Jan 25 2018

The Locrian Mode: No, It’s Not Unusable

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.

 

B Locrian with Diminished Triad

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 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.)

Locrian Stacked Fourths

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.)

 

B Locrian 1(3/5)

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).

 

B Locrian 1(4/7) chord

C Ionian 7(4/7) chord

E Dorian 5(4/7) chord

C-sharp Locrian 7(4/7) chord

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.

E Dorian 4(3/4) chord

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:

 

B Locrian CPID Model

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.

 

Tonal Progression 1

Tonal Progression 2

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 - Modal

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

 

C-sharp Locrian

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

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:

Octave collapse process

Process of collapsing octave

4-note quartal stack collapsed

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.

 

Mode Completion on C-sharp Locrian

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

C-sharp Locrian 2(4/5) chord

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.)

 

Locrian Progression 1

   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) resolution

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:

Locrian Progression 2

     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.

 

Locrian Progression 3

     AUDIO: Locrian Progression 3

Voice leading map for 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.

Locrian Progression 4

     AUDIO: Locrian Progression 4

 

Locrian Progression 5

     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.

 

Locrian Progression 6

     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

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