Anchors and Prime Lattice Coordinates

The Concept of a Local Anchor

In Prime Period Theory, a period space is a continuous bounded space mapped to a specific perceptual phenomenon (e.g., a pitch octave, a rhythmic bar). To navigate this space meaningfully using the prime lattice, we require reference points. These reference points are local anchors.

A local anchor serves as the Base for a local subperiod — the terminal, unlabelled origin of that subperiod’s own fractal descent. An anchor’s own coordinate needs no explicit digit: termination of a path at length zero is the Base declaration. (The same applies following a neighbour-frame edge re-basing; path length zero at the new anchor is simply its Base). From an anchor, the comma system navigates outward via fractal descent to locate any micro-position.

Anchors as a 12-Interval Even Grid (Base-Mode)

The twelve solfège anchors below are defined as exactly evenly-spaced divisions of the period. This provides a versatile, domain-agnostic grid: in pitch, it precisely yields 12TET (100 cent increments); in rhythm, it creates a pure 1/12th snapping grid.

Because they divide the period into equal rational fractions, these anchors are navigated entirely in Base-mode. Each position can be reached exactly using a Prime Lattice Path of fractal descent. For example, dividing the space into 1/12ths requires splitting by 2, then by 3, and then taking a step by 2 again (e.g. [0/2, 0/3, +1/2] for Ra).

This represents a conceptual shift: the anchors are the even scaffolding of the space itself. Commas and diacritics are then used to measure outward from these fixed grid lines to locate exact microtonal or Just Intonation (JI) positions.

Reduction convention: symmetric around Do

Do’s local period space is bounded on both sides by its neighbouring anchors, and Fi sits at its Boundary (Axis) — the shared edge between Do’s space and its neighbour’s, at exactly half the period. This forces every other anchor’s coordinate to be resolved by nearest-address reduction, (−600¢, +600¢] around Do, not by ascending reduction across the full [0, 1200¢) octave. An anchor whose position exceeds 600¢ has a shorter distance to Do going the other way around the period, and that shorter distance is its correct address.

Concretely, this means five of the twelve traditional ascending-solfège anchors — So, Le, La, Te, Ti — sit below Do in this coordinate system, not above it. This is a real, intended consequence of treating Fi as a true boundary rather than a convenience marker at the top of an ascending scale: the conventional ascending octave (Do up to Ti) is actually anchored starting from So — the octave “begins” a fifth below Do and Do sits inside it, not at its root. Traditional ascending pedagogical order is a register convention layered on top of this structure; it is not the structure itself. The values below describe position relative to Do; how that maps to a specific octave of absolute pitch is a separate, deliberate convention (illustrated for Do = C4 below), not a mathematical necessity.

The 12 Anchors

The table below specifies each anchor as an exact Base-mode path. Cents are precisely 12TET (100¢ increments).

Characteristics of the Map

1. Nearest-address symmetry, not ascending order. Every non-Do anchor resolves to whichever direction gives the shorter path — this is what produces the So–Ti-below-Do result, and it is the direct consequence of taking Fi’s role as Boundary literally rather than as a top-of-scale marker.
2. Perfect Mirroring. Because the grid is evenly spaced, the Base-mode paths on the negative side are exact inversions of the positive side. So is the direct negative reflection of Fa, La reflects Me, and so on.
3. Fi’s dual address is structural, not an oversight. Fi sits at exactly ±600¢ — equidistant from Do in both directions, the one point in this table where nearest-address reduction does not force a unique answer. Convention resolves Fi’s register to the positive spelling (F#4) rather than the negative one (F#3) — consistent with Axis conventionally being read as this anchor’s own boundary — but the negative spelling [-1/2] is not wrong, merely unconventional.
4. Base-mode Navigation. The Prime Lattice Path shown is the actual, exact location of each anchor. There is no residual comma and no approximation here — this is a mathematically perfect subdivision of the period space.

Solfège frames and the diacritic space

Dividing the canonical resolution constant N = 27,720 (see Prime Lattice) into twelve equal Solfège frames — the evenly spaced divisions of the octave described above — gives exactly:

27,720 / 12 = 2,310 = 2 × 3 × 5 × 7 × 11

This is a direct consequence of 27,720’s factorization, not a coincidence requiring separate justification. 2,310 is the radical of 27,720 — the product of its distinct prime factors, each to the first power — because 27,720 = 2³ × 3² × 5 × 7 × 11 needs exactly one extra factor of 2 (beyond the first power, to cover divisibility by 8) and one extra factor of 3 (beyond the first power, to cover divisibility by 9). That excess is 2² × 3 = 12 exactly, and dividing by it strips the excess and leaves the radical.

The consequence for the Prime Diacritics system: each of the twelve Solfège frames has a local resolution of exactly 2,310 points, precisely enough to give an exact Base-mode address to any squarefree (first-power-only) 11-limit adjustment entirely within that one frame — a diacritic combining ±1 steps of 2, 3, 5, 7, and 11 — without needing to borrow resolution from a neighbouring frame. This gives Prime Diacritics a clean, principled local budget rather than an arbitrary fixed precision.

This does not extend to every comma of interest. Adjustments requiring a prime to a second power or higher — the syntonic comma (81/80 = 3⁴/(5·2⁴)), the Pythagorean comma (3¹²/2¹⁹) — need more depth in a single prime than the local 2,310-point budget carries, and correspondingly draw on the “excess” 12-fold structure that separates 27,720 from its radical — i.e., they reach outside a single Solfège frame. This is the same distinction already drawn in Prime Lattice: squarefree, single-frame adjustments are what the local diacritic space is for; the classic higher-power commas are a cross-frame phenomenon, consistent with their being a cross-route (not single-target) fact about the lattice.

Pure Ratios and Cast()

While this 12-interval Base-mode grid serves as the foundational scaffolding for Prime Period Theory, certain applications may specifically require representing the anchors as exact, pure Just Intonation (JI) ratios (e.g. 4:3, 3:4, 5:4).

When an exact JI ratio is required as an anchor, the position is no longer a rational fraction of the period (Base-mode), but rather a logarithmic one (Reel-mode). In this case, one can define the anchor by wrapping a Prime Lattice step in the Cast() function (which translates a Reel-mode position back into linear space for a multiplicative operation).

For example, a true JI Fa (4:3) can be reached exactly via Cast(+1/3), and its reciprocal So (3:4) via Cast(-1/3).

However, for general Prime Period Theory applications, the even 12TET Base-mode scaffold provides a universally compatible grid from which all exact comma refinements can subsequently be measured.

This table is the exact bridge between the continuous period space and the discrete 12-anchor writing system of Uniform Solfège. Prime lattice paths and their diacritic renderings are a separate, additional layer: refinements measured outward from these fixed anchors. Absolute register (which octave a syllable sounds in for a given Do) is a separate convention layered on top of this structure, illustrated above for Do = C4 but not fixed by the coordinates themselves.

See also

• Period — the general model this page’s local-anchor concept is a pitch-domain instance of
• Prime Lattice — the comma-sequence path system that navigates and refines position relative to these anchors, and why it cannot exactly reproduce them
• Prime Period Diacritics — Overview — the writing system rendering comma-sequence refinements from these anchors
• Just Intonation — the tuning theory context for the ratios in this table