Each harmonic division, shown here as whole number ratios of the whole string, does not merely correspond to a number or a geometric division; it is, more importantly, a unique, physically-derived point of singularity and resonance within the whole of the vibrating string.^{7See a demonstration of that at science.larouchepac.com (see “Resources” for full link). Johannes Kepler also showed that those divisions which produced consonances were related to the constructible and knowable divisions of a circle (or constructible polygons), and those which produced dissonances to the inconstructible and unknowable ones} Therefore, the notes of the musical scale are not arbitrary, but are a manifestation of an ordered resonance and structure within the physical world.

Now, having seen the set of harmonic divisions of the octave (those notes which form a harmonic interval with the whole string), take a look at how these divisions relate to one another: examine the intervals between them.^{8Each harmonic division creates a geometric proportion between the whole string and the shorter length created by the division (the speeds of their vibrations being in the inverse proportion). The interval between two harmonic divisions is the geometric proportion between them. For example, ⅔ and ¾, or 8/12 and 9/12 are in the proportion of 8 to 9. Thus, 8/9 is the interval between them.}

While in their construction, the harmonic divisions are derived one by one,^{9There are several, related ways of deriving the harmonic divisions by construction. In his Timaeus dialogue, Plato takes the geometric and the arithmetic means between the whole string and its half, then takes the difference between them (8/9) and builds major and minor scales from those elements. Similarly, the 16th century music theorist Zarlino takes the arithmetic and geometric means between two strings of lengths 1 and 2, and then the geometric and arithmetic means of those. Johannes Kepler, discoverer of physical astronomy, divides the whole string into proportions which are derivable from the knowable geometric divisions of the circle (Harmonice Mundi). In each case, the harmonic divisions are not derived all at once, but in sequence.} you can see that taken together as a set they create a symmetrical division of the whole space of the octave (half the string), with the first and last intervals being identical, the second and second to last, etc.^{10In this case, equality is measured by equal proportions between lengths of string, not the differences in their lengths. Even though the length of string between the whole string and the 5/6 division is greater than that between 3/5 and 1/2, their proportional relationships are the same, and therefore the heard “distances” (or the interval) will be equal.}

However, this sequence of intervals is not complete. When the harmonic divisions (containing both the major and minor 3rds and 6ths) are separated into the major scale and the minor scale, and the (dissonant) notes of the 2nd and 7th are added to complete the scales, the perfect symmetry is lost, and a simple dissymmetry emerges.^{11For more on how this dissymmetry emerges, see Kepler’s derivation of the major and minor octave scales in his Harmonice Mundi.}

These two dissymmetrical scales are the most basic structure of the harmonic system of the octave, which the more complex, interconnected system of 24 major and minor keys is built upon. However, even these are merely shadows of something which can not be explained by the system’s structure. To get an insight into the “why”—why the musical system is constructed in this dissymmetrical way, and not otherwise—we must go to the substance, that which is prior, of which the sounds are a mere shadow: the living domain of music.

Musical Inversion

Let’s now put the system of harmonics, which we’ve so far considered only as a non-living system, into motion.

In a good musical performance (of a great—or at least adequate—composition) the notes are not chosen beforehand—a fact which will probably surprise most readers, considering that the score of a musical piece by Bach or Beethoven, for example, was written down centuries before. The music must come and the notes must be sounded anew, as if an improvisation, within the unfolding creative process of the performer. The notes, therefore, clothe and make perceptible something antecedent, which can only be found “between” the audible tones, so to speak: ironies which provoke the mind of performer and listener alike to an insight into the actual, unspoken and unutterable meaning of the piece. These types of unuttered thoughts, expressed outwardly and veiled in language, are what the poet Percy Bysshe Shelley referred to as “the interpretation of a diviner nature through our own.”

One of the ironies which appears often in Classical musical composition, not as a formal device, but rather a natural consequence of such artistic efforts, is the action of inversion. In music, as in poetry and other meaningful modes of communication, a statement and its inversion are not equivalent. This non-equivalence is built into the potential of the musical system itself, and more importantly allows the composer to create an experience for the listener which is neither the musical “statement” itself, nor its inversion, but something in between: the unhearable simultaneity of the two.

