Composer Paul Hindemith (1895-1963) contributed a great idea to the theory of music, but expressed that idea as a pair of somewhat cryptic musical diagrams, titled Series 1 and Series 2 (above). These first appeared in his 1936 book, The Craft of Musical Composition, and he continued to develop and refine these concepts while teaching at Yale (1940-53). Hindemith outlines a general theory of the character and motion of musical sound based on the features he observed in chromatic intervals. An interval, as a metric, is the distance, measured in half-steps, between any pair of chromatic tones. An interval also refers to the sound of such a distance between two tones. An interval can happen in two basic ways: either two tones sound at the same time, or one tone is followed (replaced) by another.
In Series 1, every tone is meant to be compared with the first tone, and this order ranks the intervals according to increasing tonal remoteness, since the general grouping is the perfect intervals first, the thirds, then and seconds. Understanding the meaning of the interval order in Series 1 is not as important as recognizing that each interval constitutes a particular unique presence of sound. Simple pairs of tones are heard in the same way as chords: the major, minor, diminished, and augmented triads each have their own sound, and this is true of each chromatic interval as well.
In Series 2, Hindemith indicates the root of every interval, using an arrow to identify the interval-root. Depending on the size of the interval, one or the other tone's root effect will be emphasized, and that tone is the root of the interval. Again, the same effect happens as with chords: the root of an ordinary chord acquires more authority from the other chord tones. Hindemith noted this effect in simple intervals as well as chords.
Series 2 divides the dozen intervals desribed in Series 1 into two classes, by pitch-space position in the interval: root-up or root-down. Similarly, we can describe intervals in pairs of melodic tones as either root-backward or root-forward.
Hindemith noted one exception to this scheme: the interval of three whole tones, or six semitones, which divides the octave equally, and which has an equivocal root effect. This interval is variously known as the augmented-4th, the diminished-5th, or the tritone. We will mostly use the term tritone to mean a span of six half-steps, but we must be sure not to confuse this term with either the term trichord, which means any set of three chromatic tones, or the term triad, which refers to the four traditionally-named trichords. For example, the major, minor, diminished, and augmented triads are all trichords, but only the diminished triad contains a tritone.
The trichord is literally the next step away from the interval, and is defined as any group of three different chromatic tones. In a trichord with three tones, there are three different pairs of tones, and therefore three different intervals. Each interval has a root, and, if the trichord has a well-defined overall root (or not), the roots of the individual intervals will be organized into a pattern to make it so (or not so). There are nineteen possible different trichord-classes, which include the four traditional triads, plus fifteen other possible combinations. The simple intervals of any trichord, any three different chromatic tones, fit but one of the nineteen trichord types.
The basic premise of this tonal model is that root is an inherent self-identifying property of any tone. Root is a concept referring to the prime tendency of a single tone or virtual pitch to focus our audio perception at a certain fixed pitch-height, somewhere from low to high in pirch-space. A single tone produces a root-down (pitch) and root-forward (time) musical event, by default.
From another point of view, a single tone can be thought of as a special kind of interval, the unison. The metric size is zero semitones, so there is no difference in pitch-position to accomodate, and thus the root-effect is conserved at the same unison pitch and pitch-class level.
The sound of one tone attracts our ear to its position in pitch-space. If we add a second tone of a different chromatic pitch, it, too, attracts our ear.
The pitch difference we hear in equidistant semitones reflects a logarithmic ratio of the tones' frequencies. An octave is an exact doubling of frequency (2:1 Hz). Each semitone difference amounts to the twelfth root of 2 (approximately 1.059) times the frequency. Each half-step of difference in pitch represents a frequency difference of about six percent.
If two tones sound simultaneously, they merge into a common composite sound wave, the sound of the interval (above left). If one tone follows the other in a sequence, the first tone will continue in our minds as it is recalled during the new note. Either one tone will occur after the other, or the other way around, so, even though an interval may have one root direction (up or down) as a harmony, the two tones may be arranged in sequence so the root comes first or last (above middle, right). If the root comes last, the motion is root-forward. If the root is just prior to last, then there is a root-backward effect. These are the basic discernable characteristics of interaction between tones.
