You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.
You should upgrade or use an alternative browser.
Experimental Mathematics: Finding Number Patterns
- Thread starter Natus Videre
- Start date
In February 2024, a team of researchers from the University of Pennsylvania, Yale and Princeton in the U.S. published a study describing their efforts to analyze Bach’s music using network theory and information theory.
The authors included a wide range of Bach compositions in their study, including some preludes and fugues from the Well-Tempered Clavier suite, two- and three-part inventions, a selection of Bach’s cantatas, the English suites, the French suites, some chorales, the Brandenburg concertos, and various toccatas and concertos.
Overall, their results have confirmed that Bach’s works have a high information content, and further that different subsets of works have distinct characteristics.
Methodology
Here is an outline of their methodology: After collecting digitized versions of the above musical selections, they represented each note as a node in a network, with notes from different octaves as distinct nodes. A transition from note A to note B is represented as a directed edge from A to B. Chords are represented with edges between all notes in the first chord to all notes in the second chord. A graphical representation of this process is shown below. To the right of this illustration is the result of this process for four specific Bach compositions: (a) the chorale “Wir glauben all an einen Gott” (BWV 437); (b) Fugue 11 from the Well-Tempered Clavier (WTC), Book I (BWV 856); (c) Prelude 9 from the WTC, Book II (BWV 878); and (d) Toccata in G major for harpsichord (BWV 916).
Future directions
The researchers have clearly identified a very interesting and very effective technique to analyze, classify and compare musical compositions. Numerous questions for future research could be asked, some of which were suggested by the authors themselves in their concluding section. Here are a few of these potential research questions, including some due to the present author:
The authors found that Bach’s music networks had a higher number of transitive triangular clusters, enabling them to be learned more efficiently than arbitrary transition structures.
Hmm... triangular clusters... geometry, geometry everywhere!
- Are pieces with a larger number of these triangles also more appealing to a listener?
- How effective are these techniques in analyzing other classical composers?
- How has the information content of a specific composer changed over time?
- Within a single genre such as classical music, how has the information content of the music changed over time?
- How effective are these techniques in analyzing other genres of music, such as modern jazz, hip-hop and country?
- How do these results compare for various non-Western music genres, such traditional Japanese music (ongaku), Cantonese opera, African tribal music, Tibetan throat singing and Scottish piobaireachd?
- How do human perceptions of music correlate with these measures?
- Do these results offer any insights into the human psychology of musical experience, such as the fundamental question of why humans have evolved to perform and value music?
Q: How does one utilize the energies inherent in prime numbers in this respect? Do they represent frequencies or frequency relationships?
A: Verities.
Q: Is there any formula, or any thing about prime numbers that makes it easier to find them... anything about them that is unique?
A: Pyramidal.
Q: Pyramid relationships would help one find prime numbers?
A: Graph.
Interference: A Grand Scientific Musical Theory
The Social Interference Thesis (Appendix 2, page 388)
Principles of Harmonic Interference (Appendix 3, page 389)
A preliminary axiomatic system for harmonic models (Appendix 4, page 396)
Harmonic Interference Theory represents a major breakthrough in our understanding of music and perception. Triggered by a moment of insight thirty years earlier, this theory explains how harmonics combine to form coherent geometrical patterns that our auditory system recognizes as simple shapes.
Using a spectral analysis of harmonic interference over an octave, the author shows how reflective patterns on vibrated surfaces can be found in the growth patterns of the human anatomy, particularly our ears and brain. From this simple correspondence, perception of music is then explained as the natural process of anticipating and matching harmonic interference patterns against identical structures in our auditory system. When represented visually, music becomes organic geometries floating inside a harmonically structured space - exactly as our ears and brain understand it.
But this is only the beginning. The author goes much further to show how everything in nature can be described as crystallized harmonics. Drawing on the latest scientific research and cutting-edge theories in the fields of genetics, quantum physics and cosmology, a unified harmonic model is proposed for the study of coherence on both a micro and macro scale. Out of this emerges a grand scientific musical theory that reintegrates ancient harmonic science with quantum physics to explore the deeper mysteries neither can answer alone.
