\section{The Rhythm Domain}

Music is a time-based art form. That is to say, we can only hear music (or any sound at all) if time is moving forward. Perhaps the most basic way to think about how music is organized, therefore, is to think about how it is organized in time, which falls under what we can broadly call the domain of \textbf{rhythm}\index{rhythm}. “Time” is a somewhat subjective and intangible thing, though. Time flies when you’re having fun, yet it seems to drag on forever when you’re bored! Because time can be so strongly influenced by perception, humans many centuries ago invented the concepts of seconds, minutes, hours, and so on to divide time into discrete and objective units. This is what I’ll refer to as \textit{real} time or \textit{absolute} time. As you may already know, all modern digital audio workstations (or “\textbf{DAW}”s)—like Garage Band, Logic Pro, Ableton, and Pro Tools—will allow you to set up a grid for a song based on real time (i.e., seconds, minutes, hours).

\subsection{The Beat}

In addition to real time, musicians also divide time into \textit{relative} units, which we might also call \textit{musical} time, based on the speed of the song. It’s kind of as if the song itself creates its own internal clock, and this song-based clock often ends up being more convenient when discussing the structure of a song. Relative (or musical) time uses different units than absolute (or real) time, though. Unlike absolute time, which is for the most part objective, relative time is strongly tied to perception. And when we are dealing with the perception of time, there are certain time spans that fall more within the middle of our perceptual window. For musical time, this central reference unit is called the \textbf{beat}\index{beat}.

Most people, even those without any musical training or experience, have a pretty good ability to find the beat of a song, so I hope this concept is fairly intuitive. But let’s tighten up our understanding a bit more formally. In music theory, the beat is a repeating and regular pattern of equally spaced pulses that divide up time, kind of like the second-hand of a clock. It is usually the rate at which a listener will tap their foot or bobs their head when listening to a song. To make this idea more concrete, listen to the opening of the song “Billie Jean” (Michael Jackson, 1982), and try to tap your foot (or bob your head) to the beat. Seriously, go and try it right now; I’ll wait.

If you are feeling the correct beat for this song, you should be tapping your foot (or bobbing your head) at a rate of about two beats per second. If it were exactly two beats per second, we could easily calculate the number of beats per minute. Let’s see—two beats per second multiplied by sixty seconds per minute equals 120 beats per minute, i.e.:

[latex]\frac{2\;beats}{second}\;\times\;\frac{60\;seconds}{minute}\;=\;\frac{120\;beats}{minute}[/latex]

The notion of “beats per minute” (which I will refer to hereafter simply as “\textbf{BPM}”) is the most common way to precisely describe the speed of a song, which is technically called the \textbf{tempo}\index{tempo}. Instead of saying, “This song is slow” or “That song is fast,” for example, we can simply say that tempo of the song is 120 BPM.

In truth, the tempo of “Billie Jean” is not exactly 120 BPM; it’s actually slightly slower. To find the exact value, it can be helpful to use some sort of tempo tapper software, such as the free online version available on my web site at:

\begin{center}http://www.midside.com/bpm.html\end{center}

Most DAWs (like Pro Tools or Logic) also include a tempo tapper function. Try now to determine the exact tempo of “Billie Jean” using either the link above or some other similar method. Really, go and try it now; again, I’ll wait!

When I tried finding the exact tempo for the song, I came up with a rate of around 117 BPM. Hopefully you got something fairly similar, within one or two BPM of my answer. If not, that’s OK!

\subsubsection*{Question: How do I know if I’ve found the right BPM for a song?}
\textbf{Answer:} Good question! It is actually somewhat difficult—even for researchers on music perception—to explain how humans find the beat of a song. A trained musician usually just \textit{feels} the beat, which is a skill developed after years of practice. Indeed, this is a skill we will work on throughout the book. For now, focus on the drum part. In “Billie Jean,” for example, the opening drum part should convey a strong sense of the song’s tempo. If we import the song into a DAW (like Pro Tools), it will look something like what is shown in Figure \ref{fig:kick_snare_waveform}. You hear the low \textit{whomp} of the kick drum first, then the \textit{thwack} of the snare drum, then the kick again, and so on and so on. You should be synchronizing with each of these kick and snare hits, which occur at the rate of about 117 BPM. In between each kick and snare hit, you may also hear the high-pitched \textit{tssst} of the hi-hat (HH), but we’ll ignore that for now. Just focus for the time being on the alternation of the kick and snare, which is often enough to convey a sense of the song’s beat. This won’t always be the case, as we’ll see in later chapters, but it is a good place to start thinking about the tempo of the song (assuming the song has a drum part!).

