Exploring Major and Minor
Tonality ①

The major scale is generally defined by the following values in twelve-tone equal temperament.

Major scale (12ET)

Hover over the floating ratios to hear them. The waveform and volume control are at the top right.

In practice, we rarely memorize these numbers. We are more likely to remember the layout of each key on an instrument as a pattern, or to work with intervals and the spacing of whole tones and semitones.

When I began studying music theory, I was surprised that the familiar do–re–mi–fa… (C–D–E–F… in the C-major example used here) was not an evenly spaced sequence of frequencies—or, more precisely, of logarithmic frequencies. On the piano, E and F are adjacent white keys, yet the interval between them is only a semitone.

I kept imagining that this familiar musical scale might have a simple underlying structure that science could describe.

Few people would regard its definition in 12ET as the origin of the major scale. Historically, mathematicians and musicians explored many tunings before a theory of music developed within 12ET, whose sound changes relatively little when transposed to different keys.

The major scale is commonly said to originate in simple integer ratios between frequencies. The major chord is a familiar example: the frequencies of C–E–G have the ratio

Pitches in the ratio ( $1$, $\frac{5}{4}$, $\frac{3}{2}$ ) are said to produce a major chord.

That describes “major” as a chord. For the major scale, the tuning known as just intonation is defined by

—the sequence of fractional ratios above. But I do not think this is the origin of the major scale. I favor the following just-ratio major scale, for reasons grounded more directly in its structure:

$\frac{5}{4}$-$\frac{21}{16}$-$\frac{3}{2}$ : try moving back and forth through these notes—mi–fa–sol–fa–mi…♪. I think you will hear the difference from conventional just intonation ( $\frac{5}{4}$-$\frac{4}{3}$-$\frac{3}{2}$ ) quite clearly.

I call this Harmonics Major (H-Major) on this site, and refer to the conventional just-intonation scale as Just Intonation Major (JI-Major).

Following the reasons behind H-Major offers a way to understand why humans can acquire relative pitch, and what major tonality means in terms of sound. This site will explore those questions in detail.

Before discussing the auditory phenomena that humans perceive as music, we need an important premise: the pitch-class principle.

This is what it means for both high C and low C to be C. “Disregarded” does not mean that we lose the ability to distinguish their pitches: we can recognize a shared identity despite their separation in pitch. This applies to arbitrary frequencies, not just those in 12ET, and the space of all perceivable pitch classes forms a circle. This circle corresponds to what music psychology calls “chroma.” For simplicity, and because this discussion extends somewhat beyond the usual scope of that term, I use “pitch class” here.

The next important auditory concept is harmonics. In Japanese these are usually called baion; here I use the English term “harmonics.” I define the following principle.

Harmonics 1–16. Each tone continues sounding as the next is added.

When harmonics are played successively or simultaneously, we can recognize them as a series belonging to the same group. Put simply, we can recognize harmonics as harmonics. Combining this with the pitch-class principle reveals an important auditory theorem.

Odd harmonics: 1, 3, …, 15. Pitch classes: hold the root while following 1, 3, 5, 7… slowly, at 1.4-second intervals.

When referring specifically to this group as pitch classes, I call it an odd-harmonic pitch-class series, sometimes shortened to an odd-harmonic series or an odd series. This series will be central to our discussion of pitch-class relationships in harmony.

Looking back at my proposed major scale, we can see that it consists of an odd-harmonic pitch-class series. Ratios alone may be hard to follow, so let us call it C major and assign note names:

Conventional just intonation is

The values of F and A differ between H-Major and JI-Major.

Now let us remove powers of two. Since $\frac{1}{2}$=$2^{-1}$ , H-Major becomes

—all odd integers. Every pitch class in H-Major therefore belongs to C's odd-harmonic series. JI-Major, on the other hand, becomes

Only powers of two may be disregarded, so F and A do not belong to C's odd-harmonic series. Nevertheless, these are still simple integer ratios. Let us divide the entire set by F's value and look at JI-Major with F as the reference. Dividing by $\frac{1}{3}$ means multiplying the entire set by three.

