A Primer on Arabic Numeral Harmonic Analysis in Modern Modal Music

Alexander LaFollett

If you have encountered my theoretical work on the area of modal and non-tertian harmony, such as my exploration of harmonic syntax in the Locrian mode, you would have also run across my Arabic numeral analysis system, which I use to label chords in this harmonic environment. While that particular writing does explain some of the concepts underlying this form of analysis, I feel that an essay specifically dedicated to this topic is warranted here.

What is Arabic numeral harmonic analysis, and why is it necessary?

Arabic numeral harmonic analysis is a method of analyzing and labeling chords, comprising a wide array of interval structures, and describing how they fit within the context of a prevailing modal structure. It is an approach I have been developing since 2005, when I needed new tools to analyze my orchestral pieces Hydrogen and Helium, from my master’s thesis (finished 2006).

The underlying concept is very similar to Roman numeral analysis, but was designed to overcome some of the deficiencies that one quickly encounters when analyzing modal and non-tertian harmony using that approach. It even addresses some of the issues one encounters using Roman numerals for their original purpose of analyzing tonal harmony, which is arguably nothing more than a restrictive subset within the broader field of modality. Because of this similarity to one of the most familiar concepts in music theory—one that occupies a considerable amount of time in the traditional two-year post-secondary theory curriculum—it also becomes very easy to grasp.

The Arabic numeral system also allows one to at least begin to address a matter in the analysis of modal music that I feel has been largely ignored to date—that being the matter of how the sense of mode is being established, and at an even deeper level, how the harmonies and the individual scale degrees behave within the context of the mode.

Much of the analysis to date in the area of modality has been occupied with surface-level details—often just identifying the mere fact that there is a mode in use, and what to call it, since the matter of nomenclature is haphazard and severely underdeveloped in current general usage, especially once one gets outside the familiar realm of the diatonic modes. The “battle royale” between Pieter van den Toorn, Dmitri Tymoczko, and Daniel Chua (the latter of whom called the whole matter a “riot”, fittingly enough) over the mere labeling of the modality at the opening of the “Augurs of Spring” section from Stravinsky’s The Rite of Spring is just one symptom of this. The fact that much of the general educational instruction on modality, a considerable amount of which is contained in non-academic method books, is little more than “here’s some scales (some of which are actually the same scale but with different names—a particular hallmark of some recent method books), go figure it out on your own” is another.

The general apparatus for handling non-tertian harmony—chords built out of intervals other than thirds—is also arguably quite underdeveloped, particularly in dealing with it from the perspective of its use in harmonic progressions in a centric environment (i.e. both modality and common practice tonality), not as merely an abstract collection of intervals, or as some sort of “nonfunctional” fluff that can be ignored.

The matter of behavior and harmonic syntax is also a rather critical matter for composers and songwriters wishing to explore modality and non-tertian harmony, and I have often been asked about it by students, as well as in general conversation with other musicians, as it really isn’t discussed beyond a cursory level. As someone who sees himself as a composer-theorist—with an emphasis on the dash, as in contrast to recent trends, the two sides are virtually inseparable for me—this is a matter I find particularly important.

How does it operate?

The basic syntax for the Arabic numeral labels follows the format b(spec), where the b is the Arabic numeral (i.e. 1, 2, 3, etc.) that represents which scale degree from the mode is in the bass—similar to Roman numerals—and spec is the species of chord that is built upon it. The species is defined as the interval structure built on top of that bass degree, generally condensed down to the span of an octave (i.e. simple intervals), and put in order of smallest to largest interval above the bass—such that 7 is the largest number seen in the species designation.

If, for instance, we consider the example of the chord D-G-C in the context of A Aeolian (A-B-C-D-E-F-G-A), D is scale degree 4. The G lies a fourth above D, and the C a seventh above, meaning that the species of the chord is 4/7, and the full label would be a 4(4/7).

For another example, let us consider the chord E-G♯-B-D-F♯ in the context of E Mixolydian (E-F♯-G♯-A-B-C-D-E), E is degree 1. G♯ is a third above, B a fifth, and D is a seventh above. The F♯ lies above the seventh in the voicing of the chord, and if we subtract the octave out of the resultant ninth, we are left with a second. Putting this into order, the chord would accordingly be labeled as a 1(2/3/5/7).

Structures that contain the same octave-condensed relationship to the bass, regardless of the ordering, voicing, or doubling of those non-bass notes, will all belong to the same species. All of the following chords, for instance, would be classed as 3/5 chords.

