I started studying music in college and remember struggling tremendously with rhythm. Repeating rhythmic pattern examples orally was fairly easy and I attribute this to many years of dance experience. However, sight-reading rhythm was a foreign concept, and brain and body coordination during these exercises proved challenging. With the help of books and thousands of hours of practice, overcoming sight-reading tasks became more straightforward and entertaining.
Definitions
Before diving into the topic of rhythm, it will be essential to define terms associated with this subject to provide clarity to the discussion.
Rhythm noun
The placement of sounds in time.
Time signature noun
A sign that indicates the meter of a composition. The sign contains two numbers aligned vertically. The top number indicates the number of beats per measure while the bottom number indicates the note value that receives one beat.
Meter noun
A rhythmic pattern constituted by the grouping of basic temporal units, called beats, into regular measures, or bars. In Western notation, each measure is set off from those adjoining it by bar lines.
Beat noun
The basic rhythmic unit of a measure. Accented and unaccented beats, their placement within the measure, and how often they occur form metric organization and operation. The strongest accent usually falls on the first beat after the bar line. The number of beats to the measure is indicated by the upper figure of the time signature.
Tempo noun
The pace of the fundamental beat. Tempo is Italian for “time.” The tempo of a piece of music is generally indicated by the composer but is never fixed. In performance, it is likely to vary according to the performer’s interpretative ideas or to such considerations as the size and reverberation of the hall, the size of the ensemble, and to the sonority of the instruments. A change within such limits does not affect the rhythmic structure of a work.
Sight-read verb
Read and perform music at sight, without preparation.
Note Duration noun
The length of time a note is played. This is determined by the type of note displayed.
Symbols
Each note duration has a corresponding symbol. The chart below shows note duration symbols in order from longest to shortest.
Whole note
Half note
Quarter note
Eighth note
Sixteenth note
Thirty-second note
Each note symbol has a corresponding rest symbol equal to the value of the note. The chart below shows the rest symbols in order from longest to shortest.
Whole note
Half note
Quarter note
Eighth note
Sixteenth note
Thirty-second note
Time Signature
Depending on the complexity of the composition and time signature, reading and understanding rhythm requires a certain level of mathematical skills. Some compositions illustrate extreme rhythmic notations accompanied by a deceptively recognizable common time signature. Others showcase uncommon time signatures resulting in difficult rhythmic reading while others are simple throughout. In any case, basic mathematical comprehension is a requirement to execute rhythm properly. This discussion will analyze two pieces. The first is Anne’s aria from Stravinsky’s The Rake’s Progress. The examples from this aria will include two different time signatures. The first meter is 6/8 and the second meter is 3/4. The second piece is Schubert’s German Lied titled Erlkönig. The piece is written in common time and emphasizes the triplet.
Rhythm is invariant in a piece and is influenced by the time signature. Though tempi can vary, rhythm remains constant. Cadenzas, rubato, and changes made by the performer are some exceptions. Time signatures have subcategories of simple, compound, and complex. The determining differentiation is dependent on the type of note that gets the beat and its relation to the number of beats in each measure. This discussion will focus on simple and compound meters.
In simple time signatures, each beat is divided into two equal parts. Therefore, every subdivision is divided evenly. There are many variable simple time signatures but the most common are duple (2/4), simple triple (3/4), and common time (4/4). For all of these time signatures, the quarter note gets the beat as indicated by the bottom number. The number of beats per measure is indicated by the top number. In duple time, there are two beats per measure, in simple triple there are three beats per measure, and in common time there are four beats per measure.
In compound time signatures, each beat is divided by three and equates to a variable dotted note. Popular compound time signatures include 3/8 , 6/8, 9/8,12/8, and 9/4. In 3/8, 6/8, 9/8, and 12/8, the eighth note receives the beat as indicated by the bottom number. The number of beats per measure is determined by the top number.
For 9/4, the quarter note receives the beat and there are 9 beats per measure. Therefore there are nine quarter notes. The nine quarter notes are grouped into three creating three sections per beat.
Analysis: The Rake's Progress in Simple Triple Meter
Anne’s aria No Word from Tom from Stravinky’s The Rake’s Progress, is sung in the first act. At this moment, Anne is mentally preparing herself for a dangerous journey to find her lover, Tom, who abandoned her to a life of gambling.
Simple Triple Meter
The time signature is simple triple (3/4). The quarter note receives each beat (bottom number) and there are three beats per measure (top number). Therefore three quarter-notes occupy one measure.
