The Circle of Melodic Dissonance

May 29th, 2009

Disclaimer: this post is about music theory. It will be somewhat dry. A reasonable understanding of basic musical scales and notation is presupposed.

Ed: this post includes Part 1, edited for clarity and grammar.

What’s in a scale?

Most popular music is written in either a major scale or a minor scale. Major scales sound happy and jubilant; minor scales are dark, stratospheric, melancholy. Think Deck The Halls vs. When Johnny Comes Marching Home as an example of the differences in sound. It is instructive to understand, however, that minor is simply a derivative of major; C major and A minor, for instance, both consist of the white notes on a piano. No flats, no sharps, no black keys. Even though the keys used are identical, if the song is played with C as “home base”, where the melodies in the song revolve around and end on C, it will sound happy; if A is used as the “base”, it will sound sad.

In this way I can say that minor, or Aeolian (the original Greek term), is the sixth mode of the major scale, much as A is the sixth note of the C major scale. The major mode (Ionian in Greek) is the first mode of the major scale. This is like saying C is the first note of the C major scale. A scale is like a skeleton key; its modes are the various doors that the key can open. A scale is the parent and its modes are the children. By way of example, the major scale can be represented by this specific sequence of seven notes, selected from the standard Western idiom of twelve possible notes [for why it’s 12 notes, peruse my related essay]:

C (C#) D (D#) E F (F#) G (G#) A (A#) B

The bolded notes represent the scale of C major. The notes in parentheses are the black notes on a piano; these notes are not in the C major scale. Note that the major scale follows the pattern of two steps from C to D (skipping C#), two steps from D to E (skipping D#), one step from E to F, two steps from F to G (skipping F#), two steps from G to A (skipping G#), two steps from A to B (skipping A#), and finally one step from B to C. [Aside: I don’t mean to be needlessly confusing, but traditionally a single step is referred to in most music notation as a “half step”, and two steps as a “whole step”, so I will use this terminology as I continue. Quick e.g. to eludicate: we consider the space between C and D as a whole step. E to F is a half step.] Written out in this way, the major scale can be represented by the the spatial pattern of “whole-whole-half-whole-whole-whole-half”, or WWHWWWH for shorthand. It’s a measurement of the distance between each note and how the distances are related. Distances between notes are called intervals. This pattern is consistent regardless of the note one begins on: the G major scale is simply G A B C D E F# G, for example, and E major is E F# G# A B C# D# E. It is left as an exercise to the reader to verify these scales’ patterns.

Now if I keep the scale the same, and start my step-counting on A, I get the universal pattern of a mode of major (Aeolian): WHWWHWW. This is the same pattern: the half-steps are separated by two whole steps and then three, which repeats ad infinitum. It is a mode of the major scale because the beginning point of the scale is simply shifted over, but the underlying pattern is the same. To wit:

WWHWWWHWWHWWWH

The above pattern is simply the spatial relationship between the notes of the major scale, repeated twice. If I choose to begin the pattern on the bolded W, this forms the first mode of major, the Ionian mode. If I begin my pattern-reckoning on the underlined W, this pattern is the sixth mode of major, or the Ionian mode.

The astute reader will wonder: “what about the other modes? Doesn’t the major scale theoretically have seven beginning-points, and thus seven modes?” This is exactly the case. In order, the seven modes of the major scale are called: Ionian (major), Dorian, Phyrgian, Lydian, Mixolydian, Aeolian (minor), and Locrian. They are all distinctly different in mood; some are darker, some are light and airy. They are related laterally, in that their step patterns are all derived from the major scale. Dorian is the mode that begins on the second interval of the major scale: it can be represented by WHWWWHW. Phyrgian is the third mode, beginning on the third interval of the major scale, and looks like this: HWWWHWW. And so on.

And then for example, in a given melody, I could shift from C Ionian (C D E F G A B C) up to D Dorian (D E F G A B C D), and then to G Mixolydian (G A B C D E F G), and each provides a different mood, borrowing shades of major and of minor. This requires changing the home base of my song, though. The listener will clearly note the song “moving” from a home base of C, then to D, and finally to G. What about shifting around within a given tonal center? What if I want to add color and flavor to a melody, but keep it strictly based in C?

SPMC: it’s easier than it looks

There is a second way that the modes are related; it is a melodic relationship, in that the underlying “home base” does not change. For example, I can shift from C Ionian (WWHWWWH) to C Mixolydian (WWHWWHW). The intervalic patterns make it less clear as to what is going on, so I will write it out as a key change:

C Ionian (C D E F G A B C) —-> C Mixolydian (C D E F G A Bb C)

Now I am exploring melodic color without changing the home base of the song, otherwise known as the tonic. I would recommend mapping the other intervalic patterns to the actual piano notes as a useful exercise. What’s the difference between Phrygian and Locrian, for example? What notes change? What notes stay the same?

