The Mathematical and Philosophical Foundations of Harmonics

Book I, Chapter 1: Music as an Integral Part of Human Nature and Morality

The human perception of sound is not merely a sensory experience but a deeply rooted aspect of human nature and morality. The sense of sight provides humans with the ability to distinguish forms and colors, while the sense of hearing allows for the perception of differences in pitch and harmony. The connection between the human mind and musical intervals is so profound that individuals of all ages, from infants to the elderly, are naturally affected by musical modes and rhythms. This universal susceptibility demonstrates that music is woven into the very fabric of human physiology and psychology.

The psychological impact of music is evident in how it influences human behavior, emotions, and moral character. Different musical modes possess specific qualities that can either elevate or degrade the human soul. Lascivious melodies tend to make the mind soft and effeminate, whereas harsh or vigorous rhythms can stimulate anger, courage, or intellectual clarity. Historical accounts show that the ancient Spartans used specific types of music to inspire bravery in battle, while philosophers like Pythagoras utilized music to calm agitated minds and restore emotional equilibrium.

The relationship between music and morality is further illustrated by the fact that nations and cultures often adopt musical styles that reflect their inherent character. A disciplined and orderly society prefers stable, well-structured melodies, while a decadent society gravitates toward chaotic, fragmented, and overly complex musical forms. Because the soul is governed by numerical proportions similar to those found in musical intervals, it naturally sympathizes with external sounds that mirror its internal state. Therefore, music has the power to either reinforce virtue or cultivate vice, making the regulation of musical styles a matter of great importance for the preservation of societal well-being.

“Music is so naturally united with us that we cannot be free from it even if we so desired.”

Book I, Chapter 2: The Threefold Division of Music

Music is categorized into three distinct kinds, which represent different levels of cosmic, human, and instrumental harmony. These three divisions are known as musica mundana (cosmic music), musica humana (human music), and musica instrumentalis (instrumental music). Each category operates on the principle of proportion and the harmonization of disparate elements.

Musica mundana is observed in the macrocosm, specifically in the movements of the celestial bodies, the combining of the four classical elements, and the turning of the seasons. The swift rotation of the massive heavenly spheres cannot occur in total silence; rather, it produces a highly ordered, harmonious sound that remains imperceptible to human ears due to its magnitude or the limitations of human sensory faculties. The distances between the planets and the speed of their orbits correspond directly to musical intervals, ensuring that the entire universe moves in an eternal, mathematical harmony. Furthermore, the alternation of winter, spring, summer, and autumn represents a harmonious balance between opposing qualities like hot, cold, wet, and dry, preventing any single element from destroying the others.

Musica humana is the harmony that exists within the human micro-universe. It manifests as the fitting integration of the incorporeal soul with the physical body. This internal music can be perceived in the way the rational mind governs the irrational impulses of the body, creating a balanced and virtuous state of existence. It is also found in the physical composition of the body itself, where the various organs, fluids, and limbs are held together in proper proportion. The harmonious interaction between the spiritual, intellectual, and physical components of a human being ensures health and sanity, while disharmony leads to disease and psychological distress.

Musica instrumentalis is the music produced by physical instruments. This is the audible music created by human craft through the manipulation of strings, wind, or percussion. It serves as a physical, audible imitation of the higher, silent harmonies of musica mundana and musica humana. Whether generated by the tension of strings on a lyre, the breath moving through a pipe, or the striking of a bronze vessel, instrumental music relies entirely on mathematical ratios to produce distinct pitches and intervals that please the ear and satisfy the mind.

Book I, Chapter 3: Sound and Its Definition

Sound is the primary matter of music, and it cannot exist without a physical motion or impact. For a sound to occur, there must be a collision between two or more bodies, which causes an alteration or displacement of the surrounding air. This displacement travels through the medium of the air until it reaches the organ of hearing. Therefore, motion is the efficient cause of all sound, and without motion, complete silence prevails.

Not all motions produce audible sounds, as some movements are either too slow or too small to disturb the air sufficiently. The nature of a sound depends entirely on the speed and frequency of the motion that generates it. A swift and rapid motion produces a sharp, high-pitched sound, while a slower, more deliberate motion produces a heavy, low-pitched sound. If a physical body strikes the air with great velocity, the resulting vibrations are frequent and compact, leading to a high pitch. Conversely, if the strike is slow, the vibrations are sparse and spread out, resulting in a low pitch.

A musical sound, or pitch, must be distinguished from random noise by its stability. A true musical pitch is one that maintains a constant, uniform frequency over a given duration, allowing the human ear to recognize its specific place within a system of high and low sounds. The continuous continuum of high and low sounds is broken down into discrete steps by the application of numerical boundaries, transforming raw physical noise into the structured material of musical science.

Book I, Chapter 4: The Concepts of Pitch, Interval, and Consonance

A physical pitch is called a vox (voice) when it is produced by a living being, but in the context of musical theory, any stable sound is considered a point of pitch. The difference between two distinct pitches, one higher and one lower, constitutes an intervallo (interval). An interval is the distance or spatial relationship between two sounds of differing height. It is not an empty space but a measurable relationship defined by mathematical proportions.

When two pitches of different heights are sounded simultaneously, they interact in one of two ways: they either blend together smoothly or conflict with one another. A consonantia (consonance) occurs when a high pitch and a low pitch combine to form a unified, pleasing, and coherent sound that strikes the ear as a single harmonious blend. This smooth mixture happens because the mathematical motions of the two sounds are proportional and compatible, allowing their physical waves to merge without harsh disruption.