Dotted Half Note
A dotted half note (green box) takes up one entire measure. The dot requires the addition of half the note's value. In this case, half of a half note is a quarter note. One quarter-note represents the dot. Looking at the half note before the dot, one half-note is equal to two quarter-notes. Therefore, a dotted half note is equal to three quarter-notes which is one measure. This example shows the chord tied to the next chord in the following measure. Tied notes will be discussed later.
Eighth Note
Each beat can be subdivided into even parts. The quarter note can be halved to equal an eighth note. Therefore, each beat receives two eighth notes which equals one quarter-note.
Sixteenth Note
The eighth note can be halved into sixteenth notes. Therefore, each eighth note receives two sixteenth notes. Because two eighth-notes are equal to one quarter-note, four sixteenth-notes are equal to one quarter-note.
The typical way to count sixteenth notes is 1 e & a 2 e & a 3 e & a, etc.
Analysis: The Rake's Progress in Compound Meter
The time signature of this aria begins in 6/8 and shifts to several meters throughout the piece as demonstrated in the simple triple example above. In 6/8, the eighth note receives each beat (bottom number) and there are six beats per measure (top number).
Quarter Notes, Eighth Notes, and Tied Notes
Each beat can be subdivided in numerous ways. The vocal line is a beautiful demonstration of five different note values within two measures.
The phrase begins on a quarter note on the first syllable of the word “quietly.” The eighth note receives the beat and one quarter note is equal to two eighth notes. Therefore, this first syllable will take up two beats. However, this example shows a curved line connecting the first note to the second note. Since this line connects two identical notes, F#, this indicates the first note is tied to the second note. When a note is tied, it is not repeated but instead, held for the duration of the note it is tied to. In this case, a quarter note is tied to a sixteenth note and therefore will occupy the very first part of beat three.
It is important to understand the difference between a tied note and a legato symbol. A tied note will always have a curved symbol connecting two identical notes. When the notes are not identical, the curved symbol represents a legato (connected) line.
Sixteenth Notes & Thirty-Second Notes
Following the tied sixteenth note are two thirty-second notes. The notes are G# and A#. Thirty-second notes are a subdivision of sixteenth notes and are twice as fast as the sixteenth note.
How to Count Rhythm
The eighth note is counted as 1 2 3 4 5 6. Each beat gets one count.
The sixteenth note, the subdivision of the eighth note, is counted as 1 & 2 & 3 & 4 & 5 & 6 &. Each beat gets two counts.
The thirty-second note, the subdivision of the sixteenth note, is counted as
1 e & a 2 e & a 3 e & a 4 e & a 5 e & a 6 e & a. Each beat gets four counts.
The two thirty-second notes followed by the tied eighth note, are going to be counted in the 3 - e - & - a family. They are a part of beat three since the quarter note before takes beats one and two. The first part of beat three is the sixteenth note that is tied to the quarter note. One sixteenth-note is equivalent to two thirty-second notes. Therefore the rhythm tells the singer to hold the F# for "3 - e." The singer must then sing the two thirty-second notes of G# and A# on "& - a."
Subdivisions & Mathematical Formulas
When practicing, it may be easier to subdivide the entire measure into thirty-second notes to better understand their rhythm. To do this, each eighth note receives four thirty-second notes. Since there are 6 beats per measure and the eighth note receives the beat as indicated by the bottom number of the time signature, each measure receives 24 thirty-second notes. The mathematical formula to help count rhythm is:
The number of counts in the subdivision (4) X the number of beats per measure (6)
The last three notes following the two thirty-second notes are three eighth-notes. They occupy beats four, five, and six.
The next measure begins with an eighth note on the word “night.” The eighth note takes the first beat. It is followed by a quarter note that is tied to a dotted quarter note. We must add the total value of one quarter-note plus one dotted quarter note.
One quarter-note equals two eighth notes. The eighth note gets the beat. Therefore, the first quarter note takes beats two and three (highlighted in blue).
Dotted Notes & Mathematical Formula
The dot connected to the quarter note tells the singer to add half of the quarter note’s value.
The mathematical formula to find a note’s half value is
The original note ÷ 2
Quarter note ÷ 2 = eighth note
(¼) or 0.25 ÷ 2 = (⅛) or 0.125
The half value of a quarter note is an eighth note. Therefore, the dotted quarter note equals one quarter-note plus one eighth-note. Finally, the dotted quarter note is also equal to three eighth-notes because one quarter-note equals two eighth-notes and the dot is one eighth-note.
