Chromaticism and shifting dominants

I came across something I wrote up about three years ago while studying music theory in college, so I thought I'd expand on it and share it with the world.

While studying 18th century counterpoint, my professor asserted that when using the diminished seventh chord, one can simply assert another completely diminished seventh chord and use that as a vehicle to move to another key. In thinking about the nature of dominant chords, I thought: why not extend this to dominant seventh chords? After all, the diminished seventh and the dominant seventh only differ by a half-step, and with minimal chromaticism, one can shift to the dominant seventh of another diatonic degree. Taking this concept one step further, I figured that it was possible to convincingly shift from the dominant seventh to the secondary dominant seventh of any of the other eleven degrees of the chromatic scale. After a bit of brainstorming, I came up with the configurations, using the closest possible voice leading (inversions indicated below).

DOMINANT TO DOMINANT

Aligning with the tendencies of 18th century harmony, modulations tend to travel to closely related keys, generally the major or minor key corresponding to key signatures no more than one flat or sharp away. Thus, I thought it would be a worthwhile exercise to compare secondary dominant sevenths to a primary key to consider how closely we could morph a pivot chord. For convenience, I used C major (no sharps or flats).

DOMINANTS IN MAJOR (1, 2, 3, 4, 5, 6, 7)

Summary:

• V⁷ with 0 accidentals: I
• V⁷ with 1 accidental: II (#1), IV (b7), V (#4), VI (b6), bVII (b3)
• V⁷ with 2 accidentals: bIII (b6, b7), III (#2, #4), bV (b2, b6)
• V⁷ with 3 accidentals: bII (b3, b5, b6), bVI (b2, b3, b7), VII (b2, b5, b7) 

Major mode is pretty straight forward. Not surprisingly, the dominants for the predominants (II, IV), dominant (V), and relative minor (VI, which oftentimes serves a predominant role) are just 1 flat away.

For the relative minor of A, the same chords above reappear, only for different scale degrees. To avoid confusion and allow comparison to the above chart, I'll keep the same spelling but consider the scale degrees with respect to A major (e.g. G = b7, Gb = F# = 6).

DOMINANTS IN NATURAL MINOR (1, 2, b3, 4, 5, b6, b7)

Summary:

• V⁷ with 0 accidentals: bIII 
• V⁷ with 1 accidental: I (7), bII (b5), IV (#3), bVI (b2), bVII (6),
• V⁷ with 2 accidentals: bV (b2, b1), V (#4, 6), VI (3, 7), 
• V⁷ with 3 accidentals: II (b2, 3, 6), III (b5, 6, 7), VII (b2, 3, b5)

In minor mode, the dominants for the subdominant (IV), neapolitan (bII), augmented sixth (bVI), and subdominant (bVII) are just 1 flat away. As you may know, the neapolitan and augmented sixth chord can be derived from the minor mode.

Now, the above considers natural minor (b3, b6, and b7). If we instead allow melodic minor so that the sixth and seventh are raised and considered part of the key, the counts change around quite a bit.

DOMINANTS IN MELODIC MINOR (1, 2, b3, 4, 5, b6, 6, b7, 7)

Summary:

• V⁷ with 0 accidentals: I, bIII, bVII 
• V⁷ with 1 accidental: bII (b5), III (b2), IV (#3), bV (b2), V (#4), bVI (b2), VI (3)
• V⁷ with 2 accidentals: II (b2, 3)
• V⁷ with 3 accidentals: VII (b2, 3, b5)

Suddenly, the tonic chord and subtonic chord are directly within reach. It's worth noting that the latter has the dual identity of the dominant of the relative major (oftentimes, hearing that chord in a progression is a good indication that you're heading to the relative, if you're not there already.

Looking at the expansion of the 1-class (having snagged several of the 2-class chords), we now have III, bV, V, and VI available. Ignoring the newly inducted dominant (V), that's an interesting set of foreign chords, considering (1) III and bV don't even make the cut in major, and (1) VI doesn't have much of a role in this mode (usually you'd use bVI for a "six chord"). I say you've got yourself a curious trio of chords there -- III, bV, and VI -- and that's something that I'd like to play around with in future compositions.

Of course, this new scheme isn't that simple, since melodic minor has its own rules, especially with raising the sixth and seventh while moving upward, but lowering them on the way down. Crudely speaking, this scheme only works "half" of the time, if that.

