ENRIQUE A.RAMÍREZ .Z
abtracct
Music, in its deepest root, does not arise from sound but from memory. We propose that musical intervals are perceptual manifestations of a memory architecture structured by mathematical sequences—the four fundamental sequences that give rise to music: Fibonacci (phi), powers of 2 (binary), the golden ratio (φ = 1.618 – 1.5 = 0.118, which generates the phi/binary duality). These sequences produce a quantized scale based on exact fractions (1/2, 1/4, 1/8) of each tonic, where the semitone (1/16) acts as a coding unit of 30°. In this resonant and angular architecture, each note is not isolated but accumulates memory, and when the C of the upper octave recalls its “evolutionary” history, it generates the intervals of that octave: +1/2 (fifth), +1/4 (third), +1/8 (second). Thinking is singing, singing is remembering. Just as twin particles 32 and 32 remain connected due to a shared origin (64), harmonic notes preserve this evolutionary memory—non-local.
Four sequences that give rise to music
Music results from four fundamental sequences:
- Fibonacci: 1, 1, 2, 3, 5, 8… In this series4, each “note” stores “memory” as the sum of the two preceding values.
- Binary (2ⁿ): This sequence5 generates the octaves, forming a structure of repetition where each higher octave is the sum of the two previous ones: 1, 2, 4, 8, 16, 32, 64...
- Fifths: The number 3 arises as 1+2, like in Fibonacci, but it also doubles in octaves as in the binary pattern: 3, 6, 12, 24, 48, 96…
- Adjusted fourths: These originate from 360/φ³ ≈ 85, and grow by binary multiplication: 170, 340, 480...
These four interrelated and intertwined sequences give rise to the quantized scale as a codified form of temporal memory.
1. The quantized scale: integer ratios as structural units.
Starting from C = 64 Hz, the fundamental intervals are generated by proportional additions to that base frequency:
- Fifth: +1/2 → G = 96 Hz
- Third: +1/4 → E = 80 Hz
- Second: +1/8 → D = 72 Hz
- Semitone: +1/16 → C♯ = 68 Hz
The semitone becomes the “tile” with which intervals are tessellated. This fractal structure replaces the logarithmic division of the tempered scale with a rational architecture based on 12 equal parts of the octave (each being the 12th root of 2):
64 x21/12= 68,8 C#
64 x22/12=71,8 D, etc
2. The harmonic circle: a clock of 12 semitones
Each of its “hours” measures 30 degrees:
12 × 30° = 360°
Each musical interval corresponds to an angular measure:
- Fifth: 210° (7 × 30°)
- Fourth: 150° (5 × 30°)
- Difference between them: 60° (2 × 30°)
3. A unifying code.
The language used to name the distances between frequencies is ambiguous and lacks universal standardization. We propose unifying this nomenclature using a functional scheme based on the human body: let us call the tonic the heart, the fourth the genitals, and the fifth the head.
Figure 1. If C is the heart, then 90° back is its “female heart,” a. If the tonic were D♯, its relative (90°) would be c.
In this 12-position circle, when C is placeded at 12 o’clock, its fifth is G (Sol) at 7 o’clock (210°). There is always a “relative” tone 90° behind it, at (a)—9 o’clock for C. That V-shape, with its “heart” at a, will have its 210° (head) on e and its 150° (female genitals) on d.
These two perpendicular Vs determine the harmonic notes for each such pair of “Vs.” If one of them is major (masculine) — C–F–G — its mirror pair (three hours earlier) — a–e–d — will be minor (feminine).
From this, the intervals that emerge from C are:
- Second: d (female genitals)Third: e (female head)
- Fourth: F (male genitals)
- Fifth: G (male head)5
A melody thus becomes what it often is: an expression of human sensations, chiche are triggered when these points are “touched” — generating these intervals. Almost all songs in a major key use only the dominant V (masculine); some invoke certain points of the subdominant V (feminine).