An dissonantia (dissonance), on the other hand, occurs when two pitches are sounded together but refuse to blend. Instead, they strike the ear as a harsh, rough, and conflicted pair of separate sounds. This unpleasant effect is the direct result of incompatible mathematical ratios, which cause the physical motions of the two pitches to collide and disrupt each other in the air. Consonance represents order, unity, and rest, while dissonance represents disorder, duality, and tension within the musical system.

Book I, Chapter 5: The Primacy of Reason over the Senses

In the study of music, the judgment of human reason is far superior to the perception of the physical senses. While the ear is necessary for detecting the initial presence of sound, it is highly fallible, imprecise, and easily deceived by external conditions. The senses can only perceive the surface appearance of sounds, but they cannot comprehend the underlying truth or the exact mathematical relationships that govern those sounds.

Reason acts as the ultimate judge and arbiter of musical truth. It looks past the physical sensation of hearing and analyzes the numerical ratios that constitute intervals and consonances. Through reason, the mind can determine with absolute precision whether an interval is perfectly tuned or slightly flawed, a task that the ear alone cannot perform reliably. The senses are limited to the material world and are subject to fatigue and variation, whereas reason deals with immutable mathematical truths that remain constant across all time and space.

Music is therefore treated not merely as a practical art or an entertainment for the ears, but as a rigorous mathematical science. It belongs to the quadrivium, alongside arithmetic, geometry, and astronomy. Like these other disciplines, music uses numbers to unlock the secrets of cosmic order. The true musician is not the person who simply plays an instrument by habit or intuition, but the person who understands the mathematical principles of sound through the exercise of pure reason.

Book I, Chapter 6: The Pythagorean Discovery of Musical Ratios

The mathematical foundations of musical intervals were discovered by the ancient philosopher Pythagoras through an observation of physical weights. While walking past a blacksmith’s shop, Pythagoras noticed that the hammers striking the anvil produced different pitches, some of which combined to form beautiful consonances. He observed that the intervals of the octave, the fifth, and the fourth were being generated by the various tools.

To investigate the cause of these harmonious sounds, Pythagoras had the hammers weighed. He discovered that the relationships between the pitches were determined entirely by the relative weights of the hammers. He recorded the following weights for four specific hammers that produced consonances when paired: 12, 9, 8, and 6 units of weight.

By comparing these numbers, Pythagoras found that the hammer weighing 12 units and the hammer weighing 6 units stood in a duple ratio (2:1). When struck together, they produced the interval of an diapason (octave). The hammer of 12 units and the hammer of 8 units stood in a sesquialteran ratio (3:2), producing the interval of a diapente (fifth). The hammer of 12 units and the hammer of 9 units stood in a sesquitertian ratio (4:3), producing the interval of a diatessaron (fourth). Finally, the hammers of 9 units and 8 units stood in an epogdoic ratio (9:8), which produced the interval of a tonus (whole tone).

“Pythagoras, wishing to determine whether the judgment of the ears could be reduced to a certain mathematical dependability, investigated the weights of the hammers.”

Book I, Chapter 7: Verification of Ratios via String Tension

To confirm that these musical intervals were truly governed by numerical ratios rather than the specific material of the hammers or the strength of the blacksmiths, Pythagoras replicated the experiment using strings. He stretched strings of equal length and thickness across a wooden frame, hanging various weights from their ends to create tension.

When he attached weights of 12, 9, 8, and 6 units to the respective strings, he found that striking them produced the exact same musical intervals as the blacksmith’s hammers. The string stretched by 12 units of weight combined with the string stretched by 6 units to produce a perfect octave. The combinations of 12 to 8 units and 9 to 6 units consistently yielded fifths, while the combinations of 12 to 9 units and 8 to 6 units yielded fourths. The difference between the 9-unit string and the 8-unit string consistently produced a whole tone.

Pythagoras also verified these ratios by altering the lengths of a single string rather than changing the weights. By using a movable bridge to divide a string into precise mathematical segments, he demonstrated that a string divided exactly in half (2:1 ratio of length) produced an octave above the open string. A string stopped at two-thirds of its length (3:2 ratio) produced a fifth, and a string stopped at three-fourths of its length (4:3 ratio) produced a fourth. This established that musical pitch is an exact physical manifestation of numerical proportion.

Book I, Chapter 8: The Properties of Different Kinds of Proportions

The mathematical ratios that govern musical consonances belong to specific classes of inequality. In the study of relative quantity, numbers can be related to one another in several ways. The most important classes for music are the multiple and the superparticular.

A multiple proportion occurs when a larger number contains a smaller number a certain number of times exactly, without any remainder. Examples include the duple ratio (2:1), triple ratio (3:1), and quadruple ratio (4:1). In music, multiple proportions are highly valued because they represent simplicity and clarity, allowing the pitches to blend with a high degree of unity. The octave is a duple ratio (2:1), and the double octave is a quadruple ratio (4:1).

A superparticular proportion occurs when a larger number contains the entirety of a smaller number plus exactly one aliquot part (a single unit fraction) of that smaller number. The most prominent examples are the sesquialteran (3:2, which is $1 + 1/2$), the sesquitertian (4:3, which is $1 + 1/3$), and the epogdoic (9:8, which is $1 + 1/8$). These proportions govern the foundational intervals within the octave, namely the fifth, the fourth, and the whole tone.

Book I, Chapter 9: Why Multiple and Superparticular Ratios Dominate Consonance

The human mind and the sense of hearing prefer multiple and superparticular proportions over other types of mathematical relationships because they possess a higher degree of mathematical purity and limitation. Other classes of proportions, such as the superpartient (where the larger number contains the smaller plus multiple parts of it) or multiple superparticular, are more complex and irregular, which makes them less suitable for forming perfect consonances.