The melodic shift from C Ionian to C Mixolydian is a subtle one; only a single note changes, and only by a half-step at that. It is the smallest possible melodic change. This is a useful concept; it provides subtle coloring without the listener consciously pulling away from the song and thinking “well, that was obvious.” Think of it as borrowing Latin or French words as a way of adding interesting diction to a short story. For those taught in traditional harmonic counterpoint, it is like shifting from C major to G major; only one tiny thing is changing. Here on out I will refer to this concept as SPMC (Smallest Possible Melodic Change). All of the modes of major are related by SPMC, which is spelled out in exhaustive detail below.

C Lydian (C D E F# G A B C) <—-> C Ionian [major] (C D E F G A B C)

C Ionian [major] (C D E F G A B C) <—-> C Mixolydian (C D E F G A Bb C)

C Mixolydian (C D E F G A Bb C) <—-> C Dorian (C D Eb F G A Bb C)

C Dorian (C D Eb F G A Bb C) <—-> C Aeolian [minor] (C D Eb F G Ab Bb C)

C Aeolian [minor] (C D Eb F G Ab Bb C) <—-> C Phrygian (C Db Eb F G Ab Bb C)

C Phrygian (C Db Eb F G Ab Bb C) <—-> C Locrian (C Db Eb F Gb Ab Bb C)

Each relationship above features only a single note moving by a single half-step; for example, A moves to Ab when shifting from C Dorian to C Aeolian. There also are a few interesting epiphenomena that arise out of SPMC. Lydian is the “lightest” of the seven modes of major; it is airy and carefree-sounding. Locrian, on the other hand, is the “darkest”, very sinister and primordial. The fewer SPMCs that separate a mode from Lydian, the lighter in mood; likewise the closer a mode to Locrian, the darker the mood.

(Dorian is centralized and is neutral in color. Also note that when “traveling” from Lydian to Locrian via SPMC, the note that changes always descends, never ascends. Which is an interesting synesthetic point: do we perceive darker modes as being “lower”? Is Locrian the Hell mode? Is Lydian angelic? Does it soar? Does it balance on the head of a pin? Pbbbbt.)

Additionally, every note of the major scale descends by one half-step at some point in this system, except for the tonic, C. While it is technically possible to “move” from C Locrian to B Lydian, the note that changes in this shift is the tonic itself (C moves down a half step to B; this would be C Db Eb F Gb Ab Bb C —> B C# D# E# F# G# A# B, and don’t think about all the sharps and flats too hard if you haven’t taken traditional music theory). By SPMC rules, this isn’t allowed, as moving the tonic has a jarring effect on the listener, and is not considered by us to be a melodic shift, but rather a harmonic one. And we have to draw the line somewhere. Which isn’t to say that this type of shift is “illegal” and you “can’t” do it; SPMC has no inherent value, but is a tool to understand how melody works.

Color by numbers

The sharp/flat system has been primarily designed for use in the major/minor system; when speaking modally, it’s a lotta look, to quote a famous fashion designer. All the flats and sharps tend to obscure the simplicity of SPMC. We need a different system of talking about melody. If C is assigned the number 0, C# the number 1, all the way up to B = 11, we can study the system numerically, and also universally, as any tonic can be assumed.  Zero can just as easily be G#, or E. Also, since the tonic can never change by our definition, it’s not necessary to include it. Here’s an example to show what I’m talking about.

C Mixolydian is C D E F G A Bb C. If we assume that C = 0, and continue from there, then C Mixolydian can also be described as 0 2 4 5 7 9 10 12. Then simply note that by SPMC rules, the note C will never change, so the numbers 0 and 12 can be omitted, which leaves us with a nice universal way to describe all Mixolydian modes. Mixolydian is (2 4 5 7 9 10). Reread this paragraph, then prove to yourself that Aeolian (minor) is represented by (2 3 5 7 8 10). These six numbers also stand for how many half-steps a given note sits above its tonic; for guitarists, it can also be interpreted as tablature for lateral movement up a single string, assuming that the open string is the tonic.