3. A spiral that rotates 137.7° at each time and arranges the sunflower’s seeds6
- 360/φ ≈ 222.5° (phi fifth)
- 360/φ² ≈ 137.5° (natural spiral, phi fourth)
- Their difference: 85° = 360/φ³
- We can do 360/φ4.....φ5 φ6 ....
Now we can observe that:
150° − 137.5° = 12.5° and 222.5° − 210° = 12.5°.
This reveals a 12.5° structural adjustment that transforms the phi-based fourth and fifth into their binary counterparts (150° and 210°, respectively).
We refer to this relationship as the phi/binary duality, and this duality expresses a functional correspondence between both systems.
The phi fourth (137.5°) awakens its complementary phi fifth (222.5°), and together they resonate with the traditional musical pair:
- 150° = fourth
- 210° = fifth
In music, the fourth and the fifth are interchangeable depending on the frame of reference.
For instance, C at 64 Hz reads G as a fifth, whereas C at 128 Hz reads G as a fourth.
Figure 5. Between 210° and 150°, there is a 60° difference—equivalent to two semitones of 30°. From F to G in the figure or from C to d.
Music, like sex, consists in measuring these distances—these intervals—modulating them, and resolving them into the fullness of the 360°, the tonal center.
4. Resolving the gap of the phi-adjusted fourth through fifths is, in essence, the act of making music.
The Pythagorean fifth is defined as the binary structure dependent on 2ⁿ. That fifth corresponds to a ratio of 3/2, or 1.5. If we take 64 as the unit (1), then 96 is 1.5.
According to the phi/binary duality, subtracting the binary value (1.5) from φ ≈ 1.618 — interpreted as the biological constant — yields a structural discrepancy of 0.118. Likewise, subtracting 0.118 from 1 yields 0.882.
If we multiply the fifth (e.g., 96 Hz) by this factor, we obtain the adjusted fourth:
- 96 × 0.882 = 84.672 Hz
Alternatively: - 96 × 0.118 = 11.328 → 96 − 11.328 = 84.672 Hz
This provides another method for obtaining fourths, which also appear through the expression φ² × 25= 83.77 Hz.
In a practical and functional way, the adjusted fourth, "descending" of 137.5°, acts as a biological impulse that awakens the fifth (360/φ ≈ 222.5°), and together with that 360/φ² ≈ 137.5°, produces the difference (222.5° − 137.5° = 85°). A delta that defines the analog biological mold at 360/φ³ ≈ 85°. And also: (φ – binary) ≈ (85°- 60°)= 12.5+12.5). Equivalent to
30°-5°=
semitone -5°
(5°/0.118=(360°/Phi³)/2.
This allows the interval of 150° (the binary fourth) to open the harmonic space toward the interval of 210° (the binary fifth). However, both 85° and 137.5°, as well as (360/φ³)/2—which is also a fourth (F ≈ 42.37 Hz)—function as vacuum attractors, structural spaces that drive harmonic movement within a binary trajectory that must remain proportional to the biological structure derived from φ. In making music, the harmonic void of the fourth is resolved through fifths. Due to this symmetry between fourths and fifths, the brain predicts what comes next in a sequence. Even when a melody is complex, it will always seek support in this pair (qubit).
1 / φ³ ≈2 x 0.118
(1 / φ³)/2=0,118
φ³ = 2x2 + 2 (0.118),
(φ³/2=2+0,118
In this framework, 0.118 emerges as another void or minimal structural space that aligns phi-based intervals with their binary counterparts as 12.5°+12.5° do.
In this sense, the fourth — descending from 137.5° — acts as a biological drive that awakens the fifth (360/φ ≈ 222.5°), and together with 360/φ² ≈ 137.5°, they produce the difference (222.5° − 137.5° = 85°), which defines the biological mold analogous to 360/φ³ ≈ 85°. And besides (φ - bynary) ≈ (85°- 60°= 12,5+12,5).
This allows the 150° interval (the binary fourth) to open the harmonic space toward the 210° interval (the binary fifth). However, both: 85° and 137.5° function as vacuum attractors — structural gaps that drive harmonic motion — within a binary path that must remain proportional to the phi-derived biological structure.