Works of the 20th century started to consider keys in terms of their letter along rather than their mode, especially since they so frequently went through different modes, borrowing here and there. So why not incorporate that idea here? If we were to take the "best of both worlds" approach (i.e. the smaller of the two for any given Roman numeral), we would end up with the following scheme that combines major [M] and melodic minor [m]:

COMBINING MAJOR + MINOR (1, 2, b3, 3, 4, 5, b6, 6, b7, 7)

Summary:

• V⁷ with 0 accidentals: I, bIII [m], bVII [m] 
• V⁷ with 1 accidental: bII [m], II [M], III [m], IV, bV [m], V, bVI [m], VI
• V⁷ with 2 accidentals: (none -- all promoted!)
• V⁷ with 3 accidentals: VII

It looks like once you use a major-minor key, nearly every chord (less the one tonicizing scale degree 7) can be attained using 0 or 1 accidentals. No wonder Romantic composers ushered in the age of chromaticism.

The above analysis looked purely at a superficial, chromatic relationship of V⁷ of I and V⁷ of any other scale degree. While my professor asserted one could simply go from one dominant to another, I wondered if one could associate a relationship between the two dominant seventh chords, whether directly (i.e. by serving a role in both "keys", such as an augmented sixth chord) or indirectly (i.e. by substituting for another similarly sounding chord, with modified or added tones).

Although one can simply assert the dominant to change the tonal center (without an explicit progression), it often works best to have a functional relationship between chords. By following the tonic-predominant-dominant scheme, one can often produce a convincing chord progression. The introduction of secondary dominants (which also serve a predominant role), allow one to extend the tonal palette to incorporate chords outside of the key with minimal chromaticism (oftentimes with no more than two altered chord members).

In the following cases, I assume we're traveling from the key of I (via V⁷) to the key of 'X' (via V⁷ of 'X'). In considering that we're moving to the dominant of another key, the original V⁷ will usually serve a predominant function to the dominant that follows. With that in mind, it's worth expressing the original V⁷ in both keys to see what we have available. Below '=' indicates an equivalency, whereas the '~' indicates a substitution.

V⁷ AS A PIVOT CHORD

Summary:

• V⁷ of I = "bV⁷" of bII ~ a bit of a stretch, but #IV resembles a predominant ⁽¹⁾ 
• V⁷ of I = "IV⁷" of II ~ (lower the third) iv⁷ in the key of ii
• V⁷ of I = "III⁷" of bIII ~ (lower the third) passing iii⁷ chord to V⁷ in the key of bIII ⁽²⁾
• V⁷ of I = "bIII⁷" of III ~ (lower seventh) i⁷ chord in the key of iii
• V⁷ of I = "II⁷" of IV = predominant V⁷/V in the key of IV
• V⁷ of I = "bII⁷" of bV = predominant neapolitan with added seventh in the key of bV ⁽³⁾
• V⁷ of I = "I⁷" of V ~ (raise the seventh) for tonic I^M7 in the key of V
• V⁷ of I = "VII⁷" of bVI ~ (raise the root) i^o7 (4)
• V⁷ of I = "bVII⁷" of VI ~ (raise root) dominant vii^o7 in the key of VI ⁽⁵⁾
• V⁷ of I = "VI⁷" of bVII ~ (lower the third) predominant vi⁷ in the key of bVII
• V⁷ of I = "bVI⁷" of VII = predominant Ger⁶ in the key of VII

A few considerations:

1. It's basically a substitution of #IVdim7 (a predominant that can connect IV to V) with #IV7.
2. Though non-functional, I've seen the use of iii as a preparatory chord for V since they already share two degrees (5 and 7). Now if you consider their sevenths, iii⁷ is moves easily into V⁷, now with three degrees in common (5, 7, and 2). Really, all you need is the root of the iii to move up a half-step to become the seventh in the following chord.  
3. The original chord could tonicize bV directly if you consider tritone substitution, but for the sake of connecting to the V⁷ I instead thought of it instead as the neapolitan (bII⁶), but in root position with an added seventh.
4. The diminished chord on the tonic, though non-functional, can be used as an embellishing chord for the tonic, often found in barbershop and jazz music.
5. The diminished seventh upon scale degree 7 seemed the easiest manipulation, though it is uncommon within a harmonic function (i.e. dominant) to go from a chord of more tension (diminished) to one of less tension (dominant seventh). Instead, the original V7 as "bVII7" may stand convincingly enough as a dominant, as found in contemporary music. 

So what does this all mean? Well, at this point, I'm not really sure. But given that this exploration in music theory deals primarily with dominant seventh chords, maybe it will best serve those who work closely with these types of chords.

Perhaps barbershoppers.