The Social Interference Thesis (Appendix 2, page 388)
Hypothesis 1: The tetrachord genera were a Pythagorean “mending function” for a musical octave, joining the Spiral of 5ths and octave cycle into a pentagram at two octave golden ratio proportions inside what is today known as the Tritone Function.
Hypothesis 2: The negative reputation of the tritone interval in music history is due to its association with the pentagram and its contained golden ratio, thought to reveal an error in nature and, thus, in mankind as “original sin.”
Hypothesis 3: The Medieval Catholic Church banned the tritone in the early 13th century due to its association with Pythagoreanism and other Hermetic/ Kabbalistic philosophies.
Hypothesis 4: The development of the 12-step octave and simplified system of major-minor diatonic scales resulted from the replacement of a Pythagorean pentagonal design with that of an equilateral triangular design.
Hypothesis 5: The Inquisition created a complicity of convenience in the 17th century between Western religion and science that resulted in the separation of harmonic science from natural science. This resulted in the formation of history, philosophy and music as a humanities track well insulated from the scientific method.
Hypothesis 6: Johannes Sebastian Bach was the leading proponent of the Tritone Function and popularized its use thereafter.
Hypothesis 7: A wholetone scale is a generalization of a tritone – thus, chromatic harmony can be seen as two oscillating wholetone scales derived from the generalization of the Tritone Function.
Hypothesis 8: The conventions of diatonic harmony based on major and (“relative”) minor scales are founded on the recognition of symmetry around a shared SuperTonic centered in the middle of the Tritone Function.
Hypothesis 9: Modern music theory is based on the rules and conventions of asymmetry inherited from the tritone avoidance laws of the Medieval Catholic Church.
Principles of Harmonic Interference (Appendix 3, page 389)
Principle 1: People interpret the pitch spectrum as a vertically geometric pitch space that is both circular and symmetric.
Principle 2: People interpret circularity in the frequency doubling at the octave.
Principle 3: People interpret tones in an interval having a tendency or tension to move up or down based on whether it is less than or greater than a half octave or tritone. The tritone itself is perceived as an ambiguous inflection point between opposing directions, producing what is popularly known as the Tritone Paradox.
Principle 4: People interpret movement between tones as motion between locations in pitch space, analogous to the perception of spatial location and motion of objects in visual space.
Principle 5: People interpret pitch space in hierarchical groupings that are recognized as auditory geometry. Furthermore, within this hierarchy exists a “half twist” reflective symmetry.
Principle 6: The distribution of wave interference in the harmonic series is described by a ratio between the square of the harmonic series and the Fibonacci series, otherwise referred to as the INTERFERENCE resonance function:
y = 1 / (Φ/√5), 4 / (Φ2/√5), 9 / (Φ3/√5), ..., n2 / (Φn/√5)
y = n2 / (Φn / √5) , n = {1..12}
Principle 7: The 12-step octave follows the natural distribution of harmonic interference that reaches an octave harmony at √144 = 12 and anti-harmonic center at √12.
Principle 8: Harmonic tension can be measured as a function of the REFLECTIVE INTERFERENCE pattern. Taking each of the amplitudes as a percentage of maximum resonance from the Leading Tones, we have the following order of tension (greatest to least) in a diatonic scale:
Diatonic Scale --- C Major Scale --- Percent
Leading Tone, Inverse Leading Tone --- {B, F} --- 100%
Tonic, Inverse Tonic --- {C, E} --- 97%
Harmonic Center --- {D} --- 72%
Dominant, Inverse Dominant --- {G, A} --- 66%
Principle 9: Interval consonance can be measured as a function of the INTEGRAL INTERFERENCE pattern. Taking each of the amplitudes as a percentage of maximum consonance from the octave, we have the following order of consonance (greatest to least):
Interval --- Inverse Harmonic Center --- Percent
Octave --- {G#, G#} --- 100%
P4, P5 --- {C#, D#} --- 54%
M3, m6 --- {C, E} --- 39%
m3, M6 --- {B, F} --- 34%
M2, m7 --- {A#, F#} --- 32%
m2, M7 --- {A, G}--- 31%
Tritone --- {D, G#} --- 31%
Principle 10: The Principle of Tritone Duality is the ability to perceive intervals as either consonant-dissonant or tense-resolved depending on context. During the recognition process, an interval can either be 1) measured spatially as an integral function to produce the sensation of consonance or 2) measured temporally as a differential function to produce the sensation of tension. The choice between the two is apparently determined by the degree of diatonic harmonic movement afforded in the context of the music.