These two terms have numerous shades of musical meanings, depending on the context. In terms of this discussion of intervals, we will adopt Hindemith's point-of-view, which is: "consonance" means that the lower tone of an interval is the root, and "dissonance" means that the upper tone is the root. Thus a perfect fifth is consonant, root-down, but a perfect fourth is considered dissonant, root-up. Likewise, a major seventh is consonant (as in jazz), but its nearby inversion of a minor ninth (swap pitches in the same register) is clearly more dissonant (because the root is above the bass).
This shows that degrees of consonance/dissonance can be discerned in simple octave inversions of ordinary intervals: a fifth is more consonant than a fourth (Series 2), even though, in a more secular understanding of consonance/dissonance (Series 1), the fourth-fifth pair would both be "consonant," and the second-seventh pair "dissonant." But within each complementary interval pair in Series 2, we can discern an additional measure of difference, the way that a major-third pulls toward the bottom and a minor-sixth pulls toward the top. This is the effect of root.
In the two cases where the interval produces an equality (octaves and tritones), low rules over high. In an octave, the roots identify harmonically, to the point that they can be considered as part of the same virtual pitch, or the same mod-12 pitch-class; therefore, the lower tone rules over the upper octave. In a melodic sequence of alternating octaves of equal durations, it should nevertheless be obvious that low rules over high in the most unmistakeable way, regardless of the pitch-class invariance.
The tritone, also known as the augmented-fourth or the diminished-fifth, is a special case, because the tritone's interval size of six semitones makes a tritone equal to its own octave inversion. The individual tones are in a noisy stalemate, tied with equally attractive/repulsive root balance, with neither of the member tones able to yield or overcome the other. But even in the tritone, low can (and does) rule over high, a phenomenon we will observe below in several of the trichords that contain the tritone (six half-steps) as an interval.
Root balance is not all-or-nothing. It seems to be a gradient of several alternations of direction through the octave following a diatonic pattern (see below). The equal-tempered intervals are points on this scale. The implications for other tunings are an open question. The root balance in some intervals is clearly stronger than in others, most imporantly, the major-third/minor-sixth and the minor-second/major-seventh. (In atonal set-theory, these would all be interval-classes 1 and 4.) For these intervals, the root position is more consonant (pulls the root to the bass) and the inversion is more dissonant (pulls the root to the treble) than in the other intervals. We will see just how this is the case when we examine the three-tone interactions (trichords), by adding a different extra tone to the interval and then observing the triple interval-root pattern. As declared in the key to symbols on page two of the Easy • Guide, the stronger intervals' root-directions are emphasized with bold arrows (thicker).
Hindemith extended his concept of consonance/dissonance to chords in root progressions, any sequence of chords whose roots can be thought to form an interval sequence of their own. In the broader scheme, root progressions are how the plans of songs and symphonies are shaped. Like intervals, Hindemith felt that voicings of the same chord (like the traditional root position, first and second inversion) fell into two big categories, in the final analysis: the root is either in the bass (most consonant), or it is not in the bass (less consonant). The overall root is either sounded in the bass, or it is sounded somewhere above the bass. If a chord arrives with its root in the bass, the chord has more "authority," says he, than if the same chord arrived with another of its tones in the bass. I interpret this to say that a root-position chord's arrival literally generates an inherently greater rhythmic accent than otherwise.
For a tonal example of dissonance created by a roots-above-the-bass situation, think of the use of the cadential 6/4 chord, considered an accented dissonance to the point of being relabelled (erroneously!) as V 6/4 instead of I 6/4. Why is the second-inversion 6/4 chord dissonant? Because none of the three interval-roots in such a triadic voicing point to the bass.
When we arrange the intervals of Series 2 in order of increasing size, we can see a pattern in their interval-root directions, which flip with each diatonic step the tones are apart. Each of the seven diatonic intervals comes in a pair of chromatic sizes, but the interval-root direction remains the same for each difference in diatonic step size.
The root-direction alternates three times within the octave. This means that the interval-root directions from diatonic step 1 to steps 2, 3, 4, 5, 6, and 7 are up, down, up, down, up, down. This is just another restatement of the short-cut given at the beginning: step up, third and fifth down. Next, we will look beyond the diatonic pattern to the finer level of chromatic steps; we will see how root-directionality is a gradient, and how this affects, or effects, the intervals' tonal function.