\begin{figure}[h!]\centering
\includegraphics[scale=0.15]{figures/ch01/kick_snare_waveform.pdf}
\caption{Waveform view for the opening drum part of “Billie Jean”}
\label{fig:kick_snare_waveform}
\end{figure}

\subsubsection*{Question: So is that why it’s called a drum \textit{beat}?}
\textbf{Answer:} Pretty much! Because the drum part is often such a strong and clear indicator of the song’s beat rate (i.e., the BPM of the song), people often use the term “beat” to refer to the drum part as a whole. It is actually rather typical to call the drum pattern the \textit{drum beat} or simply the \textit{beat} of the song. The term “beat” has a few other similar but slightly different meanings that you should also be aware of. For example, electronic musicians and hip-hop producers often talk about “making beats,” which means that they are combining a drum part or other percussive pattern with a bass line, a synth line, samples, or other instrumental parts into an entire musical texture. The term “’beat” is also sometimes used in a loose way to refer to music or parts of the music that generate a strong sense of pulse and body motion in the listener, i.e., the musical aspects that generate a good feel for the underlying \textit{beat} of the song. This is the meaning of the term when people say, “This song has a good beat!” All of these meanings are related, although I will generally use the term “beat” to mean the primary pulse layer of the music—the rate at which you would tap your foot or bob your head. Some people call this primary pulse layer the \textbf{tactus}\index{tactus}, which is a fancy Latin word for “beat.” That said, I will also refer to the drum part as the “drum beat,” since that is the way people usually talk about the drums. Just be clear that the drum part (or drum beat) influences where you feel the beat, but these are two separate concepts. More on that later.

\subsubsection*{Question: Any other tips for knowing I’ve found the right BPM for a song?}
\textbf{Answer:} For those who have worked in a DAW like GarageBand or Pro Tools, you know that when you open a new session, the tempo defaults to 120 BPM. (This is not true for every piece of audio software, but it is generally true.) It turns out that 120 BPM is roughly the average tempo for most popular music.\footnote{See de Clercq 2016 for an overview of empirical research on ideal tempos in popular music.} Of course, average tempo is just that—an average rate, with some songs slower and some songs faster. According to music cognition researchers, the range of viable beat rates that humans can perceive spans from about 30 BPM on the lower end to about 240 BPM on the upper end.\footnote{See Table 2.1 in London (2012, 28).} So if you find that the tempo of a song is, say, 15 BPM (too low!) or 350 BPM (too high!), you likely have the wrong number. But even though humans can theoretically perceive any tempo in the range of 30–240 BPM, remember that these are the extreme limits. For now, we will focus just on songs in the middle of this range, from 80–160 BPM (which is 120 BPM plus or minus 40 BPM). We’ll discuss tempos outside this smaller range later.

\subsection{The Bar}

You may have noticed that the kick and snare not only divide up music into regular intervals of time—that is, the beat; they also create a natural grouping of these beats into pairs: kick then snare, kick then snare. For whatever reason, we tend to perceive the kick as the beginning of this grouping, perhaps because the deep and low sound of the kick drum feels stable and grounded. In contrast, the snare drum usually feels like something that happens in between each kick. But does the snare happen \textit{after} the kick that comes before it, or does the snare come \textit{before} the kick that comes after it? I think both are actually true, although we traditionally think of the snare as coming \textit{after} the kick in the grouping structure, since the kick usually sounds like the start of something.