Thus JI C major turns out to be an odd-harmonic series of F. As a complete aside, this makes me wonder whether George Russell, who developed the Lydian Chromatic Concept, might have heard the major scale in this way.

From a perspective that emphasizes harmonic series, Harmonics Major is preferable because its root—the pitch class corresponding to “1”—coincides with the note we actually want to regard as the tonic: C in this example.

Some digital pianos offer conventional just intonation as a tuning option. When playing pieces with that setting, I often felt that something sounded uncomfortable. Looking back, I think the note I wanted to hear as the tonic did not coincide with the harmonic root.

This alone may not persuade everyone. You might be thinking, “You defined the harmonics principle to make Harmonics Major work.” But I can explain why harmonics deserve this emphasis, and why hearing supports it. The key is combination tones.

Combination tones have frequencies obtained by applying integer coefficients to sounding frequencies. For example, for two frequencies $f_1$, $f_2$($f_1$<$f_2$ ), these include

$f_2$ – $f_1$, $f_1$+$f_2$, 2×$f_1$ – $f_2$…

Zero-frequency results are excluded. Human hearing is said to be able to perceive such frequencies. There are studies supporting this, and you can try listening for them here.

In my experience, combination tones are easier to grasp as a change in the quality of a sound than as separate tones. Listen to these examples. Can you hear a difference between playing the first and second tones separately and playing them together?

To me, saying that the two tones “interfere” captures the impression, and I think it is also close to the physical phenomenon. I interpret the trembling or roughness heard when they sound together—different listeners may describe it differently—as an integration of several combination-tone components presumed to arise in the ear, or perhaps in perception within the brain.

Since combination tones have frequencies, it may also be possible, with effort, to perceive them as pitches. The difference tone corresponding to $f_2$ – $f_1$ seems comparatively easy to hear, so if you trust your ear, try listening several times. Here, taking tone ① as $1$ , tone ② is at $\frac{5}{4}$ . The difference tone is therefore

A reference sine wave. It is not added to the two-tone playback.

That is, $\frac{1}{4}$=$2^{-2}$ : a pitch two octaves below tone ①, with the same pitch class. In my experience, however, the important point is that it does not sound like a pure sine wave at the frequency $\frac{1}{4}$ . This remains a hypothesis, but I suspect that harmonics of the tone at $\frac{1}{4}$ —that is, harmonics of the difference tone—also arise, giving it a timbre unlike a sine wave. The set of frequencies calculated as combination tones does, in fact, include components corresponding to these harmonics. I have not found research that establishes how many such components arise in the ear, so I do not know whether harmonics really occur here or how far they extend. To me, though, the difference tone seems to sound faintly with a timbre resembling a triangle wave.

A triangle-wave reference at C2, the difference-tone frequency. It is not added to the two-tone playback.

Since timbre has come up, let me add a related point. This is close to the idea of a Fourier series: under suitable conditions, harmonics combined with a particular sequence of amplitudes are perceived less as a chord of several tones than as one sound with a particular timbre. Harmonics can form something like a “shape of sound.”

I suspect a deep connection between harmonics forming timbre and musical chords being built from harmonics. Not all chords consist of harmonics, of course, but the major chord mentioned near the beginning has the ratios ( $1$,$\frac{5}{4}$,$\frac{3}{2}$ ), corresponding to the odd-harmonic pitch classes $1$, $3$, $5$ .

That was a small detour. The main point is that combination tones are closely connected to the reasons for emphasizing harmonics.

There is one important qualification about combination tones. As already noted, the earlier example was

In pitch-class terms, the result is $\frac{1}{4}$~$1$ , where “~” is a simple notation for equivalence of pitch class. This is an example in which a chord's difference tone returns to one of the chord's own constituent pitch classes. Let us tentatively call this a return to a chord tone. But what happens if we use the pitch-class principle to change the octave relationship of the two notes? Starting from $\frac{5}{4}$ and removing the factor $\frac{1}{4}$=$2^{-2}$ gives

Here, $4$=$2^2$×$1$~$1$ , so the return to a chord tone is preserved. But with $\frac{5}{2}$(~$\frac{5}{4}$ ), we get

This pitch class was not in the original chord. Simply applying octave equivalence to a chord can therefore break the property of returning to one of its own constituent pitch classes.