All have C in the bass, and also have an E (a third above, or some octave multiple thereof) and a G (a fifth above, or again some octave multiple thereof) present. C-E-G, C-G-E, and C-G-E-C (canceling out the duplicate C) would all reduce down to C-E-G. (Orderings of these three notes in which C is NOT in the bass, however, like E-G-C or G-C-E, are not 3/5 chords, a matter described in more detail later.)

There are actually a total of 64 (!) different species possible within a heptatonic mode, counting the trivial “singleton” (literally just a single note, wherein no species designation would be shown in the Arabic numeral label), and the catch-all “heptachord” or “full modal aggregate”—designated as 2/3/4/5/6/7—in which all notes of the mode are present above the bass.

This effectively means that, excluding the trivial singletons, each mode has a total of 441 possible native chord structures—63 over each of the seven scale degrees.

How does this system handle chord qualities?

With respect to the matter of chord qualities, the typical application of the Roman numeral system uses upper and lower case (for “major” (IV) and “minor” (iv), respectively), along with three common modification symbols, to handle the “augmented” (+), “diminished/fully-diminished” (°), and “half-diminished” (ø) cases. Occasionally, some special situations (i.e. augmented sixth chords and altered dominants) may bring about a bit of additional symbology, but there’s not much else needed. The majority of this, however, is done within the context of a two-mode system, with a very limited set of the above species being considered—primarily just 3/5, 3/5/7, and their rotations (“inversions” in the common practice tonal nomenclature).

Once one begins to accept the idea of additional modal and species possibilities, a wider array of qualities also becomes possible, and the options presented by the traditional Roman numeral quality symbology are either quickly exhausted, or start to seem questionable to use. Even a structure as simple as the 4/5 chord D-G-A poses such a dilemma—the two intervals above the bass are classed as “perfect” under the traditional nomenclature. The fact that the Arabic numerals don’t have typographical cases in the same way as Roman numerals also factors into the equation.

The most natural and concise solution I have found is to use the modes themselves as quality identifiers, essentially as an extension of “chord-scale theory”. This bears some resemblance to how things are typically handled in the jazz idiom, though excising the vestigial remnants of common practice tonality that regularly figure into the nomenclature there. To give a familiar example, I will first examine the case of 3/5 chords.

The system that I have derived involves labeling the qualities based on the first mode in my Heptatonic Modal Catalog that contains that particular quality on degree 1. The ordering I use in the catalog for the diatonic series begins with Dorian as Mode 1 (due to historical precedent, and the fact that the mode has some rather unique properties, like its underlying palindromic structure), Phrygian as Mode 2, and so on, ending with Ionian as Mode 7.

If we build 3/5 chords on degree 1 of each of these modes, we find the following:

Dorian, Phrygian, and Aeolian all have the 3-half-step third, and 7 half-step fifth; Lydian, Mixolydian, and Ionian all have a 4-half-step third and 7-half-step fifth, while Locrian has a 3-half-step third and a 6-half-step fifth. The 3H/7H quality, given that it first arises in Dorian (Mode 1), becomes the Dorian 3/5 chord. Lydian (Mode 3) is the first to have the 4H/7H species, which thus becomes the Lydian 3/5. The 3H/6H model, accordingly, becomes the Locrian 3/5. Coincidentally enough, this Dorian/Lydian labeling ends up producing a more transparent reading of the common jazz theory chord-scale models.

In order to better show off the real benefits of this system, I will now provide an example featuring a less familiar species—the 2/5/6 chords. Following the same process as above for the diatonic modes, we get the following results:

The resultant four species, per the above example, would be the Dorian 2/5/6 (2H/7H/9H), the Phrygian 2/5/6 (1H/7H/8H), the Aeolian 2/5/6 (2H/7H/8H), and the Locrian 2/5/6 (1H/6H/8H).

The interesting thing about doing this quality analysis is that it yields knowledge about the relationships between scale degree positions between modes, which might not otherwise be initially considered—especially due to the way in which the ubiquity of common practice tonal ideas has conditioned many Western musicians to think on a tertian basis when dealing with music that has tonal center. This sort of knowledge is, I believe, of particular utility for those writing music that makes use of modality and/or non-tertian harmony.

Expanding out to the full breadth of all 462 heptatonic modes that are possible within the space of 12-tone equal-temperament (12-TET)—which also comprises the full extent of the Heptatonic Modal Catalog—there are actually a total of twenty-one different qualities available for just the 3/5 chords alone. In fact, each of the fifteen possible 3/5 species have twenty-one qualities apiece. The lone heptachord species—2/3/4/5/6/7 (under which the tertian construct of the “thirteenth chord” would be encompassed)—actually contains a total of 462 qualities, just as many as there are modes.

