Roman numeral analysis, and “four chords that made a million”

[No, I’m not dead yet — just absolutely snowed under in my day job. Here is a little blog post to keep the blog alive.]

Anybody who has ever played off a jazz lead sheet or ‘fake book’ is familiar with chord symbols like G (G major triad), Cm (C minor triad), D7 (D dominant seventh), G/D (G major triad with a D in the bass), and the like.

One ‘abstraction level’ above is the so-called ‘Roman numeral analysis’ which is found in music theory texts, particularly classical ones. It considers not the absolute chords but their relative position (and diatonic function) in the scale. For example, a 12-bar major blues in G corresponds to the progression G-C-G-D-C , and in C# to C#-F#-C#-G#-F#, but in Roman numeral notation, both would be I-I-IV-I-V-IV in their respective keys. Likewise, a minor blues would be i-i-iv-i-v-iv regardless of the key it is in.[*]

The system was invented by the eccentric classicist composer and music theoretician (teacher of Carl Maria von Weber) in the late 1700s. It is not a system that comes naturally to people with absolute pitch (since the same progression in different keys really sounds different to us) but it is an excellent ‘meta’ tool for describing commonalities between what may be very different compositions. Effectively, it is a form of notation that stresses function (as in ‘functional harmony’) — tonic (I), mediant (III), subdominant (IV), dominant (V), etc.

As already seen in the blues example above, major chords are indicated by uppercase Roman numerals, and minor chords by lowercase ones. Seventh, ninth,… chords take digit qualifiers just like in conventional chord notation, e.g., i9, Vb9, etc. A ‘+’ and a ‘°’ indicate augmented and diminished chords, respectively.

Inversions are indicated by suffixes like Ib (tonic, 2nd inversion) and V7d (dominant, third inversion) although I personally find  the ‘b’ confusing due to its similarity to a ‘flat’ sign and prefer I/V and V7/IV, respectively. (In C major, these would correspond to C/G and G7/F, respectively)

Finally, out-of-scale chords are prefixed by accidentals # and ♭. For example, a Neapolitan chord in a piece otherwise in a minor scale would be written♭II

Rick Beato has a video here about ‘the four chords that killed pop music’.

What he really means is the progression I-vi-IV-V and its permutations like vi-IV-I-V, which some producers seem to think are nearly a prerequisite for hit singles. In Rick’s video, you can hear a plethora of examples in a wide variety of keys and styles (what do Taylor Swift and Iron Maiden otherwise have in common?! Or the choruses of Roxette’s ‘Listen To Your Heart’ and The Beatles’s ‘Let It Be’?). At a higher abstraction level, all of them boil down to just that pattern I-vi-IV-V, straight up or rearranged.

To be sure, if one is willing to escape the tyranny of simple triads and power chords, even I-vi-IV-V can be made interesting… And if one is not (e.g., because on a distorted guitar more complex chords quickly get muddy), then changing the scale to a more exotic one helps…

For instance, here is a video touting the Mixolydian mode as the ‘secret sauce’ of AC/DC

Leaving aside that AC/DC, while fun to play, is hardly a model of musical sophistication: What is he talking about? Let’s compare the diatonic triads on the major scale with those on the Mixolydian scale (more correctly: the Mixolydian mode[*]):

major (Ionian):   I – ii – iii – IVV – vi – vii°

Mixolydian: I – ii – iii° – IV – v – vi – VII

Yes, the major triads in the major scale are the familiar tonic, subdominant, and dominant — but the mixolydian one has them on tonic, subdominant, and leading tone! This automatically invites riffs like A-A-A….D-D-G…D-D-G-D-D-G-D-A-A (“Highway To Hell”) or E–D-A/C# (“Back In Black”), or …

They also use the Dorian mode fairly freely (“Hells’ Bells”, “Shot Down In Flames”,..)

Dorian: i – ii – ♭III – IV – v – vi° – VII

Now the Mixolydian and Dorian modes are, of course, very common in Anglo(-American) folk music — but yes, much of the character of different scales and modes derives from the chords progressions they generate. I will elaborate in this in a future post. Meanwhile, here are the biting observations of one of my musical heroes, Steven Wilson, on the music industry:

[*] The “Nashville number system” used by some country and gospel singers (including by Elvis Presley’s backup singers the Jordanaires) is a variant that uses Arabic instead of Roman numerals, with minor chords being indicated by a dash (e.g. 6- instead of vi). It was invented to facilitate transposition to fit the vocal range of the singer being accompanied.

