Theory of Multiple Chord Roots

By Daniel White
(This article was part of my dissertation for my course at university.
I thought I'd put it online (14/04/2007) for anyone who might be interested.)

Before the 18th century, the theoretical reduction of a chord was found by simply taking the chord's bass note. This is the idea behind "figured bass". Any other notes building on top of the bass was regarded more as 'consonant decoration', with the bass taking focus as the main function of the chord. But in 1722, Rameau first realised the importance of the third as the basis of harmony, and published his findings in the "Treatise on Harmony" [60]. From this point, the "root" or "basse fondementale" ("fundamental bass") - as it came to be known - was the defining factor for a chord's representation [30], at least most of the time for most people. Like the bass note in figured bass, the root is a single note which 'represents' a chord, by searching for the explicit or implicit triad inside the chord. A triad could be implicit by only one or two of the three notes existing, or in the form of an arpeggio, or perhaps even from the presence of the seventh or ninth etc.

Traditional music theory presents a single root for any particular chord, or sonority (usually a short time span). However, to me, it seems quite conceivable that any given chord could be comprised of one, two or multiple simultaneous roots in varying proportions of diatonic degrees. If this was true, then the advantage for analysis would be to present a more accurate and complete representation of a music's harmony than a single root would provide. For example, one could speak of chord purity or complexity, or how ambiguous a section of chords are, as well as new insights into how the chords may function together.

Perhaps the simplest example would be the triad in first inversion. Take for example, the first inversion of the supertonic minor triad. Theoretically this should function as a minor triad on the supertonic degree. However, in practise it functions more like a subdominant [40], adhering more closely with figured bass! Under my theory, we can estimate perhaps 50-80% for a subdominant root, and 20-50% for a supertonic root. Here is the whole set of triads in first inversion, given a major mode:

Arguably, each of those chords functions more strongly (or at least equally) as the first note of the triad than the root would suggest. Unlike traditional theory, my theory also excludes non-chord tones such as passing/neighbour notes, appoggiaturas, suspensions or other 'relatively insignificant' or ornamental surface/background phenomena, which I will now refer to as NCT in future) to be distinguished as separate phenomena from usual harmony units (Schenker's "Stufen") [8]. That is not to say that pointing out so-called ‘non-chord tones’ aren’t signifying something important (i.e. how consonant or dissonant the interval is), but I think that the scale is a continuous, not discrete-binary type, and therefore consonance and dissonance (tendency to resolution) deserves its own dimension, and should ideally be represented below the chord root analysis, with levels of strength.

In this new light, a simple suspension can be analysed in the following manner:

Chord: Bass note: Root analysis: Appended analysis: My analysis (very roughly):
D A D F D D (minor) - D - 100%
G A D F G G (9th without 3rd?) Suspension on A... G - 50%, D - 50%
G B D F G G7 (dom. 7th) ...Resolution to B. G - 100%, (or perhaps G - 75%, B - 25%)

Above we can see how a 'suspension' could be viewed alternatively as a transition from the root of D to the root of G, with the intermediate chord containing elements of both. I believe my analysis could be a more accurate and complete portrayal of the underlying perception of harmony, and may even lead more easily to a theory of consonance and dissonance. Here is another example:

Chord: Bass note: Root analysis: Appended analysis: My analysis (very roughly):
G B D G G G (major) G - 100%
A B D G A G with 9th, or.... G - 75%, A - 25%
A with 7th, 9th & 11th? Susp. chord on B & D
A C E G A A7 (minor 7th) ...resolving to C & E A - 100%, (or perhaps A - 75%, C - 25%)

As a consequence, we can also theorize a continuous graduation from A minor to C major with careful selection of notes (proportions of A and C are intuitively guessed, and only very approximate).

(A - 100%, C - 0%)
(A - 75%, C - 25%)
(A - 65%, C - 35%)
(A - 50%, C - 50%)
(A - 25%, C - 75%)
(A - 0%, C - 100%)
The spelling of a note also makes a difference, which is partly why I researched that above. As an example:
C Eb G - (root of C)
C D# G - (mixed root of C and G. This would function as such in sharper keys such as A or E major onwards).