In a multiple proportion, the smaller number acts as a clear measure of the larger number, creating a direct and uncomplicated relationship. When two pitches stand in a multiple ratio, their physical vibrations align frequently, preventing any chaotic interference. The duple ratio (2:1) is the most perfect of these, as it preserves the essential character of the sound while merely shifting it to a higher register.

In a superparticular proportion, the excess of the larger number over the smaller number is a single, unified part. This singular difference provides a firm, clear boundary that the mind can easily comprehend. Because consonances require a sense of stability and rest, only those ratios that maintain simple, well-defined mathematical boundaries can produce a smooth, unified blend of sound. Complex or indefinite ratios result in unstable intervals that the ear rejects as dissonant.

Book I, Chapter 10: The Addition and Subtraction of Musical Ratios

In musical mathematics, intervals are not added or subtracted through simple arithmetic addition or subtraction. Instead, because intervals are expressed as ratios, the combining of intervals requires the multiplication of their corresponding fractions, and the subtraction of one interval from another requires division.

To add the interval of a fourth to the interval of a fifth, one must multiply their mathematical ratios. The ratio of the fifth is 3:2, and the ratio of the fourth is 4:3. Multiplying these fractions ($3/2 \times 4/3$) yields $12/6$, which simplifies to 2:1. This proves mathematically that adding a fourth and a fifth together results in a perfect octave.

Fifth (3:2) × Fourth (4:3) = Octave (2:1)

To subtract a fourth from a fifth to find the remaining interval, one must divide the ratio of the fifth by the ratio of the fourth. Dividing $3/2$ by $4/3$ is accomplished by multiplying $3/2$ by the reciprocal $3/4$, which results in 9:8. This demonstrates that the difference between a fifth and a fourth is exactly a whole tone. This method ensures that all musical intervals can be manipulated and analyzed with absolute mathematical certainty.

Book I, Chapter 11: The Smallest Musical Intervals and the Diesis

The whole tone, represented by the epogdoic ratio (9:8), cannot be divided into two equal integers because it is a superparticular ratio. In the mathematics of the pythagorean system, a superparticular ratio cannot be split evenly by a single whole-number ratio. Therefore, a whole tone is divided into two unequal parts, known as the maius semitonium (major semitone) and the minus semitonium (minor semitone).

The minor semitone is also referred to as the limma or diesis. It is the small interval that remains when two whole tones are subtracted from a perfect fourth. The ratio of a fourth is 4:3. Two whole tones are calculated by multiplying 9:8 by 9:8, which equals 81:64. To find the minor semitone, one divides 4:3 by 81:64, which results in the ratio of 256:243.

The major semitone, or apotome, is the remaining portion of the whole tone after the minor semitone has been accounted for. To find the apotome, one divides the whole tone ratio (9:8) by the minor semitone ratio (256:243), which yields the complex ratio of 2187:2048. Because these two semitones are unequal, the pythagorean musical system is built on a network of precise, interlocking, yet asymmetrical micro-intervals.

The Structure of the Greater Perfect System

Book I, Chapter 12: The Monochord and the Development of the Tetrachord

The monochord is a single-stringed instrument equipped with a movable bridge that allows for the precise measurement and demonstration of musical intervals. It serves as the primary tool for translating the theoretical mathematics of ratios into audible sound. By moving the bridge to specific markings on the sounding board, a theorist can isolate individual notes and observe how changes in length alter pitch.

Historically, the musical system began with a simple four-stringed structure known as the tetrachord. The four strings were tuned such that the two outermost strings formed a perfect fourth (4:3), while the two inner strings were tuned according to specific melodic intervals depending on the musical genus. The boundary notes of the tetrachord remained fixed and immutable, serving as the stable anchor points of the musical scale.

As musical art expanded, multiple tetrachords were linked together to create a broader range of pitches. This linking occurred either by conjunction, where the highest note of one tetrachord served as the lowest note of the next, or by disjunction, where two tetrachords were separated by the interval of a single whole tone. Through the combination of multiple tetrachords, ancient theorists constructed a comprehensive, ordered scale encompassing two full octaves.

Book I, Chapter 13: The Names and Ordering of the Seven Strings

In the earliest stages of Greek music, the scale consisted of seven strings, each named according to its physical position on the instrument or its function within the melodic structure. The lowest-pitched string was placed at the top of the physical instrument from the player’s perspective, while the highest-pitched string was located at the bottom.

The seven strings were named as follows:

Hypate: The lowest pitch, meaning “highest” in physical position on the lyre frame.
Parhypate: The string adjacent to the hypate, pitched slightly higher.
Lichanos: The index-finger string, used to play the third note of the system.
Mese: The middle string, which served as the central organizing note and focal point of the entire scale.
Paramese: The string next to the middle string, separated from it by a whole tone.
Paranete: The string adjacent to the highest string.
Nete: The highest pitch, meaning “lowest” in terms of its physical position at the bottom of the instrument.

These seven notes formed a complete scalar framework. Within this seven-stringed arrangement, various intervals could be navigated, providing the foundational melodic material for early classical compositions and philosophical discussions on music.

Book I, Chapter 14: Expansion to the Eight-Stringed Lyre

The expansion from a seven-stringed system to an eight-stringed system is historically attributed to Pythagoras. He realized that the seven-stringed scale did not encompass a full, completed octave from the lowest note to the highest note without gaps. To resolve this, he inserted an eighth string between the mese and the paramese.

Before this insertion, the middle string and the string above it were often tuned tightly, causing a structural irregularity in the intervals. By introducing a new string, Pythagoras created a clear separation between two distinct tetrachords. The lower four strings formed one tetrachord, and the upper four strings formed a second tetrachord.