Principle 11: The Fibonacci series converging to the golden ratio and its inverse ratio acts as natural Φ-damping proportions within the harmonic series to prevent the formation of destructive fractional wave partials.
Principle 12: The Landau damping principle in plasma waves provides a physical model for energy transfer between harmonic wave partials in a sonic standing wave. Our auditory system appears to judge interval consonance and dissonance based on the gain or loss of energy in corresponding harmonic partials. We also appear to recognize the directional energy flow between wave partials in the harmonic series as tension and resolution, creating a cognitive anticipation/reward potential for tones to move toward and across Φ-damping zones. The degrees of consonance and tension are represented as amplitude, or change in velocity, on the REFLECTIVE INTERFERENCE distribution model.
Principle 13: The Fibonacci series, converging to the golden ratio Φ, acts as a natural damping proportion within the harmonic interference pattern of an octave to prevent fractional wave partials from forming while enabling standing wave harmonics to resonate. Maximum resonance and damping locations within the harmonic series or octave may be estimated to four decimal places using these equations:
Max Resonance Ratio = Φ + (7 / 122) = 1.6666 ≈ major 6th = 5:3 ratio
Max Damping Ratio = (5 / 3) – (7 / 122) = 1.618 ≈ Φ ratio
The distance between these two extremes is equal to about 7:122, composed of the Philolaus octave comma
of (9:8)/27 = 6:122 plus an additional “free space” of 0.006966 ≈ 0.007 ≈ 1:122 of an octave.
Principle 14: Harmonic Partial 9, corresponding to the SuperTonic, is fully π-symmetric and Φ-damped relative to the fundamental (Tonic). This tone-to-octave relation is given the label of Harmonic Center as a special point of balance in the harmonic series.
Principle 15: The greater the wave symmetry in Φ-damping, particularly when weighted toward the out-of-phase cosine component, the greater is the perceived timbral dissonance. No damping alignment indicates maximum timbral consonance. In general, harmonics above the thirteenth partial are increasingly damped due to shorter wavelengths that bring them ever nearer to damping locations.
Principle 16: The Timbral Consonance Principle is the ranking of standing wave partials and their corresponding music intervals based on harmonic Φ-damping and Φ-alignment attributes. Following the order from non-damped to even to mostly odd-damped, we can rank intervals from most consonant to most dissonant:
Interval --- Harmonic Φ-damping Attribute
1.major 6th, minor 3rd --- Not Φ-damped
2.minor 6th, major 3rd --- Not Φ-damped
3.perfect 5th, perfect 4th --- Even Φ-damped
4.minor 7th --- Odd Φ-damped
5.major 2nd --- Odd/ Even Φ-damped
6.major 7th --- Near Odd/ Even Φ-damped and Odd Φ-symmetric
7.minor 2nd --- Near Odd/ Even Φ-damped and Odd Φ-symmetric
Principle 17: The greater the wave symmetry in π-alignment, particularly when weighted toward the in-phase sine component, the greater is the perceived harmonic resolution. In general, partials above the thirteenth partial are non-aligned, making them seem harmonically unresolved to the ear.
Principle 18: The Timbral Tension Principle is the ranking of standing wave partials and their corresponding diatonic music intervals based on π-alignment and π-symmetry about Partial 9 (the Harmonic Center or SuperTonic). Following the order of even to odd to symmetric alignment, the corresponding diatonic scale steps are ranked from most tense to most resolved:
Scale Step --- Harmonic π-symmetry Attribute
1.Leading Tone (major 7th) --- Even π-aligned
2.Dominant (perfect 5th) --- Odd π-aligned
3.Subdominant (perfect 4th) --- Odd π-aligned
4.Augmented 6th (minor 7th) --- Odd π-symmetric
5.Mediant (major 3rd) --- Odd π-symmetric
6.Submediant (major 6th) --- Odd / Even π-symmetric
7.SuperTonic (major 2nd) --- Odd / Even π-aligned and fully symmetric
8.Tonic (unison) --- Odd / Even π-aligned and fully symmetric
Principle 19: The Timbre/ Harmony Equivalence Principle holds that instrument timbre and music harmony are the exact same cognitive recognition process occurring at different levels in a hierarchy of harmonic interference. Intervals and chords simply amplify corresponding harmonic partials to strengthen the effect of the underlying harmonic interplay occurring in a standing wave of sound.
