1 Why did I write this article?

The purpose of writing this article is not to create a vocal tutorial, nor is it a complete review of music theory. It is more like a record of my own reflections—attempting to answer a simple yet complex question:What exactly am I listening to? And why are these sounds heard by me in this way.

If you, like me, are curious about sound and music, please join me on this journey of exploration.

In a previous article, in order to give the highly subjective judgment of “good-sounding” singing some basis for discussion and comparison, I introduced the “three-dimensional theory” framework (see article:Why a song sounds good: A three-dimensional theory analyzing the secrets of singing), which is suitable for standing atthe level of the entire song and overall performanceto analyze and judge the singing results.

But as I continued to use it to deconstruct specific singing performances, I gradually realized:the three-dimensional theory itself has very clear boundaries of applicabilityOnce we continue to break it down further, such as isolating and analyzing the “sound” focused on in the first dimension, new questions immediately arise, such as: Why do some people's voices make people feel “very clean”? Why do some sounds not sound structured, but instead present a unique “messy beauty”? These questions have clearly exceeded the scope of what the three-dimensional theory can explain.

Because at this point, the object of our discussion is no longer “whether a song is sung well,” but a deeper level—how sound itself is produced, perceived, and understoodIn other words, the question has shifted from “musical aesthetic evaluation” toacoustic structure and auditory perception mechanismsabove.

It was also in the process of constantly questioning these issues that I gradually realized: my confusion at this moment does not really belong to singing methods or techniques, nor does it even entirely belong to music theory, but points to a more fundamental, yet often ignored level—the physical structure of sound, and how humans understand “pitch”, “intervals”, and “stability” at the auditory level. For example: when we say a sound is “clean” or “muddy”, what exactly are we listening to? When two sounds have different pitches but are considered “different heights of the same note” (octaves), where does this judgment come from? Why do certain intervals naturally make people feel stable and comfortable, while others are more likely to bring tension or even harshness?

These phenomena are not rules agreed upon after the fact, but are closely related to the frequency structure of sound, internal relationships, and the way the human auditory system works. Therefore, the following discussion will deliberately avoid those common entry points that start withmemorizing rules and mastering techniquesas starting points, and instead try to follow a deeper path to re-understand the musical concepts we are familiar with:Starting from the physical structure of sound → to the formation of auditory perception → and then to why intervals and scales were established in the way they are today.

In this process, notation systems such as numbered musical notation, letter names, and key signatures will no longer be regarded as the starting point of musical understanding, but rather as an abstraction and compression of complex acoustic reality made by humans based on long-term auditory experience.

2 Sound is not a single frequency

1 Pitch is not the perception result of a single frequency

Before talking about intervals, tonality, or scales, we almost all unconsciously accept a premise:a note corresponds to a pitch; and pitch, in essence, is a frequency. This premise seems very reasonable in notation systems: whether it is “1, 2, 3” in numbered musical notation or the positions of notes on a staff, they are constantly reinforcing an intuition—music is composed of a series of clear, stable “points”, each of which can be precisely marked, named, and reproduced.

But once we return to the real auditory experience, this intuition immediately begins to shake: the same pitch, when played on a piano, drawn on a violin, or sung by a human voice, sounds vastly different; even with human voices, when different people sing on the same note, some make you feel “stable” and “clean”, while others make you feel “floaty” and “scattered”; even under the premise that the pitch is completely correct, we still instinctively judge: this sound is “not right”.

If “pitch = frequency” were truly sufficient to describe what we hear, then these differences would be difficult to explain. Because in this model, as long as the frequency is correct, it should sound like “the same note”. Reality is obviously not like this. This is also a hidden confusion that many people encounter when learning music or singing: they clearly know which note they are singing, and have not even sung it wrong in the sense of numbered musical notation or a tuner, but the imagination in their minds of “what this note should actually sound like” remains vague. This vagueness is not a problem of comprehension, but comes from an oversimplified assumption—we think we are listening to “pitch”, but in reality, hearing is never just facing an isolated frequency.

When a sound is produced, whether by an instrument or a human voice, it is not a “point”, but rather like a whole unfolding on the spectrum. Pitch is just the part of this whole that is easiest for us to name and easiest for notation systems to capture, but it is not everything.

And precisely because of this, we constantly use words unrelated to “frequency” in real listening experiences: clean, muddy, thick, thin, bright, dark, gritty, rough... These words are not rhetorical associations, but are attempts to describe some kind ofauditory difference beyond a single pitchIt's just that, before there is a suitable concept, we can only use feelings to refer to it.

Therefore, before continuing to discuss intervals, scales, and even tonality, there is a question that must be faced first:If what we hear is not just “pitch”, then what exactly are we listening to?