This structural separation is called disjunction. The highest note of the lower tetrachord (mese) and the lowest note of the upper tetrachord (paramese) were now separated by a perfect whole tone (9:8). This allowed both the lower four strings and the upper four strings to form identical intervals of a perfect fourth internally, while the entire system from the lowest string to the highest string now formed a perfect octave (2:1).

Book I, Chapter 15: The Expansion to the Fifteen-Stringed Greater Perfect System

To accommodate a wider vocal and instrumental range, later theorists expanded the musical scale further by adding more tetrachords, eventually establishing a system of fifteen notes known as the Greater Perfect System. This system spans two full octaves and is organized into four distinct tetrachords, plus one additional low note at the very bottom of the scale.

The four tetrachords are ordered from lowest to highest:

Tetrachordon Hypaton (Tetrachord of the Principals)
Tetrachordon Meson (Tetrachord of the Middles)
Tetrachordon Diezeugmenon (Tetrachord of the Disjointed)
Tetrachordon Hyperbolaion (Tetrachord of the Excellents)

The single note added at the absolute bottom of the scale is called the proslambanomenos, which translates to “acquired note.” It sits one whole tone below the lowest note of the first tetrachord, serving to complete the full two-octave span from the highest note of the system down to a true mathematical baseline.

Book I, Chapter 16: Complete Nomenclature of the Fifteen Notes

Each note in the fifteen-stringed Greater Perfect System has a specific technical name that designates its precise position within the interlocking tetrachords. The complete list of notes, ordered from the lowest pitch to the highest pitch, is as follows:

Proslambanomenos: The lowest added note, forming the base of the two-octave scale.
Hypate Hypaton: The lowest note of the first tetrachord.
Parhypate Hypaton: The second note of the first tetrachord.
Lichanos Hypaton: The third note of the first tetrachord.
Hypate Meson: The highest note of the first tetrachord and the lowest note of the second tetrachord (joined by conjunction).
Parhypate Meson: The second note of the second tetrachord.
Lichanos Meson: The third note of the second tetrachord.
Mese: The central note of the system, terminating the second tetrachord.
Paramese: The first note of the third tetrachord, separated from the mese by a whole tone (disjunction).
Trite Diezeugmenon: The second note of the third tetrachord.
Paranete Diezeugmenon: The third note of the third tetrachord.
Nete Diezeugmenon: The highest note of the third tetrachord.
Trite Hyperbolaion: The second note of the fourth tetrachord, which begins on the nete diezeugmenon (conjunction).
Paranete Hyperbolaion: The third note of the fourth tetrachord.
Nete Hyperbolaion: The highest note of the entire system, completing the two-octave range.

Book I, Chapter 17: The Three Genera of Melodic Composition

The internal tuning of the strings within each tetrachord determines the genus (kind) of music being played. There are three distinct genera: the diatonic, the chromatic, and the enharmonic. Each genus alters the placement of the two movable inner strings of a tetrachord, while the two outer strings remain perfectly fixed at the interval of a fourth.

The diatonic genus is the most natural, robust, and widely used. Within each tetrachord, the intervals progress from lowest to highest as: a minor semitone, a whole tone, and a second whole tone. This structure is stable, easy to sing, and forms the basis for standard musical education and performance.

The chromatic genus is a modification that introduces a more coloristic and expressive melodic path. It alters the intervals of the tetrachord to progress by a minor semitone, a major semitone, and a large interval consisting of three semitones (a minor third). This genus steps away from the natural progression of whole tones, creating a shifting melodic texture.

The enharmonic genus is the most complex, precise, and tightly packed of the three. It divides the lower part of the tetrachord into extremely small intervals called dieses (quarter-tones). The progression within an enharmonic tetrachord consists of two consecutive quarter-tones followed by a very large, undivided interval of two full whole tones (a major third). This genus requires immense vocal control and precise instrumental tuning, and it was highly regarded by ancient philosophers for its purity and difficulty.

The Detailed Analysis of Ratios, Tuning, and Intervals

Book II, Chapter 1: The Definition of Inequality and Its Classes

All musical intervals are fundamentally expressions of mathematical inequality. When two quantities are equal, they exist in a state of absolute identity ($A = B$), which produces no variance in pitch. True musical sound requires a departure from equality into inequality, where one value is larger than another. Inequality is divided into five primary classes, which form the mathematical foundation for all numerical sciences.

These five classes are:

Multiplex: Where the larger number contains the smaller number multiple times exactly (e.g., 2:1, 3:1).
Superparticulare: Where the larger number contains the smaller number once plus one single unit fraction of it (e.g., 3:2, 4:3).
Superpartiens: Where the larger number contains the smaller number once plus more than one unit fraction of it (e.g., 5:3, 7:4).
Multiplex Superparticulare: Where the larger number contains the smaller number multiple times plus one single unit fraction of it (e.g., 5:2, which is twice plus a half).
Multiplex Superpartiens: Where the larger number contains the smaller number multiple times plus multiple fractions of it (e.g., 8:3, which is twice plus two-thirds).

The study of music utilizes these mathematical classes to categorize intervals according to their structural simplicity. The closer a ratio is to the top of this hierarchy (identity and simple multiplicity), the more perfect and consonant the resulting sound will be when heard by the human ear.

Book II, Chapter 2: The Core Consonances and Their Numerical Values

The primary musical consonances are derived strictly from the multiple and superparticular classes of inequality, using the lowest integers possible. These intervals are the octave, the fifth, the fourth, the octave-plus-fifth, and the double octave.

The mathematical values assigned to these consonances are:

Diapason (Octave): Governed by the duple ratio (2:1).
Diapente (Fifth): Governed by the sesquialteran ratio (3:2).
Diatessaron (Fourth): Governed by the sesquitertian ratio (4:3).
Diapason et Diapente (Octave plus Fifth): Governed by the triple ratio (3:1).
Disdiapason (Double Octave): Governed by the quadruple ratio (4:1).

