A Physicist’s Perspective on Tonality

Tonality is so deeply ingrained in the way we make and perceive music that it is very hard to imagine music that is not in some form tonal. The few people that have dared to make atonal music are only peripheral events in the history of music.

We like tonality so much that most of today’s pop music is tonal to the point of caricature. The epitome of this might be the ubiquitous four chord song (as shown so poignantly in the Axis of Awesome video).

But tonality is also the bread and butter of a long ranging tradition of the best what music has to offer: tonality is equally important in diverse musical traditions, be it in classical or jazz music.

In the rest of this article, I have tried making some of these concepts accessible to readers without too much background in music theory, but I can’t really start from scratch, so I hope you carry on even if there are some things that are a little hard to grasp.

I want to adress two big questions:

What is tonality? And why do we need it so much?

But before talking about tonality, another point warrants attention: even most of the atonal classial music takes place in the narrowly confined space of the twelve tone system of Western music. So before talking about tonality, I want to first talk how we even come up with this system of tones in which tonality eventually arises.

The twelve-tone system

Now we’re not talking about the earth but about musical space, which is in itself a sort of empty potentiality. It is the space of all possible vibratory frequences (let’s ignore dimensions like sound colour at this point). We can play gods for a bit, try to bring forth order within the infinitude of possibilities, try to constrain in order to allow creativity to flourish.

This is a very abstract question to start off with. But while today, questions of science and question of art tend to belong into two different categories altogether, these fundamental questions about music and tuning are deeply intertwined with mathematics. And vice versa, in Ancient Greek, ideas such as the Musica universalis or the Music of the Spheres generalized musical terminology and ideas derived from music to theology, astronomy, and metaphysics.

The Pythagoreans were prominent advocates of these ideas, and the man himself, Pythagoras, is credited for discovering the connection between nice-sounding intervals like the fifth and the mathematical relationships between their vibrational frequencies.

Artistic rendering of Pythagoras and Phylolavs discovering the laws of music with some confusingly numbered flutes (credit for this terrible drawing goes out to Gaffurius)

These Pythagorean insights were used to put quite severe constraints on the tonal space in Western music.

Instead of the infinite variety of possible vibrations, we have picked out only a few “legitimate” vibrations in relationship to one vibration that’s also somewhat arbitrarily fixed. This is the concert pitch a’, usually coming down at a’=440 Hz in modern days. Ignoring the people that say that a’=432 Hz is superior “because of it makes some chakras within our body resonate with some magical Schumann frequencies”, that tone choice is somewhat arbitrary. Fun fact: different European cities in the 19th century were actually battling for who would have the higher concert pitch, as elaborated in this video by Adam Neely).

Built on this pitch are twelve tones (a,b,c,d,e,f,g, a flat/g# , b flat/a#, d flat/c#, e flat/d#, g flat/f#) that have designated vibrational frequencies.
In just intonation, the relationships between their frequencies are given by small whole numbers, and their intervals correspond to the harmonic overtone series of the tone at its basis (which in turn corresponds to the key signature). This is the idea that sort of goes back to Pythagoras.

Taking e.g. the tone C as a basis, we can build upwards to get to the harmonic overtone series. This is pretty important. I quote:

The overtone series is the basis of tonal music. The order of pitches in the series has established the progression of Western Music over the last 300 years.

This overtone series is relatively simple to form. You take a string that vibrates with a certain frequency (here, with a C), and hold it fixed at different points of its length, causing it to vibrate with a different frequency which relates to the point of the string at which you fix it. That‘s pretty much the entire idea behind a guitar (the grey dot being the guitarist’s body part of choice, be it the finger or the Hendrixian tooth).

Taking a string and partitioning it by whole number fractions. The underlying series 1/n is called the harmonic series in mathematics.

When we do this for something whose rest pitch (the undisturbed string) is a C, we get this series:

This is the harmonic series over the tone C. The notes correspond to the whole number partitions in the above picture. The numbers above the notes show how much the tones of the pure overtone series of C differ from the tones used in modern tunings (the unit used is called “cents”). This difference is due to the impossibility of achieving a nice even tuning for all keys by taking the pure pitches fromeach individual tone’s overtone series as a basis.

The point is that when we play a C on an actual musical instrument, this instrument is based on an accoustic resonator. And this resonator oscillates at multiple modes simultaneously. So you never actually play a pure pitch in music. A pure pitch sounds very unnatural and unmusical (people that have played around with oscillators know this). So when using actual instruments, physics causes the overtone series to be naturally present to some degree. You can actually hear this on the piano: it is kind of like we are already hearing a soft octave and fifth being played over every tone we play, and then the following tones more and more subtly (for a demonstration, listen here).

The intervals of the octave and the fifth appear earliest and most frequently, then comes the fourth, then come the third and the seventh, then come the “flavour” tones (like the 9th, sharp 11th or 13th). The higher up the series we go, the more harmonically dense it is and more complex it sounds.

In other words: when we play an octave or a fifth, it fits in very nicely and uncontroversially into the overtone spectrum, a tritone or a seventh not so much.

