When you first start to learn the rudiments of music theory, everything seems pretty cut and dry. Less of music theory, and more of music fact. C to E is always a major third, G minor always has two flats, there are four dotted quarter notes in a bar of 12/8. When you get to the first level of harmony, things expand significantly. You start to learn what chords can precede and succeed what other chords, and you start to find room for creativity. Depending on what your learning environment is, some places will give you significant amounts of freedom, while others will be more restrictive (here’s looking at you, RCM). But even here, when it comes to analysis, it seems like there are answers that are completely right or completely wrong. Tonic goes to pre-dominant goes to dominant goes to tonic. Even when there are countless examples from the composers who the theory is based on that break their own rules (let alone other, more obscure and/or more interesting composers), you’re expected to follow these rules.
A less experienced student may wonder, then, what do professional music theorists actually research? Surely they don’t just comb Mozart’s catalogue for a piece that hasn’t been yet given a harmonic analysis and put one on it, an analysis that has only one right answer? Of course not! Imagine what a sad field that would be.
Music theory as a whole is a surprisingly underdeveloped field: for two centuries, it focused so narrowly on the vocabulary of 18th century German composers, and yet there are still people in this world who think it’s worth arguing over the Tristan chord some more. You know how far into the Real Book I had to go to find an enharmonic Tristan chord? Eleven songs.
Because one of my favourite things to do is share niche knowledge of things I’m interested in, I thought what better way to expand peoples’ ideas of music theory than to show some of the things I think are really neat. Here are three of my current favourite topics in music theory, that are relevant to the music we play today.
Topic Number One: Intonation and Temperament
Most music that is composed and performed in the modern Western world uses a system called 12-tone Equal Temperament, or 12TET. This is a direct result of the prominence of the piano, as it is the only tuning system in which all twelve major and minor keys can be made to sound identical, without increasing the amount of keys per octave by several times. Because sound is logarithmic and not linear, the ratio in frequency between two adjacent tones is 12√2. This system is why notes like F-sharp and G-flat are considered the “same” note.
Just because most of our music is WRITTEN in 12TET, however, doesn’t mean all music is, and it doesn’t even mean music written in 12TET is necessarily PLAYED in 12TET either! For string players, wind players, and vocalists, tonal music written in 12TET is almost always more effectively played in a system called Just Intonation. Just Intonation is based on using very basic ratios that appear in nature to tune your intervals. For example, a perfect fifth in 12TET has a ratio (12√2)7 ≈ 1.498, while a perfect fifth in Just Intonation has a ratio = 1.5. Thirds are even more noticeable, and when a brass ensemble plays a justly tuned major chord, it has a kind of magic that an equally tempered major chord lacks. You never want to go back.
Why do we use equal temperament? Because if you tired to tune a piano in Just Intonation, you’d reach a wraparound point where you would be unable to tune all the intervals of a given type. Let’s use the perfect fifth as an example: say you wanted to tune just perfect fifths in your piano. You have to decide where to start with the black keys, so for the sake of argument, let’s tune them F-sharp, C-sharp, B-flat, E-flat, and A-flat. The perfect fifths F-sharp to C-sharp and E-flat to B-flat would sound just fine, but the diminished sixth C-sharp to A-flat would sound atrocious. If you tuned all the black keys as flats, B to G-flat would sound atrocious. If you tuned all the fifths to sharps, A-sharp to F would sound atrocious. You can’t win without dramatically increasing the number of keys per octave.
12TET isn’t the only attempt that has been made to properly tune a piano. Other systems, like the various kinds of Mean Tone Temperament, have been employed in the past, and have been expanded on more recently by people trying something new. Even equal divisions of the octave can be experimented with: 31TET is surprisingly favoured for its nearly perfect major thirds, despite having sub-par perfect fifths. And of course, with the advent of technology in music, you can change tuning on a dime, like so:
Topic Number Two: Useful Analysis of Chord Loops
Recommended Reading: Everyday Tonality, by Philip Tagg
Probably the most notable feature that separates Western Classical music apart from folk and other world music is the long, structured journey through keys and chords over time. They way harmony is currently taught gives the student progressively more and more chords the longer they study, and then expects them to use all of them frequently. Of course, analyzing this music highlights this journey of chord changes.
Popular music, however, tends not to feature this slowly evolving journey of chords, and is made up of loops: usually four chords, sometimes even less. For something that’s so deceptively simple, traditional theoretical analysis is REALLY BAD at describing or explaining these loops, because our analysis relies on finding goals and ways of getting there. When there’s no longer a goal, traditional analysis is unable to explain why chords work the way they do.