The diatonic root-direction is the same for both chromatic sizes. Both of the seconds have a root-up character, but in the minor-second (the small second), the root force is stronger and resists being reversed. The major-second (the large second) is weaker in that its root direction can actually be reversed under some harmonic circumstances (e.g., when it acts as a chordal ninth).
The two thirds have a contrasting situation: the major (large) third strongly defines the root direction, but the minor (small) third can be reversed, and might act as an added (major) sixth in some other interval circumstance. However, the minor sixth, as the inversion of the major third, cannot be consonantly "added" since it will strongly draw the root effect from the bass to itself.
The perfect fifth could be thought of as strong, since it is unambiguous in its root direction. When used as a finalis, it is prima facia proof of the concept of root effect in an interval. But in combination with other intervals, it often yields to the effects of stronger intervals discussed above. In three-part counterpoint, sequences of seventh chords never include chord fifths, and no one misses them. In many multi-toned harmonies, the fifth above the root is the most expendable note, as far as omitting a tone without diminishing a chord's tonal effect.
So, how strong is a perfect fifth really? I think this is where Hindemith went a little astray, giving too much credit to the fifth as a "best" interval in determining chord roots. The strength of the fifth is over-rated because of its Series 1 quality: its timbral weight is lightest of any (except the octave), and the root-direction can even be reversed, as we will show below. As we explore the nineteen trichords, we will notice the less-dominant (or more egalitarian) influence of the fifth, and its routine deferral to stronger intervals like major thirds and half-steps.
Thus, there are different degrees of interval-root strength, which draw a distinct pattern across the octave, roughly outlining the interval regions of the diatonic steps. The interval classes with the most strength are the minor-second/major-seventh (1 semitone difference) and the major-third/minor-sixth (4 semitones difference).
When two tones are at an octave, the tones yield to each other, and it is as if there is but one tone, and the "interval" seems to disappear. When two tones form a tritone, six semi-tones apart, they are also equal: equally non-yielding. The tones resist combination to the point that the interval almost refuses to appear. If we are trying to determine the root directions of intervals in a chord, the tones at the distance of the tritone are non-participants, although each may display root effect as part of other intervals present. The tritone is a noisy (Series 1) standoff (Series 2) between the two tones. Hindemith uses the term "indeterminate," but I feel this term is misleading because it implies a lack of effect. (He uses the same term to describe the root of the fourth chord, however; but we will show how the interval-root structure of the fourth chord is actually cyclic -- a circumstance that permits a perfect fifth to get root-reversed.)
Thus the root direction of the tritone is a draw, a tie, a stalemate. What happens when we add a tone to a tritone, making a trichord with a tritone as one of the three intervals? We will look at the root structure of five possible three-note combinations with tritones as one of the intervals. One thing that is clear about the tritone, true in melody as well as harmony, is that only the addition of some third tone (in one of five possible ways) can force the tritone into any definite root pattern.
Keep in mind that, just as with octaves, the low rules over high rule applies to the tritone. If the member tones of a tritone are separated by more than an octave, the root effect of the lower tone will be favored.
Knowing the root-directions and strengths of each interval, we can calculate the root effects of the intervals among the member tones of any chord. By a careful accountings, we can figure out the overall root of some pitch collection by seeing which tone accrues the most root-favor among the intervals. Then we can compare the result of our analysis to that of other music theories, and, most importantly, our own ears as well.
To prepare samples of melodic intervals for research on their root properties, there are a few experimental controls that must be considered. First, we must isolate the melody, and its intervals and root directions, from any bias from harmony or homophony, to observe the pure monophonic effect of the interval-root succession. Second, for consistent comparison of the interval paths, we must normalize the point of arrival, so that all examples arrive on the same pitch. Last, we must minimize any differences in durations, which might generate accent patterns unrelated to interval content. The limits are: to consider only the contours of three different chromatic tones, to set each trichord in a simple, non-biased quarter-quarter-half rhythm, and to transpose the respective tones so that the point of arrival, the final tone, is the same in all cases (C), but the interval-root contours are preserved.
To keep track of the root effects within melodic movements, we will draw an arrow showing the interval-root direction between pairs of sequential tones: forward (→) or backward (←) or a tie (≠) for a tritone. The arrows clearly and concisely show the pattern of root directions within each three-note melodic example.