Notably, 1/ϕ3≈2 x 0,118, and φ3=2x2+2(0,118), in this framework, 0.118 emerges as other vacuum or minimal structural gap that aligns phi-based intervals with their binary counterparts as 12,5°+12,5° do it.
and besides:
(85° − 60°) = 12.5 + 12.5. Equivalent to 30° − 5° = semitone − 5°.
(5° / 0.118 = 42.37 ≈ (360°/φ³)/2).
360° × 0.118 = 42.48° ≈ 42.37°.
But also, 5 Hz or 5° separate F from F#. It seems consistent to say that 85° = 85 Hz, but F# at 90 Hz coincides with 90°, just as F# at 180 Hz, 360 Hz, or 720 Hz coincides with angular quadrants proportional to frequency in Hz. Thus, this 5 appears to be the coupling or phase shift between binary quadrants and φ, introducing the geometry of phi. (72 × 5 = 360).
Figure 6. In a pentagon with side = 1, its diagonals measure φ, and in the inverted pentagon all its sides measure 1/φ².
The nested pentagons measures: 1/φ4 ,1/φ6... generating a self-similar rhythm: Pentagon interior (side = 1/φ2→ diagonal = 1/φ≈0.6181. next =
1/φ4 → diagonal = 1/φ3≈0.236.
So if the side length is 1/φ2n the diagonal length is=1/φ2(n−1)
-
one from the even indices (sides),
-
one from the odd indices (diagonals).
5. Pythagoras theorem and phi/binary duality confirmation.
The phi–binary duality becomes explicit in right triangles with legs 1 and 2, where the hypotenuse is √5. Remarkably, √5 can be written as 2 + 2×0.118, revealing the binary component ‘2’ corrected by the golden residual 0.118. Moreover, the golden ratio itself emerges: the larger leg (2) is in golden proportion to the sum of the smaller leg (1) and the hypotenuse (√5).” (1+2,236)/2=1,618 φ.
6. Perfect squares, perfect cubes, and 0.118
The binary sequence 2ⁿ generates values such as 4, 16, 64, and 4096, which resemble φ² as perfect squares—just like 12 × 12 = 144. Notably, 144 is the only Fibonacci number (aside from 1 × 1) that is also a perfect square.
Similarly, φ³ acts as a perfect cube, just like 8, 64, 512 (8³), or 4096 (16³). In this context, the binary structure also attains harmony, musical or even sexual, because frecuencies are the code of the body, and they do it when compared to its mirror: 2 × 2, 4 × 4, 8 × 8 like when a body is in front of its mirror (fig. 1).
However, it still lacks a small factor — n × 0.118, just as 1.5 falls short of φ ≈ 1.618 — to fully align with biological structure (binary/φ dulity).
There is other coincidences like the B (Si) at 120 Hz which represents 0.118 encoded 1000 times, plus a residual of φ/100 ≈ 0.016
Figure 7. Music creates symmetric binary structures, while biology creates fractals ones. The product N × 0.118 brings them into alignment.
Base 2 x2 square x3 cubic
2 4 8
4 16 64
8 64 512
16 256 4096
32 1024 32768
64 4096 262144
Table 1. Powers of 2 and Their Square–Cube Expansions.
7. Fifths as octaves and their alternative origin from phi.
The sequence of fifths can be interpreted using the rule of adding the memory of its past + 1/2 to generate the next fifth:
96 + 48 = 144 Hz (D₂)
Analogously:
96 + 1/4 = 96 + 24 = 120 Hz (B)
96 + 1/8 = 96 + 12 = 108 Hz (A), and
- 108 Hz × 4 = 432 Hz (A₄) distinct from the commonly used 440 Hz, which may represent an “approximation by ear” to this more mathematically symmetrical quantized system.
This 120 Hz B (7th Interval, absent in Figure 1) can also be obtained by summing:
64 + 1/2 (32) + 1/4 (16) + 1/8 (8) = 120 Hz
This suggests that the seventh (B) tone 7th functions as a collector of all previous octaves.