Principle 20: Within the calm Landau parameter space between neighboring amplitude and frequency Φ-damping locations, wave partials transfer energy as a phase/frequency modulation. This is perceived as an auditory sensation of temporal movement in the direction of energy flow. Within a standing wave and interference pattern of tonality, energy exchange produces an “anticipation/reward potential” in the progression of melodies, intervals and chords in music harmony.
Principle 21: The Harmonic Hierarchy is defined as an equivalence class of 5 identical levels of harmonic
interference that is generated from a single tone, aligning at different resolutions over pitch space:
TwelfthTone =2^(2/ 3456) = 2^(12^-3)2^(12^-2)/121.000401207
Tone =(TwelfthTone)^122^(12^-1)/121.004825126
Semitone =(Tone)^122^(12^0)/121.059463094
Octave =(Semitone)^122^(12^1)/122
TwelfthOctave =(Octave)^122^(12^2)/124096
which may be represented as the recursive exponential functions:
4096 = TwelfthOctave (Octave ( Semitone ( Tone ( 2^(2/3456) )^12 )^12 )^12 )^12
2^(2/3456) = TwelfthTone (Tone ( Semitone ( Octave ( 4096 )^(1/12) )^(1/12) )^(1/12) )^(1/12)
or as a finite power series of 2 beginning with n = -2:
f(n) = 2^(12^n)/12 , n={-2..2}
These five layers act as a harmonic projection screen from which the musical geometry of melody, intervals
and chords can emerge. Any property found at one level of this hierarchy will apply at all levels.
Principle 22: The equal temperament system, based on the multiplied semitone ratio of 2^(1/12), is a natural proportion within the recursive structure of the harmonic series generated by a standing wave. Therefore, contrary to any argument that equal temperament is man-made, it is the one true natural tuning.
Principle 23: The cognition of music harmony is defined by the proportional interference of wave resonance and Φ-damping in the natural harmonic series following the hierarchy of 2^(12^n)/12. Specifically, the proportions recognized in the standing wave interference pattern of a single tone are the same across the hierarchy of a semitone, octave and 12-octave frequency spectrum. In this way, the cognitive spatial and temporal qualities of timbre, harmony and spectra form a cognitive equivalence class.
Principle 24: The development of the 12-step octave originates in the natural recognition of the tritone interference pattern of Partials 5 and 7 against the fundamental.
Principle 25: The organizing property of the golden ratio is the central cognitive principle of music harmony.
Principle 26: The Fibonacci Series acts as a vortex-like temporal damping function at each level of the auditory hierarchy of 2^(12^n)/12. This is the cognitive “gravity” of music harmony.
Principle 27: The oscillating ratios of the Fibonacci series represent increasingly calm areas within an octave and semitone where energy may be exchanged between harmonic wave partials. Common practice music theory and preferred voice leading was a direct result of a natural cognitive awareness of this energy transfer.
Principle 28: Common practice use of the tritone and Tritone Function in music harmony follows the oscillating behavior of the Fibonacci Series as it temporally damps any harmonic standing wave. The universality of this principle in Western history suggests the human brain is itself organized like a standing wave.
