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can be viewed as the natural basis of musical harmony in all cultures that have it. The earliest polyphonic (multiple-voiced) music that has come down to us deals mainly in the first 4 notes of the overtone series, that is, with the octave, fifth, and fourth. In music for three or more parts, a preference gradually arose to fill in these intervals until, in the Dunstable-Dufay period (early fifteenth century), a style arose based on the triad, a 3-note chord [C-E-G or C-Eb-G]. This is tantamount to expanding the harmony to include the first 6 notes of the overtone series—the senario, as Zarlino pointed out in the sixteenth century.
This state of affairs held throughout the Renaissance (through 1600), after which point Monteverdi and his fellow innovators felt the need for denser chords to better express the gamut of human pain. Among the favored chords in the new Baroque style was the dominant seventh chord, built from the first 8 notes of the overtone series, and its relatives, all the seventh chords. By constant use, the seventh chord lost its edge until, with the styles of the early twentieth century—impressionism, blues, jazz—a style became possible with the seventh chord as the basic sonority.
But at this time, not only the seventh chord but all the upper harmonics of the overtone series were explored: the 9th, 11th, and 13th, until we reach a “chord” that is an entire scale: GBDFACE = CDEFGAB! Adding more notes only resulted in a tonal mush. Moreover, with the eclipse of jazz by rock in the 1960’s, the listening public decisively reversed the trend of harmonic densification of the preceding centuries in favor of simple chords. Moreover, rock has gone global, overwriting and intermingling with the musics of other cultures. So have we reached the end of the history of harmony?
Enter Pedal Matrix Harmony.
II. Pedal Matrix Harmony
II. Pedal Matrix Harmony
“Pedal Matrix Harmony” is my term for a system of chord construction that has been on the ascendancy in the late 20th and early 21st centuries. The principle is simple but at the same time revolutionary: add notes to a root note, based not simply on their intervallic relationship to the root note, but also based on their importance in the tonality.
The name deserves explanation. The term “pedal” (or “pedal point”) denotes a sustained tone, almost always the tonic or dominant of the key, that is a backdrop for changing harmonies that are often dissonant with the pedal point. In the 16th century, this technique was common in organ music (hence the name) and quickly found its way to other genres. The pedal point is usually in the bass, but one can also find classical examples where it is in a middle or upper voice. In modern pedal-matrix harmony, the pedals are most commonly in inner voices, sometimes touched upon by the melody.
“Matrix” is used to emphasize the interweaving of vertical coincidences (chords) with horizontal coincidences (the pedals as they are sustained from one chord to the next or drop out).
The choice of pedals is to some extent malleable by the composer; but in modern use, the most common pedals remain the tonic and dominant. When attached to the six consonant triads of a diatonic key (the mainstay of pop harmony, and of Renaissance harmony before it), this technique produces the following chords:
In contrast to styles based on the triad or 7th chord, etc., this gives a chord with a different interval content on each scale degree. Note the following peculiarities:
I is the only chord that does not gain dissonances when adding the pedals. This reinforces its tonal stability, but there are times when the bare triad is too plain for its surroundings. Then we add another note: the most popular choice is the 9th, 2̂ in the scale. This has the virtue of mimicking the structure of the IV(add9) chord, and it is also true to the overtone series, an essential for stability.
IV and VI gain a 9th and a 7th respectively, forming familiar sonorities. But the 9th and 7th lack the downward tendency traditionally attached to those chord tones: instead, the dominant note wants to stay fixed, or it can be freely included in a melody heading either upward or downward.
Likewise, II gains both a 7th and an 11th, both without their downward tendencies. Either added note alone is enough to sustain the Pedal Matrix style, but there are instances where both occur (see below).
The III chord jeopardizes its root function with the addition of the tonic, becoming an inversion of the tonic seventh (I6/5). For this reason, III is rare in major in Pedal Matrix settings, and if the bass needs to hit the note 3̂, then the harmonic support will most often be I⁶, the first inversion of the tonic triad. Likewise, if the bass is on 7̂ (omitted from the above example because of the troublesome diminished triad), an attractive harmonic realization is G(add4)/B, the Lauridsen chord (see below).
With the V (dominant) chord, the most revolutionary developments arise. Here, the chord shown in the example is a V(add4) [GBCD]. In jazz discipline, the 4th (C) above the root of a major triad is classed as an avoid note: even in that famously lax style, the note is used only in passing because of its strong clash with the 3rd of the chord (B). Modern composers are not so shy of this clash, but there will be situations where it is unduly harsh. In that case, the Pedal Matrix composer has the choice between V (GBD) and Vsus4 (GCD), but the relationship between these two sonorities is not as it was in previous centuries. Classically, Vsus4 resolves to V: so in an example like the following (from Pitoni’s Laudate Dominum, which incidentally is an example of the stile antico—the Renaissance style which lingered, especially in church music, for another century after 1600 when secular music had followed other trends):
the trained ear picks up the G-sharp as the fundamental chord tone of the penultimate chord, even though it lasts but half a beat: we would feel cheated if the soprano were to sustain the A all the way to the end.