Celes and Locke's Themes in Final Fantasy VI

Final Fantasy VI has several memorable themes, but two of my favourites are the leitmotifs (or character pieces) for Celes and Locke, two of the more fully-developed characters.

Below is Celes' theme, whose opening notes continue to remind me of Aerith's theme (FFVII). It's a light, sweet, and somewhat introspective theme, with a couple of leading tone accidentals that make it quite interesting.

Besides the accidentals, I also enjoy how the strings move in contrary motion with the melody in every other measure. That bit of "complexity" is complemeneted with a rather straightforward harmonic analysis. There's the tonic (I), predominants (IV, ii) and dominant (V) chords with a handful of iii's. The iii (the minor chord built on the third degree appears quite a bit here and function as a substitute for the dominant V.

That secondary dominant in the penultimate measure also catches the ear since it's so foreign. After all, a diatonic (i.e. "in the key") chord appearing on scale degree 7 is usually diminished (even more "minor" than "minor"), but instead we have a dominant seventh chord built on that same degree -- which means plenty of extra accidentals. Yet it leads right back to that familiar iii chord, so it's not all that strange in the grand scheme

Now, contrast this with the theme of Locke, the energetic, adventurous treasure hunter, who plays the role of "hero" for our damsel in distress.

This upbeat theme has an even simpler melody, but it also has an extra element with its rhythmic driving force. Though the other theme does some stepwise motion, this theme includes some more in the bass accompaniment and the trade-off during the second half. Though it doesn't do much with contrary motion, it does have some interesting chords (and a key change) thrown in the mix.  The bIII and bII, in particular, stand out, as do those few close-harmony chords played by the horns.

What I like most about this theme is its shift on tonal center; it starts off in G major and shifts to F major (the next portion goes into Eb major, and it eventually looping back to G). What it lacks in contrasting motion, it certainly makes up with motion in rhythm and between keys.

Each of these themes have contrasting rhythm, characters, and overall sound, which is probably why it fascinates me when they appear in counterpoint during the end credits theme. Listen for yourself: 

First, I thought it's worth mentioning that this final key, Bb with 2 flats, is the "average" of the original two keys, Eb with 1 flat and G with 1 sharp (or -1 flats). A striking coincidence? Perhaps.

Upon closer inspection, the reason why the counterpoint harmony works so well for those overlapping measures is because (1) several of the chords are actually common (though expressed differently) and (2) for the half that aren't exactly the same, their scale degrees overlap (it also helps that the notes between the melodies line up to form appropriate harmonies). To break it down, here is how the chords compare:

Measure 1 2 3 4

Celes I iii IV iii

Locke I V IV I

Ending I iii7 IV I

If you aren't as familiar with chords and their scale degrees:

• A triad built on scale degree 1 (I), contains 1, 3, and 5 ("do, "mi", and "so")
• A triad built on scale degree 3 (iii), contains 3, 5, and 7 (or "mi", "so", and "ti")
• A triad built on degree 5 (V), contains 5, 7, and 2 ("so", "ti", and "re").
• A seventh chord built on 3 (iii7), contains 3, 5, 7, and 2

So if you line up a I against a iii, you're sharing 3 and 5. If you line up a iii against a V, you're sharing 5 and 7. As long as your melodies don't have too many disagreeable notes when superimposed, you're in for a treat (especially when the independent lines form chords when they meet).

No wonder it sounds so good. Of course, that's the whole "point" to counterpoint.

If you're into games and music theory, check out Cruise Elroy.

How to transcribe music

Recently, someone emailed me asking for some pointers on how to approach the "B" section of "Where in the World is Carmen Sandiego?" (the "Nashville to Norway" part).

I responded to his email with a lengthy explanation of my thought process when I transcribe in general. I thought I'd share the contents of that email (with notes that pertain specifically to the Rockapella song). I know some you already transcribe (and quite well, much better than me), so for those of you who have been curious yet timid might find my approach useful. Below, I've included song-specific notes in italics.

Here's a breakdown of the approach, which you requested earlier. The trick to transcribing it is a mix of awareness of the individual and awareness of the ensemble. The process starts off with the picture, then divide & conquer, and finally bring it back together.

Transcription Skill #1

First, you need to listen through as a whole and identify chords - their types (e.g. triads, tetrads, major, minor) and relationships (e.g. functional, predominant, dominant). It's all relative here so don't worry about precisely giving it a name (though it sure helps if you know when you have a tonic triad since that's the home base). You luck out here because it's just parallel triads (mostly major) without a true chord progression (but still trying to find their way back to the home Bb chord). Frequently you'll usually find some tetrads (e.g. seventh and ninth chords) and the ever-popular V-I movement (e.g. G7 to C chord).