Does that make combination tones incompatible with the pitch-class principle? Do they contradict one another? No. The harmonics principle comes into play again, precisely to resolve this issue.

Look more closely at the last example. With $\frac{5}{2}$ , the resulting tone was $\frac{3}{2}$ . In pitch-class terms, this is an odd harmonic of $1$ ( $\frac{3}{2}$~$3$ ). In other words, the difference-tone calculation is closed in the following sense: a chord built from odd harmonics of the root ( $1$ ) produces combination tones that return to the same odd-harmonic series. This can in fact be proved mathematically for general combination tones.

Under the harmonics principle, humans can recognize harmonics as a group—as one series. Taking octave equivalence into account, it therefore poses no problem if a combination tone falls outside the pitch classes actually sounding in the chord: it still belongs to the root’s harmonic pitch-class series. The chord and its combination tones, considered together, still belong to the same harmonic series.

This is why I came to think that harmonics deserve special emphasis. Harmonics draw even their combination tones into their own series.

The next section gives a mathematical proof of this closure.

I said “the same series” because an odd-harmonic series is defined relative to a reference frequency, $f_{\mathrm{ref}}$ . The harmonics of C and those of F, although they do share some tones, differ as a whole and can be regarded as different series. Among such series with different reference frequencies, the claim is that combination tones return to the harmonic series of the same reference.

For an odd-harmonic pitch-class series with reference frequency $f_{\mathrm{ref}}$ , an arbitrary member can be written as

Here, $k_i$ is the exponent of the power of two that determines octave placement, and may be any integer, including a negative one. $q_i$ is odd, so $f_i$ can be regarded as an odd multiple of $f_{\mathrm{ref}}$ placed in an arbitrary octave.

The combination tones arising from a chord built from this odd-harmonic series can then be written as

$m_i$ is an integer coefficient. The expression gives the combination-tone frequency $f_{\mathrm{comb}}$ obtained by assigning integer coefficients to frequencies from the odd-harmonic series. The absolute value is used because a negative frequency has no meaning here.

Since $f_{\mathrm{ref}}$ itself is positive, we have

Some values of $k_i$ may be negative, so the quantity inside the absolute value need not be an integer. Let $k_i$ have minimum value A = min $k_i$ among the terms under consideration. Then

The absolute-value expression is now a nonnegative integer. Call it P. Since we exclude zero-frequency combination tones, $f_{\mathrm{comb}}$ ≠ 0 implies P ≠ 0. Thus P is a positive integer and can be written as

Here, q is odd and l is a nonnegative integer.

Consequently,

In pitch-class terms,

The combination tone therefore belongs to the original odd-harmonic pitch-class series.

For the odd-harmonic pitch-class series generated by the reference frequency $f_{\mathrm{ref}}$ , every combination tone arising within it belongs to the same series.

The preceding proof used frequencies directly for clarity, but the subject is pitch classes. The odd-harmonic pitch-class series based on $f_{\mathrm{ref}}$ and that based on $2^k$ ×$f_{\mathrm{ref}}$ are identical. This suggests that the reference of an odd-harmonic pitch-class series should itself be considered as a pitch class. Let us examine that idea.

Let us call the frequencies discussed so far physical frequencies, denoted by f, a positive real number. A standard representation of pitch class is

I call this a pitch-class coordinate. Taking a base-two logarithm turns an octave upward into +1 and an octave downward into −1. Disregarding integer differences (mod 1) extracts the pitch-class information alone. The values of h lie in [0, 1), with 0 and 1 identified, so the resulting space is a circle.

$f_0$ determines the origin of the pitch-class coordinates. In practice, f will usually be a frequency in hertz from an audio spectrum, so setting $f_0$ = 440 Hz, the standard A4 tuning frequency, is convenient. The origin is then A in 12ET at A4 = 440 Hz, and ordinary 12ET pitch-class positions are simply

Here, k is an integer.