Excluding arpeggios (broken chords), the calculation of root weights can theoretically be absolute for each chord. However, one could say that relatively speaking, in the key of C, that GCE feels dominant when compared to CEG, but relatively tonic when compared to GBD, so it seems that consonance and dissonance should always be taken in context when studying how a piece of music really works. One should also bear in mind that the strength of a harmony may also depend on whether it takes place on a weak or strong beat [43]. The other factor to bear in mind is how a chord (especially with a small number of notes), can easily blend into the surrounding (horizontal) chords/notes. Typical examples include pieces with broken chords such as Beethoven's Moonlight Sonata, Fur Elise, and Mendelssohn's Fingal's Cave (e.g. bars 9-12). Should one blur the harmonies after analysis, or look for ways to combine sonorities during the initial analysis? It seems that for one or two part harmonies, that the ear makes up for the missing note by perceiving an (unheard) extra note of the key’s center.

Often, you'll find that even traditional theory will have problems in distinguishing NCT from harmony [27][28]. Instead, I think that NCT can be looked upon as shades of grey between two or more different roots. I theorise that their effect isn't generally recognised as having much significance in structural harmony because of the following reasons:.

1: The melody or passing/neighbour notes are usually very brief, so they don't hold as much influence compared to long held notes.
2: All NCT usually appears above the bass, and it is true that the bass note weighs more heavily than the upper notes when finding the root (see later).
3: NCT is often much thinner in texture than the rest, so the root is defined more by the thicker texture of the 'bigger part' of the chord.
4: NCT influence and control voice leading, but I think the workings of voice leading is a separate matter.

Despite the above four reasons, it doesn't mean we can't discount NCT as exerting influence to the underlying harmony/root, no matter how small their effect may be. Indeed, if instead we assign harmonies to every single chord and note, we can unify the previously separate analysis of NCT, and the root analysis of normal chords, into a single broader concept.

Many or all of these are transposable along the diatonic scale, to produce the equivalent roots. For example, CEGA can be diatonically transposed to FACD to produce the roots of F and D, or to EGBC (roots E and C). Speaking of which, Walter Piston and Mark Devoto, in the book 'Harmony', interestingly present this chord on the seven degrees of the scale, and seem to indicate the 'dual rootness' of the chord. However, their perception of the root from many of these chords may disagree with my perception and theory. Unfortunately, he does not give musical examples from which to base their findings on.

Traditional Theory Walter's perception My own perception (matches my theory too)
C E G A A (min 7th) Mainly C (+A) Mainly C (+A)
D F A B BB Mainly D (+B)
E G B C C C Mainly E (+C)
F A C D D F (+D) Mainly F (+D)
G B D E E E Mainly G (+E)
A C E F F F Mainly A (+F)
B D F G G G Mainly B (+G)

A possible explanation for the contradiction is that the surrounding chord/s from which he is basing his findings on, are themselves mixed root chords - which he has represented by their theoretical root, rather than his own perception. My evidence for this rather speculative theory is not without merit. An example would be the B diminished chord (which is usually in first inversion - DFB), which he may have used to compare against DFAB. Certainly, DFAB sounds quite like the B (diminished) triad in first inversion, perhaps even more so than a D minor triad. However, the B (diminished) triad in first inversion feels itself like a D minor triad to start off with! (or at least, that is its perception to me - compare DFA with DFB). This is why in all my aural experiments, I use the simplest possible triads to compare with a potentially complicated chord.

Possible problems and issues with determining roots

1: My research doesn’t take into account accidentals influencing the sonority of the triad. For example, perhaps a diminished triad inside a larger chord should have less of an influence towards that root than a major triad. There is also the issue that I couldn’t find (in the key of C) Db being functionally equivalent to D, and F# to F. Maybe these chromatically altered chords have different functions to any of the seven diatonic notes (apart from obviously hinting at key change).

2: Assuming harmonies with at least three notes simultaneous notes (so at least one triad can be determined), is the root of a chord solely determined by itself, or is it also influenced by the key and/or previous and successive chords?

3: How are prolonged harmonies handled? Arpeggios, and any other type of broken chords need to be analysed in context before a harmony can be assigned to a section (unless there is a background harmony to thicken the texture, so that triads are heard again). For example, the repetition of the notes; CEDG could be taken as a single chord of C, or a switch between C on the first note and G on the third, or a state of 'undetermined' with either as 50% chance each (perhaps needing more notes for conclusion), or my current favourite theory - a 50% mix of both. It would seem that repetitive metrical time - with note groups of 2, 3, 4, 5 etc. - are used to separate each arpeggio harmonically.