Grouping beats into larger units turns out to be very useful, in the same way that we group seconds into minutes and minutes into hours to talk about longer spans of time. But although we could use the kick-snare pair as this longer reference unit of musical time, musicians tend to use a more abstract concept called the \textbf{bar}\index{bar} to group musical time into larger and longer units. You may also hear the term \textbf{measure}\index{measure}, which means exactly the same thing as bar, but I will use the term bar exclusively since it is much more common among commercial musicians.

\subsubsection*{Question: So what exactly is a bar?}
\textbf{Answer:} Another good question! Like the term \textit{beat}, it is also difficult to give an exact definition of what a bar is or how we identify how much musical time constitutes a bar. And like the concept of a beat, the bar is something that musicians develop a feel for after years of musical training. That said, the vast majority of popular music has four beats in a bar. This means that, given a normal drum pattern in the range of 80–160 BPM, two sets of kick-snare pairs will form a bar of musical time. In many cases (but certainly not all), the amount of musical time that constitutes a bar is thus fairly straightforward. We will see many exceptions to this rudimentary rule-of-thumb in later chapters, but for now let’s assume that four beats equals one bar.

\begin{figure}[h!]\centering
\includegraphics[scale=0.15]{figures/ch01/kick_snare_bar.pdf}
\caption{Two bars of a normal kick (K) and snare (S) drum pattern, with four beats in each bar.}
\label{fig:kick_snare_bar}
\end{figure}

To make the idea of the bar more concrete, listen back to the beginning of “Billie Jean” and say “one” on the first kick drum of the song, “two” on the next beat (the first snare), then “three” on the next beat (the second kick), and then “four” on the next beat (the second snare). But instead of saying “five” on the beat that follows, restart your counting back at one again, as shown in Figure \ref{fig:kick_snare_bar}. So you will just be saying continuous loops of “one, two, three, four” every four beats. Each loop of four beats constitutes one bar of musical time, which are delineated in Figure \ref{fig:kick_snare_bar} by the vertical lines. These vertical lines are called \textbf{bar lines}\index{bar lines}, and they help visually separate the content in one bar from the next.

Notice also that there are thus two bars (eight beats) in “Billie Jean” of only drums before the shaker and bass line enter the mix. When the bass comes in, it plays in a one-bar loop. That is, the bass line repeats itself every four beats. Many musical elements in a song will repeat or change based on units of a bar, such as every half bar, every bar, every two bars, every four bars, and so on. For example, the bass line in “Billie Jean” lasts eight bars before the synth strings come in. How long is the synth string loop? Unlike the one-bar bass loop, the synth strings take two bars to repeat (a two-bar loop). This two-bar string loop happens twice (four bars total!) before Michael Jackson starts singing. Note that the grouping structure keeps increasing: kick and snare making a two-beat group, the bass making a four-beat (one-bar) group, and then the strings making a two-bar (eight-beat) loop. This lengthening buildup process in common for song introductions.\footnote{See Attas 2015 for more details on buildup processes commonly found in song introductions.}

So how many bars total from the beginning of “Billie Jean” until Michael Jackson starts singing? Fourteen bars total! Counting bars like this is an important skill for keeping track of where you are in the song overall. Counting bars (like anything in music or life) gets easier with practice and should eventually become second nature. A skilled musician can actually just \textit{feel} how many bars have passed, without necessarily counting every single beat and every single bar.

\subsubsection*{Question: So how can I get better at just \textit{feeling} a bar?}
\textbf{Answer:} Just like tapping your foot or bobbing your head is helpful to feel the beat, it can be helpful to physically perform the bar and its beat cycle as you listen to music. To do this, you can use \textbf{conducting}\index{conducting} to internalize the beat structure of a bar. The conducting pattern for a bar with four beats is shown in Figure \ref{fig:conducting_pattern_44}. If you are conducting with your right hand, Figure \ref{fig:conducting_pattern_44} shows the direction you would move your arm. (It would be the mirror image if you are using your left arm.) Either way, you bring your \textbf{Down} on beat 1, move your hand across your chest (\textbf{In}) on beat 2, move your hand away from your body (\textbf{Out}) on beat 3, and then move your hand \textbf{Up} on beat 4.

\begin{figure}[h!]\centering
\includegraphics[scale=0.15]{figures/ch01/conducting_pattern_44.pdf}
\caption{Conducting pattern for a bar with four beats.}
\label{fig:conducting_pattern_44}
\end{figure}

These hand/arm motions make the cycle of beats feel real, so much so that musicians often refer to these beat locations according to where they occur in this conducting cycle. The first bear of a bar, for example, is called the \textbf{downbeat}\index{downbeat}, and the last beat of the bar is called the \textbf{upbeat}\index{upbeat}, since that is the motion your hand is doing at those points in the cycle of beats.