Transcription Skill #2

Before you can tackle individual notes, remember that they won't be helpful useful unless you know when the chords change, chord progression or not. When you listen, I'm sure you recognize that despite the frequent motion, the chords are basically set for every two beats (i.e. each set of three notes) except for a few instances of chromatic ascension, and the bassline clearly support this. It's not always that easy, but generally this harmonic rhythm is consistent for popular songs.

Transcription Skill #3

With the general scheme mapped out, it's type to break it down and observe the nuances. During the "B" section, the top three parts follow the words, while the bass has this "dmm dmm" part that lines up with the words but doesn't follow the exact same jumps between notes. So you definitely want to figure out the bass since the harmonies are based upon it . Well, the bass notes are: Bb (3x) - A (3x) - Ab (3x) - G Ab A - Bb (3x) - C (3x) - Db (2x) - "and back!"

Transcription Skill #4

With the lowest line set, you need to map out the chords above it, with the awareness that whenever you have a triad (i.e. all of the chords), you can bet that on each chord you can expect another part to share the bass on each chord. If we refer to the parts in order of descending range as "high tenor", "tenor", and "baritone", then you'll observe that in those sets of three (I'd refrain from calling them "triplets", since their rhythm is actually dotted-eighth, dotted-eighth, eighth and not a quarter note triplet), the baritone aligns on notes #1 and #3, with the tenor aligning on #2. It just keeps going back and forth except for the chromatic ascent, when the baritone moves in parallel octaves with the bass.

Transcription Skill #5

Once you've mapped out critical bassline, and identified the overlapping notes, you can listen for individual parts with respect to the entire chord to determine which direction they move between notes/words. In general, during the initial listening, I like to write arrows in place of notes to assess their vertical placement as the song moves forward horizontally. Then I go back and deductively fill in the notes for the other parts. I think of this as bringing it all together, since they have to make sense with respect to the ensemble.

A Key Thing To Remember

There have been many times with more difficult music when I was convinced that a part moved by, say, a whole step, but the new note didn't fit with the next chord and in actuality it moved by a minor third. What this all boils down to is trust your instincts. When you're writing the line for a specific voice besides the melody, you might want it to move in a certain way, but it turns out that another voice has claimed the note you expected because the music demands it. Voices can cross with small groups, especially if the melody jumps around (i.e. the tenor isn't always above the lead and the baritone isn't always below). While listening for the parts, remember that you want to fill in notes that make sense with what you decided earlier (chords) and not what you want to hear now (single line).

I'm sure some of you might disagree or have your own approaches. Feel free to share them for the benefit of anyone who reads.

Super Mario Bros. Underwater Theme analysis

I was playing the "Underwater Theme" from Super Mario Bros., and in noticing how "pretty" the chromaticism sounds, I decided to do a harmonic analysis.

Here it here: https://www.youtube.com/watch?v=zc28zcBIX7U

Read the analysis below:


Conclusion? Eh, just use of secondary dominants and passing augmented chords.

For sake of argument, I'll consider a major chord built on scale degree 7 a "secondary dominant" since it's usually interpreted as V/iii, though it's not exactly tonicizing iii here.The ostinato G and voice leading in general is pretty smooth (Note: I also put all nonharmonic tones in parentheses, including passing tones, neighboring tones and ostinato).

Gotta' love them cute Koji Kondo harmonies.

Music theory chord chart

In my own time, I decided to create a chord chart that helps bridge the gap between theorists (who use Roman numeral analysis) and contemporary musicians (who use chord names).

Below are my result, with a bit of color-coding to make make things clearer.




So what does it all mean? Well, here are some guidelines for starters:

• Keys on the left-hand side, with logical chords in that row.
• Functions are written across the top row, with corresponding chords in the columns.
• Light gray chords are unique to major; dark gray to natural minor. But that doesn't have to stop you from borrowing chords from the other mode.
• In general, for those of you who didn't know, chord progressions tend to follow a pattern of tonic - ... - dominant - tonic, with other chords interspersed. "Predominant" chords can prepare a dominant chord, and others can precede a predominant.
• Pay attention to the indicated color, which generally match the name of the function. Some intermediary chords have a different color, and that's because depending on their usage, they fall under either category (to the left or right).
• In general, chords within the same category can be substituted with another, regardless of accidentals (chord alteration from the norm).
• That slash notation (i.e. X/Y) are secondary dominants, noted for convenience. What that means is that the indicated chord (call it "Z") functions as a dominant "X" of (i.e. prepares or expects) the chord "Y". So if you used that "Z" chord, the next chord you expect is "Y". But as I mentioned in that last point, instead of "Y" you could use another chord in Y's category.