Taking the equal-tempered major scale introduced at the very beginning, with A as its tonic, gives

In these coordinates, C major becomes

The apparent difference in form is only due to the shifted origin. Define

to move the coordinate origin to the pitch class $h_{\mathrm{ref}}$ .

To compare scales in different keys, we can fix h and rotate the viewpoint using u. Alternatively, we can choose the desired pitch class as the origin from the beginning. The essential point is that pitch class is the fractional part of frequency on a base-two logarithmic axis.

More generally, n equal divisions of the octave can be calculated simply as

Suppose we fix an absolute pitch-class coordinate system and choose a reference pitch class $h_{\mathrm{ref}}$ . For a positive integer k, the pitch-class position of its kth harmonic is

To select only the odd-harmonic pitch-class series, let l be a positive integer. Then 2l − 1 runs through the odd integers starting at 1, giving

The graph below uses C as the origin of h and plots the harmonics for $h_{\mathrm{ref}}$ = C.

This makes it visually clear that, under octave equivalence, the odd harmonics provide the distinct positions. For example, a C-major chord lies close to positions 1, 3, and 5 of the odd-harmonic pitch-class series. As promised, H-Major also lies on this series. Highlighting its positions with dotted lines gives the following:

I have included deviations in cents to make the differences between H-Major and the equal-tempered major scale easier to see. Compare these as well.

Section 6 proved closure using a physical reference frequency, $f_{\mathrm{ref}}$ . However, all frequencies of the form $2^k$ × $f_{\mathrm{ref}}$ have the same pitch class, $h_{\mathrm{ref}}$ . We can therefore restate the result: the odd-harmonic pitch-class series with a given pitch-class reference, $h_{\mathrm{ref}}$ , is closed under combination tones.

Recall that Harmonics Major on C was

Since every scale tone belongs to C's odd-harmonic pitch-class series, all combination tones arising from chords built on this scale also return to C's harmonic series. If we replace the reference of H-Major, $h_{\mathrm{ref}}$ , with another pitch class X, we obtain a scale whose combination tones return to the series on X. Since pitch-class space is circular, the freedom to choose X also forms a circle. This is the primitive circle of tonality I envision. More elaborate circles, such as the circle of fifths, may also exist, but I see this circle as preceding them.

A further point about Harmonics Major: for clarity, let us keep the key—the pitch-class reference—at C. You may already have noticed gaps in the harmonic numbers used by H-Major. In order of harmonic number, the notes are

These numbers were not skipped merely to approximate the equal-tempered major scale. There is a structural reason. Divide the entire set by 3, which corresponds to G:

Retaining only the integers gives

Thus G–D–B–F–A form the consecutive odd harmonics of G. In C major, G is the dominant of C. This is therefore a dominant chord reinforced by harmonics.

By closure under combination tones, chords made from these notes produce combination tones in G's odd-harmonic series. You might now wonder: wasn't this C H-Major? The point is that

Returning to G's odd series also means staying inside C's odd series. And since G is

, the odd harmonic immediately after C, the C H-Major scale harmonically emphasizes both the root C and its next neighbor G. In music-theory terms, it brings out the characters of tonic and dominant.

G includes the odd harmonics through 9, but C has 1, 3, 5, and 9, leaving out 7. Adding 7 gives

It falls at the flattened seventh degree, ♭VII. In music theory, this just interval is known as the harmonic or natural seventh; as an odd harmonic, it is literally number 7. Here is how the scale sounds:

Does that ascending sound feel familiar? I call Harmonics Major with this additional seventh harmonic of C “Harmonics Major including 7,” or simply treat this as the full form of Harmonics Major. In H-Major, 7 is a note to bring in as needed.

C Harmonics Major including 7 contains the odd series through 9 for both C and G.

A dominant progression using all of these harmonics sounds like this:

The G reference lies below C. In each series, 1, 3, 5, and 7 are placed in the root's octave, and 9 in the octave above.