These sheer numbers may seem daunting, though unless one ventures into the wilds of highly-eccentric heptatonic modes (think B♯-C♯-D-E♭-F♭-G♯-A♯-B♯), one will likely only encounter a more manageable subset of all of these possibilities. That said, the biggest thing that the presence of all these species, qualities, and modes bring to the table is a heightened ability to control musical expression, and one that starts from a familiar base.

Another interesting fact to note is that the alteration of a single note in a mode—i.e. changing C Ionian into C Mixolydian by lowering scale degree 7—will result in 255 harmonic structures within the mode (out of 441—about 57.8%) changing qualities. That is a statistically significant figure.

How does this system handle the concept of inversions?

Simply put, it doesn’t.

That may seem like a shocking statement, and an admission of what one would initially imagine to be a gaping hole in the system. However, there are a number of significant logical, musical, and historical inconsistencies with the concept, at least as it is usually thought of in established tonal theory. As such, its exclusion is very much by design. The nature of working in a modal harmonic environment, in which both non-tertian harmony and seemingly warped forms of tertian harmony are common, often magnifies these inconsistencies to an extreme, causing the entire concept to break down with considerable frequency.

To review the basic premise as is traditionally taught in tonal theory, the notion is that the chords C-E-G, E-G-C, and G-C-E, by virtue of containing the same notes (C, E, and G), are all effectively the same chord, “C Major”, and have the same root—C—despite the fact that the latter two have a different note in the bass. Because of the “non-root” bass in the latter two, they are considered to be “first inversion” and “second inversion” of the former, which is in “root position”. This notion is based on the idea of tertian primacy, and Jean-Philippe Rameau popularized the idea of tying it all to the overtone series in his Traite de l’harmonie (“Treatise on Harmony”) in 1722, using the fact that the inversions of the major triad (or Lydian 3/5) can be derived from higher partials off the fundamental of the apparent root, as shown in his well-known “triangle diagram”:

Technically, my approach with inversion is to replace the concept with a new related one, which happens to be derived from an already integral part of modal theory—rotation. If we examine the basic process used in common practice chord inversion with a triad, for example, C-E-G, we find that the first step moves the root (C), up to the top of the structure, making E be the lowest note—E-G-C. Following this process again, the E gets moved up to the top, leaving the G at the bottom.

This is, of course, parallel to the relationship between the different modes. Taking the unique pitches of D Dorian, ignoring the usually included octave repeat (D-E-F-G-A-B-C), and moving them around such that the first pitch, D, gets moved to the back of the order, with the other pitches moving forward, the result is E-F-G-A-B-C-D. This result is E Phrygian—centered on E, not on D. If we stack D Dorian up in thirds, to create the traditional concept of a thirteenth chord, as D-F-A-C-E-G-B, the result of an “inversion” is F-A-C-E-G-B-D—yet another thirteenth chord, which corresponds to F Lydian (F-G-A-B-C-D-E-F) stacked in thirds. Again, this structure is centered on F, not on D.

Beyond this, there is the fact that Rameau himself was not consistent with the concepts he set up in 1722. Under the paradigm from Traite de l’harmonie, the chord C-E-G-A would be regarded as the first inversion of the seventh chord A-C-E-G, with A as the root. However, in one of his later treatises, Génération harmonique from 1737, Rameau begins theorizing about the concept of a sixte ajoutée, or “added sixth chord”, and possibility that the structure C-E-G-A might actually be rooted on C after all, at least in certain situations. There’s also the matter that some more recent tonal theorists—especially those of a Schenkerian bent—have been regularly questioning the concept in various fashions, primarily with second-inversion triads, as evidenced by David Beach (1967), as well as noted theory textbook authors Edward Aldwell and Carl Schachter. Despite Rameau’s later reconsideration—and the fact that the sixte ajoutée has found favor in jazz and popular idioms—the earlier, uncorrected version of his theory is still almost taught as gospel in the traditional, classically-oriented theory coursework.

And this does not even begin to touch the matter of non-tertian chords. The most notable attempt to apply this same sort of inversion logic to non-tertian harmonies came from Paul Hindemith, in his 1937 treatise, The Craft of Musical Composition. His rationale is based on difference tones—by-product tones produced by the frequency difference between two other pitches—generating theoretical non-bass “roots” for certain structures (or in some cases, rendering them “rootless”). William Thomson, however, riddles Hindemith’s theory with holes in his 1993 Music Perception article “The Harmonic Root: A Fragile Marriage of Concept and Precept”, for its impracticality and imperceptibility.

