Advanced Mathematics 4: Notes inégales

Whew, we’ve recently gone from a very new concept, microrhythms, to a fairly advanced discussion on polyrhythms, polymetres, and polytempo. I hope that this season of Advanced Mathematics is enriching to you! Today, we’re going to talk about notes inégales – uneven notes –, what kids today call “swing feel”. It goes much deeper than what you probably imagine, and I hope you’ll learn something today!


The Triplet Feel

Most prevalent in jazz and blues music, the swing feel usually involves altering the length of two consecutive and equal notes. The most common example of this is the triplet feel, named thusly because it gives the listener the impression that the piece is written in triplets, but that doesn’t mean there are actually three notes in a beat. So, what happens here? Say we play in \(\frac{4}{4}\) time. The notes you will play inside these measures will likely be eighth notes. Each eighth note here has the same length value: half a beat. However, in regular swing feel, this changes.

A triplet feel for eighth notes.

If your music sheet has the notation above inscribed on it, then you must do something different. Indeed, it means that each first eighth of a group of two needs to be played as if it were a quarter note in triplets, and each last of the group should be played as an eighth note in triplets. This creates uneven notes – notes inégales in French –, and this is by far the most common example of this nowadays. I will often use ratios, in this post, to denote a swing feel. In this example, it would be \(2:1\) since the first note is twice as long as the second note.

It’s present in most, if not all musical genres to varying degrees, most prominent in blues and jazz, but also making appearances in rock, pop, and classical music, for example. If you know The Diablo Swing Orchestra, then you’ve probably heard triplet swing in metal music as well!

However, the history of uneven notes is quite ancient…


Rhythmic Modes

Rhythmic modes was a concept widely used in Medieval music. One of the most important aspects of this concept is that a note’s length value was not so much attributed by its form (eighth, quarter, half, etc.), but by its position within a ligature. A ligature is not used much nowadays, but it’s rather similar to note groupings, where the overhead beam delimits notes that are grouped together. Today, this is mostly used to denote the beat of the time signature, or to highlight accents underlying the measure.

The same motif grouped according to the beat (upper part) or the accents (lower part).

During the 1200s, there was six rhythmic modes, one of which being the aforementioned triplet feel widely used today, which was called a trochée. Back then, these modes were meant for vocal pieces, and at least one of them is still quite famous nowadays: the iamb. Known primarily through Shakespeare’s beloved “iambic pentametre”.

The iamb is a rhythmic mode where the first syllable is weak and the second is strong. If we make the analogy with swing, the first note gets shortened and the second gets lengthened, and it can be written here with a \(1:2\) ratio. If we use them vocally, we will tend to put the second note on the beat, because it is the stronger one. However, this would then make it similar to the trochée, where the longer note starts on the beat. Therefore, for this post, we will always consider that the first note of a group starts with the beat, and that a beat consists of as many notes as the group contains.

The short-long mode, also known as reverse swing.

The iambic mode is also known as the Lombard rhythm, or the Scotch snap for its prevalence in Scottish traditional music, but ratios between the first and second notes vary. While I used a triplet feel for the illustration above, the exact note value of the mode is not specified. It could as well be a sixteenth followed by a dotted eighth (a \(1:3\) ratio). However, since we’re trying to make these ancient concepts useful today, we need clear notations. We’ll talk later about different feels, other than the triplet feel, which could be applied to these rhythmic modes.

The rhythmic modes also included the “short-short” and “long-long” metres, for two weak or two strong syllables, but this can hardly apply to music swing, since the ratio between those two is \(1:1\). So, I’ll just skip them for now.

Three-Note Groups

The rhythmic modes included groups of three, and even four notes. Since the same concepts apply no matter the amount of notes in your group, I will only write about the three-note groups. You should have all the tools you need to work with larger groups after this. First, the dactyl represents a three-note grouping where the first note is elongated while the other two are shortened. Historically, it was meant for a strong syllable, followed by two weak ones. Here is one musical interpretation of this, in a \(2:1:1\) ratio.

One interpretation of the dactyl rhythm, where three equal notes receive uneven values.

The anapaest, on the other hand, is the opposite, representing two short notes and a long one (\(1:1:2\)).

One interpretation of the anapaest rhythm, where three equal notes receive uneven values as “short-short-long”.