2 Why the same pitch can sound different

If the previous section addressed an intuitive question—that we are not just listening to pitch—then next we must face a more practical reality: even when the pitch is completely identical, the differences between sounds remain real and vast.

This difference is not some “advanced perception” that only emerges after studying music; even without any background in music theory, most people can instantly tell that some sounds are more “stable” while others are more “drifting,” or that some sounds seem focused and clear while others have blurry boundaries and are hard to grasp. More importantly, this judgment often occurs before we determine whether a sound is “out of tune”—that is, before the pitch is explicitly judged as right or wrong, our hearing has already evaluated the sound on another level.

This indicates that while listening to pitch, we must also be receiving other information, and it is precisely this information that determines the overall perception of the sound. If “pitch = frequency” were sufficient to describe everything we hear, then as long as the frequency is the same, there shouldn't be such a stark difference in how they sound. Yet reality constantly reminds us that the difference does not stem from the pitch itself, but from what lies beyond it. In everyday discussions, people often attribute this difference to “different timbres,” but this term is more of a label for the result rather than an explanation—it fails to answer a more fundamental question: if the pitch is identical, what exactly makes these sounds hear so differently?

When two sounds are completely identical in the dimension of pitch, any difference perceived by our hearing can only come from other internal characteristics of the sound. These characteristics are never marked by numbered musical notation or note names, yet they always truly exist during the process of sound production: some sounds seem concentrated in a clear location with a well-defined outline, while others appear diffuse, with blurry boundaries that make it difficult to form a stable auditory image.

This difference is often intuitively described as “stable” or “unstable.” The sense of stability here is not a psychological feeling of reassurance or relaxation, but a very direct auditory experience: whether the sound presents a clear, sustainable, and cohesive form. When the internal variations of a sound are too frequent—with certain components fluctuating unpredictably in strength—it becomes difficult for our hearing to establish a stable object. Consequently, even if the pitch is correct, the sound is still perceived as “unstable” or "unable to hold its ground."

Conversely, when a sound exhibits a certain internal consistency in both time and structure, the auditory system can quickly accept it and treat it as a complete acoustic unit. Even if the sound is not “beautiful”—or even carries a distinct roughness—it can still be perceived as stable and credible. This explains why, in real-world listening, some sounds with a pronounced “grittiness” or rough texture are actually easier to accept than others that are clean on the surface but drift unpredictably.

This also leads to another frequently mentioned but rarely analyzed phenomenon—the perception of noisiness. As irregular components in a sound gradually increase, our hearing often perceives the pitch as becoming blurry, the boundaries of the sound beginning to loosen, and the overall listening experience becoming more fatiguing. The issue is not simply whether the sound “contains noise,” but whether these irregular components have begun to interfere with our auditory judgment of the overall structure. Once this interference reaches a certain threshold, the auditory system can no longer stably grasp the part that should serve as an anchor, and the perception of pitch naturally begins to waver.

By this point, one thing has become clear: the reason the exact same pitch can sound completely different is not because we are mishearing the pitch, but because we have never been listening to pitch alone. The differences stem from the internal structure of the sound, and these structural differences are the very root of perceptions like “stable,” “diffuse,” “clean,” or “rough.”

The question thus becomes more concrete. If these perceptual differences are not subjective illusions but stem from the actual internal structure of the sound, then what exactly are these structures? And how do they work together to form the “acoustic whole” that we perceive?

3 Overtones: The reason why sound can be “heard as a single note”

In the preceding discussion, we have repeatedly touched upon a fundamental fact: real-world sounds are never a “single frequency.” Whether it is a human voice, a piano, or a guitar, as long as it is not an electronically synthesized pure sine wave, it is physically and inevitably the result of multiple frequencies coexisting. It is just that most of the time, we do not consciously perceive this.

And this is precisely where the problem begins to arise. If every real sound internally contains a vast number of different frequencies, then how on earth does the human auditory system, amidst such a complex acoustic reality, still stably determine that “this is a single tone”, “roughly how high it is”, and “whether it is stable and pleasant to hear”? In other words, what the auditory system truly needs has never been more detailed information, but rather a structure that can be unifiedly understood—and “overtones” are precisely the core of this unified structure.

From a physical perspective, when a sound source vibrates at a fundamental frequency, it is almost impossible for it to undergo only a single, ideal vibration. The actual vibration process naturally generates a series of additional vibrational components, the frequencies of which do not appear at random, but are strictly arranged in integer multiples of the fundamental frequency: 2 times, 3 times, 4 times... These frequency components that coexist with the fundamental frequency and appear according to regular patterns are overtones:

And any single tone we usually hear is actually a complex tone containing both the fundamental and overtones.

However, if we only remain at this definitional level, overtones can easily be misunderstood as a “factor that makes sound complex,” as if pitch, which is already difficult enough to distinguish, were stuffed with a bunch of extra frequency information. But in reality, the existence of overtones is precisely the key to making the auditory world comprehensible.

