scaling, coherence across time |
If one layer dominates completely, imbalance appears:
An octave shift does not require music or mysticism. It simply means doubling scale.
Each level preserves structure but increases magnitude. That is octave scaling.
The page earlier described discrete digits approximating continuous drift. This appears in modern life as:
Spiritual does not mean ceremonial. It means structural coherence across scales.
Eating food does not require blessing. But digestion still follows thermodynamics. Writing code does not require chanting. But architecture still follows invariants.
The Z-X-Y system describes invariants: balance between execution, structure, and direction.
In a technological era:
A system is stable when all three are synchronized.