Automatic calculation of roots

It was only towards the end of my studies that I had enough knowledge to create an algorithm to determine the approximate roots of any arbitrary chord. My previous attempts at an algorithm resulted in many incorrect classifications, with dubious root weightings for many others. In my own experience, the results are now close to 100% (based on my own perception of chords), but obviously, the subjective judgement of roots will vary from person to person.

The algorithm is written in C, and is detailed at the end of this thesis, along with full code and documentation (Appendix 6). Far from simply counting the triadic components for each degree of the scale, real perception of root/s in a chord is more complicated. For example, extra weight is given to the bass, or to a full triad. The fourth of any note reduces the effect of its first, and a note, according to its position, does not necessarily become the root of another triad, but may become its upper note. Only repeated listening, along with chord substitution in real music allowed me to gather a feel for the saturation of each root in a chord, thus enabling me to create the algorithm. Along with the program, you will see root analysis on actual music in Appendix 2-5.

Here are some results from the program. I agree roughly with all of them, although some chords are very difficult to verify for sure. Others are slightly questionable, such as CEGB given a weight of C=3 and E=2. I would think the E root is weaker than 2/3rds of C’s strength. Same may go for CEGA (maybe A should be weaker than shown).

Chord 01: C G B D         C: 2   D: 0   E: 0   F: 0   G: 2   A: 0   B: 0
Chord 02: C D             C: 2   D: 0   E: 0   F: 0   G: 1   A: 0   B: 0
Chord 03: C D E G         C: 3   D: 0   E: 0   F: 0   G: 1   A: 0   B: 0
Chord 04: C D E F A       C: 1   D: 2   E: 0   F: 3   G: 0   A: 1   B: 0
Chord 05: C D F G         C: 1   D: 0   E: 0   F: 2   G: 0   A: 0   B: 1
Chord 06: C D G           C: 2   D: 0   E: 0   F: 0   G: 1   A: 0   B: 0
Chord 07: C D G B         C: 2   D: 0   E: 0   F: 0   G: 2   A: 0   B: 0
Chord 08: C E             C: 2   D: 0   E: 0   F: 0   G: 0   A: 0   B: 0
Chord 09: C E D           C: 2   D: 0   E: 0   F: 0   G: 1   A: 0   B: 0
Chord 10: C E D F         C: 1   D: 0   E: 0   F: 2   G: 0   A: 0   B: 1
Chord 11: C E D A         C: 2   D: 1   E: 0   F: 0   G: 1   A: 1   B: 0
Chord 12: C E F           C: 1   D: 0   E: 0   F: 2   G: 0   A: 0   B: 1
Chord 13: C E F G         C: 2   D: 0   E: 0   F: 2   G: 0   A: 0   B: 1
Chord 14: C E F A         C: 1   D: 0   E: 0   F: 3   G: 0   A: 2   B: 0
Chord 15: C E F B         C: 1   D: 0   E: 1   F: 1   G: 0   A: 0   B: 1
Chord 16: C E G           C: 3   D: 0   E: 0   F: 0   G: 0   A: 0   B: 0
Chord 17: C E G A         C: 3   D: 0   E: 0   F: 0   G: 0   A: 2   B: 0
Chord 18: C E G B         C: 3   D: 0   E: 2   F: 0   G: 0   A: 0   B: 0
Chord 19: C E G B C       C: 3   D: 0   E: 2   F: 0   G: 0   A: 0   B: 0
Chord 20: C E G B D       C: 3   D: 0   E: 2   F: 0   G: 2   A: 0   B: 0
Chord 21: C E A           C: 2   D: 0   E: 0   F: 0   G: 0   A: 2   B: 0
Chord 22: C E A B         C: 2   D: 0   E: 1   F: 0   G: 0   A: 2   B: 0
Chord 23: C F             C: 1   D: 0   E: 0   F: 2   G: 0   A: 0   B: 1
Chord 24: C F G           C: 1   D: 0   E: 0   F: 2   G: 0   A: 0   B: 1
Chord 25: C F A           C: 1   D: 0   E: 0   F: 3   G: 0   A: 1   B: 0
Chord 26: C F B           C: 1   D: 0   E: 1   F: 1   G: 0   A: 0   B: 1
Chord 27: C G A F         C: 1   D: 0   E: 0   F: 3   G: 0   A: 1   B: 0
Chord 28: C G B           C: 2   D: 0   E: 1   F: 0   G: 0   A: 0   B: 0
Chord 29: C A             C: 2   D: 0   E: 0   F: 0   G: 0   A: 1   B: 0
Chord 30: C A D F         C: 1   D: 2   E: 0   F: 3   G: 0   A: 0   B: 0
Chord 31: C B             C: 2   D: 0   E: 1   F: 0   G: 0   A: 0   B: 0
Chord 32: C B D           C: 2   D: 0   E: 0   F: 0   G: 1   A: 0   B: 0
Chord 33: C B D F         C: 1   D: 0   E: 0   F: 1   G: 0   A: 0   B: 2
Chord 34: C B F           C: 1   D: 0   E: 1   F: 1   G: 0   A: 0   B: 1
Chord 35: C B F A         C: 1   D: 0   E: 1   F: 2   G: 0   A: 1   B: 1
Chord 36: C B G           C: 2   D: 0   E: 1   F: 0   G: 0   A: 0   B: 0
Chord 37: C B A           C: 2   D: 0   E: 1   F: 0   G: 0   A: 1   B: 0
C was chosen as the bass note for the chords above, but the weights seem to be diatonically transposable to any other degree. For example, a major triad+6th on the dominant, has the root weights transposed up a diatonic fifth compared to the same chord on the tonic.