One can imagine an extreme case: if the frequency components within a sound are completely chaotic and irregular, it will only be perceived auditorily as noise. It has no clear pitch center, cannot bring a sense of stability, and is even less likely to be naturally integrated into a musical system. Overtones are important not because they are “many,” but because they are “ordered.”

This order is reflected in a very simple yet extremely crucial feature: all overtones are integer multiples of the fundamental frequency. It is precisely this highly regular proportional relationship that allows the auditory system to compress and understand what would otherwise be a complex set of frequencies as a whole. We do not perceive one by one, “Now I hear 440Hz, 880Hz, 1320Hz...”, but instead directly form an intuitive judgment: this is a sound with a clear pitch and a stable internal structure.

And precisely because the overtone structure is unified and predictable, the auditory system naturally anchors its attention to a certain “core,” which is what we commonly refer to as the sense of pitch. In other words, pitch is not the result of a single frequency being “heard” in isolation, but rather the perceptual center pointed to by the entire overtone structure.

Looking back from this perspective, the meaning of what we often call “a complete tone” becomes much clearer. So-called completeness does not mean that it contains only one frequency; on the contrary, it is because it contains a whole set of regular overtone structures internally that it appears focused, stable, and solid.

This also explains an often-overlooked fact: whether a sound is “pleasant to hear” does not depend on whether it is simple enough, but on whether it is unified enough. When overtones are clearly arranged and stably proportioned, the auditory perception appears clean and focused; when overtones exist but are chaotically distributed, the perception becomes rough and blurred; and when overtones are deliberately distorted or destroyed, the sound presents a strange, tense, or even distorted feel.

These differences are not pure aesthetic preferences, but rather the natural response of the auditory system when facing different internal structures. Therefore, when we truly put “overtones” back in their rightful place, they are no longer an additional acoustic term, but a key to understanding auditory order. It is with the help of this key that we can further understand why musical instruments, as the source of modern music, can produce such a diverse range of sounds.*

From a physical level, the sound of any musical instrument originates from vibration; as long as there is a stable perception of pitch, it must be accompanied by a complete overtone structure. Because different instruments have different sounding principles, their vibration methods also differ, thereby forming their own unique overtone distributions. These differences ultimately manifest as the differences in timbre we perceive, and constitute the fundamental reason why “the exact same note can sound completely different.”

In most musical practices, people do not need to explicitly understand the concept of overtones. But when we attempt to explain differences in timbre, interval stability, and even why musical scales exist in the form they do today, overtones become an unavoidable underlying mechanism.

Here, it is necessary to make a brief supplementary explanation regarding terminology—the terms “fundamental” and “overtone” used previously are more common expressions in the field of music; whereas in physical acoustics, the same phenomenon is usually described asharmonics。

In a physical context, the lowest frequency produced by a vibrating object is called the first harmonic, and the subsequent second and third harmonics correspond to higher multiples of the frequency components in turn. It is precisely because of the coexistence of these two naming systems that, in the field of music, in order to more clearly distinguish the “pitch center” from the “subsidiary structures,” the first harmonic in the physical sense is often referred to separately as thefundamental, while starting from the second harmonic, they are sequentially called thefirst overtone, second overtone, and so on:

Regardless of which naming convention is adopted, what they point to is actually the same objectively existing frequency structure. Once this is understood, the previously discussed issues such as the perception of pitch, the sense of stability, and “why a sound is heard as a single tone” are no longer abstract or artificially defined concepts, but rather natural results built upon this ordered vibrational structure.

3 The order between sounds: From the perception of a single note to multiple notes

1 Why octaves are heard as “different pitches of the same note”

In everyday musical experience, there is a phenomenon that almost everyone accepts by default, yet is rarely questioned: when the frequency of a note doubles, for example from 440Hz to 880Hz, and then to 1320Hz, we say without hesitation—“this is the same note, just an octave higher.”

To make the auditory concept of the octave more intuitive on a physical level, the diagram below shows the actual frequencies corresponding to each note name (C to B) in different octaves in twelve-tone equal temperament:

This judgment seems like a rule of music theory, as if artificially prescribed. But if we return to the previous discussion about overtones, we will find that this auditory “categorization” actually has a very clear physical and perceptual basis.

Looking first at the internal structure of a sound, suppose there is a sound with a fundamental frequency of f. Its overtones will naturally align at integer multiples like 2f, 3f, 4f... (this has been repeatedly mentioned in the previous section). And when this sound as a whole is raised by an octave, with the fundamental frequency becoming 2f, its overtone structure does not “reshuffle,” but instead shifts upward as a whole: the new overtones will appear at 4f, 6f, 8f...