Unlike the Easy • Guide, the annotated diagrams of melodic permutations given below all use one style of root-direction arrow only, omit using a bolder style of arrow for the strong interval-classes (1 and 4 semitones, with inversions). This simplifies the visual presentation of the root directions and the common patterns that emerge. But the text will discuss whatever level of effect such intervals contribute, and the readers are encouraged to be aware of such distinctions on their own.
In the first set of six examples below, the Major Triad trichord-class is subjected to every possible melodic rearrangement, six permutations of the same three pitches, set in the plain rhythm described above. The equally-spaced rhythms keep the focus on the variations in root-forward or root-backward effects, while time itself always steps forward at a constant rate. Below the staff, arrows point toward interval-roots of adjacent tones. Above the staff, the arrows show the non-successive root-direction formed between the final tone and the initial tone. We can see almost immediately that, with as few as three tones over time, the three intervals form a two-level hierarchy of musical structure.
In the second set of six examples, everything is exactly the same, including the interval-root directions, except that the melodic permutations are transposed as necessary, so that they always arrive on C. By establishing a common point of arrival, we improve our ability to compare how the cumulative root effects of melodic permutations may vary, giving us a better sense of the capacity of such interval-root combinations to propel or restrain the sense of arrival. Later in this essay, each individual trichord-class is discussed, and the same transposed format is used, so that each respective set of permutations is normalized.
Hindemith says if a chord arrives with its root in the bass, it will arrive with more "authority" than if it had arrived with some other tone in the bass. We can simply regard this difference in "authority" as an accent, an emphasis that is added to the rhythm of time alone, giving rise to the phenomenon we commonly understand as harmonic rhythm. A difference in sound, chord, interval, or tone will create an accent greater than time alone. But, a difference with root-forward directionality as well will create more accent.
If the arriving tone is the root of the interval it forms with some preceding tone, the perceived accent will be greater than otherwise.
So far, we have outlined the subjective musical properties of the individual simple intervals, beginning with Paul Hindemith's idea that intervals have roots; that a single tone is its own root; that in "consonant" root-down intervals the lower tone is the root, and "dissonant" root-up intervals the higher tone is root; that the tritone is equivocal regarding root direction; that, in a sequence of tones, if an arriving tone is the root of the new interval, this root-forward energy is felt as a kind of accent. Harmonic rhythm is a form of accent which appears to be correlated with interval-root direction. Interval strength influences the accent effect.
A trichord is any three different chromatic tones. Any member tone of a trichord has an interval relationship with each of the other two tones. Depending on what the two intervals are, the tone will be: both intervals' roots; neither interval's root; one root and one non-root; and a tritone plus one root or non-root. Every tone in the nineteen trichords is connected with the other two tones by a pair of intervals whose root pattern will be one of these four alternative ways.
In our context, a tetrachord is any four different chromatic tones. We obtained the nineteen trichords by adding a single different chromatic tone to the various chromatic intervals. By adding a new tone to an existing trichord, we find that this can give rise to 43 possibilities of tetrachord types, sets of four different chromatic tones with common interval patterns. Also, adding a fourth tone produces a new interval with each of the three old tones, so that is three plus the three intervals of the original trichord for a total of six intervals in any tetrachord.
A good framework to conceive of the four tones and six intervals is first to understand that each tetrachord contains four (4) embedded trichords: the original trichord, plus a new trichord for the new note plus one of the old intervals.
It is possible to use our triangular diagram to diagram the root relations of any trichord, as in the Easy • Guide. To visually represent all the interval paths in a tetrachord, put the new note into the middle of a trichordal triangle and connect the middle to the vertices. This will give you an outer trichord and three inner trichords, and you can diagram the root directions for analysis, etc. A tree diagram may be a helpful model to evaluate the vectors of root effects in tetrachords.
Since one of the goals of this study is to become familiar with the sound of each of the nineteen trichord types, it is helpful to frame your perception by seeing that these trichordal sounds do not disappear in tetrachords; instead, one can think of any tetrachord as sounding like four more-or-less familiar trichords at once, if you will. Also, all interval-root directionalities will likewise continue to bear influence.
Based on distinct interval structures, any set of three different chromatic tones will fall into one of these nineteen sets of three-interval patterns. The interval-root pattern of each trichord-class reveals a particular melodic and harmonic character. Even highly familiar musical structures are better understood when the intervals' actions within are revealed.