This is the kind of tonal character I mean. I favor Harmonics Major over conventional just intonation because it organizes the notes around the two characters of tonic and dominant. This division still lies entirely inside one series, the C series. Within it, an upper layer reinforces the harmonics of the root's next neighbor: 3 beside 1.

Simply taking every odd harmonic through 27 brings in many notes not intrinsically related to C major. Many of these can, however, be obtained from the odd harmonics of 5, the next neighbor after 3, and then of 7, and so on. They might therefore be understood as higher layers. I will not pursue that here: the present aim is to extract the skeleton, or roots, of the structure. The important idea is that reinforcing harmonics with further harmonics reveals the skeleton of a scale's major tonality—which, to me, means its harmonic character. Below is a scale made by simply folding the odd harmonics through 27 into one octave and sorting them upward. Hearing it may make the importance of taking harmonics of harmonics clearer.

How was it? We have considered “combination tones” arising in the human ear and the odd-harmonic series as a series consistent with them. I believe I have now shown that, within the “earliest part” of those odd harmonics—the odd harmonics of 1 and of 3—lies hidden a tuning remarkably similar to the major scale used in contemporary music.

I also argue that conventional just intonation is inconvenient in this respect: the reference of its encompassing odd-harmonic series is F, the fourth degree of the key. The note we want to perceive as the tonic is therefore displaced from the reference of the series in which the combination tones align.

If harmonics appear to be the origin of the major scale, what changes for music? Scientifically, I see a clue for studying human perception of tonality: why does a piece feel bright or dark, and how does that work? I imagine that research into the harmonic aspects of music could help build a better mathematical space for musical expression—perhaps a theory of all the “colors” of sound humans can perceive, analogous to color science.

Within musical culture, people have long sought ways to produce just intervals. I have heard that musicians playing acoustic instruments still practice and perform with these intervals in mind; knowledge of harmonics should be useful there. In practice, though, just intonation may often be understood as the possibility of more beautiful harmonies beyond equal temperament, while musicians rely on the refined ears and sensibilities of teachers and artists rather than strict numerical tuning. I do not reject that; I think it is a very healthy attitude. After all, I favor Harmonics Major over conventional just intonation. If H-Major were taken as “true,” the conventional just-intonation settings of digital pianos would themselves be problematic. Either way, it may be better to trust human harmonic hearing than to force every note onto a tuner. There is also the issue of modulation. My aim is not to impose rigid mathematical answers on music, but to explore what kind of mathematical map we can draw across the world of musical expression.

These findings could also be used directly in electronic music. Equal temperament can be understood as a tuning designed to preserve workable harmony through every transposition on instruments with fixed tuning, such as pianos. With electronic instruments or music-production software, suitable tools could let us move the tonic freely while tuning melodies and chords to just ratios according to context. Microtonality and equal temperaments with arbitrary numbers of divisions seem to be attracting interest, but fractional intervals can also be specified directly, without going through an equal-division system. I would welcome more composition that works directly with just intonation. Such experiments might reveal more about just tuning and harmonics, deepening the exploration of human musical sensibility.

I also suspect that much music composed in modern equal temperament is a projection of its composer's harmonic sensibility onto that tuning. Odd-harmonic intervals do not coincide with equal-tempered ones. Repeated pure fifths, as in Pythagorean tuning—repeated multiplication by 3 under octave equivalence—never close into an exact finite cycle. If equal temperament approximates the harmonic world while making it finite and closed under transposition, what kinds of movement become possible within a tuning that approximately reproduces tonal and harmonic feeling? That is how I think about composition in equal temperament.

This article is called “Exploring Major and Minor Tonality,” but I have not yet discussed the minor side. The basic minor chord has the frequency ratios

I agree with this description. Dividing these values by $\frac{3}{2}$ gives

In pitch-class terms, this is

The chord is thus made from a sequence of reciprocals of odd integers. These are called subharmonics—specifically, odd subharmonics.

Minor tonality emerges by following subharmonics. Together with the theory of harmonics, this opens into a more general world of the varied tonal expressions that humans can perceive.

I plan to discuss this in detail next time.

Wooly