How does this modal idiom handle the case of 3/5 chords, then? Following mode rotation logic with C-E-G, this would imply that C-E-G, E-G-C, and G-C-E are all “modes”, so to speak, of the same series, produced through the same rotation process. The first would have C as its “center”, the second E, and the third G, rather than all being rooted on C.

If we are in the context of C Ionian, the Arabic numeral symbols would be 1(3/5), 3(3/6), and 5(4/6), respectively.

In this situation, the supposed “root weakness” of the latter two chords, rather than being explained by Rameau’s original inversion theory from 1722 (which he himself started to contradict 15 years later), is instead explained by a concept I refer to as solidity. The idea of solidity, which loosely takes inspiration from the perception research of Richard Parncutt (1988) (and by extension, the earlier work of Ernst Terhardt (1979)), posits the notion that various intervals have different tendencies to strengthen or weaken the gravity projected by the bass note.

My theory of solidity currently is two-pronged, with hierarchies based on both generic intervals (i.e. seconds, thirds, etc.) and specific intervals (in half-steps), which provides the system more nuance in handling extended modal contexts, where positioning on both continua is important. With respect to generic intervals, the hierarchy, from most “solid” to least, goes as follows: fifths, thirds, sevenths, seconds, fourths, and finally, sixths. The first two are strongly solid, the second two are moderately so, and the latter two are not particularly solid at all. Accordingly, at this general level (which is true for the diatonic modes), 3/5 is an incredibly solid structure at the general level, while 3/6 and 4/6 are particularly weak and ineffective in establishing a tonal center in a modal context at this same level. The solidity approach helps provide a more logically consistent explanation of these chords, and of some phenomena in Renaissance counterpoint, such as the curious treatment of the perfect fourth.

How does this system handle altered chords?

In cases where notes have been altered from those present in the mode, my solution is to simply use plus (+) and minus (-) symbols, showing the direction of the alteration—this allows for the unambiguous identification of the alteration. In the context of using the chord E-G♯-A in D Dorian (D-E-F-G-A-B-C-D), one would first see that E is degree 2, and—ignoring the accidentals/inflections following the upper pitches—that G(♯) and A are a third and fourth above, resulting in a species designation of 3/4. The 2(3/4) chord native to D Dorian would be E-G-A, so comparing this to E-G♯-A, we see that the third of the chord has been raised by 1H, giving us a symbol of 2(+3/4).

Staying with D Dorian, if one were to encounter the chord B♭-D-F, one would see that the root, B♭, does not exist in the mode, but can be produced by lowering degree 6 (B) by 1H. The other two notes in the chord, D and F, do exist in the mode, and therefore, are not altered. The resultant symbol would be -6(3/5).

In the case of a situation where there is pervasive mixture of two forms of a scale degree—something I term a polymodal complex, where no single component mode from the complex can be clearly identified as the prevailing mode (and thus, no way of definitively determining which option is native to the mode, and which is an altered degree), then it may become useful to mark these different forms by using up (↑) and down (↓) arrows to prefix instances of these different options for this degree. One of the most applicable cases would actually be with “minor” in a common practice context—which, if all the supposed “forms” are merged together, can very easily be described as such a polymodal complex , in which degrees 6 and 7 have upper and lower options (i.e. A-B-C-D-E-[F-F♯]-[G-G♯]-A, where either F or F♯ can be used for degree 6, and either G or G♯ for degree 7). The familiar “vii°” from Roman numeral reckoning in this context would be rendered as ↑7(3/5) (and be of Locrian quality), while the subtonic “VII” would be ↓7(3/5) (and be of Lydian quality).

What does this mean with respect to the notion of “non-chord tones”, and harmonic salience?

In the comfortable confines of common practice tonality, the notion that “if it doesn’t fit in a stack of thirds, it’s probably not a chord tone” makes things virtually automatic in many situations. As soon as one drops the assertion of tertian primacy embedded in tonal theory, however, things start to become much more subjective—and much more complicated. With all of these chord species now existing as valid entities, theoretically, one could actually dispatch with the idea of non-chord tones altogether, and analyze every single vertical pitch structure in a piece of music using this system. (For the record, I have actually done this, with two of my orchestral pieces, no less. The full harmonic breakdown ended up taking the form of a spreadsheet.)

I also attempted to do this note-by-note spreadsheet analysis with Darius Milhaud’s String Quartet No. 2. (Note this is with an early prototype of the Arabic numeral system.)