Taking a little excursion (that might be a little difficult to get without too much theory background, I’m not sure), this little fact is, I believe, one of the keys to understanding the history of harmonic developement in Western music. Simplifying as much as possible, music started out with the pure fifths of the early Gregorian Chants, where even the major third was thought to be a dissonant sound (!), moved on to music built principally on triads (so on thirds) in the Baroque and Classical era (except maybe the odd insane Bach piece with its 20th century jazz chords that defies all logic of chronology).

The dominant seventh chord that we will look at in more detail later plays a crucial role here, but in this the seventh is used mainly functionally (as admittedly, everything serves a function in German music), and not really as a colour. As the harmony is pushed and chords become larger, we see them more often in the late Romantic style. Then come some ninths, early on in some of the daring pieces by Schumann (as in the glorious “Im Wunderschönen Monat Mai”, starting on a ninth), but developing into a trademark sound only with guys like Ravel (the Dominant 7add9 is sometimes referred to as the Ravel chord, and Debussy used it extensively as well). At that point pretty much everything is allowed: see for example the first chord of the Suite Bergamesque by Debussy, adding all three “flavour” tones (9,11 and 13) on an F base tone.

The beginning of the suite bergamesque by Debussy: F, C, G, Bflat and D making up a F 9 11 13

There are many more examples from the late Romantic Era, be it Scriabin’s mystic chord or the Elektra chord of Richard Strauss, where the whole harmonic series is explored in rich and colourful chords and harmonies.
Further down the road, Jazz becomes defined by seventh chords, and chord extensions go rampant.

Lead sheet of ‘Round Midnight by the great Thelonius Monk. The 7 in the chord symbols is omnipresent. In line 3, extensions like the b9 can be seen. Further extensions (like the ninth, eleventh, and thirteenth), are added by the players at liberty.

Seven-tone chords are standard in modern jazz, which are literally a whole scale played at the same time. It can be difficult to understand how people back in the day thought these chords sounded weird or ugly.

But the ears did grow. It is one of the achievements of centuries composers and listeners alike that this territory became more fully explored, and we who relish in the glory of a well placed sharp eleven should be grateful.

Pop music, on the other hand, trying to be as uncontroversial as it could possibly be, limits itself again to thirds and fifths, the tones that are most subconsciously uncontroversial to the ear.

Coming back to the basic harmonic series, there is a problem, because you can’t have these nice whole number relationships for every key. In Western music, tunings like the well tempered or even tempered tuning are used instead of taking just intonation (where you take the actual overtone series of a tone), because it makes building instruments technically feasible. This is because

Just intonation has the problem that it cannot modulate to a different key (a very common means of expression throughout the common practice period of music) without discarding many of the tones used in the previous key, thus for every key to which the musician wishes to modulate, the instrument must provide a few more strings, frets, or holes for him or her to use. When building an instrument, this can be very impractical.

Things like the well-tempered tuning are much more practical, and were established in the eighteenth century as pretty much the standard tuning. Of help might have been what can be considered as the greatest piece of advertising in the history of mankind: Bach’s two Well-Tempered Pianos, featuring 96 pieces in all 24 keys.

So that’s how we ended up with 12 tones that have somewhat nice frequency relationships with each other that create a relatively even hierarchy between all twelve key signatures (as not one scale is a better major scale than any other). It can be argued that the symmetry is only real on the paper: much can be said about which key signature gives what kind of feeling (I have had hour-long conversations about what colour c minor feels like, or why a sharp eleven on a D major chord is the brightest note there is). There is obvious correlation between certain moods and certain key signatures, which the composers themselves admitted to openly.

Look at the C minor of Beethoven’s fifth, of Mozart’s piano sonata, or of Bach’s Prelude in the first WTC. I have always wondered how much of that is confirmational bias (when all the C minor pieces you know sound grave and strong and like destiny knocking at your door, no wonder you think the key itself “sounds” that way), and how much of that is actually “noticeable” by non-perfect-pitch listener. I heard a talk recently by the members of the Emerson quartet about Sibelius’ String Quartet, in which they talked about a modulation from E flat minor (which is thought of as the darkest key) to D major (which is thought of as the lightest key). These tones are only a half-tone apart, the smallest possible difference there is. If one then assumes there to be variation with respect to the concert pitch, this difference seems quite unsubstantial and hard to ground in any objective fact.

“Distances” in mood and character are thought of as being related to the distance of keys in the circle of fifth, which is very large for those two (from 6 B’s to two sharps). These colour-association with the keys is pretty much the common narrative in the classical music community. But considering the degree of arbitrariness that goes into constraining musical space, I find this a little hard to swallow. This would make fascinating ground for a study (play a piece once in C minor and then in A minor, and judge whether people rate its emotional content differently, or play these pieces to a large number of people before and after tuning to a different concert pitch etc.), but studies on these subjects are hard to control, because we all grew up with music already ingrained with these values.