The key point here is:A large number of overtones in the original sound will highly overlap with the overtones in the new sound: the original 2f becomes exactly the new fundamental; the original 4f corresponds to the new second overtone; and the original 8f still exists in the new structure. That is to say, from the perspective of spectral structure, these two sounds are not two objects unfamiliar to each other, butthe same type of structure with highly similar internal compositions, just on different overall scales。

It is precisely this highly consistent arrangement of overtones that causes the human auditory system to make a very stable judgment: “They belong to the same class of pitch, just at different levels of height.” Because from the perspective of auditory perception, the human ear does not analyze “what specific frequencies this sound contains” one by one, but instead quickly grasps the stable patterns in the structure. When the overtone structures of two sounds are highly consistent in their proportional relationships, the auditory system automatically classifies them into the same category, and on this basis, adds a simple dimensional difference—high or low.

This also explains an interesting phenomenon: the reason why the octave is special is not because the mathematical relationship of “doubling” the frequency is inherently mysterious, but becauseamong all interval relationships, only the octave can preserve the overall shape of the original overtone structure to the greatest extent. Other intervals, even if they also follow simple integer ratios, inevitably introduce more non-overlapping frequency components, thereby producing a more pronounced “sense of separation” in hearing.

Therefore, when we say “an octave is different heights of the same note,” we are not using a metaphorical expression, but describing a real auditory compression result. When faced with two highly similar overtone structures, the auditory system chooses to ignore the difference in absolute frequencies and instead emphasizes the unity of the structure itself.

Looking back at the design of “note names” from this perspective, it also seems logical. The reason why C2, C3, C4, C5... share the same note name is not to simplify notation, but because on an auditory level, they are inherently perceived as different ways of unfolding the same class of sound.

Furthermore,the octave is not the starting point of the musical system, but an answer actively provided by the auditory system. Before humans began to construct scales and name pitches, hearing had already taken the first step in classifying sound structures. And music theory merely followed this already existing perceptual path to fix, abstract, and record it.

Once this is understood, the “sense of octave” is no longer a rule to be memorized by rote, but becomes a phenomenon that can be intuitively understood: when two sounds are sufficiently unified in their internal structure, humans naturally hear them as “the same note.”

2 When multiple sounds appear simultaneously: How hearing establishes a sense of unity

In real musical scenarios, it is almost impossible for us to hear only an isolated sound. Whether listening to a song, playing a piece on the piano, or simply accompanying a voice with a guitar,multiple sounds appearing simultaneously, is the more common and natural state in music.

When multiple sounds are heard simultaneously, the situation faced by the auditory system is no longer just “what sound is this,” but has quietly changed:Will these sounds blend together, or will they interfere with and pull against each other?

On a physical level, this meansmultiple sets of complete, independent frequency structuresare superimposed on the same timeline: each sound contains its own fundamental frequency and a whole set of overtones; when they exist simultaneously, these frequency components act together on the auditory system. From a purely quantitative standpoint, this is obviously more complex than a single tone. But for hearing, the real challenge is not that “there are more frequencies,” but rather:whether these frequencies can be organized into a comprehensible whole。

The auditory system does not analyze the structures of these sounds one by one as completely independent sources of information; instead, it quickly judges whether there is some kind ofrelationship that can be unifiedThe core of this judgment lies not in any specific frequency, but inwhether there is a regular correspondence between the multiple sets of overtone structures。

When a clear, stable ratio can be found between the overtone structures of multiple sounds (such as the 2:1 ratio between octaves, or the frequency relationships of thirds and fifths in common chords), the auditory system tends to perceive them as different components of the same whole. Even if these sounds are not identical in pitch, they will sound natural and stable, as if they “were meant to exist together.” This ratio, much like the integer-multiple rule among overtones, provides a reliable structural anchor for the auditory system.

Conversely, if these overtone structures lack such a correspondence, it is difficult for the auditory system to establish a unified framework of understanding. In this case, the sounds do not truly blend, but instead present a sense of separation, tension, or even clutter. In other words, when multiple sounds appear simultaneously, the human ear is not simply “hearing many tones,” but is subconsciously comparing:whether these structures can be placed into the same perceptual system。

This comparison is almost instantaneous. The auditory system automatically searches for alignable frequency components and simple, stable ratios. As long as enough anchors are found, the perception of sound tends to stabilize; once these anchors are insufficient, a sense of tension arises.

From this perspective, an interval is not an abstract distance between several pitches, but ratherthe degree of similarity and coordination between multiple sets of overtone structuresThe so-called “consonance” and “dissonance” are, in essence, the auditory system's direct feedback to this structural relationship. It is precisely under this comparison mechanism that humans have gradually developed stable intuitions about certain intervals. These intervals of sound are not a coincidence to appear so frequently in music, but because their overtone structures are easily integrated by the auditory system, making it easy to form a sense of unity.