This, however, raises the questions as to whether or not this is advisable, and to what extent certain chords/vertical pitch structures can be considered “structural” or “salient” in the music. If one were to try to assign priority of some chordal structure over a given duration, it would likely revolve around features such as:

(a) harmonic rhythm and the durations associated with the various pitches (i.e. pitches/structures that are sustained longer, and those that land at regular and/or metrically stronger points), as well as the frequency of their appearance and register.

(b) the nature of the prevailing musical texture and its activity levels (i.e. whether one is dealing with a chorale, a drone-accompanied melodic line, or a fugue).

(c) the types of harmonic structures that seem to be most prevalent—effectively “pattern-matching”, akin to the “stack of thirds” rule from tonality, but with more flexibility in determining what the pattern can be—for instance, a stack of sevenths.

(d) any sort of arpeggiation or chord outlining within the musical lines.

Taking these sorts of features into consideration can lead one to perhaps come up with some sort of generalized reduction, perhaps bearing some superficial resemblance to Schenkerian analysis (though used in a way that Heinrich Schenker himself certainly would not have approved, given his anti-modal sentiments), which could then be used to hone the targets of an Arabic numeral harmonic analysis to the most essential pitches.

On the opposite end of the spectrum—should there be some reason to zoom in on a particular harmonic happening in a segment of music, minus certain factors, to pick up some finer detail that would perhaps go unnoticed otherwise, the prospect of doing so is also viable.

Ultimately, analysis often becomes a highly subjective matter, and this is especially true once one begins to deal with techniques such as these on complex pieces of music. That is, for me personally, one of the great joys of it—the fact that there often is no single “right answer”, and the fact that music theory, in many ways, is the art of creating some form of scientific logic about a piece of art, which requires both rational and creative thinking.

Alexander LaFollett, Ph. D.

16 December 2019

REFERENCES

Aldwell, Edward, Carl Schachter, and Alan Cadwallader. 2018. Harmony and Voice Leading, 5^th ed. Boston, MA: Cengage Learning.

Beach, David. 1967. “The Functions of the Six-Four Chord in Tonal Music.” Journal of Music Theory 11/1, 2-31.

Chua, Daniel K.L. 2007. “Rioting with Stravinsky: A Particular Analysis of The Rite of Spring.” Music Analysis 26, 59-109.

Hindemith, Paul. 1937 [1945]. The Craft of Musical Composition, translated by Arthur Mendel. London: Schott.

LaFollett, Alexander. 2006. “Analysis of Hydrogen and Helium for Large Orchestra.” Master’s thesis, Central Washington University. 

LaFollett, Alexander. 2019. Heptatonic Modal Catalog, 2019 edition. Forest Grove, OR: Sonata Enterprises.

Parncutt, Richard. 1988. “Revision of Terhardt’s Psychoacoustical Model of the Root(s) of a Musical Chord.” Music Perception 6.1, 65-93.

Rameau, Jean-Philippe. 1722. Traite de l’harmonie. Paris: J.B.C. Ballard.

Rameau, Jean-Philippe. 1737. Generation harmonique. Paris: Prault.

Terhardt, Ernst. 1979. “Calculating virtual pitch.” Hearing Research 1, 155-182.

Thomson, William. 1993. “The Harmonic Root: A Fragile Marriage of Concept and Precept.”Music Perception 10.4, 385-415.

Tymoczko, Dmitri. 2002. “Stravinsky and the Octatonic: A Reconsideration.” Music Theory Spectrum 24, 88-102.

van den Toorn, Pieter and Dmitri Tymoczko. 2003. “Stravinsky and the Octatonic: The Sounds of Stravinsky.” Music Theory Spectrum 25, 167-202.

The Locrian Mode: No, It’s Not Unusable

As a modal composer and theorist, known for geeking out about harmonic technique, I have always been fascinated by the way others perceive the Locrian mode. From my perspective, it is simply a mode, and one that I have been known to use on occasion in my own music—more frequently than Ionian, in fact. To others, however, it tends to be viewed with a sort of befuddlement, as the “black sheep” of the seven diatonic modes, due to the fact that its first and fifth scale degrees are a diminished fifth apart, instead of the perfect fifth contained in the other six modes in the series.

With perhaps the exception of the death metal community, it often tends to be treated as merely a theoretical curiosity for that reason. The oft-cited reason from common practice-inclined theorists is that Locrian’s “diminished tonic” is unstable and requires resolution, supposedly rendering the mode effectively unusable.

Locrian and its alleged “problem”

     AUDIO: B Locrian Scale and Diminished Triad

This trope is oft-repeated in theory texts, method books, and general discussions of the (diatonic) modes.  Some textbooks even try to pretend it doesn’t exist. People have been bagging on Locrian ever since Heinrich Glarean (who called it “Hyperaeolian”) rejected it from his expansion of the Renaissance-era modal system in his well-known 1547 treatise, Dodecachordon.