Of course, the use of such notation today seems quite pointless: you can very easily write an eighth followed by two sixteenths. Yet, I believe these notations convey more information with less ink than always changing the notes each time. When more complex ratios are described, however, it might be much more interesting to write a three-note swing notation. A \(4:3:2\) swing, for example, would be more concise than to use “quarter note – dotted eighth – eighth”. I recognize that these instances must be rare, but when they happen, it would be nice to have a vocabulary for it.

Lower in this post, I’ve included all different rhythmic modes in one table, up to four-note groups, since they are the ones that are classically recognized, but feel free to make up groups of your own with more notes!

This concept of having rigid note lengths is very modern, as the notes inégales concept was used to create surprise and a non-monotonous rendition. But times have changed, and most practical uses of this today would be in a more rigid setting, so I’m trying to make the concept more appropriate for current use by speaking with more rhythmic rigour. However, if you play or make music in a freer context, feel free to take these concepts as they were originally intended!


Other Tuplet Feels

During last class, I wrote a lot about tuplets for the polyrhythm section of the post. I’m going to talk about those again here, so be sure to be up to date on the topic before going along.

Let’s go back to groups of two notes for a second. Like I said earlier, the most popular swing feel is the triplet one, in \(2:1\) ratio. It is so commonplace that it serves as the line separating soft and hard swings. Soft swing represents a feel where the difference of value between the two notes is smaller than it is in triplets. Basically, everything with a ratio between \(2:1\) and \(1:1\) exclusively can be categorized as soft swing. On the other hand, everything with a more striking discrepancy will be categorized as hard swing. Theoretically, this could go as high as \(∞:1\) (effectively omitting every second note), but, practically, say we go up to \(7:1\).

A standard quintuplet feel swing, a “soft swing” with a 3:2 ratio.

A standard 3:1 swing feel, a “hard swing”.

There seems to be more and more experimentation with the quintuplet feel (\(3:2\)), especially in jazz and electronic styles of music. I think it confers an amazing vibe to a song. Some even softer feels include the septuplet (\(4:3\)) and octuplet (\(5:3\)). Some DAWs (digital audio workstations) have a swing slider, or feel slider, where you can move the slider to decide a certain swing feel that isn’t a simple ratio, and you can get some really light swing feels that would be almost impossible to play correctly and consistently by live musicians, but that can give electronic music a super special feel.

Just as with light swing, you can have many (infinitely many) other ways of creating hard swing. For example quintuplet hard swing, with a \(4:1\) ratio, but I have a hard time finding real-life examples of hard swing music. Another example of hard swing could be made using septuplets, with a \(5:2\) ratio, or using sextuplets with a \(5:1\) ratio. Here’s a graph showing all possible swing feels from two- to eight-note feel.

All possible ratios between two notes from 2-tuplet to 8-tuplet feel. The green line represents a straight feel (1:1), and the vertical grey lines represent regular swing (1:2 and 2:1). The red lines represent the limits (1:∞ and ∞:1). The left side of the graph shows “short-long” swings while the right one shows “long-short” swings. The area between 1:2 and 2:1 represents the “soft swing” zone, outside of that is “hard swing”.
(Click to enlarge.)

These infinitely-many ratios can be affixed to the aforementioned rhythmic modes, or metrical feet. For the sake of simplicity, I stated that a trochée was similar to a \(2:1\) swing ratio, but the only part of it that holds true in any scenario for that particular mode is that the first note should be longer and the second one shorter. Therefore, it would add to your vocabulary to refer to it as a “triplet trochée”, or “2:1 trochée”. If you want it lighter, call it a “light trochée”, “3:2 trochée” if you need to be more specific. This also applies to all other modes, including those containing more than two notes. For example, even if I notated the dactyl as a \(2:1:1\) rhythm, it is classically written out as a \(3:2:1\) sequence – a dotted quarter, followed by a quarter and an eighth note. However, you could theoretically use any ratio you can think of, even creating parallels to our previous discussion about harmonic rhythm by deciding to play a “6:5:4 dactyl” swing, mimicking a just intonation major chord using 25-tuplets. More realistically, now, a \(4:3:2\) complex swing in 9-tolets could foreseeably be played and offer a new and alien-sounding basic rhythm pattern.

A table with all rhythmic modes, or metrical feet, with their format of short (s) and long (l) syllables, and the simplest musical ratio that can be applied to it.