Some other helpful hints and observations:

• Seventh chords weren't considered, but generally they serve the function of the triad on which you're stacking that seventh.
• The chords "vi" and "bVI" are orange (not yellow) because they can be substituted for the tonic on occasion in what's referred to as a "deceptive cadence".
• Just because a chord has funky symbols like flat signs or slashes, doesn't mean you should shy away from it. They actually sound pretty cool and can really add flavor to your music.

And now for my own comments. It's theory related, so feel free to skip the rest:

• Common practice / classical theory is a bit more rigid than contemporary music (primarily using only the notes within the key with subtle alterations). I think this chart helps show how "foreign" chords (i.e. beyond the key) connect to other chords.
• I didn't realize that the "bII" chord can also be a dominant ("tritone substitution"). Usually it appears in first inversion (i.e. with the third in the bass) as the "neapolitan".
• The "bVII" isn't used much functionally in old-school music theory, but I've heard enough examples of it as a dominant. Oh, and speaking of the dominant, just because something's called a dominant doesn't mean it has to be the penultimate chord. Have a ball.
• I tried to figure out a way to fit "bV" (or any chord built on the tritone), but I couldn't find it. I also wasn't sure where to fit the minor "v", but I guess it would be a dominant.

Note: This isn't an exhaustive chart, by any means. Just a basic guide to how to easily use chords. I hope someone out there finds this useful. Perhaps a person not as versed in theory but still eager to compose.

Hyrule Field vs. Good Egg Galaxy

A while ago I made a remark about two Nintendo themes, one from The Ocarina of Time and the other from Super Mario Galaxy. Well, I finally decided to set some time aside and analyze the fragments. Prepare yourself for a little bit of music theory (excuse the hasty transcriptions).

Hyrule Field - The Legend of Zelda: Ocarina of Time

Good Egg Galaxy - Super Mario Galaxy

Chords are given above the staffs, with Roman numeral analysis below. Now since they're in two different keys, I transposed the Hyrule chord progression to the simplest key of C major so you could compare the chords within the same key.



Now, even if you don't know music theory and can't hear the chords, I think you can at least agree that they look similar. Add to that the basic melodic contour and it seems like Mahito Yokota (composer of SMG) ripped off of Koji Kondo (composer of OoT)

Now, let's dissect the progressions and compare the chords against one another (vertically):

• Coinciding chords. Chord #1, 3, 7, and 8 are the same. That's half of the eight chords and traditionally the the framework of any basic tonic-predominant-dominant progression. Another way of looking at it is that these are the main chords, while everything else embellishes them.


• Substituted tonic. Chord #5 in the first progression is a "I" or tonic chord (C-E-G), so the chord of the key (C major). The second progression has a "vi" chord in its place, which is A minor (A-C-E), which shares two of the three notes. In general, substituting a vi for a I is called deceptive cadence since you be expecting the original C major chord but instead hear a chord of a different quality, yet it differs by one note (as a triad). Add to it the G from the arpeggiated melody and this seventh chord (A-C-E-G) actually contains the other progression's tonic chord  (C-E-G). How's that for overlap?


• Foreign predominant chords. Chord #6 in the first progression is a D major chord, a chord outside of the key of C major. The corresponding chord in the second progression is the A-flat chord, also outside of the key (accidentals are the first hint). The D major chord functions as a secondary dominant (V of V), as it sets up the G major in the following two measures. The A-flat chord, when thought of as an augmented sixth chord without the seventh also sets up the G major chord in the following measures. Even if you don't understand a word of that, the point is these two major predominant chords serve the same function yet clearly lie outside of the key.


• And as for that second chord. Well, that's a bit tougher to explain, and I could easily cop-out and say it's a throwaway (we've already accounted for 7 out of the 8). Here, I want to say that the strength of the fifth in the melody (in the scores, that's D with respect to G and G with respect to C) dominates the measure as it stands out as the highest note of the passage. And as I suggested above, it is an embellishing, passing chord that isn't as significant as the others.

When you consider it all together, you can see how they're structurally similar, especially with the primary chords. I actually thought they were the same progression the first time I heard the Good Egg Galaxy theme. Then again, that could just be because I fondly remember the Hyrule Field theme and pick up on anything that bears any resemblance.

So what do you think? Have I convinced you? Do you think Mahito Yokota ripped off of Koji Kondo?