However, it still lacks the final 1/16 step (4 Hz) or a value that most closely approximates the minimal harmonic vacuum that remains before reaching the next octave (128 Hz). Its subtle absence marks a threshold of transition, reinforcing the idea that the quantized scale encodes memory, where each note—or its void—contains a record of its harmonic ancestry. Once again, this confirms that memory is not simply a sequential accumulation of frequencies (I) with their sensations (II) and associated interpretations (III).
Figura 8. Cone of sensorial interpretation.
Once again, this confirms that memory is not merely a repository, but rather a structural code (Table 3), and memory itself is a function of self-consciousness—or perhaps self-consciousness could even arise from this frequency-based architecture.
Table 2. The same intervals emerges by subtracting fractions from the upper C. For example, D (72 Hz) is obtained as 128–(32+16+8), and D# (68 Hz) as 128–(32+16+8+4).
8. Another way to derive fifths (G) from phi
If we take the binary secuencie 2n: 1, 2, 4, 8, 16 ..(x) and subtract from each x its value divided by φ³ ( x / φ³), we obtain a sequence of G notes (Sol): 6, 12, 24, 48, 96… (approximately adding +0.11 × 2ⁿ).
This structural constant 0.118 continues to reappear in connection with phi.
X 2n Fx=x/ φ3 (x-Fx)
8 1,8885 6,111
16 3,7777 12,223
32 7,5554 24,446
64 15,109 48,841
128 30,2171 97,782
Table 3. Division of binary sequence by x/ϕ3: and x-Fx=Gs
x/ φ3 /≈ 360/ φ3 =85° ≈ (222,5-137,5) ≈ the vacuum of fourth that generate fifths. But: 0,11,0,22,0,44 is binary+ the structural constant 0.118, that repeatedly appears in connection with φ, acting as a harmonic unit that links binary and phi-based intervals. Each step in this sequence exhibits a consistent ratio ½ (8/16) or 2/1 (16/8), between the transformed value and its original, reinforcing the fractal and proportional nature of the system.
That A (La) at 432 Hz can also arise from: (2⋅2⋅2)2 x (3⋅3⋅3)⋅⇒432= cube of 2×2 x cubo de 3. Or 2 x cube of 3 x cube of 2. Like 2(23, 83 or 163) riching the superior octave,like the figure 2.
It may seem forced, but it helps illustrate how cubes (2×2×2) can reflect through a mirror (×2), and how that reflection generates another echo: 3×3×3 or ϕ3 evolving toward 2ϕ3
Figure 8. Like a mirror or a cube of mirrors, this 2ϕ3is a cubic mirrored structure of the 2n sequence enables self-awareness.
That same A at 432 Hz in cubics expresions represents not just a tuning reference, but a convergence of fractal, harmonic, and geometric structure. That same 432 Hz A also emerges if we subtract Mi (80 Hz, two octaves below) from high 512−80=432.
This implies that in the quantized scale, many high notes find their support in lower tones, just as the body does: shoulders carry the feet, and the shoulders support the jaw, which in turn supports notes from the throat—while the ear, by listening (×2), stabilizes them through copying and reflection.
This kind of support also explains how a fifth can be found by:
- Adding the lower Do to the base one:
64 + 32 = 96 - Or by subtracting two-octave-lower Do (32) from high Do (128)
→ again yielding 96.
96 = 128 – 32.
Tabla 3 Mirror scale: descending encoding from the igher C.
https://sensoterapia.org/sonidos/escala cuantizada.m4a
From Do₂ = 128 Hz, each note can be derived as a subtraction of proportional values from the lower Do:
- G (Sol): 128 − 32 = 96 (1/4)
- E (Mi): 128 − (32 + 16) = 80
- D (Re): 128 − (32 + 16 + 8) = 72
- C♯ (Do♯): 128 − (32 + 16 + 8 + 4) = 68
This demonstrates that each higher note contains structural memory, and can be read as a summary of its past.
Each step is a cumulative descending reading. The semitone is the condensation of all previous levels.