\subsubsection*{Question: So then is beat 2 called the “inbeat” and beat 3 called the “outbeat”?}
\textbf{Answer:} That would certainly make sense! But no, musicians really only use nicknames for the first beat (“downbeat”) and last beat (“upbeat”) of the bar, perhaps because those beats are the most important beats in a bar. (Sorry beats 2 and 3; we still love you!) That all said, there is one other important nickname for beat locations that musicians commonly use, which is the \textbf{backbeat}\index{backbeat}. The backbeat refers to beats 2 and 4 (the even-numbered beats), but specifically when those beats are accented or emphasized in the musical texture. The snare drum in a normal four-beat drum pattern is the most common instrument that plays the backbeat, but other things—like finger snaps, a tambourine, or hand claps—can play the backbeat (or “on” the backbeats).

It might take some time to get used to conducting a pattern of four beats. There is a certain amount of coordination that you need to develop so that you can sing along with music and conduct at the same time, kind of like patting your head and rubbing your stomach. But it is really important that you become comfortable with conducting so that you don’t have to even think about it. Conducting should eventually become something like breathing. It’s happening in the background without you thinking about it; you can focus on it if you want, but it should become so natural and intuitive that you do it consistently while focusing on other things. And the only way for something to feel so natural is just to do it over and over again. When you are listening to music on headphones while walking down the street, practice conducting; when you are listening to a song on the radio while relaxing at home, practice conducting; maybe just don’t practice conducting while you are driving a car. If you can sing along with a song while conducting, then you’ve got it!

As we try to understand more and more complicated rhythm patterns, conducting will be a very helpful tool to help untangle the location of each note in time. In essence, conducting is the physical manifestation of the \textbf{grid}, which is used in a DAW like FL Studio or Logic to divide music into relative units of bars and beats. The technical name for the grid in music is the \textbf{meter}\index{meter}. Meter describes how many beats are in a bar, how those beats are organized, and how those beats are subdivided. Be aware that meter is different than rhythm. Meter is like the seconds, minutes, and hours, but instead of dividing time into absolute units, it uses relative units referenced to the beat. In contrast, rhythm is the actually pattern of musical events in time.

\begin{figure}[h!]\centering
\includegraphics[scale=0.15]{figures/ch01/please_please_me_melodic_rhythm.pdf}
\caption{Rhythm of opening melody for “Please Please Me” (The Beatles, 1963).}
\label{fig:please_please_me_melodic_rhythm}
\end{figure}

For example, sing the opening line of the 1963 song “Please Please Me” by the Beatles (“Last night I said these words to my girl”) while conducting (at a tempo of about 142 BPM). Notice that the first word (“Last”) does not occur until the second beat of the bar. After that, there is a string of words on every beat. But the last two words (“my girl”) are longer, such that the word “my” is on the downbeat (beat 1) while the word “girl” is on the third beat, as shown in Figure \ref{fig:please_please_me_melodic_rhythm}. (Remember, the vertical lines in Figure \ref{fig:please_please_me_melodic_rhythm} are the bar lines.) The word “my,” therefore, lasts two beats, unlike most other words in the melody. The important thing to realize here is that the rhythm of the vocal melody (shown in the top half of Figure \ref{fig:please_please_me_melodic_rhythm}) is different and distinct from the underlying meter (or grid) of the song (shown in the bottom half of Figure \ref{fig:please_please_me_melodic_rhythm}). As we think about rhythmic organization in songs, it will be very important to distinguish between the meter (or metric framework) and the rhythm of some musical layer (like the melody or chords or drum part). This is why learning to conduct and getting used to internalizing the metric framework is critically important.