Once this is understood, looking at the chords and intervals frequently used in music reveals that they seem special not because of human convention, but because they happen to align with the positions where the auditory system most easily achieves unity—which is the very root of musical harmony.

3 Fifths and scales: How ratio relationships construct a musical framework

In the previous section, we discussed how the auditory system searches for clear, stable ratios when multiple sounds appear simultaneously, thereby integrating complex vibrations into a comprehensible whole. Among all interval relationships,the fifthis precisely one of the most typical and easily perceived examples of this stable ratio in musical practice.

The so-calledperfect fifthrefers to a very simple and stable frequency relationship between two pitches:namely, the frequency of the upper note is approximately 3/2 times that of the lower noteThat is to say, if the fundamental frequency of a certain note is f, then the frequency of the note a perfect fifth higher is approximately f × 3/2.

This ratio is not artificially prescribed, but comes directly from the overtone structure of sound. Taking C and G as an example, in the overtone series of C, the third overtone happens to be highly close to the fundamental frequency of G; in the overtone series of G, there are multiple components that can form simple integer-multiple correspondences with the overtones of C. Precisely because there is a large amount of overlap and matching between the two sets of overtone structures, the auditory system, when processing these two notes, requires almost no extra “computation” to naturally integrate them into a stable, harmonious whole.

In other words, the reason the fifth sounds “stable” is not because we have studied music theory, but because at the overtone level, it is inherently astructural relationship that is easiest for the auditory system to understand。

If we look at this relationship in terms of actual frequencies, it becomes even more intuitive. Taking the common middle C (C4) as an example, its frequency is approximately 261.63 Hz. Stacking a perfect fifth upward, the corresponding frequency should be: 261.63 × 3 / 2 ≈ 392 Hz, and this value falls right within the frequency range of G4 (approx. 392 Hz). That is to say,C4 → G4 this pair of pitches naturally approximates a 2:3 relationship in physical frequency, rather than being a result of “tuning” after the fact.

Continuing to stack perfect fifths upward following the same logic yields a continuous chain of fifths:C → G → D → A → E → B → …

In this chain, the meaning of each step is consistent:the frequency of the next note is approximately 3/2 of the previous oneFor example, stacking a fifth above G yields D, and stacking another fifth above D yields A. If you compare this with the frequency table of each note listed earlier and perform a simple calculation, you will find that the relationships between these notes are numerically continuous and consistent, rather than scattered or accidental.

It should be noted that this logic of “continuous fifths” does not require all pitches to fall within the same octave. In actual music, pitches are often brought back into a suitable range by shifting them up or down octaves, but when the auditory system judges a fifth relationship, it always focuses onthe ratio matching between overtones, rather than the absolute height of the frequencies. Therefore, even if two notes are separated by one or more octaves, as long as they are close to 3/2 in frequency ratio, the sense of stability of the fifth can still be clearly perceived.

From this perspective, the importance of the fifth is not just an “interval name,” but an underlying logic that runs through single notes, multiple notes, and even scale construction. It is precisely this highly stable and easily integrated ratio relationship in the overtone structure that makes the fifth an irreplaceable foundational puzzle piece in musical perception, and provides a clear and natural starting point for the subsequent formation of the scale system.

As a side note, the fifth relationship describesthe auditory closeness and distance between pitches, for example, C and G, and G and D are naturally close in hearing; whereas the piano keyboard does not arrange pitches according to this auditory order, but rather takes these note names (C, D, E, F, G, A, B)in a way that suits two-handed performance and memoryand projects them into a linear key structure—the keyboard prioritizes retaining the seven most commonly used and easily auditorily integrated note names as the basic skeleton of the white keys, while inserting the remaining pitches as black keys, thereby simultaneously satisfying the needs of playability and scale completeness within a limited physical space:

The two seem to conflict, but in fact, each serves completely different cognitive and practical needs. It is precisely in this tension between auditory structure and expressive tools that, when humans attempt to construct acomplete and closed scale system, new problems truly arise.

Additional Knowledge: 12-Tone Equal Temperament and the Overflow of Fifths

In the previous section, we discussed how the continuous stacking of fifths naturally constructs a stable sequence of pitches. Each time a perfect fifth is stacked upward, a highly harmonious interval in hearing is obtained; the matching between overtone structures makes this relationship perceivable and acceptable with almost no learning. From the perspective of a single interval, this chain composed of fifths is almost “perfect.”

But this is precisely where the problem arises:This perfect chain of fifths cannot naturally close into a finite scale system in reality—if we start from C and repeatedly stack perfect fifths upward, we get a sequence like this: C → G → D → A → E → B → F# → C# → G# → D# → A# → F → > C′ (octave higher)。

Throughout this process, the logic of each step is consistent: the frequency of the next note is approximately 3/2 of the previous one, and aurally, these adjacent notes still maintain a familiar and stable relationship of fifths.