Here’s the SPMC major scale order, in this new easier-to-look-at (hopefully) format:

( 2 4 6 7 9 11) Lydian

(2 4 5 7 9 11) Ionian

(2 4 5 7 9 10) Mixolydian

(2 3 5 7 9 10) Dorian

(2 3 5 7 8 10) Aeolian

(1 3 5 7 8 10) Phrygian

(1 3 5 6 8 10) Locrian

It’s worth comparing this list to the earlier SPMC list that uses sharps and flats. The relationship becomes much clearer; it is easy to see, for example, that Dorian and Phyrgian differ by two SPMCs, or that 7 exists in all of the modes except Locrian, or that the half step between 9 and 10 is present only in Mixolydian and Dorian. It is, to me, a primarily functional system. It’s a quick way of comparing two modes and determining how they are different, and why.

(Quick aside: to avoid confusion, and continue to use words the way that traditional music theory uses them, we will refer to the unwritten zero as the first note of the mode. Just mentally insert it into your understanding. After all, it is physically the first note of the mode, the tonic. The six notes shown in our numeric shorthand are the second through seventh notes, respectively. The unwritten twelve is the eighth note.)

Experiment: noodle around in the Ionian and Mixolydian modes, and end your phrases on the seventh note (the 11 and 10, respectively). Prove to yourself how 11 functions differently in Ionian from the 10 in Mixolydian. They both hold the same position in the mode as the seventh note. Yet 11 often sounds “unhappy” and “needs” to be resolved by moving up to the 12, the tonic. Conversely, the 10 of Mixolydian doesn’t have that same tonicwards pull. It’s a little happier. There’s a little bit of a pull towards the 9, but not as strong; it can also move up to the 12, or just hang there and build some juicy tension.

And that brings us to the next big section. What is tension? How can we talk about it?

How I learned to stop worrying and love tension

When looking purely at melodic structure, we can consider two types of tension: internal and external. Internal tension is the “happiness” of notes within a mode, not to put too fine a point on it. External tension, which I will also call stability, is the “happiness” of modes within a scale. This happiness isn’t just theoretical; you can easily hear the tension of a given note or scale, which of course is related to various cultural norms and socioeconomic upbringings and squishy qualitative “life experiences”, ad nauseum. E.g. how Ionian just sounds more “finished” at the end of a phrase than Locrian. Or e.g. just play the tonic for a given Locrian, and then hit that nasty sounding 6. It’s dissonant and filled with tasty tension. It’s hard to describe; it’s qualia. Try it for yourself.

But there are basic rules that we can talk about.

Internal tension of a mode is generally centered around that mode’s two half-step intervals. In the case of Ionian, those intervals are 4-5 and 11-12. For the closely related Lydian, it is 6-7 and 11-12. Most of the time, one of the notes in this tension-pair is happier than the other, and pulls its mate towards it. 11 really pulls towards 12; Lydian’s 6 is in love with its stable mate, the 7. Ionian’s 5 is moderately enthused about the 4, but not as strongly as the 11-12 match. The intensity of each match varies from mode to mode; speaking extremely vaguely, the closer a mode to the “end” of the SPMC continuum, the more internal tension its half-step matches carry. Lydian and Locrian have strong pairings. Dorian’s 2-3 and 9-10 are hardly attracted to each other at all. Remember of course that this explanation is descriptive, not prescriptive; go ahead and end that phrase on Lydian’s 6, if that’s what’s called for. Tension is a tool.

And don’t forget about the nonmatched notes; there are three of them in each mode. These singletons don’t have a strong pull, but conversely they don’t pull other notes towards them. Ionian’s singletons, the 2, 7, and 9, therefore exist in a tension continuum somewhat between the pullers and the pullees. We could roughly assign an order to Ionian tension, from the most pull to the least pull; this is necessarily informed a bit by our own cultural upbringing, and thus will vary from person to person. My personal evaluation of this mode goes like this: 11, 5, 9, 2, 7, 4, 0. The bolded notes want to resolve; the normal-texted notes are moderately stable; the italicized notes have a lot of pull, and feel good to end a phrase on.

I realize it’s dangerous to come out and say “this is how it is”, so bear in mind there are some serious caveats lurking all about this concept of tension. But internal tension does exist.

External tension, or stability, is a little trickier to describe. It is partly a description of where the tonic lies within a mode’s tension continuum. The less pleasing it is to finish that phrase on the tonic of the mode, obviously the less stable the mode itself will sound. For example, I could argue that Lydian is less stable than Ionian, because Lydian’s 6-7 is so tension-filled that 7 sounds like more of a release than the zero. Whereas Ionian’s 11-12 is top hog, and 11 pulls towards that 12 like nothing else can. (Also keep in mind that the tonic can be approached from both directions.)