\subsection{Rhythm Notation}

We could continue to notate rhythms as shown in Figure \ref{fig:please_please_me_melodic_rhythm}, indicating the beat locations for each musical event with respect to the underlying grid. But a long time ago (literally centuries!), musicians developed a shorthand for notating rhythms that can be more convenient and efficient (at least once you get used to it!). In this section, I will introduce some of the basic elements of this traditional rhythm notation, since this notation scheme is so widespread.

Although it may seem natural to talk about musical events in terms of where they occur in relative time (like on beat 3 of bar 2, or the downbeat of the last bar), traditional rhythm notation focuses on the \textit{duration} of each note. From this duration information, we can deduce the location information. For example, in music with four beats per bar, a note that lasts a beat (and only a beat) is usually called a \textbf{quarter note}\index{note!quarter}. It’s called a quarter note because, well, if there are four beats in a bar and a note lasts one beat, then that note lasts one fourth (i.e., a quarter) of the bar.

\subsubsection*{Question: So if a note lasts two beats, is it called a half note?}
\textbf{Answer:} You got it! If there are four beats in a bar and a note lasts two beats, then it lasts one half of the bar, and so we call it a \textbf{half note}\index{note!half}. Along the same lines, if a note lasts a full four beats, we call it a \textbf{whole note}\index{note!whole}, since it lasts the equivalent of a whole bar! Of course, this is all assuming we have four beats in a bar.

When there are four beats in a bar, musicians typically refer to the meter as \textbf{4/4}\index{meter!4/4} (pronounced “four four”). It is also called \textbf{common time}\index{meter!common time}, indicated with a big letter “C” since it is the most common meter in music. The 4/4 notation is called the \textbf{time signature}\index{meter!time signature}. As we will see later, the time signature is only one part of the overall metrical organization of a song, but it is a very important part nonetheless. For simple meters, the number before the slash indicates how many beats there are in a bar (four in this case), and the number after the slash indicates what note value gets that beat. So in 4/4, the quarter note (or 1/4 note) is the beat, and four of these quarter notes equal a bar of musical time. You will also see time signatures written with the first number on the top and the second number on the bottom with no slash in between, like this: $_4^4$.

Figure \ref{fig:notes_and_rests_in_44} shows the traditional notations for these note durations (assuming we are in 4/4). Each note type consists of a \textbf{note head}\index{note!head}, which is the filled-in circle or the open circle, as well as a \textbf{stem}\index{note!stem}, with the exception of the whole, which does not have a stem. The dot next to the note head on the dotted half note (which lasts three beats) is all the distinguishes it from the half note (which lasts only two beats), so be careful to check also whether a note is dotted or not, because it affects the duration. (We’ll talk about other dotted notes later.) Figure \ref{fig:notes_and_rests_in_44} also gives the notations for rests, which are periods of musical time during which no notes occur.

\begin{figure}[h!]\centering
\includegraphics[scale=0.15]{figures/ch01/notes_and_rests_in_44.pdf}
\caption{Rhythm notation for durations of simple notes and rests in 4/4 meter.}
\label{fig:notes_and_rests_in_44}
\end{figure}

To be clear, these traditional rhythmic notations indicate \textit{duration only}; you have to figure out on what beat the actual notes occur based on the context of the other notes shown. This may take some getting used to. Let’s see how rhythmic notation works with an example. Consider, for instance, the opening melody of “Please Please Me” (originally shown in Figure \ref{fig:please_please_me_melodic_rhythm}). Figure \ref{fig:please_please_me_melody_trad} shows a rhythmic \textbf{transcription}\index{transcription} of the vocal melody. The act of transcription is what musicians call turning the music into written notation. Notice that Figure \ref{fig:please_please_me_melody_trad} begins with a time signature of 4/4 (written as $_4^4$), indicating that there are four beats in each bar and that the quarter note gets the beat. Since the first word of the melody (“Last”) begins on the second beat, we need a quarter rest at the beginning of the bar to indicate that we shouldn’t do anything for a beat. (Take a break!) After this quarter rest, there are three quarter notes in this first bar, occurring on beats 2, 3, and 4 respectively. In the last bar, the word “my” lasts two beats, indicated by the half note, which makes the last word (“’girl”) occur on beat 3. The duration of the word “girl” is actually somewhat unclear in the original recording. I have shown it as lasting two beats (via a half note), although another person might think it lasts only a beat. If you thought it lasted only a beat, you would just write a quarter note for the word “girl” followed by a quarter rest to complete the bar. (It turns out to often be somewhat tricky to say exactly how long any particular note lasts in a song, as we will see.)

