PROJECT 02 · MATHEMATICAL MUSIC
Development of the Myerthall–Owens Theorem
A General Theory Developed Using Flowers in the Rain
September 2, 2026
1 / INTRODUCTION
The paper develops two related constructions for proposing salient locations in a piece: an exponential sequence based on the golden ratio, and trigonometric candidates built from a duration-adjusted oscillation. It then combines them, applies a separate circle-based index filter, and compares the retained candidates with score changes read by the author.
The interactive figures below make those steps inspectable. They show how the formulas generate positions and how sensitive the reported agreement is to the selected duration, filter, and error rule. Numerical resemblance motivates the model; the score annotations provide exploratory comparisons rather than proof of a general law.
2 / OWENS’ THEORY
The first move is to adjust a piece’s bar count by a factor built from the golden ratio and a nearby unit-circle angle. Repeated division by ϕ then proposes progressively earlier musical landmarks.
Start with ϕ = (1 + √5) / 2. Rewriting ϕ as a/b gives a₁ = 1 + √5 and b₁ = 2. Rewriting it once more gives ϕ = (3 + √5) / (1 + √5), so a₂ = 3 + √5 and b₂ = 1 + √5.
The construction compares a₁ and a₂ with angles spaced every 30°. Move the selector to see the distance from each candidate.
At θc = 5π/3, a₂ differs by about 0.000080 radians. This is the paper’s closest match on its chosen π/6 grid.
2.1 / TRUE DURATION OF THE PIECE
The adjustment is almost neutral in length; choosing the factor above 1 is a modelling convention, not a meaningful change to the score duration.
2.2 / EXPONENTIALLY OCCURRING SALIENT POINTS
These values reappear because the same golden-ratio construction defines the sequence. The paper presents them as consistency checks, not independent evidence.
The paper points to melody changes near bars 8, 14, and 36, and a major-to-minor shift around bar 23. Those are the author’s score readings used to interpret the predicted positions.
See all reported score readings and matches ↓2.4 / MEASURED MOVEMENT-LENGTH CORRELATIONS
The paper links its scale choices to a golden triangle, a unit-circle projection, and the angles of a pentagon. Adjust the measurements to see where the numerical coincidences hold.
The three movements are 59, 95, and 95 bars. In an ideal 36° / 72° / 72° triangle, the sine rule gives b/a = sin 72° / sin 36° = 2 cos 36° = ϕ.
Measured angles: 36.182° / 71.909° / 71.909°. The ideal triangle is 36° / 72° / 72°.
A pentagon divides a turn into 72° sectors. Bisecting gives 36°, where twice the cosine equals ϕ; halving again gives the reciprocal identity.
At exactly 36°, x = cos 36° ≈ 0.809 and y = sin 36° ≈ 0.588; x² + y² = 1. The diameter is 2, and 2 sin 18° = 1/ϕ ≈ 0.618034.
3 / MYERTHALL’S THEORY
Owens’ divisions identify only a few points. Myerthall starts from a second near-match and asks whether sine and cosine can supply more candidate positions between them.
ϕ ≈ 1.618 radians and π/2 ≈ 1.571 radians, a gap of about 0.04724. Since sine reaches its maximum at π/2, their sine values differ by only about 0.00112.
3.1 / TRUE DURATION
Here the scale is about 103.007% of the notated length. As with tO, this is an explicit construction choice.
The first attempt uses one full cycle across the adjusted duration. Intersections, maxima, and minima become candidate landmarks.
3.2.2 / REFINEMENT WITH THE THREE-GAP THEOREM
Wrap an irrational step around the unit circle. The points keep arriving in a new order, while the spaces between neighbours settle into at most three lengths.
The model adds a separate filter. It retains candidate index m when {m/ϕ} ≤ ½. That semicircle cutoff is a modelling choice, not a consequence of the Three-Gap Theorem.
THE SAME ROTATION IN PITCH SPACE
Take C = 256 Hz and a just-intonation perfect fifth G = 384 Hz, so ρ = 3/2. Repeating the interval multiplies by ρ; taking log₂ turns that multiplication into a repeated step, then the fractional part removes complete octaves.
The factor 2 is one complete octave. Removing it leaves the just-intonation whole tone 9/8, at circle position {log₂(9/8)} ≈ 0.16993.