Before getting carried away completely, I actually have a point: in the classical music of Haydn, Mozart or early Beethoven, this symmetry between the key signatures is reflected in the initial tonal choice of the piece: it can be in A minor, or F major, or C sharp minor, but after deciding on that key signature, the symmetry is broken, and the piece is usually pretty obviously in one key or the other. The sonata form, maybe the most important structural device in all of classical music, gives rules of how the tonal scheme of the piece will usually develop. Nevertheless, rules are there to be broken, and the masters played around with alternative harmonic layouts, fake reprises etc. a lot, and especially Haydn loved taking all the cliche devices of classical music and re-presenting them with a twist.

The tonal center is used to show the listener in relationship to which basis the music is to be understood. And this is still true in Romantic music, despite modulations being more common and daring.

But what does it actually mean to be in a key? And going back to my initial question, why do we need tonality so much?

Tonality is kind of like the place that feels like the home of a piece of music, or the home of a part of the piece of music (don’t take my word for it, I got it from Jacob Collier, maybe the most overall talented musician of our age). Everyone likes going home, so the tonal home of the music is something that pulls the music towards it. In a different picture, the tonal center is kind of like a gravitational center. And this pull has something to do with tension, with curving the initially homogeneous manifold of musical space in a way that makes things fall a certain way.

Enter the cadence, and the tritone that makes it all work.

The interval of the tritone, which splits the octave in half, and is therefore symmetrical (c and f sharp and f sharp and c are both a tritone apart).

Playing these two notes together will sound kind of tense (listen here). This tension is resolved within the cadence by something called leading tones: in the cadence going to C, the tritone consists of a F and a B natural that move downwards/upwards by a half tone step to E and C. This already sounds quite satisfying, but what we can further do is add a bit of structure in the form of a bass tone (here, it is a G), that moves upwards to C. This movement by a fifth is the second quintessential ingredient we need.

A V-I movement from a G dominant seven to a C major seventh.

Combining these two things, we have a very simple chord movement that can “make you feel at home” in C. If it’s not resolved, this feels very unsatisfying.

Famously, the old sick Bach was pranked by his children who played unresolved Dominant Seventh Chords in the room next to his. Although he was at that point almost blind and lying on his death bed, he still got up and resolved them properly.

We can further add an intermediate step ( in the classical cadence, this is a F major, in jazz theory, this becomes a D minor 7) to round off the voice-leading. Without needing to understand why that is in too much detail, listening to it intuitively makes lot of sense (which you can here). It can be thought of as a preparatory step so we don’t jump right in on the tension, but come in a little more prepared.

In total, this gives us the famous ii-V-I progression, which, despite its simplicity, remains the most heavily used progression in jazz, even in crazy harmonic tour de forces like Coltrane’s Giant Steps.

A ii-V-I in C, the single most common chord progression in jazz.

This cadence creates harmonic movement that establishes us in a key, and helps us get back to it after we strayed off.

Of course this is not everything people came up with in an incredibly rich tradition that spans hundreds of years, and there are many many alternatives like Deceptive Cadences, Backdoor ii-V-I’s, Tritone Substitutions etc. that can also make us feel like home and lead us somewhere, but in more subtle ways.

As noted in the beginning, pop music takes this desire to feel uncontroversially at home perhaps a little too seriously. The four chord songs (an astoundingly large list of which can be found here) use nothing more than a variation of this basic cadence.

As a last point, we can look at how classical music stepped away from tonal music, and how the definitiveness of key signatures was slowly dissolved in the late Romantic period. This is perhaps nowhere as noteable as in Wagner’s famous Tristan chord:

The ambiguous Tristan Chord, heralding in the beginning of a tonal music.

While the chord itself is not that special (it is simply an F half-diminished) and naturally occurs as the II in every minor II-V-I, it does not take the usual route of the II-V-I, but opens up to an E 7 (with a flat five that resolves to the five), which is still full of tension. So in which key are we in exactly? This is impossible to answer before the final resolution at the end of the Tristan opera three and a half hours later.

I think this can help answer the question why we feel that we need tonality in music. Music that is not tonal feels arbitrary, strange, full of tension. There’s nowhere to belong to. If employed with a master’s skill, it can make us experience the unbearable tensions that lead to the Liebestod in Wagner. But if it isn’t, it becomes hard to deal with and hard to enjoy. Tonality gives a something to hold on to, somewhere to arrive.
It reminds me of the great lines from Yeat’s Second Coming:

Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world

While uttered in a very different context, these words were written down in 1919, where tonality had indeed become problematic for many serious musicians, reflecting the general state of the Zeitgeist.

So while music is not a real language, it nevertheless follows its own logic, which is interwoven with the logic of our emotions. We enjoy structure and the physicality of resonance.We enjoy tensions that are created only to be resolved, be it in sexual pleasures or when riding a rollercoaster. We like returning to a place we feel at home in after being out exploring the unexplored.

And much of this, if one looks closely, is reflected in the way music is made, and in tonality and the cadence.

I love new ideas. My science podcast: https://anchor.fm/acit-science Connect via LinkedIn: https://www.linkedin.com/in/manuel-brenner-772261191

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