Here are other examples of chords which have been commented on:

a) G C E - (mostly C with a hint of G ("the fifth gives a suggestion of dominant feeling without an actual harmonic change" [41]))
b) C E G A - (like above, but stronger feeling of C root. Walter speaks of the contradiction between traditional theory and practise for this chord [55])
c) F A C D - (F+ hint of D - transposition of above. "The subdominant chord, which is a triad with added sixth, likewise becomes a seventh-chord through Rameau’s concept of the double emploi of this added sixth (the second degree of the scale) as a fundamental" [61]).
d) F Ab C D - (as above, but feels even more like F because the D component contains D, F and Ab, a weaker functioning diminished triad. "The subdominant feeling of this chord is rather stronger" [56])
e) F Ab B D - (a diminished 7th. Ab is chosen over G# because of the assumption of a C harmonic minor key, but the diminished 7th can have four different spellings. Often perceived as a dominant [59], but Walter suspects a subdominant for this inversion [54]).
f) B D F Ab - (B+D since Ab could be treated as a (weaker?) natural sixth. Theory says that the diminished seventh can't be pigeon-holed [13][14]).
g) Ab C E - (a root of A since Ab is naturalized to A. Quote: "The augmented triad has a certain vagueness due to the fact that the inversions sound like root position" [57].)
h) E G# C - (a mixed root of C and E, since G# naturalizes to G)

Other research on root calculation

Though not quite the same method I use, I have found so far a couple of authors who at least hint at the idea of perceiving multiple roots simultaneously. David Temperley for example has created the Melisma program which hopes to weight chords to find which single root is most appropriate. Any notes in the chord which are then not part of that finalized root are counted as "ornamental dissonance". This is in contrast to my algorithm which takes all components into account, no matter how weakly. Also, instead of necessarily calculating the root for a single chord, his program will trek over a defined amount of time for each root produced. He assigns certain ‘penalties’, and ‘rewards’ for each pitch according to their position in the line-of-fifths away from the potential root under review. I quote: "For example, the value for b5 is -5.0; this means that, given a root of C, the pitch Gb (b5 of C) will contribute a score of -5.0 (per second) to the score." [63]. Molly Gustin also devised a rather elaborate theory based on the harmonic series to determine the root/s of a chord [65], but she was rather heavily criticized by Arthur Komar in the "Perspectives of New Music" journal [64]. Still, she did mention how a chord could contain two simultaneous roots with one possibly predominating over the other. I have yet to study her theory in depth.