2:1 = Octave
3:2 = Fifth
4:3 = Fourth
3:1 = Octave + Fifth
4:1 = Double Octave

Other combinations, such as the octave-plus-fourth, do not form a simple multiple or superparticular ratio when considered from the base note in the traditional pythagorean calculation, and therefore their status as pure consonances is a subject of rigorous debate among harmonic theorists.

Book II, Chapter 3: Proof That the Duple Ratio Belongs to the Octave

The relationship between the duple ratio (2:1) and the interval of the octave can be proven through both empirical observation on the monochord and deductive mathematical logic. If a string is divided exactly in half, the sound produced by the half-length is universally recognized by the ear as the exact same note as the open string, merely higher in pitch.

Mathematically, the duple ratio is the primary and most elemental form of multiple inequality. It represents the first step away from unity. Similarly, the octave is the primary consonance of music, acting as the framework within which all other intervals are contained. When an octave interval is doubled to form a two-octave span, the ratio scales from 2:1 to 4:1 (a quadruple ratio), preserving the multiple nature of the relationship.

If the octave were governed by any other ratio, such as a superparticular ratio, doubling the interval would lead to complex fractions that destroy the natural symmetry of the scale. The perfect blending of the octave, where the two pitches seem to merge into a single sound, is the direct physical counterpart to the clean, uncomplicated mathematical doubling of a value.

Book II, Chapter 4: The Invisibility of Smaller Parts in Superparticular Ratios

A fundamental principle of superparticular ratios is that they cannot be divided into equal middle ratios using whole numbers. This is a crucial mathematical proof in pythagorean music theory, demonstrating that an interval like a fifth (3:2) or a fourth (4:3) cannot be split into two identical smaller intervals.

To prove this, let us look at the numbers themselves. There is no integer that sits exactly halfway between 2 and 3, or between 3 and 4, in terms of multiplication or geometric proportion. If we attempt to find a mean value between the terms of a superparticular ratio, we are forced to move into fractional parts, which breaks the unity of the original ratio.

Because of this law, a whole tone (9:8) cannot be divided into two equal half-tones. The space between 8 and 9 cannot be split by an integer. This means that any attempt to create a “half-tone” results in two unequal intervals: one slightly larger, and one slightly smaller. The smaller part is hidden within the structural asymmetry of the tuning system, creating a necessary complexity in the division of musical space.

Book II, Chapter 5: The Calculation of the Major and Minor Semitones

Because a whole tone (9:8) cannot be divided equally, it is split into the minus semitonium (minor semitone) and the maius semitonium (major semitone). The exact mathematical calculation of these semitones is achieved by analyzing the relationship between a perfect fourth and two whole tones.

A perfect fourth has a ratio of 4:3. If we expand these terms to larger numbers to find common factors, we can express the fourth using the values 192 and 256. A whole tone above 192 is 216 (since $192 \times 9/8 = 216$). A second whole tone above 216 is 243 (since $216 \times 9/8 = 243$).

The distance from 192 to 243 represents two full whole tones. The distance from 243 to the top of the fourth at 256 represents the remaining interval, which is the minor semitone. The ratio of this minor semitone is therefore 256:243. To find the major semitone, we divide the full tone ratio (9:8) by this minor semitone ratio, which yields the ratio of 2187:2048. The major semitone is clearly larger than the minor semitone, as verified by these exact numbers.

Book II, Chapter 6: The Comma and Its Numerical Manifestation

The comma is an extremely small musical interval that represents the discrepancy between six whole tones and one perfect octave. It is the microscopic building block of tuning theory, showing that a series of pure mathematical steps does not perfectly close a circle.

If we calculate the value of six full whole tones by multiplying the ratio 9:8 by itself six times, we arrive at a complex ratio: 531441:262144. An octave, expressed in the same mathematical scale for comparison, can be represented by the ratio 2:1, which scales up to 524288:262144.

Six Whole Tones = 531441 : 262144
One Perfect Octave = 524288 : 262144

By comparing these two values, we see that six whole tones exceed a perfect octave by a tiny margin. The ratio of this margin is found by dividing the two fractions, resulting in 531441:524288. This minute difference is the pythagorean comma. It is an inescapable mathematical reality that prevents a scale constructed of pure whole tones from aligning perfectly with a pure octave.

Book II, Chapter 7: The Interlocking Ratios of the Greater Perfect System

The entire Greater Perfect System can be mapped out as a continuous series of interlocking mathematical ratios. Every note from the proslambanomenos to the nete hyperbolaion is connected to its neighbors by precise fractions, ensuring that the entire two-octave structure is perfectly unified.

Starting from the bottom, the intervals move upwards through a pattern determined by the diatonic genus. The step from proslambanomenos to hypate hypaton is a whole tone (9:8). Within the Tetrachordon Hypaton, the steps go from a minor semitone (256:243) to a whole tone (9:8), and then another whole tone (9:8). This brings the system to the hypate meson.

This pattern repeats across all four tetrachords, with the disjunction between the mese and the paramese introducing a fixed whole tone (9:8) in the middle of the scale. Because every interval is derived from either 9:8 or 256:243, the entire system forms an interlocking mathematical grid where every single note has a fixed, calculable value relative to every other note.

The Arguments and Theories of Aristoxenus versus Pythagoras

Book III, Chapter 1: The Philosophy of Aristoxenus and the Judgment of the Ear

The musical theorist Aristoxenus proposed a philosophy of harmonics that stood in direct opposition to the pythagorean school. Aristoxenus argued that the human sense of hearing should be the ultimate authority in determining musical intervals and consonances, rather than abstract mathematical calculations. He believed that music is an art perceived by the ear, and therefore, what sounds correct to the ear must be accepted as true.