However, when we complete twelve such stackings, the note we theoretically return to, the C an octave higher (C′), does notcompletely coincide in frequency with the exact octave relationship of the starting CThis phenomenon of “being just a little bit off, yet impossible to align” is known as thePythagorean commaIt reveals a very crucial fact:A chain of fifths relying purely on perfect ratios cannot form a self-consistent closed loop within a finite scale：

In other words, if we want to both retain the stability of fifths and have the scale cycle completely within an octave, we must make a compromise somewhere.

Twelve-tone equal temperament is precisely the compromise born out of such practical limitations—it no longer attempts to keep every interval at its most “pure” integer ratio, but instead chooses to approach it from a holistic perspective:dividing an octave (the interval where frequency doubles) into twelve equal geometric steps, so that starting from any note, after twelve identical interval changes, it can accurately return to the starting point of the octave, as represented by a clockwise circle in the figure below:

The cost of doing this is obvious: in twelve-tone equal temperament, fifths are no longer an absolutely pure 3:2, and thirds no longer perfectly fit the natural overtone ratios. Every interval has been “slightly adjusted.” However, these deviations are distributed evenly across the entire scale, small enough that the auditory system can still smoothly integrate the overtone structure without producing obvious discomfort or conflict.

From the auditory result, this is avery clever engineering compromise: octave relationships remain clear; most fifths are still very close to ideal ratios; yet the entire scale system gains unprecedented unity and flexibility.

Because of this, twelve-tone equal temperament is not meant to be “purer,” but rather “more usable.” It allows music to modulate freely, keeps keyboard instruments playable in all keys, and provides a stable foundation for complex harmony and polyphonic writing.

From this perspective, twelve-tone equal temperament is not a negation of natural ratios, but a rational choice after acknowledging practical limitations. It compromises and connects human auditory preferences for harmonious intervals with the needs for unity and operability in musical practice, thereby laying the foundation for the modern musical framework we are familiar with today.

4 The auditory logic from octaves to chords

In the previous sections, we focused on the uniqueness of octaves and fifths: they maintain a high degree of integer-ratio relationships in the overtone structure, making them extremely easy for the auditory system to integrate into stable, natural intervals. Besides octaves and fifths, there are many other commonly used intervals in music, such as fourths, thirds, and various major and minor thirds, each with its own characteristics in overtone matching.

Taking the fourth as an example, its frequency ratio is about 4:3, while thirds are about 5:4 or 6:5 (major and minor thirds). Although these ratios are not as perfectly neat as octaves or fifths, they still form relatively simple integer relationships. Aurally, this means that some corresponding points can still be found between their overtone series, thereby producing a certain sense of stability and fusion. It is precisely this degree of matching that allows fourths and thirds to appear naturally in harmonic construction and melodic progression without causing obvious tension or clutter.

Furthermore, the perception of harmony and chords is essentially an intuitive judgment of the matching of multiple sets of overtone structures. When multiple notes sound together, the auditory system automatically searches for integer ratio relationships between frequencies, thereby forming a unified perceptual whole. The simpler the ratio relationship and the clearer the match, the more stable and harmonious the perception; when the ratio is complex or the overtone conflict is severe, the perception will seem tense or unstable. This is why certain chords are widely used across different cultures and musical systems, while some extreme interval combinations sound “harsh” or difficult to integrate into the whole.

In summary, octaves and fifths are merely the most extreme and intuitive examples in the world of intervals, making it easy for us to understand how our hearing integrates overtones. The existence of other intervals, however, demonstrates the flexibility of the auditory system when faced with complex sounds: it does not require perfect, neat integer-ratio matching, but instead relies on relatively clear proportional relationships to form comprehensible harmonic structures. This mechanism not only explains the sense of stability in commonly used intervals but also provides a physiological and physical foundation for the perception of chords, modes, and melodies.

4 The history of vocal music and scale exploration

1 From singing practice to scale systems

In the vocal practices of early humans, there were no fixed note names, scales, or systematic theoretical descriptions. Whether in singing or the use of instruments, there was always only one core basis—whether it sounded rightOver long periods of practice, people gradually discovered that certain combinations of intervals sounded stable, natural, and were easy to accept and remember, while other combinations brought tension, discomfort, or even harshness. This seemingly “empirical” judgment was not a pure aesthetic preference, but was closely related to how the human auditory system perceives the internal structure of sound.

From the perspective of auditory mechanisms, when the frequency relationship between two sounds presents a relatively simple and stable ratio, the overtone structure they produce is more easily integrated by the human ear into a single whole, thereby creating a “harmonious” and “stable” auditory perception. Conversely, combinations with complex ratios and more overtone conflicts are more likely to trigger tension and instability. It was precisely through this repeated auditory feedback that early vocal practices gradually formed a set ofunnamed yet highly consistent criteria for judgment: what kind of sound was “right” and what kind of combination was “smooth.”