However, in the framework of Pedal Matrix Harmony, V ↔ Vsus4 is a free-floating progression, neither chord a resolution of the other. This is reminiscent of the change in meaning of the 6th and 6/3 chord at the dawn of the Renaissance. In medieval France the 6th was classed as a dissonance and actually so treated, so that the 6th in such an example as the following well-known piece,
is a dissonant passing tone just like the 7th. But within a couple of centuries, Dufay could write 6/3 chords in fauxbourdon,
with no need for the 6th to resolve in any particular direction, and Josquin played heavily with the effects of 5-6 and 6-5 motions over a held bass. It might seem inconceivable that a dissonance as strong as the suspended 4th should undergo such a change in meaning: but on inspection, our perception of this chord is culturally conditioned. If the major triad [CEG] is stable, and if the minor triad [CE♭G] formed by altering its least fundamental note by a half step is also stable, then what of the chord [CFG] formed by altering the same note in the opposite direction? In Chinese traditional music the lute (pipa) is tuned to a suspended chord [ADEA], and this chord, called in Chinese music theory the shang, zhi, or yu harmony according to its position in the pentatonic scale, is used as a basic consonance. Is Western music headed toward a similar aesthetic?
III. Examples of Pedal Matrix Harmony
III. Examples of Pedal Matrix Harmony
For those looking for it, pedal-matrix harmony is on the ascendancy in a variety of genres.
Say Something by A Great Big World, featuring Christina Aguilera
This 2013 hit is a simple but touching example of the pedal-matrix as a compositional device. Stubbornly, the piano reiterates the tonic (D) through a harmonic pattern including VI, IV, I, V (a ubiquitous progression) and (in the verse) III⁷. In the middle of the song, both tonic and dominant pedals are added, thickening the sonority. The tonic alone returns for the closing segment.
Piano Sonata, op. 90, 2nd mvt. by Ludwig van Beethoven
If we search for the roots of pedal-matrix harmony, we are led to the explorations of Beethoven, for whom pedal chords seem to be merely one of many effects turned up by his relentless search for new methods in harmony, texture, and form. Observe that the 3rd full measure of this movement has an unconventional dissonance where the B, carried over from the preceding measures, clashes sweetly against an A-major chord in first inversion. This can be viewed as a passing chord
that was also much favored by Mendelssohn, but it is more reasonable to think that Beethoven was thinking of an inner pedal here, especially in view of what he does for the second theme, where a written-out trill thickens into a double pedal of F♯ and its neighbor note G♯ sandwiched between a melody and bass in B major (seen on the still image at left before clicking the play button).
Piano Sonata, op. 109, 3rd mvt. by Ludwig van Beethoven
In this piece we see the discoveries of the preceding one applied on a grander scale. The Aadd9/C♯ chord recurs in m. 10, but note how the sustained B pedal in the left hand also affect the interpretation of m. 9. Without it there is a clear-cut secondary dominant resolving to F♯ minor; with them, all of mm. 9-10, including the C♯ seventh, sounds like an ornamentation of a B7 chord in various inversions.
Since this movement is in theme-and-variations form, we can see how Beethoven thought of this theme by the way he varied it. For the first 5 variations, he disregards the pedal and tonicizes F♯ minor when mm. 9-10 come around. But in the the last variation, the pedal comes back with a vengeance, with trills on the dominant alternately in the different voices for a dizzyingly dramatic return to the original theme.
(Note that both this and the preceding sample are in E major. In my experience, the best keys for resonant pedal-matrix chords lie in the narrow range D♭–D–E♭–E, as will be seen in the examples on this page.)
Transcendental Etudes, No. 11: Harmonies of the Evening, by Franz Liszt
Liszt pushed the boundaries of harmony in many directions. Woven through his Transcendental Etudes (so called because they are notoriously difficult to play) are various explorations of pedals and pedal harmony, none more pervasively than in No. 11, which is set in the rich and peaceful key of D♭ major. The opening pits an A♭ (dominant) pedal against chords as remote as E♭♭ major, the Neapolitan sixth. The second theme in E major (3:40) has both a tonic and a dominant pedal (in the bass and an inner voice respectively) for a daring chord progression including the characteristic Aadd9/E chord. The theme is reprised twice more; note how the texture and the location of the pedals is ingeniously shuffled each time.
O Magnum Mysterium by Morten Lauridsen
Lauridsen was trained in the 20th-century post-tonal sound world, but his mature compositions show him integrating dissonance into a decidedly diatonic norm. His starting point appears to be a family of personal chords (a typical 20th-century mode of self-definition), consisting of a major triad, archetypically in first inversion, with an added natural 9th and/or 11th. On IV, the 9th is preferred (to remain within the key); on V, the 11th is preferred, in both cases making (consciously or not) a pedal-matrix sonority.
In this piece, his most famous work, note that at the triumphant return of the main theme (4:23), the bass brings in a low D, a pedal point used in a classical manner. (See here for the score.)
[Curiously, this piece does not use any chords rooted on the 2nd degree (E) and can be performed in 5-limit just intonation, thereby lessening the roughness of the minor 2nd dissonances. This performance seems to steer a bit in this direction. However, Lauridsen does not seem to have paid attention to this feature, and in his larger works in a similar harmonic vein (e.g. Lux Aeterna) the II and V/V chords are used liberally.]