9. Training and Vibrational Reversibility
Figure 9. Two was to read the frecuencies, from up or from down.
By training, the self at C₁ (64 Hz) builds upward through addition (+1/2, or +1/4, or +1/8..
Later, the self at C₂ (128 Hz) can already read its history through subtraction.
The same applies from 256, 512, 1024, or 4096 Hz—the system becomes commandable.
10. Fractality on Fibonacci and 2n
The binary sequence 2n reproduces a self-similar pattern in which 4096 contains scaled reflections of the previous tones.
1 4096
2 2048
4 1024
8 512
16 256
32 128
64 64
128 32
256 16
512 8
1024 4
2048 2
4096 1
Table 4. Binary sequence 2ⁿand its inverse mirror.
Specifically:
- 4096 reads to 2048 2 times.
- 1024 4 times,
- 512 8 times
- 64 exactly 64 times,
- and the value 1 exactly 4096 times.
In this sequence, it is notable that 8 cubed equals 512, and 4096 is 64 squared,
In the binary sequence 2n each frequency—such as 4096—contains scaled reflections of its predecessors. In 4096 + ½ or +1/4 … , each one of these values forms its own intervals — for example, 4096 + 1/16 is its semitone. 4096 +256. This is fractal7 because each tone creats, in this same way, yours own intervals.
In the binary sequence 2ⁿ, we observe that 4096 scans its preceding frequencies as a mirror sequence of itself. At the top, 4096 Hz can summon harmonics like a general commanding his soldiers.
Likewise, in the Fibonacci sequence, the number 144 acts as a structural reader of its own memory:
- Dividing 144 by each prior term generates a progression converging toward powers of ϕn
- The inverse ratios x/144 converge toward 1/ϕn: 89/144=1/ ϕ,
- 55/144=1/ϕ2, 34/144=1/ϕ3…
The φ-based scaling factor (both φⁿ and 1/φⁿ) acts as a geometric attractor of resonance, akin to the binary self-similarity and fractality of 2ⁿ. Both binary and Fibonacci frameworks, memory is not additive but proportional.
The golden ratio ϕ acts as a geometric attractor, guiding the resonance across scales.
Each tone or term remembers its past, preserving the identity of the whole within every part.
11. Conclusion: The Binary–Phi Duality as Harmonic Memory Architecture
The binary sequence (2ⁿ) and the phi-based sequence (φⁿ) represent two complementary structures: one based on exact doubling and the other on proportional growth. Binary nodes generate their own harmonic intervals (1/2, 1/4, 1/8…), forming self-similar resonant structures. Phi introduces a minimal structural delay (≈0.118), allowing evolution without loss of coherence.
Together, they define a dual system where binary ensures identity, and phi introduces memory and transformation. This duality encodes not just music, but time, resonance, and self-awareness8.
1.Ramírez E. Enrique A. “placer o dolor”funcionalidad y manejo del cuerpo energético. Publicaciones Maticez Medellín. 19988.
2.Ramírez E. Enrique A. “Todos somos uno”, con una sola enfermedad. Publicaciones Maticez Medellín. 1998.
3.Penrose, R. (1994). Shadows of the Mind. Oxford University Press.
4.Livio, M. (2003). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. Broadway Books.
5.Canción que es su propia partitura. In preparation.
- From phi to alfa: angular quantization and musical structure of the universe. Enrique A. Raamírez. https://orcid.org/0009-0005-4995-8599
- Mandelbrot Benoit. The fractal geometry of nature. San Francisco: W. H. Freeman. 1982.
9 Briggs John y Peat David. Espejo y reflejo: «Del caos al orden». Gedisa Editorial. Barcelona 1994.
10 Sheldrake Rupert. La presencia del pasado. Editorial Kairos. Barcelona. 1990.
- Nogier Paul C. De l’Auriculothérapie à l’Auriculomédecine. Maisonneuve. 1981.
12 Guyton C. Artur. Tratado de fisiología médica. Interamericana. 1977.
13 Javier Romañach. Algo pasa con PHI https://www.youtube.com/watch?v=r1OJj8LWb5I