To begin, examine inversion first as it is built into the structure of the harmonic system.

Take the dissymmetry which appears when we invert the major scale. Below is the major scale, with the intervals between each note indicated by a W, whole step, or H, half step.

The intervals of the ascending major scale are: W W H W W W H. What happens if we take the same intervals, in the same order, beginning from the same note (in this case, C), but invert the direction to create a descending scale?

Inverting the major scale thus, we get the notes of a minor scale, built on the the fourth note of the original key (in this case F). The minor and major are not actually separate, but are inversions of one another. This is seen (among countless other places) in Bach’s Musical Offering.^{12Bach was the first to use inversion as a principle of development in music in a major way, and his The Art of the Fugue in its entirety is a detailed pedagogy.}

In this example, A is the puzzle posed by Bach, and B is its solution. In the puzzle, Bach indicates for two voices to play the lower line, entering in canon (one coming in later than the other, at the segno). However, he writes two clefs, one upside down and displaced.

The puzzle is solved as shown in B, with the second voice reading the music upside down, and so the descending C minor scale of the first voice is transformed an ascending G major scale in the second. (This is the same relationship between major and minor keys as the example above, where the C major scale inverted to F minor. The note F is the 4th above C, just as C is the 4th above G.) Set in canon, it is as if one voice is having a discussion with its reflection. The voice and the reflection play back and forth contrapuntally (i.e. in their rhythmic and melodic interactions) as two characters engaged in conversation, and come together harmonically in a beautiful duet, with something said by the two that couldn’t have been said by either individually. Bach’s joining of the major and minor together, in harmony and by inversion, shows in practice that they are not actually separate modes.

Take another, related example of inversion, which hints at the major/minor relationship—the division of the fifth. The interval of a fifth (the relationship between the whole string and ⅔ of it) can be divided harmonically^{13“Divided” meaning that another note is placed in between the two notes which span the fifth, and “harmonically” meaning that the three resulting notes are consonant when sounded together.} in only one way: into the two thirds, major and minor. Said differently, a major third plus a minor third equals a fifth. Moving from one note via the major third, followed by the minor (to reach a fifth above) generates a major triad (the skeleton of a major scale),

and inverting that motion, starting from the fifth above and moving back down to the starting note, generates a minor triad.

Once again, the major and minor prove to be inversions of one another, and while the fifth remains the same, the inversion gives the two results very different meanings, musically (i.e. when you hear these two triads, they are quite distinct)! The same space (the fifth) has been traversed with the same intervals, but something completely different was said. The musical system is a space in which A+B ≠ B+A!

Now look at a more general example, where the same action taken in different directions generates something new.

In this example, C is the tonic (or basis) of the scale. Motion upwards in pitch by the interval of a fifth generates G, the fifth note of the C scale. The same action downwards generates a different note, F, which is not the fifth, but the fourth note of the C scale (as counted when moving upwards).

In order to invert the motion, but arrive at the same note of the scale (G) below the tonic, the interval of action must be truncated from a 5th to a 4th.

When we move up by the interval of a 4th, we arrive at F. When we move down by a 4th, we arrive at the G below, the 5th of the scale. Thus, the 4th and the 5th are inversions of one another.

Inherent to the musical system there are complementary pairs, like the 4th and 5th, which together add up to an octave, or said differently, are inversions of one another.

In practice, in the composing and experiencing of great music, these pairs are actually inseparable. One does not exist without, implicitly, its inversion. As we’ll see in a few examples below, composers from Bach through Brahms use the ambiguity of these pairs and their inversions to create a type of transformation and development which cannot be heard by the sensuous ears, but only by the “ears” of the mind.

Begin with Bach

Take the example below from Bach’s The Art of the Fugue.