As illustrated in the Easy • Guide pamphlet, every trichord-type is shown in a notated example, along with several symbolic captions, which provide some things to notice about each trichord type. Here is a brief review of the elements covered in the pamphlet:
Key to Depiction of Trichord Properties in E • Z Guide
Above each example trichord is a triangular diagram with the example's note-names connected by arrows showing the root directions of the three intervals in a clear and simple way. In each triangle the note C is in the 12:00 (clockface) position, and the other note-names are placed in ascending order clockwise at the other two corners (4:00 and 8:00). The note-names are spaced evenly regardless of the actual interval sizes, so that the interval-root direction may be clearly shown. In all triangular diagrams, moving clockwise ascends higher in pitch, and moving counter-clockwise descends lower. In the case of the five trichords with tritones, a double-slashed line (≠) shows the tritone (dis)connection. Arrows are either bold, to show the strongest interval classes 1 and 4, or are plain for the others.
There is a descriptive label, which associates that interval pattern with a simple musical descriptor you may already be familiar with, such as "do re mi" or "incomplete minor-seventh". Communicating information about such an ethereal and non-verbal phenomenon like interval roots in clear and plain language is difficult, but such communication is nevertheless a prime purpose of this study, to strike an effective verbal balance, avoiding (or maybe just balancing) both oversimplification and overcomplication.
One other useful category, suggested by Howard Boatwright, is whether the trichord could be a subset of the pentatonic, diatonic, or chromatic gamuts. All trichord classes are of course included of the chromatic set. However, all but four of the nineteen possibilities turn out to be subsets of the diatonic set, and six of these are included in the pentatonic set. This means that purely diatonic music may contain fifteen of the trichord classes, but purely pentatonic music will contain but six trichord classes.
Every trichord example is rendered into music notation, over a common tone, bass clef C, for comparison. Each trichord's realization is arranged so that the interval-roots are pointing down, as much as possible. This means that instead of half-steps (root-up) there are major-7ths (root down), or perfect fifths instead of perfect fourths, even if this causes the voicing to exceed an octave. Even among the cyclic and tritone-inclusive trichords, I arrange interval-roots toward the bass when possible. Also, instead of drawing interval-root arrows over the notation itself, I have placed all that information above each staff as part of its triangular diagram.
To aid thinking about these intervals in half-steps or semitones, the pitch-class integer for each tone notated on the staff is given beside it. Since the constant bass C is always zero, the numbers beside the two higher tones give the size of two of the three intervals as well, in semitones. For example, with C=0, the pitch-classes for the C-major triad are 0, 4, and 7, representing C, E, and G, respectively. The major-3rd interval is 4 semitones, the fifth is 7 semitones, and the remaining interval is a minor third, 3 semitones (7 minus 4). Although the terminology of this text uses the traditional interval and diatonic scale-degree names, one should always bear in mind the size of each interval in chromatic semitones.
When we lay plain and compare the interval-root patterns of all the nineteen trichord types, we find that they group themselves into three general design categories, based on trichords' similar interval-root patterns:
The most common interval-root pattern the trichords exhibit is where one tone is the root of two intervals, includin unambiguous root, in a two-to-one interval-root orientation, where two tones point to the same third tone, with the interval-root between the first two tones points to one or the other, each already captured by the third tone. There are nine trichords of this tonally-focussed type, from the tone-cluster of half-steps to the major triad.
The second interval-root pattern is cyclic, which means that each interval-root points to a different tone. For example, in the augmented triad, the tones are all major-thirds (4 semitones) apart; thus, each tone is the root of a major-third. Cyclic trichords may have two or even three different but plausable orientations, supported by low rules over high, all things equal.The five cyclic sets include the three sets of diatonic steps, the 4th-chord, and the augmented triad.
The final category is the set of trichords that contain a single tritone. A trichord containing a tritone cannot be cyclic, and must be diatonic. The tritone itself can be thought of as a cyclic interval. A trichord can have only one tritone, and the other two intervals' roots will determine the root character of the trichord. The tritone itself is simply going to loudly abstain from all commitment. Yet, depending on what third tone is added (5 choices), the root pattern may be definitive or ambiguous.