External stability/tension can also be a description of the “distance” between two modes; in other words, it is a measure of how many SPMCs separate their intervalic patterns. Moving directly from Locrian (1 3 5 6 8 10) to Lydian (2 4 6 7 9 11), for example, is a much more sudden and tension-filled shift than the comparatively bucolic Dorian-Aeolian variation (where all that changes is a half-step descent from Dorian’s 9 to Aeolian’s 8). I admit that this concept of intermodal stability is still burgeoning; I will leave it at these vague beginnings and look forward to future dialogue on the subject.

Segue: When moving from Ionian to Dorian, for example, if I want to incur only slight external tension, I could shift from Ionian (2 4 5 7 9 11) to Mixolydian (2 4 5 7 9 10), and then from Mixolydian to Dorian (2 3 5 7 9 10). There is a second SPMC-approved route, however; note that the 11 drops to 10 first, and then the 4 drops to 3, yielding Dorian. But what if this order is reversed? Then the intermediate, as yet unnamed mode would look like this: (2 3 5 7 9 11). Here, the 4 drops to 3 first, and then the 11 drops to 10 and becomes Dorian. Question: what prevents us from using this “modal pathway” instead of the intramajor shift?

A fucking can of goddamn worms

Answer: nothing. It’s a valid mode in its own right. Which beggars the follow-up: what prevents our theory from simply including all six-note permutations, like (3 4 5 8 9 10) and (1 4 5 6 7 11), and thus crashing under its own exhaustive potential? The qualitative response is that most “technically possible” modes sound like utter shit. A quantitative response is a bit more subtle. For those not trained in a bit of Western harmonic theory, skip the following italicized paragraphs, and just trust me that there are three other scales, each of which contain their own unique seven mode systems, that qualify for inclusion under our melodic analysis. And for those readers who seriously don’t have other things to do:

[The question stands. What makes a “valid mode”? First, remember that the Western harmonic system is based upon thirds. Even jazz 13th chords are a theoretical grouping of thirds upon thirds. A C major chord, for example, is C E G. The interval between C and E is a major third (four half-steps); between E and G, a minor third (three half-steps). Stack ’em in reverse order and you have a minor chord instead. C Eb G. Add another third on top of a C triad and you have a variety of possible sevenths. Or alternately, expand the triad to C E G#, and call it augmented (two major thirds). The point is that all of these chords are systems of major and minor thirds.

And then realize that the major scale is simply a grouping of thirds, stacked up to form a cyclical pattern. C —> E —> G —> B —> D —> F —> A —> C. This scale is created by following the pattern of major third, minor third, major third, minor third, minor third, major third, minor third; we represent this by MmMmmMm. A scale has to have exactly three major thirds and four minor thirds to arrive back at the starting point, thus making a complete and enclosed harmonic system. So how many other unique combinations of thirds are available? There are three. There is mMMmmMm, or C —> Eb —> G —> B —> D —> F —> A —> C, commonly known as melodic minor. There is mMMmmmM, or C —> Eb —> G —> B —> D —> F —> Ab —> C, commonly known as harmonic minor. And there is MmMmmmM, or C —> E —> G —> B —> D —> F —> Ab —> C, which is (un)commonly known as harmonic major.

Note that three major thirds stacked in order yield C —> E —> G# —-> C, and four minor thirds yield C —> Eb —> Gb —> A —> C, both of which are degenerate. Also note that this “third stacking” method insures we leave out theoretical scales that include multiple half-steps in sequence, like C-C#-D, or large jumps, like C-D-F. We find that ethnic scales that contain degenerate sequences, like Hungarian Minor, generally function in a chromatic way, their multiple half-steps used as passing notes only.

It is surely possible to include many more scales. Most of those scales are harmonically inaccessible, and melodically incomprehensible. As our inclusion of scales expands, the power of this type of melodic analysis diminishes. It is a judgment call, but one well thought out and justified by virtually all Western music and attendant theory.]

Still with me?

There are three other scales, besides the major scale, that are worthy of inclusion in this system. They are:

Major Scale: WWHWWWH (for completeness)

Melodic Minor Scale: WHWWWWH

Harmonic Minor Scale: WHWWHTH

Harmonic Major Scale: WWHWHTH

The symbol T, of course, stands for “three half steps”. The three intervalic patterns shown above are unique and non-superimposable upon both each other and the major scale. Also, for completeness, here are all 28 modes (from 4 parent scales), listed in lateral order, written in the format (unique numerical pattern) Name Of Mode – INTERVALICSTRUCTURE. This is mostly for reference and you can skip it; many of the nonmajor modes are not used in Western music very often and so have only working names.