At this stage, the transmission of music did not rely on abstract rules or explicit concepts, but was built more uponauditory consensus and physical imitationSingers would not explain that “this is a certain interval” or “this conforms to a certain ratio”; instead, through demonstration, singing along, and repeated correction, they allowed the sound to gradually approach a stable result mutually recognized by the group. Precisely because the human auditory system possesses high consistency in judging stable overtone structures, this oral transmission method centered on auditory perception was able to endure effectively over the long term even without theoretical support.

The ancient Chinese system of Gong, Shang, Jue, Zhi, and Yu (pronounced gōng, shāng, jué, zhǐ, yǔ, which are the names of the five different notes in the Chinese pentatonic scale, corresponding respectively to 1, 2, 3, 5, and 6 in modern numbered musical notation) is precisely the result of the long-term accumulation of such vocal practices centered on auditory stability:

A seemingly simple yet extremely crucial fact is that:this system only retained 1, 2, 3, 5, and 6, while deliberately omitting 4 and 7. This was not because the ancients “did not know” these two notes, but because, in vocal practices where auditory stability was the primary goal, they were not the optimal choices.

If we start from the physical structure of sound, this selection process is actually very natural. Early pitch relationships were not constructed starting from abstract scales, but originated from the continuous stacking and comparison ofperfect fifth relationships (frequency ratio of 3:2)When a note is repeatedly stacked with perfect fifths upward or downward, it preferentially generates a set of pitches that highly match in overtone structure and easily blend with one another. Compressing these notes back into the same octave naturally yields a set of stable notes:1, 2, 3, 5, 6。

In contrast,4 and 7 are not notes naturally generated through this “perfect fifth path”They have a lower degree of matching in overtone ratios with the tonic and core stable notes, making them more likely to generate tension and instability. Especially 7, which auditorily “points” strongly to the tonic, forcing the melody to resolve; and 4, which also tends to create a pull between 3 and 5. While these types of notes became important driving forces for melodic development in later tonal music, they were not suitable for prolonged dwelling in early vocal practices centered on smooth flow and auditory comfort.

Therefore, Gong, Shang, Jue, Zhi, and Yu is not a “simplified scale,” but rather anactive selection of the most auditorily stable pitch relationships: it retains those notes that are easiest to integrate in overtone structure and create the least internal conflict, thereby forming a sound space that can flow freely without forcing resolution. It is precisely in this kind of space that melodies can unfold naturally without being held back by excessive tension.

This system did not originate from precise calculations of absolute frequencies, but rather, through countless trial singings, corrections, and imitations, allowed the sound to gradually approach those “smoothest and most stable” positions. In this sense, what Gong, Shang, Jue, Zhi, and Yu records is not some abstract rule, but rather the human auditory system's recognition, through long-term practice, ofovertone stabilityThe collective choice made.

In the development of Western music, we can also see a path highly similar to *Gong, Shang, Jue, Zhi, Yu*, but leading to a different destination. Early church modes and folk melodies were likewise built on the stable relationship of perfect fifths and overtones: by continuously stacking and resolving fifths, they formed a set of pitches highly blended and harmonious to the human ear. This process is essentially no different from the Eastern generation of pentatonic scales through fifths.

The real divergence appearedafter polyphonic music became the core mode of expression. With the emergence of harmony, polyphony, and functional accompaniment, music was no longer just about “how the melody flows smoothly,” but began to take on a new task:how to construct a sense of direction and belonging when multiple voices exist simultaneously. At this point, the pentatonic system consisting only of 1, 2, 3, 5, and 6 was no longer sufficient, because it lacked a strong enough “pull” and “return mechanism.”

It was precisely under this demand that 4 and 7, which were originally aurally unstable, were systematically introduced. They formed more complex ratios with the tonic in the overtone structure, generating obvious tension: 4 tends to resolve to 3 or 5, while 7 points strongly to the tonic 1. It is precisely this “instability” that provided a clear sense of direction for polyphonic music, allowing harmonic progressions to form a “departure-deviation-return” structure. This mechanism was eventually fixed in the major and minor systems, forming the backbone of modern tonal music.

From this perspective, the Western heptatonic scale is not a simple “extension” of the pentatonic scale, but rather astructural choice made for harmony and tonality: it preserved the stable core generated by fifths (1, 2, 3, 5, 6) while introducing 4 and 7 to create tension and direction, thereby forming a sound system that can both rest stably and be continuously propelled forward.

Therefore, whether it is the Eastern *Gong, Shang, Jue, Zhi, Yu* or the Western major and minor keys, they are not products of theory-first design, but rather two different “balanced solutions” naturally selected under the dual constraints of the human auditory system and musical practice—one favoring the stability of melodic flow, the other favoring the directionality of harmonic structure.