My program for automatically finding the root can be found here for download:
Chord Root Finder


HiC = Miguel Roig-Francoli, "Harmony in Context", pub: McGraw-Hill, January 1, 2002
Grove-tonality = Brian Hyer: ‘Tonality’, Grove Music Online ed. L. Macy (Accessed 14/03/2007)
HiWM = Richard Franko Goldman, "Harmony in western music", pub: Barrie and Jenkins Ltd, 1968
Harmony = Walter Piston and Mark Devoto, "Harmony", pub: Victor Gollancz Ltd, 1978

[1] HiC - p38
[2] Grove-tonality - "Musicians agree that there are two basic modal genera, major and minor, with different but analogous musical and expressive properties.".
[3] Grove-tonality (Historiography): "... As a metaphysical principle, then, tonality does not itself evolve, but rather remains invariant and universal, true for all people and for all time. He thus regarded what he felt to be the undeniable historical progress of Western music as a series of discrete advances toward completion, the ever more perfect realization of a musical absolute."

From the same page, contrast this with:
"Now that popular and commercial music has overwhelmed and displaced 'serious' music in cultural attention, and in view of an ongoing re-emergence of tonal idioms within the postmodern avant garde, the narrative of continuous tonal evolution no longer seems as credible as it once did and has begun to loosen its grip on the music-historical imagination. In the absence of the musical and cultural polemics that were responsible for the tremendous prestige of the concept, musicologists, whether historians or theorists, will turn to the description of tonal music in terms of contingent harmonic practices rather than invariable laws that inhere in or arise from the musical material and determine its ultimate historical fate."

[4] Grove-tonality - "Schoenberg, for instance, conceptualised the firm sense of closure in this music in terms of 'monotonality', the idea that, no matter how extended in duration, pieces of music retain their allegiance to the original tonic from beginning to end (Structural Functions of Harmony, 1954). Roger Sessions [25], and Schenker, who elaborated the same basic idea, heard modulations as temporary 'tonicizations' of non-tonic scale-degrees rather than permanent departures from the original tonic. This allowed him to regard entire pieces as recursive hierarchies of harmonies, progressions within progressions."

[5] Grove-tonality - "Scale-degree theories accounted for chromaticism by means of what Schenker called mixture (Mischung), which refers to contexts in which the music gains access to or borrow harmonies from the parallel major or minor. In order to increase the harmonic resources of C major, for instance, one can replace A minor (or VI) with A major (or VI), borrowed from the parallel minor. In Harmonielehre (1906), Schenker goes on to describe how, in the music of late Romanticism, major and minor fuse together: he combines the notes of both the major and minor scale into a single chromatic scale and then places, as in ex.5, major and minor triads (via mixture) on each degree. Similarly, Schoenberg heard late Romantic music in terms of 'a transition from 12 major and 12 minor tonalities (Tonarten) to 12 chromatic ones', a historical transition 'fully completed in the music of Wagner' (Harmonielehre, 1911)." - grove?

[6] Heinrich Schenker, "Theory of Harmony", (1906), p395

[7] - "The first tenet of Schenkerian harmony is that nature, through the harmonic series, gives us the triad as the ultimate (and only possible) basis for musical composition. (See Harmony pp 20-29 which includes Schenker's defence of the 5-limit). In fact, Schenker's explanation only secures "naturalness" for the major triad, whereas Schenker describes the minor triad as an artificial construction of musicians. (Harmony pp 49-52) Despite this difference, in practice the major and minor triads are treated equally in Schenkerian analysis."

[8] - "A chord in a piece of music may represent the Stufe corresponding to its root. However, many surface phenomena in music that appear to be chords are not actually representative of Stufen themselves but are voice-leading constructions of a passing nature whose real function is the prolongation of some other Stufe. In short, not all chords represent Stufen."

[9] - "Furthermore, in terms of Schenker's mature theory, the question of whether a given triad possesses scale-step status depends on the structural level under discussion. Indeed, it follows from Schenker's concepts that, at the highest level, a tonal composition possesses only one scale step, since the entirety of the work may be understood as an elaboration of its tonic triad (i.e. scale-step I)."

[10] - "Where others consider the music to have changed key (into the dominant or relative major), Schenker suggests that all keys other than the tonic are scale steps. He hears them in the same way other analysts might hear a dominant chord - as part of the tonic key. The only difference is that Schenker hears these relationships on a much larger scale."