Principle 29: Anticipation-reward potential in music harmony can be measured using Partial 5 as a “coherent pathway” through the interference pattern of the harmonic series. This pathway is hypothesized to be recognizable by the auditory system in two concurrent and opposing phase states of oscillation either side of the Harmonic Center. At the octave level of the interference pattern, the “cognitive cue” for which phase to recognize can be defined by which member of the Tritone Function is in play and its oscillation state. The phase indicators around the Harmonic Center are:
Diatonic Phase 1 = tritone = {up, down}
Diatonic Phase 2 = major 3rd = {down, up}
The auditory system may then measure and anticipate the potential direction of movement as an averaged direction in each half octave around the Harmonic Center, ideally using the Diatonic Phase indicators as cues. This principle applies over time within a memory context to predict melodic direction and overall musical momentum. In the most general form, anticipation-reward potential follows the oscillation and energy exchange in a standing wave.
Principle 30: Any temperament (tuning method) that provides exclusivity and cyclic closure in pitch space enables the recognition of harmonic shape to some degree against the proportions of the harmonic series.
Principle 31: Music cognition results from the pattern matching of auditory shapes against the same harmonic shapes evolved into the structure of the inner ear and auditory cortex. The degree to which auditory shapes can be recognized and predicted is defined by how closely the musical scale conforms to a harmonic standing wave, especially following the energy transfer across Φ-damping locations.
Principle 32: The REFLECTIVE INTERFERENCE structure of the eardrum and Fibonacci action of the basilar membrane of the inner ear is the essential coupling mechanism between the physics of sound and the Brodmann Area in the auditory cortex of the brain. We can predict from this that the Brodmann Area itself is also organized as a REFLECTIVE INTERFERENCE neural network. Principle 33: Holonomic Music Cognition is defined as a spatiotemporal coherence pattern matching operation as follows:
1. Sensory perception of music harmony begins as a neurological Fourier transform from spatial
frequency to spatial position and proportion.
2. Spatial coherence in sound is a cognitive pattern matching of harmonic proportions against fixed
proportions of the natural harmonic series within a range of tolerance.
3. Temporal coherence in sound is a cognitive pattern matching of melodies, intervals and chords
(following Partial 5 and the Fibonacci series as a coherent pathway) phase shifted and/or frequency
modulated against a fixed spatially coherent reference scale.
4. The proportions of the reference scale are instantly and economically recognized as neural pathways
(like Partial 5) in the auditory cortex.
Holonomic brain theory [Pribram (1991)] offers the best explanation for the cognitive functions required to
recognize the standing wave interference pattern produced by the natural harmonic series. From this, we
might also predict that the fundamental organizing principles of the brain will follow the INTERFERENCE
functions.
A preliminary axiomatic system for harmonic models (Appendix 4, page 396)
Axiom 1: Transposition Tn and Inversion In operations for orbits under Z/12Z can be defined as:
If n, m ∈ Z/12Z such that n is the pitch to be transposed and m is the transposition interval, then:
[Tn = (n + m) mod 12] [In = 12 – n]
Axiom 2: An initial set definition of affine orbits for cyclic ring Z/12Z can be defined as:
m2 / M7 Orbits:[Tn, In : n, m = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}]
M2 / m7 Orbits:[Tn, In : n, m = {0, 2, 4, 6, 8, 10}]
m3 / M6 Orbits:[Tn, In : n = {0, 3, 6, 9}, m = {0, 1, 2}]
M3 / m6 Orbits:[Tn, In : n = {0, 4, 8}, m = {0, 1, 2, 3}]
P5 / P4 Orbits:[Tn, In : m, n = {0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5}]
TT Orbits:[Tn, In : n = {0, 6}, m = {0, 1, 2, 3, 4, 5}]
Major Scale Orbits:[Tn : n = {0, 2, 4, 5, 6, 7, 11}, m = {0..11}]
Minor (Relative) Scale Orbits: [Tn : n = {0, 2, 3, 5, 7, 8, 10}, m = {0..11}]
Derivation of Axiom 3: oscillation of the Tritone Function
If n ∈ Z/12Z and z ∈ 2Z = {...,-6, -4, -2, 0, 2, 4, 6, ...}, we can define a generalized oscillation function ψ “Psi” between the dual wholetone scales over time using t ∈ N = {0, 1, 2, 3, ...}:
ψtz = [Tn, In : n ∈ {0, 4}, m ∈ Z] // Major 3rd = Even: 2Z
ψt+1z+1 = [Tn, In : n ∈ {-1, 5}, m ∈ Z] // Tritone = Odd: 2Z + 1
The Tritone Function can then be defined as an oscillating subset of ψ by pairing up members of the tritone and major 3rd sets over time under the same transposition m. This produces the axiom:
Axiom 3: The Tritone Function is defined by the harmonic oscillation of orbits ψ ∈{-1, 0, 4 5} taken from Z/12Z. This is represented by the dihedral relation {ψ 2z ψt+12z+1} as it occurs over a time t and between 2Z (even) and 2Z+1 (odd) cycles.