2 The presentation of major and minor keys in hearing

In the previous section, we mentioned that major and minor keys are not designed systems, but rather the result of gradual stabilization through long-term singing practice. So, when these structures are actually sung and heard, what exactly is the ear perceiving? To answer this question, we need to temporarily set aside history and naming itself, and return to the most direct auditory experience.

In modern vocal music, major and minor keys are almost ubiquitous.When a melody is sung and heard, the pitch arrangement in a major scale is usually integrated by the ear into a bright, stable, and open overall feeling; whereas the minor scale, due to changes in some interval relationships, is more likely to present a soft, slightly introspective auditory color.These differences do not come from the attachment of emotional labels, but stem from the differences in how well the pitch arrangements match the overtone series, as well as the auditory system's natural response to stable structures.

Taking the most common C major as an example, the scale consists of eight notes, “C D E F G A B C”, where semitones occur between E-F and B-C, and the rest are whole tones, corresponding exactly to a set of white keys on the piano:

This arrangement of pitch relationships allows the ratios between octaves, fifths, fourths, and even thirds to highly match the overtone series, thus sounding stable and natural.

In contrast, C minor consists of eight notes, “C D E♭ F G A♭ B♭ C” (where the notes with ♭ form a semitone with the preceding note). This scale arrangement slightly alters the ratios between intervals, making it sound soft, slightly melancholy, or introspective, in sharp contrast to the major key. This subtle difference is also an important tool for composers to evoke emotions in melodic and harmonic creation.

The charm of major and minor keys is reflected not only in the progression of simple melodies, but also in the construction of harmony and chords. By choosing different combinations of intervals, vocal and instrumental performances can create rich emotional layers within the same tonality: simple triads can convey a bright or warm feeling, while extended chords and modal changes in progression can bring tension, anticipation, or mystery. All of this relies on how the notes in the scale match the overtone structure.

Because of this, major and minor keys have become the primary framework for modern Western vocal composition and performance: whether in opera, art songs, or popular music, most melodies and harmonies are based on these two modes. They are both the fruit of historical exploration and the natural product of the combination of auditory perception and musical practice.

5 Summary and reflections

Looking back, we started all the way from the physical properties of sound, gradually extending to how hearing perceives pitch, intervals, and the relationships between multiple sounds. The overtone structure within a single sound allows us to “hear” a whole set of complex vibrations as "a single tone"; while special intervals like octaves and fifths rely on stable and simple frequency ratios, allowing the auditory system to integrate different pitches into a natural whole with almost no extra judgment. It is precisely this structural stability that makes certain intervals sound exceptionally harmonious, bringing a near-instinctive sense of comfort.

When bringing the perspective back to historical vocal practices, you will find that this mechanism is not an abstract theory, but an empirical result repeatedly verified by humans. Whether it is China's *Gong, Shang, Jue, Zhi, Yu*, or the major and minor systems that gradually formed in the West, humans, through thousands of years of singing and instrumental experimentation, have always been filtering thoseinterval combinations that are easily integrated by hearing, easily remembered, and more easily passed down.The modern vocal system is not a set of rules designed out of thin air, but the result of gradual settling along the natural preferences of the auditory system over long-term practice—it is not a matter of “who decided what sounds good,” but rather our ears making similar choices time and time again.

It is worth noting that these laws do not depend on cultural background or theoretical knowledge, but stem from the physical structure of sound itself and the way the human auditory system works. Because of this, certain intervals and structures can transcend regions and eras, repeatedly appearing in different musical systems, and being perceived and understood in similar ways by different people.

Therefore, what this article really wants to convey is not a new set of vocal conclusions, but a perspective on understanding music: harmony and beauty in music are not accidental aesthetic preferences, but are deeply rooted in the physical properties of sound and our own auditory mechanisms. When you realize this, and look back at practicing your voice, listening to songs, or even composing melodies, those subtle differences that could originally only be judged by “feeling” will often become clearer and more credible—not because feeling has been replaced, but because you begin to know,why it feels that way.

Having written this article, I would like to reiterate: it is not a vocal tutorial, nor is it an encyclopedia of music theory, but rather like a personal path of exploration—an attempt to figure out what exactly we are listening to when we listen to music. For professionals, some of the content here may be too basic; for general readers, perhaps some details are slightly obscure. But in any case, if you are willing to read through it patiently, you will see that the development of sound, intervals, and scales is actually closely linked to physical structures, auditory psychology, and historical practices.

The value of this article may not lie in making you sing better immediately, or making you understand all music theory right away, but in providing a way of thinking: trying to connect scattered musical phenomena and knowledge into a more complete and “understandable” whole. Hope it can provide some clues or inspiration for your curiosity about sound.