At the top is the opening of Contrapunctus I, with the four-measure opening theme (outlined in blue). The theme rises by a 5th from D to A, then descends, outlining a D-minor triad (D-F-A) and continues down by a half-step to the leading tone C#. It then rises back up to the minor 3rd (F) before finally returning to the tonic, just as the second voice enters. Look at a few things that Bach does to transform and develop that theme which play with the relationship of the 5th and 4th.

First is the way that the second voice enters in measure 5 (outlined in red). This second voice echoes the opening theme, but rather than beginning on the same note as the first voice, D, it enters a 5th higher, on A. If the second voice rose by a 5th, in exact imitation of the first voice, it would land on the note E and would declare itself as being in a different key entirely (the key of A). Therefore, it rises just a 4th, reaching the D an octave above where the first voice started, and the space of the fugue is defined as the octave D-d, built on the D-minor arpeggio D-F-A-d, with the element of the C# leading tone below. Ironically, if this voice had risen a full 5th (as it does later in the fugue’s development, in measures 29-31), it would sound more distant from the original theme than does this slightly transformed version, because of the modulation in key.

The effect of the second voice rising a 4th rather than a 5th, due to the fact that two fifths cannot fit within the space of an octave, is that the space in which it unfolds the theme is compressed, and the theme is slightly changed by operating within a smaller space. The second interval becomes a 2nd from D to C♮, rather than a full 3rd, as in the first voice. The changed theme of the second voice introduces an element of sameness-and-difference right at the opening of the fugue.

Now look at what Bach does in Contrapunctus III, where his subject is an inversion of the theme. (See image above.)

As we saw with the simple inversion of the major scale, the musical system is naturally ordered and structured such that it is not possible to move in perfect inversion^{14A perfect inversion is where every interval is exactly the same, only inverted in direction.} and stay in the same key. Here, as with the entrance of the second voice in Contrapunctus I, Bach chooses to keep the theme’s inversion in D-minor, and so the first interval is not the descending 5th, D to G, but is truncated to a descending 4th (making this more like an inversion of the second voice’s entrance in Contrapunctus I).

Comparing this inverted theme to the second voice in Contrapunctus I, where that voice descends a 4th from D to A (measures 5-6), the inversion rises a full 5th, via the minor triad A-C-E. Bach then utilizes the rising half-step within the scale from E to F to mirror the descending D-C# half-step of the original theme. From F, it then descends down to C#, before resolving to D in the first beat of measure 5.

The second voice, entering in measure 5, descends a 5th, and outlines the D-minor triad D-F-A, followed by the half-step up to B♭, a near-exact inversion of the first voice in Contrapunctus I.

There are a few things which can be pointed out here about the ambiguities which Bach makes use of in the opening measures of the Contrapunctus III inversion. First is the already stated one, that upon inverting the theme, Bach alters the first interval from a 5th to a 4th, thus staying in the key of D-minor, rather than a perfect inversion which would change the key to G-major.

However, already by measure 4, the minor has been inverted to the major, further erasing the distinction of major and minor.

Adding to this ambiguity, after introducing the inverted theme in Contrapunctus III, the first voice goes into a completely “keyless” chromatic passage, played against the second voice’s inverted theme.

Note that the use of inversion in Classical composition is not a formulaic thing! Bach does not simply invert what is already there; he is constantly transforming and developing the theme beyond what would be given by a simplistic or mechanical inversion. It is not the notes which dictate the music—first there must be creative insight into the implications of a musical idea, and the ironies to be unveiled, which demand the use of inversion as a driver of the musical development.

Beethoven and Brahms

Now look at how Beethoven and Brahms use these complementary pairs, in inversions, in a way which more dense than the previous example.

While there is only one type of 4th and one 5th within the scale (with little flexibility in tuning^{15The range of how much the 5th or 4th can vary from the exact, geometric divisions of the string and still be in tune (this is determined by the musical context) is much more narrow than that of the 3rds and 6ths.} ), the 3rds and 6ths are less determined—they can be either major or minor, and have much more variability in tuning.
Here we’ll look at two related examples which are based on the complementary pairs of thirds and sixths, in which the interval and its inversion are not treated separately, but as one unfolding process of development, such that each heard interval implies the unheard presence of its partner.