Major Scale Modes

1. (2 4 5 7 9 11) Ionian – WWHWWWH

2. (2 3 5 7 9 10) Dorian – WHWWWHW

3. (1 3 5 7 8 10) Phrygian – HWWWHWW

4. (2 4 6 7 9 11) Lydian – WWWHWWH

5. (2 4 5 7 9 10) Mixolydian – WWHWWHW

6. (2 3 5 7 8 10) Aeolian – WHWWHWW

7. (1 3 5 6 8 10) Locrian – HWWHWWW

Melodic Minor Scale Modes

1. (2 3 5 7 9 11) Melodic Minor – WHWWWWH

2. (1 3 5 7 9 10) Dorian b2 – HWWWWHW

3. (2 4 6 8 9 11) Lydian Augmented – WWWWHWH

4. (2 4 6 7 9 10) Lydian Dominant – WWWHWHW

5. (2 4 5 7 8 10) Mixolydian b6 – WWHWHWW

6. (2 3 5 6 8 10) Locrian #2 – WHWHWWW

7. (1 3 4 6 8 10) Altered – HWHWWWW

Harmonic Minor Scale Modes

1. (2 3 5 7 8 11) Harmonic Minor – WHWWHTH

2. (1 3 5 6 9 10) Locrian Natural 6 – HWWHTHW

3. (2 4 5 8 9 11) Ionian #5 – WWHTHWH

4. (2 3 6 7 9 10) Dorian #4 – WHTHWHW

5. (1 4 5 7 8 10) Spanish Phrygian – HTHWHWW

6. (3 4 6 7 9 11) Lydian #2 – THWHWWH

7. (1 3 4 6 8 9) Altered bb7 – HWHWWHT

Harmonic Major Scale Modes

1. (2 4 5 7 8 11) Harmonic Major – WWHWHTH

2. (2 3 5 6 9 10) Dorian b5 – WHWHTHW

3. (1 3 4 7 8 10) Phrygian b4 – HWHTHWW

4. (2 3 6 7 9 11) Lydian b3 – WHTHWWH

5. (1 4 5 7 9 10) Dominant b2 – HTHWWHW

6. (3 4 6 8 9 11) Lydian Augmented #2 – THWWHWH

7. (1 3 5 6 8 9) Locrian bb7 – HWWHWHT

The circle of melodic dissonance

Relating only the major modes to each other by SPMC is easy enough. It travels linearly, from Lydian to Locrian, one SPMC at a time. Relating all 28 valid modes, however, is considerably more challenging. One of the simpler ways to organize such a description is in a circle; thus the circle of “melodic dissonance” was born. Music thrives on dissonance; and melodic color within a given line is acheived by subtly (or not, as is your wont) reaffirming and breaking various tension barriers. I see the circle as a “how-to” manual on all the various ways, both orthodox and unusual, to shift between acceptable modes; I will get into that briefly. First, this is how the circle is put together.

Here is a symbolic representation of how the major modes are related by SPMC. The numerals correspond to the numbers in the previous section; for example, Dorian is the 2nd mode of major, so Dorian is represented by a number 2. The circle I have arbitarily designated as symbolic of belonging to the major scale. A line between numbers shows the SPMC connection; the line between 7 and 3 indicates that Phrygian and Locrian are separated by a single SPMC. Not included is the six-note pattern of each mode: they can be referred to in the table above.

Then, we will add the melodic minor modes to the circle; they are represented similarly. The symbol of a square indicates that a given mode belongs to the melodic minor scale. For example, look at the relationship between mode 1 of major (Ionian) and mode 1 of melodic minor (Melodic Minor). They share an SPMC relationship: (2 4 5 7 9 11) <–> (2 3 5 7 9 11). And so it is with the rest of them.

Here is the full circle of melodic dissonance (don’t mind the slightly blue square; picked the wrong color). Click to enlarge.

Parting thoughts

So what we have arrived at is a system that describes how to shift between modes of several types of scales, and (roughly) the kind of tension that such shifts entail. A great many things are implied from this circle, too many to discuss in this post. I will make just one example. See how 7-triangle, the 7th mode of the harmonic minor scale, isn’t directly linked to the main major scale series? Not only that, but that it can only be “reached” by “traveling” though Locrian? This distance is explainable by looking at the mode itself: (1 3 4 6 8 9) Altered bb7. Note that the tonic is reached from one side by 1-0, and the other side from 9-12, a three-half-step jump. This jump really increases the internal tension of the tonic; and 9 is a half-step itself away from 8, which it prefers over 12. This tension is graphically described by the mode’s distance from the stable modes, like Dorian and Mixolydian. I’m not sure why this is, but I’m still working on it. And of course, all of this is still a work in progress. I will post more thoughts about melodic theory as I decipher them. Enjoy!

Graham

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One Response to “The Circle of Melodic Dissonance”

  1. […] Here we are then with example one: […]

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