That reputation, however, is not deserved. Indeed, before the anonymous authors of the tenth-century treatise known as the Alia musica started applying the familiar Greek demonyms to the medieval catalog of diatonic modes, initiating our modern usage of them, the Greeks of centuries prior actually applied the names in a completely different order. In fact, in that system, the name “Mixolydian” did not refer to the mode we would typically think of today (i.e. G-A-B-C-D-E-F-G, and its transpositions), but what we now know as “Locrian”. (The modern use of the term “Locrian” to refer to B-C-D-E-F-G-A-B and its transpositions, per musicologist and mode scholar Harold Powers in The New Grove’s Dictionary of Music and Musicians, did not seem to come about until sometime after the eighteenth century.) It was not merely a “theoretical curiosity” to the ancient Greeks, either—we see evidence in the writings of Plato, Aristotle, and other philosophers suggesting active usage, with Aristoxenus (by way of Pseudo-Plutarch) crediting its invention to Sappho.

While there is one obvious way to enforce the tonal center in the Locrian mode—the “brute force” method, by using a drone (as is the typical death metal paradigm)—this approach is very unlikely to fully dismiss some of the skepticism directed at the mode, particularly with respect to its use in the context of larger progressions, and especially ones that can serve as an analogue to the tonal idea of “functionality”. This essay is my response to this challenge. While there are actually multiple ways of achieving success here, I will detail one particular approach, which provides a framework that is of considerable use in many other modal situations.

First off, I’ll remark on arguably the most biggest mistake that pretty much everyone in the “unusable” crowd makes: tertian bias. That’s a term I coined some time ago to refer to the fact that, outside the realm of post-tonal theory, there are still many that think strictly in terms of tertian (third-based) harmony, such as triads and seventh chords. Even jazz theorists resort to tertian-based kludges when confronted with the non-tertian harmonic constructs of the fusion sub-genre (the “sus”), due to the strong bias built into the ubiquitous lead sheet symbol system.

While my “modal advocacy” is fairly well-known, I am also an avid supporter of non-tertian harmony and its harmonic potential, particularly within the realm of modality. Indeed, once we remove the unspoken restriction of sticking to tertian harmony, the Locrian mode’s potential comes into focus. Taking such an approach allows us to completely circumvent that supposedly problematic “diminished tonic” by finding another harmonic entity to serve as a larger gestalt representation of the tonal center in the mode.

To hone in on what I mean by “non-tertian harmony”, I use the term to refer to any sort of harmonic structure that appears to be, wholly or in part, constructed of intervals other than thirds. The most commonly considered types of non-tertian harmony are quartal harmony (constructed of fourths), quintal harmony (constructed of fifths), and secundal harmony (constructed of seconds), but mixed and other structures are possible as well. The distinction between these types does start to break down once enough notes are accumulated—for instance, a chord consisting of all seven notes of a heptatonic mode could be considered a tertian thirteenth chord, but it could just as easily be stacked up all in fourths, as a secundal cluster, or any other combination of intervals, for that matter. The distinction between these types tends to make more sense when dealing with smaller chords, particularly in the range of three or four notes.

On the subject of stacking all seven notes of a heptatonic mode, Locrian becomes particularly interesting in this regard. If all the notes of the mode are stacked in fourths over the tonal center, the result consists entirely of perfect fourths. We will return to this thought shortly. (Interesting side note: its inverse, Lydian, can do the same thing with fifths.)

The result of stacking the Locrian mode in fourths, over its tonal center.

Now, it may be useful to get into a little bit of terminology, so we can get a little more of an idea of what we are doing within this expanded harmonic realm. My personal system I use for labeling harmonies in this environment is something I call “Arabic numeral analysis”. It bears some resemblance to the more familiar Roman numeral system, but strips out the tertian bias that permeates that system through and through.

To give a brief overview of the system, let us first consider a very familiar harmony—a triad built on the tonal center. In Arabic numeral analysis, it would be symbolized as 1(3/5). The initial “1”, of course, indicates that the bass of the chord is scale degree 1 in the prevailing mode, while the “3/5” in parentheses shows that the upper notes of the chord form a third and a fifth (or some compound version of one or both intervals) above the bass. If we are using B Locrian (B-C-D-E-F-G-A-B) as a prevailing mode, 1(3/5) would be an indication that we are dealing with the chord B-D-F, or any voicing of those three pitches, where B is in the bass. (The ⁶B symbol is a shorthand for “Mode 6 on B”, as Locrian is the sixth mode in my Heptatonic Modal Catalog, since I begin the ordering with Dorian, not Ionian.)