Enter Philip Tagg, in his book Everyday Tonality, which strives to give a nomenclature to the functions of chords in popular loops. Tagg casts aside the conventions of pre-dominant and dominant chords, and instead uses the labels tonic, outgoing, medial, and incoming. The tonic and medial chords have equal and opposite weight, rather than one building tension and releasing to the other, and the outgoing and incoming chords resolve by either a step, or a leap of a fifth, traditionally, and motions by third tend to be reserved for leaving the tonic or medial and heading to the next outgoing or incoming chord (but there are many, many exceptions!). These loops eschew the traditional thought of a journey, and reinforce a concept Tagg calls the extended now, where the entire song is centred around the weight of the Tonic and Medial chords.
For a practical example, let’s look at Don’t Stop Believing, by Journey. The chords are fairly simple: E – B – C#m – A. Or, in roman numeral analysis, I – V – vi – IV. Would it do to call this a tonic chord, deceptive cadence from V to vi, and then plagal cadence from IV to I? No, of course not. Cadences imply stopping, because that’s what the word cadence means. The loop implies continuous motion. Using Tagg’s theory, we have a tonic E, strong move up to the outgoing chord B, that resolves by step to C#m, and is followed by an A that resolves by fifth (or descending fourth) to E. The progression looks much, much stronger when using a theory that’s actually designed for that genre of music, rather than trying to shoehorn it into a different genre’s theory, don’t you think?
Topic Number Three: The Lydian Scale
Recommended Reading: The Lydian Chromatic Concept of Tonal Organization, by George Russell
Our music is based predominantly on what we call tertial harmony; that is, chords built by stacking thirds on each other. Our basic three note chord, the major chord, is built by stacking two thirds, a major third and a minor third. If we add another third, we get a major seventh chord. If we add another third, we get a major ninth chord. And we can add two more thirds before we loop around again to the tonic, meaning every chord’s “maximum” form is about a thirteenth chord.
But something funny happens at the eleventh. Most people across most genres of music agree that the normal eleventh (fourth of the scale) just doesn’t sound right. Perhaps because it clashes with the third, the second-most important note in the chord? Probably not, because the seventh clashes with the root, the MOST important note in the chord, and we don’t mind that. For whatever reason, it’s nearly unanimous that people agree the sharp eleven (raised forth) sounds much more befitting of being in the extended major chord. If we take a C13 #11 (C E G B D F# A) and rearrange it as a scale, we no longer get a major scale, but what we call a lydian scale, C D E F# G A B C.
In his book The Lydian Chromatic Concept of Tonal Organization, George Russell states that every two notes have a certain gravity between them: every pair of notes will resolve to one of the two notes. If we look at all the intervals built off of the root of a major scale (C-D, C-E, C-F, C-G, C-A, and C-B), we find they all resolve to C except for one: C-F resolves to F. However, if we were to look at the lydian scale, every interval would resolve to the C. Thus, according to Russell, the lydian scale is the most resolved of the scales, and is the natural baseline for music.
Popular musician and harmony buff Jacob Collier would say the lydian scale is the brightest of the scales, even moreso than the major scale. However, he doesn’t stop there. Lydian is more than a scale to Collier: it is a sound, and it is a sound that can be extended. Take a look at this monstrosity:
This, my friends, is the Super Ultra Hyper Mega Meta Lydian scale. A regular lydian scale is like a major scale, where the fourth note is raised. The Super Ultra Hyper Mega Meta Lydian scale is like a regular major scale, but EVERY fourth note is raised. Or, another way to think about it, is that it’s the first four notes of EVERY lydian scale, around the circle of fifths. It’s a sound that continuously gets brighter ad infinitum. Note how it’s bookended enharmonically (G-flat on the bottom and F-sharp on the top), but you don’t have to start or end there! This can go around the circle of fifths forever. Of course, the effect is even more dramatic in non-equal temperament, where the F-sharp is literally sharper than G-flat instead of being tuned the same, and the scale REALLY spirals out of control.
In conclusion
There are tons and tons of neat fringe concepts in music theory, more than I can even get into here. I’ll definitely do more of these, as while writing this, I’m already imagining what I want to say about time signatures that aren’t powers of two (4/6 time is real!), or about the extreme physical process notation of artists like HK Gruber. I hope you learned something new and cool, and let me know about some of your own favourite topics!
Matt Richard 