As example, a cadence of the Tritone Function for the C major scale could be specified like this:
{ψ00 = T-1}, {ψ00 = T5}
{ψ00 = (-1 + 0) mod 12, {ψ00 = (5 + 0) mod 12}
{ψ00 = -1}, {ψ00 = 5} // Tritone interval F – B at t = 0
{ψ 11 = T0}, {ψ11 = T4}
{ψ11 = (0 + 0) mod 12}, {ψ11 = (4 + 0) mod 12}
{ψ11 = 0}, {ψ11 = 4} // Major 3rd interval C – E at t = 1
With respect to time T, we can then construct the oscillation sets for each clock tick t in harmonic progression:
C Major Tritone Function Oscillation Set: {ψ00 = {-1, 5} ψ11 = {0, 4}}
And the union set of all members over time t:
C Major Tritone Function Union Set over Time: {F, B, C, E}
Axiom 4: The Wholetone Function is a generalization of the Tritone Function to represent chromatic harmony as the oscillation between the two wholetone scales:
WTtz = {ψt2z = {0, 2, 4, 6, 8, 10}
ψt+12z+1 = {1, 3, 5, 7, 9, 11}}
Axiom 5: The Dominant and Inverse Dominant Function is defined by the harmonic oscillation of orbits {ψt0 = {5, 7, 9, 11} ⋃ ψt+11 = {0, 2, 4}} contained in the harmonic series as divided by Z/12Z. Cancellation or resolution of oscillation occurs upon introduction of a union set of ψt+n = {0, 4, 7} of the major Tonic triad or ψt+n = {9, 0, 4} of the minor Inverse Tonic triad. Other resolving set intersections are possible, though resulting in lesser degrees of standing wave cancellation and cognitive resolution.
Axiom 6: The Diatonic Cycle of 5ths is defined as a symmetrical movement across the harmonic series following a path of alternating downward perfect 5th phase modulations between sine (odd Tonic) and cosine (even Dominant/ Inverse Dominant) components. This is defined as a harmonic oscillation of odd-even orbits {ψt0 = {0, 4, 2} ⋃ ψt+11 = {{5, 11}, 9, 7} contained in the harmonic series as divided by Z/12Z. Note that tritone orbit {5, 11} acts as an equivalence class within the oscillation set.
Axiom 7: A 7-step Diatonic Key is defined by a perfect 5th/ 4th axis of symmetry between the Tonic sine and Dominant/ Inverse Dominant cosine groups, phase shifted with a reverse “half twist” of 180 degrees within a standing wave interference pattern. For a given Harmonic Axis, a diatonic key is represented by the orbits ψt0 = {{0, 2, 4} ⋃ ψt+11 = {{5, 11}, {7, 9}} contained in the harmonic series described by Z/12Z. Its complementary tritone substitute key is given by ψt0 = {6, 8, 10}} ⋃ ψt+11 = {{1, 3}, {5, 11}.
Axiom 8: A Simple Rule of Thumb for Chromatic Harmony: Any progression of scales, intervals or chords that alternate one or more tones between the wholetone scales will create a sense of tension. The sensation of resolution occurs when the oscillation is canceled with any interval that straddles the two wholetone scales, such as a perfect 5th.
Axiom 9: A Simple Rule of Thumb for Diatonic Harmony: Under Axiom 8, incorporate the Tritone Function and/or Dominant (or Inverse Dominant) cadences to strengthen recognition of a single Harmonic Center and coherent pathway of the 7-step diatonic scale. Continued recognition of diatonic harmony is then directly proportional to persistence of just one Harmonic Center.