1 (3/5) chord in B Locrian

This approach immediately pays off when we start looking at non-tertian chords. Let us consider, for instance, the three-note quartal stack B-E-A. If we are in B Locrian, since B is scale degree 1, E is a fourth above B, and A is a seventh above B, the Arabic numeral symbol would be 1(4/7). If we were in C Ionian instead, since B would be scale degree 7 there, the chord would be 7(4/7) in that particular case. In E Dorian, it would be 5(4/7), and in C♯ Locrian, it is also a 7(4/7).

Analyses of the 4/7 chord B-E-A in several modal contexts.

Similarly, the mixed chord A-C♯-D, in the context of E Dorian, would be 4(3/4)—A would be scale degree 4, C♯ is a third above A, and D is a fourth above A.

4(3/4) chord in E Dorian – an example of a “mixed” harmonic species.

There are a number of other nuances to consider with this system in order to gain a full understanding of it (particularly with dealing with alterations, and the whole matter of how this system wreaks havoc on the common practice notion of chord inversions), but this information should be enough to process my demonstration below.

One other concept to consider is the placement of the other scale degrees in relation to the tonal center, which can play an important role in creating a sense of harmonic motion. To generalize this and strip out some of the common practice baggage, I use something called the “CPID model” (pronounced like “Cupid”). The “C”, in this case, refers to the central degree (degree 1). The “P” refers to “proximal” degrees, those adjacent to the center (degree 2 above and degree 7 below). The “D” to “distal” degrees, those farthest from the center (degrees 4 and 5), while the “I”, or “intermedial” degrees, lie in between (degrees 3 and 6). Each heptatonic mode will have a single central degree, and a pair each of proximal, intermedial, and distal degrees. Here is the CPID model on B Locrian:

CPID (“Cupid”) Model on B Locrian.

It is instructive to first take a look at the norms of common practice tonality through the CPID model. In cadential gestures that return back to a tonic (1) chord, we most often find the dominant (5) chord, which exists in a distal relationship against the tonal center. Occasionally, however, one may also encounter a leading-tone chord (7) used in a similar “dominant function” capacity, which would exist in a proximal relationship with the center. The next most common pre-resolution chord (“pre-dominant” in tonal terminology) in a common practice environment would be the subdominant (4) chord (in the case of the plagal cadence), which would also be distal to the center. From this, we can surmise that our tonal center is generally preceded by a chord built on a distal or proximal scale degree—go big or go (not far from) home. Going farther back to pre-dominant function chords in a common practice tonal environment, we find that those are also typically either proximal (ii/ii°) or distal (IV/iv).

Additionally, there is the matter of pitch inventory between the chords. In many cases in the common practice era, particularly when the chord before the tonic—the dominant function chord—has a seventh, the parent scale structure is actually completed or nearly completed between just it and the resolution chord. Counting the tones in both chords, the V7 resolution to I/i includes all notes in the scale except scale degree 6, and the vii°7/viiø7 resolutions include every note. This element of contrast also plays a role in the sense of harmonic motion.

Examples of “mode completion” in a stock common practice tonal context

We have already noted the fact that the Locrian mode produces a tritone-free stack of fourths, up through seven notes. This gives us a good starting point for how we might produce seemingly stable progressions in the mode, and avoid the aforementioned pitfalls that have prevented many from seeing the mode’s viability. Speaking as a modal composer, familiarizing oneself with the harmonic potential of an unfamiliar mode in a sort of “sandbox” setting—including its vast array of non-tertian resources—can be a very useful way to get one’s bearings. For myself, this acclimation has usually come from some combination of “theorizing on paper”, and actually playing around with that theorizing through improvisation, such that the sound and idea of the mode can start to be internalized. This becomes increasingly useful—and rapid—once one has an educated sense of what to expect from a given mode under different harmonic scenarios.

Taking into consideration the use of (a) non-tertian harmonies, (b) the CPID model, and (c) mode completion, it now becomes possible to look at some potential progressions in the Locrian mode, which actually create some form of modal analogue to functional tonality. For the sake of variety, I will be using C♯ Locrian (C♯-D-E-F♯-G-A-B-C♯) here, instead of B Locrian. (And for those who don’t yet realize it, all of the diatonic modes fit on the familiar circle of fifths, filling it with the equivalent of 84 “keys”, rather than a mere 24.)

Circle of fifths with all the diatonic modes (click to enlarge)

C♯ Locrian with CPID Labels

     AUDIO: C♯ Locrian Scale

First off, let us begin with a four-note quartal stack for our chord on degree 1 (C♯), giving us C♯-F♯-B-E.