An octave has log₂(2) = 1 and returns to the same pitch-class point; an equal-tempered semitone has step 1/12 and repeats after 12 steps. The exact 3/2 fifth has an irrational log₂ step, so its circle rotation never closes exactly.
4 / UNIFIED THEORY
The unified function multiplies Owens’ exponential envelope by Myerthall’s trigonometric oscillation. Its zeros and stationary points are ordered first, then filtered by the same circle-index rule.
Each cₘ is the next positive zero or stationary point after sorting both families together. Only cₘ ≤ L belongs to the score. Since tM > L, c₁₆ = tM is already outside the domain; the paper’s retained set through c₁₅ is {2, 4, 5, 7, 10, 12, 13, 15}. The damping uses absolute bar position, so the candidates are recalculated for each piece length rather than copied as fixed percentages. The circle theorem does not imply the separate filter {m/ϕ} ≤ ½.
UNIFIED FUNCTIONExponential decay scales a sine wave. Its positive zeros and stationary points form the ordered candidate sequence cₘ.
f(n) = tO/2 · ϕ⁻ⁿ · sin(8πn/tM)PAPER VALUES / SELECTED RULE
Signed offset = chosen integer candidate − reported score boundary. The paper reports the exact offsets below.
| INDEX m | MODEL POSITION cₘ | SCORE BOUNDARY r | TABLE OFFSET d | CONTINUOUS e = cₘ − r | WITHIN ±1? |
|---|---|---|---|---|---|
| Loading the paper’s reported values… | |||||
Two residuals, two readings. The earlier tables choose either floor(cₘ) or ceil(cₘ) according to which integer is closer to r, then report d = chosen integer − r. Section 5.2 deliberately drops that integer step and evaluates e = cₘ − r directly. Use the selector above to see how the totals change.
The paper’s stated headline uses the continuous residual at a one-bar tolerance.
The outcomes come from only four pieces, and candidates within a piece may not be independent. The paper treats this interval as descriptive, not a precise estimate of performance on music generally.
5 / THE MYERTHALL–OWENS THEOREM IN PRACTICE
The paper applies the retained-candidate rule to four pieces ranging from 59 to 264 bars. The three additional works were chosen for recognition and varied duration. Its final precision calculation compares each continuous point directly with its matched score boundary.
5.1 / APPLICATIONS
At ±1 bar, the change in comparison rule shifts the apparent precision. Section 5.2 explicitly discards the earlier integer choice for its aggregate calculation.
The model carries m = {2, 4, 5, 7, 10, 12, 13, 15} across every score. Continuous residuals produce these one-bar results:
For 18 true positives in 32 candidates, the paper retains a 95% Wilson score interval of 39.3% to 71.8%. The considered Wald interval was 39.1% to 73.4%; the paper uses Wilson for the small sample.
The paper cautions that the 32 candidate results come from only four pieces and may not be independent within a piece. It treats the interval as descriptive, not a general performance estimate.
The paper includes detailed score excerpts for these two works. For Rondo alla Turca and Ballade No. 1 it reports candidate tables but omits corresponding score-excerpt verification; the page keeps those cases visible in the interactive table without inventing musical annotations.
6 / CONCLUSION
The equations construct predictions. The score annotations are exploratory observations. The gap between those things is where future testing belongs.
Continuous bar positions from a defined function. The retained Myerthall–Owens candidates are zeros and stationary points whose index passes the semicircle filter.
Score changes identified by the author: phrase shifts, changes in melody, key, tempo, dynamics, or mood. Boundary placement is interpretive, and the matching procedure can assign nearby predictions to the same boundary.
With continuous residuals and a one-bar tolerance, 18 of 32 retained candidates match: 56.3%. The earlier rounded-offset tables produce a different, more permissive count.
Four examples cannot establish a universal musical law. The paper calls for broader, more varied testing and treats mismatches as evidence to investigate.
James Myerthall, Mathematical Coincidences in Musical Structure: Development of the Myerthall–Owens Theorem, 2 September 2026. Concept originator: Ryaed Owens. The paper’s appendices include Owens’ whiteboard demonstration and score material for the four examples.
RETURN TO MODEL ↑Appendices A–E, original paper pages 34–47.