[**] Technically, a scale is a sequence of notes/intervals covering an octave, a mode a different choice of tonal center (‘starting point’) among them. For example, the G mixolydian mode is generated from the C major scale simply by starting at G rather than C.

On consciously and unconsciously knowing


A Facebook friend of mine, very articulate, a sharp thinker, and with multiple academic degrees in “hard” subjects, was discussing his frustration with only speaking one language, and even so, “don’t ask me about the rules of grammar. On good days, I know what a gerund is.”

Now his written communication is always flawless in spelling and grammar, so he clearly knows how to apply grammar — which illustrates the difference between knowing something and knowing the words for it. Or, if you like, between having internalized a skill and being able to explain it.

Richard Feynman, in “Surely you’re joking, Mr. Feynman!” recalls how his father taught him that knowing the name of, say, a species of bird in several languages still doesn’t teach you anything about  the bird. That is true enough, of course, except for one thing — if I know what the bird is called, I can go look it up — trudge to the library for the Britannica or a handbook of ornithology when I was young, or just search in Google or Wikipedia nowadays.

I write a fair amount of highly technical nonfiction in my day job — well enough that I’ve been asked to teach others — and frankly didn’t consciously know any of the grammatical rules until I realized I was able to teach people how something was done, but not why. “This is how it goes, it just sounds wrong otherwise, don’t ask  why,” isn’t how thinking people like to be taught. Consequently, I was forced to hit the textbooks myself just so I could “tell people what the bird was called so they could look it up”. I imagine this is a similar situation to people who are self-taught as jazz or rock musician but need to go learn theory just so they can be more effective teachers.

In an interview shortly before he passed away, the legendary jazz trumpeter and bandleader Miles Davis reminisced about a meeting with Jimi Hendrix, planning a recording session that sadly never came to pass due to Jimi’s untimely death. He recalled mentioning a “diminished seventh chord”, and Jimi looking blank. He then took his trumpet and arpeggiated the four notes — Jimi of course immediately played the chord that he’d never known the name of. In fact, Jimi would have stared the same way at the mention of a “major-minor chord”, a.k.a. “dominant seventh-sharp ninth chord” — even though it’s nowadays often referred to as the “Purple Haze chord” or “Hendrix chord” due to its prominent use in one of Jimi’s best-known compositions.

Hendrix “spoke music like a native”, but didn’t consciously know the grammar, if you like — he just could apply it in his sleep. A very different intuitive musician, Evangelos Papathanassiou — world-famous among electronica and soundtrack lovers by the Greek nickname Vangelis — had classical piano lessons but never properly learned to read music: blessed with a prodigal ear and memory, he could reproduce what his teacher showed him just fine. While he apparently took some college classes in music (as did his more meditative German college, Klaus Schulze), he kept an intuitive, “feel” attitude toward music his whole life. When an interviewer in Keyboard magazine asked him how he composed, he answered tellingly: “it’s like breathing: if you think about how to breathe, you choke”.

Now while some of Vangelis’s more ambitious compositions (such as “Heaven and Hell”) clearly draw inspiration from Western classical music (Klaus Schulze even wrote a brief orchestral fugue in the studio version of “Ludwig II”), it would be hard for a musician to “function” in the classical world without the musical equivalent of “knowing your grammar”. (To be sure, at least one famous classical organist needed to learn most of his repertoire by ear — Helmut Walcha had been totally blind since age twelve — but he surely knew his theory, and taught for many years at the Frankfurt Conservatory.) Likewise, in some of the more ambitious, through-composed realms of jazz and progressive rock, a thorough conscious knowledge of music theory is a great asset—though you may be able to get by just fine with an unconscious one, as long as your fellow band members are comfortable learning by ear.

Conversely, knowing the rules without being able to apply them in real time may get you a job as a critic, but won’t get you far as a musician — or a writer.



The “Magical Mystery Chord” finally revealed?

The classic Beatles song, “A Hard Day’s Night”, opens with a complex ringing chord that has had songbooks (and musicians) arguing among themselves for decades. Complicating the answer is that even Paul McCartney can’t exactly remember what was done.

Full disclosure: I relate to the Beatles much the way I relate to Mozart: I recognize their musical genius but much of their most popular music does not ‘move’ me either intellectually or emotionally. But I love a good musical puzzle as much as can be.

In principle, given modern computer technology, the problem of transcribing a piece of music should be simple: digitize the audio, carry out a Fourier analysis, and convert the resulting frequencies to note names. Right?