Aristoxenus rejected the idea that intervals must be defined by rigid numerical ratios like 9:8 or 4:3. Instead, he treated musical space as a continuous line that can be divided into equal parts based on sensory perception. In his view, a whole tone is simply an interval that can be split into two perfectly equal half-tones, and an octave is made up of exactly six equal whole tones.

According to this perspective, the mathematical complications of the pythagoreans—such as major and minor semitones, or the existence of the comma—are irrelevant to the actual practice of music. If the ear perceives a sound as a perfect half-tone, then it is functionally a half-tone. Aristoxenus focused on the functional relationships between notes in a melody, prioritizing the lived experience of the listener over the theoretical calculations of mathematicians.

Book III, Chapter 2: The Pythagorean Refutation of Aristoxenus

The pythagoreans vigorously refuted the theories of Aristoxenus, arguing that relying on the physical sense of hearing leads to error and intellectual blindness. They maintained that because the ear is a physical organ, it is inherently imperfect and cannot perceive the minute differences that separate true harmony from discord.

The core of the pythagorean argument is that musical intervals are not arbitrary divisions of acoustic space, but real physical facts governed by immutable numerical laws. A superparticular ratio, by its very nature as a mathematical entity, cannot be divided into two equal whole-number ratios. Since a whole tone is proven to be a superparticular ratio (9:8), it is geometrically and arithmetically impossible to split it into two identical halves using integers.

“To believe the ears alone without the aid of reason is to wander in darkness, for the senses are crude and easily deceived.”

Therefore, the pythagoreans declared that the “equal half-tone” of Aristoxenus is a myth born of sensory imprecision. What the ear of an Aristoxenian believes to be two equal halves is actually two unequal intervals that the untrained sense fails to distinguish. True science must rely on the unshakeable certainty of mathematics rather than the shifting, unreliable impressions of the human ear.

Book III, Chapter 3: Mathematical Proof That Six Whole Tones Exceed an Octave

To completely dismantle the Aristoxenian claim that an octave consists of exactly six whole tones, the pythagoreans provided a definitive arithmetical proof showing that six whole tones are actually greater than a perfect octave.

An octave is defined by the duple ratio (2:1). A single whole tone is defined by the epogdoic ratio (9:8). To find the value of six whole tones combined, we must multiply the ratio 9:8 by itself six times. This calculation is performed as follows:

$$ (9/8)^6 = \frac{9 \times 9 \times 9 \times 9 \times 9 \times 9}{8 \times 8 \times 8 \times 8 \times 8 \times 8} = \frac{531441}{262144} $$

To compare this directly with an octave, we express the octave ratio (2:1) with the same denominator of 262144. Multiplying 2 by 262144 gives us 524288. Therefore, a perfect octave is represented by the fraction 524288:262144.

When we place the two fractions side by side, the result is undeniable:

$$\frac{531441}{262144} > \frac{524288}{262144}$$

Six whole tones equal 531441, while a perfect octave equals 524288. The six whole tones are clearly larger than the octave by a difference of 7153, which is the exact value of the comma. This proof demonstrates that Aristoxenus was mathematically incorrect, and that an octave cannot be cleanly divided into six equal tones.

Book III, Chapter 4: The Geometric Impossibility of Equal Superparticular Division

The debate between the two schools also involved geometric proofs regarding the division of lines and spaces. The pythagoreans used the laws of geometry to show that a superparticular space cannot be halved by a single rational boundary point.

If we represent an interval as a physical line whose length corresponds to a specific number, dividing that interval into two equal musical parts requires finding a geometric mean ($A:B = B:C$). In the case of superparticular ratios, where the difference between the two numbers is a single unit ($n + 1 : n$), no such integer mean exists.

For example, in the ratio of the fifth (3:2), if we double the terms to 6:4 to look for a middle number, we find 5. However, the ratio 6:5 is a sesquififth, and the ratio 5:4 is a sesquiquartan. These two resulting intervals are completely different from one another. The interval is not split into two equal parts; instead, it is split into a larger interval and a smaller interval. This geometric reality confirms that the basic building blocks of harmony are inherently asymmetrical.

Book III, Chapter 5: The Division of the Tetrachord According to Both Systems

The practical application of these opposing theories is most clearly seen in how each school divides the interval of a perfect fourth within a tetrachord. The two systems produce different tuning structures for performance.

In the diatonic system of Aristoxenus, a fourth consists of exactly two and a half tones. He divides the tetrachord into three parts: a half-tone, a full tone, and another full tone. The sizes of these intervals are fixed by raw fractions of a tone ($1/2 + 1 + 1 = 2.5$).

In the pythagorean system, the tetrachord is divided using precise mathematical fractions. It begins with a minor semitone (256:243), followed by a whole tone (9:8), and ends with a second whole tone (9:8). When these three ratios are multiplied together, they return exactly to the ratio of the fourth (4:3):

$$\frac{256}{243} \times \frac{9}{8} \times \frac{9}{8} = \frac{4}{3}$$

The pythagorean division ensures that every note is locked into a perfect mathematical relationship, whereas the Aristoxenian division relies on a simplified approximation that ignores the physical laws of acoustic numbers.

The Advanced Mathematics of Harmonics and Proportions

Book IV, Chapter 1: The Three Types of Mathematical Means

The internal structure of musical intervals is analyzed using three primary types of mathematical means: the arithmetic, the geometric, and the harmonic. A mean is a value that sits between two extremes, dividing a larger space into smaller intervals.