The most prevalent type of interval-root pattern in a trichord is the case where one tone is the root of two intervals between itself and the each of the other two tones, with the remaining interval's root favoring one of the two tones. This shows why C is the root of the C major triad: E and G both yield directly to C as the root; G also yields to E, which has already yielded to C. Therefore, the tonal center of C dominates the pitch collection and is recognized as the root.
Because each tone is the root of one interval, the root directions wrap around. Among other things, this means that no matter which note of the trichord is in the bass, some higher tone will be dissonant with it by some degree, since dissonance is defined as having an interval-root above the bass, and there must be an interval root above the bass in cyclic trichords because the interval roots have to wrap around.
Three of the cyclic trichords are the diatonic scale-steps: Do-Re-Mi, Re-Mi-Fa, and Mi-Fa-So. You can start anywhere, but the basic interval pattern is always up a step, up another step, down a third. These three patterns account for all seven possible sets of three adjacent steps in the white-note scale: Do-Re-Mi is equivalent to Fa-So-La and So-La-Ti; Re-Mi-Fa is equivalent to La-Ti-Do; and Mi-Fa-So is equivalent to Ti-Do-Re. As Clough has shown, the same diatonic set-class of cardinality three (012) is expressed in three chromatic varieties (013, 023, 024).
The cyclic nature of all three of the scale-step trichords has many implications for seeing how interval-roots engage in melodic passages. First, we must observe the viability of scalar passages, up or down, filled with intermittant turns, steps or skips of thirds: the churn of cyclic root movements can be so ambiguous in such lines that almost anything is possible, and such processes can be seen spun out almost indefinitely in the melodic details of the sonata style, filling out a form maintained by the slower interval movement in other parts, or, free homophony. many).
The fourth-chord can also be a fifth-chord plus a whole step which wraps the larger intervals around. Just as the cycles of scale steps spin out
The final cyclic trichord is the augmented triad, dividing the octave into three equal major thirds.
Half-Step/Whole-Step - half-step up, whole-step up, plus a minor-third down. Also, Mi-Fa-So is equivalent to Ti-Do-Re.
Whole-Step/Half-Step - Whole-steps up, half-step up, plus a minor-third down. Re-Mi-Fa and Mi-Fa-So are chromatic inversions of each other. Re-Mi-Fa is recognizable as the first three steps of the minor scale, just as Do-Re-Mi is the first three steps of the major scale. Re-Mi-Fa is equivalent to La-Ti-Do.
Whole-Steps - Two whole-steps, plus a Major-3rd down. This trichord includes three-fifths of the entire pentatonic scale, and it introduces the strong interval of the major-third into the steps. In the diatonic system, Do-Re-Mi is equivalent to both Fa-So-La and So-La-Ti.
4th-Chord - two perfect fourths and a whole step up, or two perfect fifths up and a whole step down, with a whole step. The most famous example of this trichord is the classic I-IV-V-I root-progression. One permutation is the classical secondary dominant: I-II-V-I, going to II instead of IV. Another variation is I-IV-bVII-I, so that the cadential resolution is a step up! These examples demonstrate the equivalence in harmonic motion between descending a perfect fifth and ascending a whole step. Even though the order of intervals is different, the root-motion is always forward.
When the root-progression is reversed, and the chain of interval-roots are backward, the pattern is a common progression in numerous rock songs, for example, ||:E-/-D-A-:|| (Theme from "Ghostbusters") or a little of both ways, ||:E-/-/-/-|/-D-A-D-:|| (Led Zeppelin's "Whole Lotta Love").
As a sonority, if we stack two fourths, the roots link to point up; if we stack fifths, the roots link downward. Fourths are used to voice modal harmonies, as brilliantly demonstrated by the jazz pianist McCoy Tyner. The fourths conduct interval-root motion upward, so the topmost voice will be reinforced.
When we link the interval-roots of two fourths or fifths, either way we are left with an extra whole-step up. We could call this a "(major) ninth" in chordal terms. The composer Frank Zappa often uses exactly such "fifth" chords as power chords, but he simply calls them "2-chords": The bass, the fifth, and the fifth above the fifth, which is a step above the bass, plus an octave. It has come to pass that the major-second degree can be freely added to both major or minor triads for "expressive" purposes with virtually no change in the chords' functionality, and no need to have a "seventh" in the chord to "explain" the presence of the "ninth."