Therefore, all sorts of swing can be applied to any rhythmic mode, and it would be very interesting to hear them used more widely as part of the common vocabulary of musicians in jazz and other genres of music. But clear ratios are not the only way to see swing, and some would argue it’s not the right way either…


Swing as a Spectrum

In reality, swing is more than simple numbers and ratios. In fact, when playing “swung” parts, it seems that the \(2:1\) ratio is but a slice of the vast range of natural swing feels in music and jazz. Some scholars found a trend where the swing ratio was highly dependent on the tempo of the piece. Thanks to composer and youtuber David Bruce for the heads up!

According to Friberg and Sundström (reference), most jazz drummers will swing notes at around a \(3:1\) ratio at 100 beats per minute, but that diminishes to around \(1:1\), or straight feel, at 300 bpm.

They also pointed out that the second note of a beat, the off-beat note, had a more-or-less consistent 100 milliseconds duration, especially at tempi higher than 150 bpm. That could explain the change of ratio with tempo, as, if the off-beat value is constant and the on-beat value depends on the tempo, then, the slower the tempo, the higher the ratio.

That trend was much less pronounced within soloists, who played at around \(2:1\) at 100 bpm and \(1:1\) at around 350 bpm. The way in which the soloists lock on with the drummer is fascinating. They don’t play in the same swing ratios, but they synchronize based on the off-beat; the shorter note. That has the side effect of delaying the onset of the on-beat note for soloists, who usually play it after the drummer’s beat.

A notation and time analysis of a 1961 recording of “Down Under”. You can see how the onset of the on-beat note (bottom part, in green) start after the beat (solid vertical lines), but the off-beat (in red) is consistently on or close to the drummer’s off-beat (dotted vertical lines).
(Taken from Dittmar et al. (2018).)

A more recent and robust study, by Dittmar, Pfleiderer, and Müller (reference) generally backs up these claims, but with a much wider set of audio recordings and drummers.

At the same time, there seems to be an alternative school of thought, led by Honing and de Baas (reference), and backed by Marchand and Peeters (reference), that states that swing ratio is not dependent on tempo.

The most recent paper I could find, from 2018 by Dittmar et al. (reference) demonstrates with even more robust data that swing ratio is dependent on tempo. However, it is not as clear cut as it previously was, as the following graph shows.

Statistical model of tempo vs. swing ratio from Dittmar et al. (2018).

Indeed, it seems that there is a wider palette of swing feels that opens up as the tempo slows down, but the data still sticks close to the 100 ms off-beat hypothesis (shown by the red curve on the graph).


Conclusion

In conclusion, while there is no strong consensus on how swing behaves – whether it changes with tempo or maintains a steady ratio. The current data seem to favour a modulating ratio based on a somewhat constant 100 ms off-beat note value, at least in jazz music. As demonstrated by studies that contradicted this version, other musical genres seem to stay closer to a \(2:1\) swing ratio at any tempo.

The current common knowledge about swing is mostly the trochée mode. This mode is usually written or known by its triplet form, the 2:1 trochée, but it’s also quite widely known in both soft- and hard-swung versions, mostly with the quintuplet (\(3:2\)) and sixteenth note (\(3:1\)) feels respectively. However, these notations don’t fully reflect the variety and humanity of actual swing feels, which can hardly be represented or paralleled mechanically by simple ratios, and which can vary at any moment for any number of reasons.

Nevertheless, these simple notations are a useful tool to convey a musician’s or a composer’s intentions. So, they should not be overlooked, but rather serve as a basis around which to play more instinctively. Moreover, the extended vocabulary offered by the rhythmic modes, or metrical feet, could be learned and used in conjunction with adjectives or ratios to convey more accurately the feel you are looking for. In addition, this vocabulary offers an opportunity that has not been tapped into – at least not extensively – in recent music: complex swing, or swing feel with groups of more than two notes.

The use of DAWs with swing or feel sliders could make a nice parallel to the human world of jazz for electronic music. It’s been used already to create swung feels that don’t use a simple ratio but rather a percentage value, which is much less discrete and more continuous. However, it would be interesting to make that feel vary ever so slightly within a composition, just as human players do, perhaps using some form of envelope or automation in the DAW.

Complementary Material

Youtuber Samuel R. Howard recently made an in-depth video on uneven beats and notes with a clear focus on Balkan traditional music. Be sure to watch it by clicking this link to be redirected on Youtube.


Advanced Mathematics

101: Morse Code in Music
102: Microrhythms
103: Polyrhythm, Polymetre, Polytempo, and More
104: Uneven Notes, Swing, Rhythmic Modes, and More

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