An arithmetic mean occurs when the difference between the terms is equal. For example, in the series 2, 3, 4, the difference between 2 and 3 is one, and the difference between 3 and 4 is one. In music, an arithmetic division of an interval results in unequal ratios, because smaller numbers represent larger physical spaces than larger numbers.

A geometric mean occurs when the ratio between the terms is equal ($A:B = B:C$). For example, in the series 1, 2, 4, the ratio of 1 to 2 is a duple ratio, and the ratio of 2 to 4 is also a duple ratio. A geometric mean splits a musical interval into two perfectly equal parts, but it often requires irrational numbers that cannot be expressed as simple whole-number fractions.

A harmonic mean is a more complex relationship where the terms are proportional to their differences. It occurs when the ratio of the first term to the third term is equal to the ratio of the difference between the first and second terms to the difference between the second and third terms. In the series 6, 8, 12, the ratio of 6 to 12 is 1:2. The difference between 6 and 8 is 2, and the difference between 8 and 12 is 4. The ratio of these differences (2:4) is also 1:2. This mean is highly valued in music because it naturally generates the intervals of the fifth and the fourth within the framework of an octave.

Book IV, Chapter 2: The Harmonic Division of the Octave

When a perfect octave (2:1) is divided using both the arithmetic and harmonic means, it reveals the inner structure of the entire scale. Let us take the numbers 6 and 12 to represent the boundaries of an octave.

If we apply the arithmetic mean to 6 and 12, we find the number that sits exactly halfway between them by addition, which is 9 (since $6+3=9$ and $9+3=12$). The relationship between 6 and 9 is a sesquialteran ratio (3:2, a fifth), and the relationship between 9 and 12 is a sesquitertian ratio (4:3, a fourth). Thus, the arithmetic mean divides the octave into a fifth followed by a fourth.

Octave (6 to 12) divided Arithmetically by 9:
6 to 9 = Fifth (3:2)
9 to 12 = Fourth (4:3)

If we apply the harmonic mean to 6 and 12, we find the number 8. The relationship between 6 and 8 is a sesquitertian ratio (4:3, a fourth), and the relationship between 8 and 12 is a sesquialteran ratio (3:2, a fifth). Thus, the harmonic mean divides the octave into a fourth followed by a fifth.

Octave (6 to 12) divided Harmonically by 8:
6 to 8 = Fourth (4:3)
8 to 12 = Fifth (3:2)

These two divisions show how the core intervals of the fifth and the fourth are contained within the octave, depending on whether the larger or smaller interval is placed at the bottom of the structure.

Book IV, Chapter 3: The Properties of the Whole Tone and Its Parts

The whole tone, expressed as 9:8, is the foundation for scalar movement. Because it cannot be split into equal integers, its division into major and minor semitones creates a subtle texture within the musical system.

The minor semitone (256:243) is smaller than a true mathematical half-tone, while the major semitone (2187:2048) is slightly larger. The difference between these two semitones is an interval known as the schisma. The presence of these micro-intervals means that when a musician moves up by a minor semitone and then by a major semitone, they arrive exactly at a whole tone, but the pathway is asymmetrical.

This asymmetry affects the tuning of instruments like the monochord. When marking out the positions for notes, one must calculate whether a step requires a major or minor semitone. A mistake in choosing the correct semitone shifts the pitch away from its proper mathematical place, causing the entire scale to lose its harmonic alignment.

Book IV, Chapter 4: The Diagram of the Greater Perfect System on the Monochord

To construct the Greater Perfect System on a physical monochord, a theorist must use a precise mathematical sequence to mark the locations of all fifteen notes along the length of the string. This process involves dividing the total length of the string into specific fractions.

First, the entire string is treated as the proslambanomenos, representing the lowest pitch. To find the octave note, the mese, the bridge is placed exactly at the midpoint of the string, dividing it in a 2:1 ratio. To find the highest note, the nete hyperbolaion, the string is divided into four equal parts, and the bridge is placed at the one-fourth mark, creating a 4:1 ratio relative to the open string.

The individual tetrachords are then marked out by applying the sesquitertian ratio (4:3) and the epogdoic ratio (9:8) to these primary divisions. Each note is established by measuring from either the open string or from a previously verified point. The result is a complete physical map of the scale, where every note corresponds to a precise geometric measurement along the monochord line.

The Theory of the Eight Musical Modes

Book IV, Chapter 5: The Definition and Origin of the Modes or Tropes

The musical modes, also known as tropi or modi, are complete scalar structures that differ from one another by their pitch register and the arrangement of their intervals. They are not merely individual keys, but entire frameworks for melody that possess unique emotional and moral characteristics.

Historically, the modes arose from the need to shift the Greater Perfect System into different pitch ranges to accommodate different voices and instruments. By shifting the entire two-octave scale higher or lower, theorists created distinct musical environments. Each mode allows a melody to utilize a different part of the overall vocal range, changing the expressive qualities of the music.

The ancient Greeks recognized several primary modes, naming them after the regions and tribes that originally utilized them. These modes were associated with specific psychological effects. For instance, some modes were considered calm and educational, while others were seen as wild, ecstatic, or mournfully sorrowful.

Book IV, Chapter 6: The Seven Primary Modes and Their Species of Octave

The structure of the modes is deeply connected to the different ways an octave can be divided into whole tones and semitones. Within a two-octave system, there are seven distinct species of octave, each characterized by where the semitones fall relative to the whole tones.

The seven primary modes are constructed based on these seven species of octave:

Hypodorian: The lowest mode, built on the first species of octave.
Hypophrygian: Pitched one whole tone higher than the Hypodorian.
Hypolydian: Pitched one whole tone higher than the Hypophrygian.
Dorian: Pitched one whole tone higher than the Hypolydian, forming the central mode of the system.
Phrygian: Pitched one whole tone higher than the Dorian.
Lydian: Pitched one whole tone higher than the Phrygian.
Mixolydian: Pitched a minor semitone higher than the Lydian, forming the highest of the original seven modes.