[11] HiC, p26
[12] HiWM, p174
[13] HiC, p39
[14] HiC, p57
[15] Grove-tonality

[16] We find then that, that modulation, like so many other aspects of harmony, is largely dependent on context for its interpretation. The formula V-I (or I-V) in a new key does not automatically establish a new key centre. For the ear to accept a definite change of tonal area, the progression needs to be prominently placed, preferably in root position at the end of a phrase or section, and the new tonality retained for an appreciable length of time......... Indeed, the effect may differ from one performance to another, and even for one listener rather than another. In short, there is no clear dividing line between chord progressions and key progressions; a short phrase based on the harmonic outline I-V-I contains the nucleus of a complete piece built on the *key* scheme I-V-I.

[17] HiC, p35/36
[18] HiWM, p9
[19] HiWM, p22
[20] HiWM, p84
[21] HiWM, p89
[22] HiWM, p113
[23] HiWM, p114
[24] HiWM, p128
[25] HiWM, p129

[26] HiWM, p130 - "and it should be said at once, that there is not always - perhaps not even often - an objective test for determining modulation. thus, we may above a short piece, a typical A-B-A form, in which the first phrase begins and ends on the dominant. Is the second phrase in a new key (that of the dominant) or is it simply a phrase on the dominant of the original key? In other words, when, or under what circumstances, does such a dominant become a tonic in its own right? This is a question for which, in the abstract, it is almost impossible to provide an answer."

[27] HiWM, p149 - "Whether one hears these as separate harmonies, or as aspects of one harmony, or as a single harmony with a passing tone, will depend on many variable factors, including performance, or on subjective aspects of hearing."

[28] HiWM, p150 - "There are in one sense not many notes in this passage that can be characterized with certainty as passing tones, and yet, in another sense, all except the pivotal harmonies are passing."
HiWM, p151 - "Does the major IV change to a VII, or do we hear and construe the G in the top voice as a passing tone, and the E natural in the third voice as a simple neighbouring tone?"

[29] Harmony, p12
[30] Harmony, p20
[32] Harmony, p21
[35] Harmony, p54
[36] Harmony, p55

[37] Harmony, p59 - "The tonal strength of the tonic is greatly enhanced by a dominant preceding it. Thus, one might expect to be able to determine whether a piece, is in C major or a minor by several tests; by examining initial and final chords, by comparing the relative abundances of C major and A minor triads, and by looking for the presence of G sharps, which would ordinarily suggest the leading-tone, the third of V, in A minor"

[38] Harmony, p61
[39] Harmony, p60
[40] Harmony, p72
[41] Harmony, 179
[42] Harmony, p191
[43] Harmony, p208
[44] Harmony, p214 "...the principles described in Chapter 5 on tonality should be observed. It is not essential that the tonic chord should appear, but the dominant must be made to sound as such."
[45] Harmony, p219
[46] Harmony, p220
[47] Harmony, p232
[48] Harmony, p249
[49] Harmony, p282
[50] Harmony, p283
[51] Harmony, p302
[52] Harmony, p311
[53] Harmony, p313
[54] Harmony, p314 - "The diminished seventh chord in this position, therefore, seems to combine dominant and subdominant functions into a single sonority"
[55] Harmony, p359
[56] Harmony, p360
[57] Harmony, p436 "The most important means of defining a tonal center, in the absence of a preceding dominant, became and remained the tonic itself, either as a triad or as a single pitch used somewhat like a pedal point, asserted vigorously or subtly but always definitely."
[58] Harmony, p? - " many different keys within a short time without ever employing a V-I cadence, but we have seen that in all such pieces there is most of the time a clear sense of some kind of a tone center, even if that center is constantly shifting and even if there is no apparent single background tonality.".
[59] My comments: Goldman argues that the diminished seventh has no root at all, unless perceived in context, where it has only a single root (for Bach's music, he would claim this usually to be the dominant) [22]. Likewise, Walter believes the diminished 7th to most often act as a dominant [52], but also says that strictly speaking, it has no actual root because it is not present [53].
[60] Thomas Christensen: ‘Rameau, Jean-Philippe (part 5)’, Grove Music Online ed. L. Macy (Accessed 14/03/2007)
[61] Deborah Hayes, ‘Rameau’s "Nouvelle Methode"’, Journal of the American Musicological Society, Vol. 27, No. 1, (Spring, 1974), p67
[62] Harmony, p186
[64] Journal: "Perspectives of New Music", Vol. 8, No. 1 (Autumn - Winter, 1969), p. 146-151.
[65] Journal: "Journal of Music Theory, Vol. 6, No. 2. (Winter, 1962)", p. 178-198