As a root progression, the intervals of this trichord have been employed forward, as a stylistic characteristic of jazz (ii-V-I cadences in the "dominant" direction), and backward as a characteristic of the subsequent era of rock (bVII-IV-I cadences from the "sub-dominant" direction).
In the augmented triad, where all the intervals are major-thirds, one can see that whatever is in the bass will be joined by both a consonant major third and a major third above that, forming a complementary but dissonant minor sixth above itself. In traditional harmony, the augmented triad can be used as a dominant harmony in the minor key, although its traditional mis-"spelling" would define the chord root not on V, but on III (6/#3), in first inversion, with an "augmented" chord fifth (the leading tone), so V "appears" in the bass). The traditional analysis simply follows common-practice chord-spelling rules, but let us be clear about what interval-root analysis shows the root to be (V).
These five are trichords in which two of the tones are exactly a half-octave apart, six semitones, which means the other tone is one of five other possibilities. The character of each trichord is determined by the two other intervals and their relationship. These two intervals compensate for the uncommitted tritone among them, and shape the root tendencies of each of these types through the two intervals' interactions with (or without) each other.
Lydian-5th - A tritone and a half-step, spanning a perfect fifth. When we use the term Lydian, we are talking about the large (augmented) fourth quality of a major mode, the Lydian mode of antiquity, and the harmonic relation it bears to a tonic harmony, a relationship that the small (perfect) fourth cannot maintain. In this case,
Lydian-7th - A tritone and a perfect fourth, spanning a major-7th.
Dominant-7th - The root, third, and seventh of a dominant-7th chord, missing the non-essential fifth. Of the trichords with a tritone, this is the only case where a tone which is not part of the tritone is the clear root.
The interval structure of this trichord can be interpreted in two different ways, depending on which member of the tritone is in the bass. For this reason, it has The third tone yields to both members of the tritone, so half-diminished-7th chord, with a diminished-fifth (tritone) and a major-3rd up, spanning a minor-7th. Or, a major-3rd and a major-2nd up, spanning a tritone.
Diminished-Triad - Two minor-thirds up spanning a full tritone.
As we have shown, complex patterns of interval-root directions emerge from even three sequenced tones. Theorist James Tenney has pointed out, three elements are the minimum number necessary for the perception of meaningful contour (which he refers to as a klang) in a musical sound or passage. Tenney's interesting and worthwhile theories develop a parametric model of events progressing over time, and hierarchichal relationships which emerge due to the pattern of changing parameters, and how we perceive these levels progressing over time.
In Tenney's model, there remains the problem of the unknown "weight" factor for each parameter, with no clear path to the correct values to use for tonal elements like pitch height or interval length. Where do those values come from? It is beyond the scope of this paper to go further into Tenney's theory here, except to say that the entire Tenney theory might explain tonal motion very nicely if the "weights" were simply the adjusted root-diractions of intervals that Paul Hindemith noticed and quantified in Series 1 and 2.
The purpose of this essay is to explain the underlying idea behind Hindemith's Series 1 and 2 in the plainest possible language, to a wider group of musicians, music lovers, listeners, scholars and others, through a close re-examination of the premises of music, the relationship of one tone with another. If there is a music theory that explains or predicts(!) any universal tendencies regarding humanity's practice and appreciation of musical arts, it needs to begin by accounting for the effect of the chromatic intervals in a way that consistently corresponds with experience and example.
At this point, I am at the end of writing my expanded syllabus, which means that whatever else I tell you, I will assume that you have actually read and understood the essay thus far. I also assume that the Easy • Guide pamphlet has been your good companion throughout, of course. It was intended to be "self-explanatory," but if that were true, then why would I have bothered to write an 8,000-word essay explaining it just a bit more?
Because this theory is actually predictive regarding the general effects of intervals, then what I am asking of the good reader is to test it yourself, on any significant intervals or musical examples you may encounter in your future listening or playing. Do we have a natural, subtle but measurable bias toward root-down harmonies and root-forward motion? And does an awareness of interval-root directionality and strength change or improve our understanding of music we already know?
Thank you for reading and considering the ideas in this essay. Please send me your own suggestions for examples, your thoughts or questions.