Each mode presents a unique sequence of intervals to the composer. For example, in one mode the semitones may appear early in the scale, while in another they appear late, altering the melodic contour and emotional resonance of the music.

Book IV, Chapter 7: The Insertion of the Eighth Mode (Hypermixolydian)

As musical theory became more systematic, an eighth mode was introduced to complete the mathematical symmetry of the modal system. This mode is known as the Hypermixolydian.

The Hypermixolydian mode was added above the Mixolydian, sitting a full octave above the lowest mode, the Hypodorian. Its introduction allowed the modal system to mirror the complete octave structure of the Greater Perfect System. With eight modes, the system achieves a state of balance, where every lower mode has a corresponding higher counterpart.

The eight modes are organized into pairs: four primary modes (Dorian, Phrygian, Lydian, Mixolydian) and four plagal or subordinate modes pitched a fourth lower (Hypodorian, Hypophrygian, Hypolydian, Hypermixolydian). This comprehensive system of eight modes provides a complete framework for organizing all musical compositions according to clear mathematical and register-based principles.

The Classification and Role of the Musician

Book V, Chapter 1: The Three Classes of People Who Engage with Music

The practice and study of music involves three completely separate classes of people, each operating at a different level of understanding, skill, and purpose. These three classes are the performers, the composers, and the true musicians.

The first class consists of those who play physical instruments, such as the lyre, the pipes, or the organ. This group includes all instrumentalists and singers. They possess physical dexterity, muscle memory, and the practical ability to produce sound. However, this class is considered the lowest of the three because its members act primarily out of physical habit, imitation, and bodily skill rather than intellectual understanding. They are like servants who execute commands without knowing the reasons behind them.

The second class consists of the poets and composers who invent melodies and write songs. They possess a natural, intuitive talent for arranging notes and rhythms in pleasing ways. However, their engagement with music is guided mostly by instinct, inspiration, and emotional drive rather than a conscious grasp of mathematical laws. While they create beautiful works, they cannot explain the scientific principles or the numerical proportions that make their melodies harmonious.

The third class consists of the true musician (musicus). The true musician is not someone who performs or creates physical sound, but someone who studies the science of music through pure reason and intellect. This person understands the mathematical ratios that govern intervals, the properties of modes, the nature of consonances, and the cosmic harmony of the universe. The true musician possesses speculative knowledge, allowing them to judge and evaluate the work of performers and composers with absolute scientific certainty.

“The class of musicians is that which has gained the faculty of judging, according to speculation and reason, the rhythms, the melodies, and the entire song.”

Book V, Chapter 2: The Intellectual Supremacy of Speculative Knowledge

In the philosophy of music, speculative knowledge—the purely intellectual understanding of theory and mathematics—is infinitely superior to practical execution. Physical performance belongs to the material world, which is temporary, flawed, and subject to decay, whereas theoretical knowledge belongs to the realm of eternal, immutable truths.

A person who relies solely on their hands or voice to make music is limited by their physical body. If their fingers stiffen or their voice cracks, their connection to music is disrupted. In contrast, the true musician who possesses speculative knowledge holds the complete science of harmony securely within their mind. This knowledge is independent of physical ability and remains perfect regardless of physical circumstances.

Furthermore, the ability to judge and evaluate is an intellectual act that requires a higher faculty than mere production. A critic or a theorist can see the structural flaws in a composition or a performance because they possess the mathematical template of perfection in their reason. Therefore, the highest achievement in music is not the creation of beautiful sounds for the ears, but the contemplation of numerical harmony by the rational mind.

Book V, Chapter 3: Music as a Scientific Component of the Quadrivium

Music is firmly established as one of the four mathematical sciences that comprise the quadrivium, alongside arithmetic, geometry, and astronomy. Each of these disciplines uses numbers to investigate a different aspect of reality, and together they lead the mind from the material world to the understanding of higher truths.

The four sciences are organized by how they apply numbers:

Arithmetic: Studies numbers in themselves, as abstract entities ($1, 2, 3$).
Music: Studies numbers in relation to other numbers, specifically as ratios and proportions that manifest as harmony.
Geometry: Studies numbers at rest as physical dimensions, shapes, and spaces.
Astronomy: Studies numbers in motion, tracking the paths, speeds, and cycles of celestial bodies.

Music is therefore a science of relative quantity. It demonstrates how numbers interact to create balance, order, and beauty. By studying the mathematical laws of sound, a student prepares their mind for the contemplation of philosophy and the cosmic order that holds the universe together.

Book V, Chapter 4: The Cosmic Order and the Final Scope of Music

The ultimate purpose of studying music is to comprehend the grand cosmic harmony that governs the entire universe. The physical, audible sounds produced by strings and instruments are merely a doorway leading to the understanding of the silent, eternal music of the cosmos.

Every aspect of creation is held together by the same mathematical proportions found in the musical scale. The movements of the stars and planets, the changing of the seasons, the balance of elements in nature, and the integration of the human soul with the physical body are all expressions of cosmic music. When a human being studies the ratios of the octave, the fifth, and the fourth, they are learning the very language that the creator used to fashion the world.

True musical science brings the human mind into alignment with this cosmic order. By moving past the physical pleasure of hearing and focusing on the underlying numerical truths, the soul purges itself of disorder and disharmony. The final scope of music is to elevate human reason to the contemplation of the perfect, immutable, and divine mathematical unity that structures all of reality.

INTRODUCTION TO MUSIC -Boethius-