In Beethoven’s Opus 106, Piano Sonata No. 29 (“Hammerklavier”), there’s a short but gripping passage in the “Adagio sostenuto” movement where Beethoven alternates falling 3rds with rising 6ths, each pair separated by the interval of a third. The surprisingly moving “theme” of the pairs is carried through a series of transformations, driving to a point of dissonance^{16A double lydian against a dissonant pedal point.} from which a new line of development emerges. As the passage unfolds, and the 3rds and 6ths evolve into one another, the inversions are no longer understood as separate entities, but a unified process of change.

Underneath the 3rds and 6ths is a sequence of sweeping arpeggiated sixteenth notes of the left hand which, while ordered, cannot be pinned to simple sequence of major or minor keys, further adding to the transcendental quality of this passage.

It is known that Brahms had access to some of Beethoven’s sketchbooks (or at least copies of them) through his friend and Beethoven scholar Gustav Nottebohm, and studied the sections from those sketchbooks where Beethoven worked out the passage of 3rd and 6ths for the “Hammerklavier”.

Brahms opens his 4th Symphony with this exact passage of alternating falling 3rds and rising 6ths, with two (minor) changes: 1) while Beethoven has falling minor 3rds (for the most part) and rising minor 6ths, Brahms uses the “exact” complementary pair of falling major 3rds and rising minor 6ths; 2) after the first two pairs, Brahms quickly transforms the theme into pairs of a descending octave, and a rising major 3rd.

Finally, take Brahms’s incredible “Vier Ernste Gesänge” (“Four Serious Songs”). Brahms composed this set of songs at the very end of his life, after his lifelong friend and dearest collaborator Clara Schumann suffered a stroke. She would die a few weeks after the composition’s completion, as would Brahms himself just a year later. The third song is set to the following text taken from Ecclesiastes (Sirach):

In the song, the singer directly address death, first in its effect on the man who “has good days…and a sorrow-free life,” and secondly, in the relief brought to one who is “beset by all sorrows.” Both sections begin with: “O Tod,…” (“O, death,…”). With the first, Brahms uses two descending 3rds as the piano moves from E-minor down to A-minor.

He reverses that motion with a rising minor 3rd on “wie bitter…”, as the vocal line shifts to the high register of the singer’s voice.

When the speaker (singer) shifts, to address the contrary quality of death, Brahms inverts the “O, Tod…” from a descending 3rd to its companion, a rising 6th, against the piano’s E major.

At the singer’s final line, Brahms again sets “O, Tod,” to a rising 6th, not once, but twice, moving in just a few moments from the very low register of the singer’s voice to the highest on the “wie” of “wie wohl.” He then reverses the motion with a beautiful falling sixth on “wie wohl…”, before finally coming to rest in E-Major.

Music, which deserves the name, is not composed note by note, each tone caused by the one which came before, or selected because of how it sounds when heard against its neighbors. The mind hears and creates both in and across space and time, with the entirety of the composition implied in the unfolding of every moment. The singer of Brahms’ “Vier Ernste Gesänge” hears in the mind, in the moment of silence before he or she begins, the simultaneity of both statements of “O Tod,” each full of the potential of the other. In this way, the intervals which appear to us as inversions of one another are unfolded as an interconnected manifestation of a single idea.

Pipe to the Spirit Ditties of No Tone

^{17John Keats had it right.}

Hopefully these few offerings have given the reader a glimpse into the soundless domain of ideas which lies behind and beyond the audible tones of the music, per se. The use of inversion in Classical composition provides an example of that soundless domain, in which the mind can hear not only what is there, the interval or statement, but also what is not there, its inversion, as an integrated one—a one which the ears will never hear. As with the major and minor, the two, which seem separate from the standpoint of the notes, are resolved into a unity, subsumed by a single generating concept which only exists in the closing silence following the final, audible tone.

Any competent composition or performance of music must originate from its proper place: the soundless realm of the imagination, of which the ears receive but a passing shadow.