Appendix 1 - Mozart’s Symphony No./40 in G, K550 (transposed to C minor)

Appendix 2 - ABBA’s Dancing Queen (excerpt)

Mixed Root Analysis explanation:

To start with, here is a weighted root analysis of ABBA’s Dancing queen. The roots I’ve assigned to the various chords in this piece can often be mixed and weighted. For example IV-5, I-3, VII-2 means the subdominant is given a 50% weighting, the tonic a 30% weighting and the seventh a 20% weighting. If one were to substitute that chord for the subdominant, that would prove most effective, though the other two roots are hinted at too, just to a lesser degree. Likewise, all chords in the song can be substituted with one of the basic diatonic triads I have given below each chord (to lesser or greater degrees). It should however be mentioned that all weightings are only approximate, and reflect my own (possibly flawed) judgement.

The other notable item in this piece is the accidental on bar 6. Ab is allowed in the context of key C, but not G#. However, the piece progresses to a temporary key of A (tonicization), and therefore E major acts as a secondary dominant, so G# is acceptable in this sense.

Appendix 3 - Handel’s Arrival of the Queen of Sheba

(Transposed to C, and piano reduction)

See Appendix 2 for explanation on my mixed root analysis (shown in this piece as blue). Here, I have given the bass note, the root, as well as my own mixed root analysis to compare and contrast. Any time you see a figuring such as "+IV-5", the + symbol means the said harmony holds for that single note, and is only hinted at.

Once again, weightings are my own opinion and only approximate, but it is interesting to note how the mixed root analysis is often a conjunction of the bass note and root lines. In beat 2 of the first bar, the bass note B could in traditional theory count as a passing note. However, as I’ve said earlier, for my idea mixed roots, every note is harmonic, and even if it is only to a small degree, the non-chord tone will still influence the overall sonority. Although non-chord tones are useful for analysis in the sense that they imply a dissonance in need of resolution, I’m beginning to think that the attribute of dissonance and consonance should be in a separate dimension than the root of the chords, labelled separately, and in context to the immediate harmony, intermediate levels, and also the global harmony.

Appendix 4 - Robert Schumann’s "Song, Ich kann‘s nicht fassen, nicht glauben, from Frauenliebe und -leben, Opus 42"

In this piece, the red circled notes show how they can be spelt differently according to what key they are under. This also affects the root in each case. I will now use all four systems of spelling to check each note:

Bar 8: F# or Gb?       System 1a: Gb (since Gb lowers to F)	   System 1b: F# (since we’re still in the key of C)
                       System 2: F# (we are/were in the key of C)  System 3: F#/Gb (both sound okay)
Bar 18: System 1a: Cb    System 1b: Cb                 System 2: B         System 3: Cb
Bar 20: System 1a: F#    System 1b: Gb                 System 2: F#                System 3: F# slightly
Bar 24: System 1a: F#    System 1b: F# (future key=G)  System 2: Gb (past key=Eb)  System 3: F# slightly
One may wish to relate the accidental not just to the previous or future key, but right back to the home key. On bar 9, it is possible to interpret the natural seventh as a mode change to the harmonic minor, and the same goes for bar 3, albeit very temporarily.

I have also compared my results with Goldman, and often we agree, but obviously, I try to split the atom (example: compare our analysis on bar 20) for a potentially clearer picture.

Appendix 5 - Franz Schubert’s Impromptu, Opus 142, No.3 (Theme and closing section)

This final piece (Appendix 5) above introduces a complication - the broken chords in the bass. It is reasonable to believe that the broken chord should be moulded into one single chord before root analysis takes place. Alternatively, perhaps we can analyse as normal, and then ‘smooth’, or ‘defocus’ the analysis afterwards for simplicity, so that each half-bar is blended into a single sonority. This would have the effect of losing detail, but the advantage of seeing the overall picture. More study needs to be done before a conclusion can be made either way.

The sonority in bar 9 surrounded by red stars suggests a spelling of Ab rather than G# (because we move to A minor though, and start on a rhythmically strong bar/beat, it is mostly heard as G#). Ab would completely change the root analysis (approximately A-6, D-2, E-2).


Skytopia > Music > Theory of Multiple Chord Roots    (article created 14/04/2007

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