Well… Feed in unaccompanied flute and this will work fine. (As anybody who’s owned an analog synth knows, a triangle wave is a pretty decent starting point for a flute sounds — and while a triangle does have some harmonics, the fundamental is very strong and there are only odd harmonics so you can tell apart the fundamentals pretty easily from the rest in the Fourier spectrum.) Feed in a Hammond organ with just a single drawbar open: ditto. Feed in a more complex sound but with restricted harmony (e.g., a violin playing only single notes), no problem. Feed in a complex chord played by multiple instruments on top of each other, and things get hairier. Have some of the multiple instruments not be quite in tune, or let some be in equal temperament and others in just intonation, and things gets even worse.

An applied mathematician at Dalhousie University did a Fourier analysis on the opening chord some time ago and turned that into a paper.  Does this sound like an academic with too much time on his hands, “partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada,” no less? Well, to me it sounds like a good “torture test” for the robustness of a musical transcription code. And where it comes to science popularization, this definitely hits the spot with the musically minded: only yesterday I saw another popular article about the now a decade old analysis being linked on Instapundit.

Just retaining all frequencies with relative amplitudes above 0.02 still gave him 48 frequencies, from which he squeezed a solution that looks good in theory but just doesn’t sound “quite right”.

A musical transcription site run by somebody with the delightful pseudonym “Waynus of Uranus” points out a fly in the ointment that people who grew up with digital recording wouldn’t even have thought of. Back in the day, loud bass tones meant pushing against the limitations of vinyl singles and lo-fi audio equipment alike, so the deep end of the bass (about 80 Hz and lower) was routinely rolled off with an equalizer or a highpass filter during mixing or mastering. What this means, for example: if Paul were to strike an open D string on his bass guitar (or an A string at the fifth fret) his fundamental would be below the filter cutoff, and the Fourier spectrum would instead have the second harmonic much stronger — leading to claims like “Paul played a D3 and a soft D2 at the same time”. I know bass players like Geddy Lee or Rush or Steve Harris of Iron Maiden play lots of double-stops, but this really is a progressive rock or metal thing to do, not a pop thing.

Applied mathematician Kevin Houston takes it from there and digs further in a very geekish way. While the original record was mono, it turns out there is a stereo mix made for the movie—and in the early days of stereo, it was not unusual for recording engineers to just put some instruments all left and others all right, with the vocal in the center. (This is, pretty much, how I used to jam along with Deep Purple records: Jon Lord’s organ and Ritchie Blackmore’s guitar were usually at opposite end of the stereo image, so you could single out their parts by listening to one stereo channel at a time.) In the stereo

In the stereo mix of AHDN, Paul (bass) and George (12-string guitar) are off to one side, and John (acoustic guitar) off to the other, together with producer George Martin on piano. Better still: after subtracting the left channel from the right (i.e., “phase-inverting”), it becomes clear that the acoustic is playing an Fadd9 chord. (That means: an F major chord with an added ninth, a.k.a. a “Steely Dan chord“. It differs from a major ninth chord F9 in that the seventh is omitted.)

To cut a very long story short (some mathematicians can get quite verbose ;)), this is the solution (which relies on a good dose of Occam’s razor/the Law of Parsimony as well):

  • Paul just plays a low D2, but because of EQing off the deep end, the D3 overtone/second harmonic comes through louder than the fundamental, hence the acoustic illusion that the bass note played is D3
  • John plays F2 A2 F3 A3 C4 G4 (in standard tuning, frets 1-0-3-2-1-3)
  • George plays the same chord, but on a 12-string in standard tuning—where the bottom four “courses” have the second string one octave higher. Hence aside from the slight tuning discrepancy with John, he adds F4 A4 as new pitches
  • Finally, George Martin on the piano, with the sustain pedal down, plays D2 G2 D3 G3 C4, which one could call a Gsus4/D chord. Sympathetic resonance from the undamped piano strings adds the wash of low-level extra pitches that befuddles the Fourier analysis.

Not only does this not require attributing instrumental acrobatics to the Beatles that are out of character for them, but actually playing those notes on the respective instruments does produce a sound quite like the record. (Listen at 7:17 in the video below.)

Kevin and his collaborators could not readily find an electric 12-string, so they simulated that by layering two six-string electric chords: once fretted 1-0-3-2-1-3, the second time 13-12-15-14-1-2 with an extra hand. “Fake Nashville Tuning“, if you like.)

If this isn’t  the solution, it sounds much closer than anything else I’ve heard. Enjoy the above video!