Defying Gravity from Wicked, by Stephen Schwartz
Contemporary Broadway musicals make liberal use of the chords of pedal-matrix harmony. In the verse of this piece, the ♭VII chord (common in popular music) appears with an added 9th, the tonic. In the chorus, the upper part plays a three-note ostinato B-C-G, marking 3/8 bars rhythmically offset from the 4/4 meter of the song. Now C and G are the usual tonic and dominant pedals, but what about that leading tone B? It can be heard as an appoggiatura or neighbor note to C, but when the other parts are on the dominant chord, this is contrary to the usual mode of hearing C as a suspension or neighbor note resolving down to B.
On the last word (“down”), another expressive use of the pedal occurs: the voice sustains a single note through a number of harmonies.
Da Pacem Domine by Arvo Pärt
Pärt is another example of a composer trained in the 20th-century avant-garde who came by circuitous ways to a style with marked pedal-matrix characteristics. His so-called tintinnabulary (i.e. bell-like) style pits melodic voices (here the alto and bass) with smooth lines against leaping voices (here soprano and tenor) that arpeggiate a triad, here D minor. At first this would seem to yield merely a static harmony, but the rich 10ths between alto and bass, thickened to Renaissance-style 6/3 chords at cadences, give the effect of classical harmonic progressions colored by a triple pedal of tonic, mediant, and dominant. Pärt habitually calls for very slow tempi, allowing the ear to fix on the varied sonorities generated by these means. The same technique is employed on a massive scale in his Passio (a setting of the Passion narrative from the Gospel of John).
Blest are They by David Haas
From the 1960’s till today, there has been a wave of “praise and worship” songs that aim to liven church services by pop-based rhythms and use of the piano and guitar (often dubiously reading the spirit of the rock-’n-roll era into the documents of Vatican II). Most of these songs use very simple harmonies, but not “Blest are They,” which begins on an idiosyncratic chord described in various hymnals as A♭sus4, D♭/A♭, or D♭2/A♭. (For simplicity we treat the song as being in A♭ major although it appears in G in some hymnals.) In full, the chord would be called D♭add9/A♭. It is in a highly resonant spacing with the frequency ratio 3:5:6:8:9, and resolves to a triad, also in a resonant spacing:
Here, in a manner analogous to the classical 6/4 chord, the F and D♭ are dissonances resolving downward. The chord recurs several times in the piece, in different inversions. At the return from the refrain to the verse, the ninth (E♭) is placed in the bass, forming a more familiar “rock dominant” chord.
A more complex, but related, harmonic effect occurs in the transition between the 3rd and 4th lines of the refrain. The chord progression is as follows:
Here the hymn books label the starred chord as Fm7/E♭, ignoring (sensibly enough) the unusual note B♭, the supertonic, in the left hand. As a tone set it is an Fm11/E♭ or a pentatonic scale or a quintal chord (A♭-E♭-B♭-F-C are all perfect fifths), but with the chosen spacing it is something of a polychord (Fm and E♭5). Thus it hearkens back to the classical use of the grosse quinte, an organ stop that doubles the bass notes at the 5th above, so that the auditory phenomenon of the missing fundamental gives the illusion of one octave deeper, as if the chords were (as, indeed they are in some arrangements):
If we apply the same logic to the opening chord, we see a 2nd-inversion triad (D♭/A♭) with the 5th above the bass as an added tone.
(As one of the millions who have been consoled by Blest are They and other Haas compositions, I would like to present the above analysis as one argument to continue printing and singing his music, regardless of whether the sexual misconduct allegations against him turn out to be true.)
IV. The Future of Pedal Matrix Harmony
IV. The Future of Pedal Matrix Harmony
In my compositions (except those that rest comfortably in historical styles), I aim to integrate Pedal Matrix Harmony with the virtues of the styles that have come before it. When mixed with the principles of chord construction inherited from earlier times, Pedal Matrix Harmony yields vast possibilities, such as the following:
Articulation of each of the six church modes (not counting Locrian) by using its tonic and dominant notes as pedals. Popular music surprisingly has not ventured beyond Ionian (major) with any frequency thus far, but here is a Mixolydian example (based on the traditional tuning of the bagpipe) and here is a minor example where the tonic, at least, appears as an added note in chords to which it is foreign.
Modulation among relative modes by changing the pedals while keeping the same scale.
Creating and resolving polytonality by chord progressions that venture chromatically beyond the scale(s) implied by the pedals.
Phrase and piece endings on chords other than the tonic. Medieval and Renaissance composers could be capricious about what tone they ended their pieces on, but since the cadence structures are similar on each step of the scale, the root of the last sonority of a piece tends to be heard as, post facto, the tonic. Rock musicians have relished the rebelliously expressive technique of ending a piece on a chord other than I, and with Pedal Matrix Harmony, the notion of ending a piece on (say), the IV of C major can be defined more precisely; indeed, IV(add9) or IV(sus2) is a popular closing chord.
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