\begin{figure}[h!]\centering
\includegraphics[scale=0.15]{figures/ch01/please_please_me_melody_trad.pdf}
\caption{Rhythmic notation of opening melody for “Please Please Me” (The Beatles, 1963).}
\label{fig:please_please_me_melody_trad}
\end{figure}

Be aware that every bar in 4/4 \textit{must} contain four beats, even if there are not four beats worth of music that happen. You can’t, for example, just have two quarter notes in a bar of 4/4 and nothing else (unless it is a pickup bar, which we’ll talk about later). The bar represents span of musical time, and you have to indicate when things are and are not happening. That’s where the rests come in handy, because they show where music is \textit{not} happening, which is sometimes as important as where music \textit{is} happening.

\subsubsection*{Question: Can I see a more complicated example of rhythmic notation?}
\textbf{Answer:} Sure, although let’s not get too complicated for now! How about another Beatles song? Figure \ref{fig:michelle_melody} shows the rhythm of the opening melody for “Michelle” (The Beatles, 1965) in a few different versions. Try singing along with the song while conducting (at about a tempo of 117 BPM). As you sing, make sure the notes of the melody are lining up with how they are shown in Figure \ref{fig:michelle_melody}a. To help you do this, I have put a simple grid underneath the notation. When notes occur, I have written the beat of the note, also called the \textbf{rhythm syllable}\index{rhythm syllable}. The beats that don’t have a melodic note are shown in parentheses; those beats are there—you should still feel them—but nothing in the melody begins on those beats.

\begin{figure}[h!]\centering
\includegraphics[scale=0.15]{figures/ch01/michelle_melody.pdf}
\caption{Rhythmic notation of opening melody for “Michelle” (The Beatles, 1965).}
\label{fig:michelle_melody}
\end{figure}

As mentioned above, it is often difficult to tell how long notes last in a vocal melody, even if we are pretty sure where they begin. For example, you might hear the second syllable in the word “Michelle” as lasting three beats. After all, the next word (“ma”) does not occur until three beats later (on the second beat of the following bar). If you think the syllable “-chelle” lasts three beats, you might be tempted to notate the beginning of the melody as in Figure \ref{fig:michelle_melody}b, with a dotted half note indicating that the note should last three beats. (See Figure \ref{fig:notes_and_rests_in_44} for a review of rhythmic duration symbols.) Unfortunately, this causes the first bar to be \textit{overloaded}, in that it now looks like it contains five beats total (a half note plus a dotted half note). In addition, the second bar now looks \textit{underloaded}, in that appears to only contain three beats (a quarter note and a half note). But this is incorrect, since we know in 4/4 that every bar must contain four beats.

\subsubsection*{Question: OK, so how do we avoid overloaded or underloaded bars?}
\textbf{Answer:} The solution is to use a \textbf{tie}\index{tie}, as shown in Figure \ref{fig:michelle_melody}c. The tie is the curved line that connects the half note at the end of the first bar to the quarter note at the beginning of the second bar. The tie makes what looks like two separate notes into a single sounding musical event. It \textit{ties} the two notes together in time, literally! We will encounter ties often when thinking about the rhythms of melodies in popular music, so it’s important you understand this concept. The notation in Figures \ref{fig:michelle_melody}a and \ref{fig:michelle_melody}c are equivalent in terms of the number of notes in the first two bars and when these notes occur. The difference is only the duration of the second note, which lasts a little bit longer in Figure \ref{fig:michelle_melody}c. In summary, the note at the beginning of a tie is played or sung, but the note at the end of the tie only adds duration to the original note. The note at the end of a tie is not a new note!