4-note Quartal Stack on C♯

     AUDIO: 4-note Quartal Stack on C♯

By collapsing the tenth (C♯-E) down to its simple equivalent, the third, we end up with C♯-E-F♯-B:

Process of collapsing octave

End result

     AUDIO: Collapsed 4-note Quartal Stack on C♯

In our Arabic numeral analysis system, using the most condensed method (in which compound intervals are collapsed down to their simple equivalents, and duplicates are ignored), both of these chords would be 1(3/4/7) chords. The latter, curiously enough, looks like a traditional seventh chord, in which the fifth has been swapped out for a fourth.

If we subtract the tones of this chord from our parent mode, the resultant negative space consists of scale degrees 2, 5, and 6. If we simply stack these in order, using scale degree 2 as the bass, the result is a 2(4/5). Again, this happens to bear some superficial resemblance to a triad, but with the third swapped out for a fourth. Having the fourth in place of the third goes a long way toward avoiding the previously mentioned pitfalls here as well.

Notes needed for mode completion shown with filled-in noteheads.

2(4/5) chord in C♯ Locrian, one realization of mode completion

The 2(4/5) chord is, of course, proximal to our tonal center, and at this point, it is worth checking the result of our initial groundwork. Progression 1 below is the result of using this 2(4/5) as a “dominant analogue”, sandwiching it between two 1(3/4/7) chords. (In case anyone is wondering what the bracketed “[Dor]” means, it is related to my concept of chord quality—it is not necessary to understand for the purpose of this demonstration, and is merely there for reference.)

   AUDIO: Locrian Progression 1

As you can see (and hear), the voice leading between these two chords is very smooth, and there should be some sensation from the listening example of the 1(3/4/7) chord, and its C♯ bass, seeming rather stable in this context.

Voice leading map on 2(4/5) – 1(3/4/7) in C-sharp Locrian (numbers in parentheses are scale degrees)

The feeling of centricity—or “tonic-ness”, for those coming from a tonal background—on the C♯ can be further amplified by using octave bass reinforcement, as Progression 2 demonstrates:

     AUDIO: Locrian Progression 2  

Bass reinforcement makes a fairly substantial difference in bringing focus and definition to this particular progression, namely due to the matter of bass clarity with the constituent intervals in the chords. That concept, due to its breadth, is best reserved for a future write-up. (In brief, my particular conception of intervals actually rejects the simple “consonance-dissonance” dichotomy for a two-dimensional interval characterization scheme, taking some inspiration from the work of Ernst Terhardt and Richard Parncutt.)

If we return to the CPID model, and the idea of chords with proximal and distal bass notes as potential dominant analogues, we have some other options to consider as well. Perhaps the most promising of these is by swapping out the proximal 2(4/5) for a distal 5(2/5) chord by changing the bass note from D to G, coupled with the bass reinforcement on the 1(3/4/7). Progression 3 examines this possibility.

     AUDIO: Locrian Progression 3

Voice leading map for 5(2/5) to 1(3/4/7) resolution

Progressions 4 and 5 explore the alternate possibilities for proximal (7) and distal (4) motion, respectively, while also offering scale completion (albeit with tones in common with our 1(3/4/7) chord). The alternate distal option (Progression 5) has a dull, rough sound, in large part due to its inclusion of a displaced cluster (a compound second and third from the bass) and the presence of the sixth.

     AUDIO: Locrian Progression 4

     AUDIO: Locrian Progression 5

After evaluating these possibilities, I felt most drawn to extending Progression 3, with the 5(2/5) chord, into a larger progression that evokes an analogue to functionality. If we consider a rather common tonal progression—I-vi-ii-V-I—we find that under the CPID model, it produces the pattern C-I-P-D-C. Progression 6 also follows a C-I-P-D-C bass pattern, but in the Locrian mode and with non-tertian harmonies.

     AUDIO: Locrian Progression 6

The use of the Locrian mode and non-tertian harmonies as in Progression 6 would be pretty far off the radar of conventional thinking,  for a situation where “functionality” can exist.  However, it should seem as if there is a strangely familiar sense about these progressions. That is, ultimately, the goal of an effort such as this one—to show one of many accessible ways of traveling into the (surprisingly) still open musical frontier of modal and non-tertian harmony.

Alexander LaFollett, Ph.D.

25 January 2018

UPDATE 13 Mar 2020: Hear the techniques described in this essay, in my Symphony No. 1 in C-sharp Locrian, Op. 58, 1st movement.