We must now add the tied notes in the last measure. Since the eighth note receives the beat, we will perform addition with eighth notes. The first note is a quarter note and is equal to two eighth notes. The tied note is a dotted quarter note and is equal to three eighth-notes. Therefore the singer must hold this note for a total of five eighth-notes. These tied notes occupy beats two through six.
Analysis: Erlkönig Simple Meter & Polyrhythm
Erlkönig is a work with music by Franz Schubert and a poem set by Johann Wolfgang von Goethe. The storyline draws from a Scandinavian folktale from the Romantic era. A boy is riding a horse with his father when he suddenly sees a supernatural being tormenting him. As the story progresses, the son continues to hear and see things that his father seemingly does not. The father comforts his son by assuring him that he only hears and sees nature and must remain calm until they arrive home. Before their final arrival, the Erl-King harms the boy and the father notices his dead son when he arrives home.
Erlkönig has an intermediate rhythmic analysis and an advanced level of playability. The music is in the key of G minor and has a common time signature (4/4). The common time signature means there are four beats per measure and the quarter note gets one beat. The tempo is set to presto agitato which translates to fast and agitated. Triplets are immediately introduced in the right hand in the first measure on the first beat and this pattern continues for the majority of the piece.
In this composition, one triplet is made up of three eighth-notes divided in one beat. Therefore, one measure of triplets for four beats would equal twelve eighth-notes. Incorporating triplets as the primary division for this piece in common time is difficult because you need to fit an odd number of notes (triplets) within an even number beat (quarter note). The even subdivisions of a quarter note include eighth notes, sixteenth notes, thirty-second notes, and so on. These subdivisions can be evenly divided within the beat to produce a stable sound. As a result, it takes practice feeling the triplet within a quarter note’s time.
Another challenging aspect of this piece lies within the vocal line when 2 against 3 is highlighted in the phrase “Mein Sohn, was birgst du so bang dein Gesicht?”
Polyrhythms
Though we are in common time, the triplet has dominated each beat establishing the presence of three notes per beat. Notice the dotted quarter note followed by an eighth note in the vocal lines in measures 38 and 39. The dot connected to the quarter note means to add half the value of the note it is connected to. In this case, half of a quarter note is an eighth note. Therefore, a dotted quarter note equals one quarter-note plus an eighth note. This in total takes up one and a half beats (three eighth-notes). The challenge lies in counting in even subdivisions for the vocal line (1 & 2 & 3 & 4 &) while hearing triplets (123-223-323-423) for each beat in the accompaniment.
The even subdivisions in the vocal line require the eighth note (followed by the dotted quarter note), to fall in between the three notes of the triplet. It does not sync with any of the triplet notes. It must fall between the second and third eighth note triplet.
When practicing, it is easy to fall victim and sing the eighth note with one of the eighth notes of the triplet. It takes dedication to go against the accompaniment and solidify even subdivisions.
Conclusion
Music cannot exist without rhythm. The arrangement of varying rhythmic symbols is an identifying characteristic of a piece and can provide symbolic meaning. The evolution of music notation shows an important shift from oral retention and repetition to written symbols and scripts. Organizing sounds into patterns through written notation provides accessibility and allows performances to flourish throughout time. Rhythm is a key factor in music’s operational success.
Learning music is like learning a new language. The different components that comprise music have their own functions but are interdependent. The beauty of music rests in the relationships between each part and how it is interpreted in performance. Musicians have the incredible responsibility of music execution and providing a line of communication with viewers. To be successful, it is recommended to study musicianship and understand its complexities. There are several books I recommend that I have read multiple times and attribute to my love for music. You can browse through my Amazon Storefront for guidance.
Sources
Britannica https://www.britannica.com/art/rhythm-music
Britannica https://www.britannica.com/art/metre-music
Britannica https://www.britannica.com/art/beat-music
Oxford Languages
Music Theory https://www.musictheory.net/lessons/11
Jade Bultitude https://jadebultitude.com/time-signature-9-4/
The Listener’s Club: The Rake’s Progress https://thelistenersclub.com/2022/03/18/stravinskys-the-rakes-progress-no-word-from-tom-dawn-upshaw/
Lumen Learning https://courses.lumenlearning.com/suny-musicapp-medieval-modern/chapter/der-erlkonig/
Duke Press Journal of Music Theory Style Sheet https://www.dukeupress.edu/Assets/Downloads/JMT_sg.pdf
Omgosh, there is so much great information in this blog post! As someone who is studying music, but to whom the maths of it does not come naturally (or easily, tbh), I can already foresee this post being a go-to resource when I'm trying to figure out rhythms and meters going forward. I like how you've laid everything out with examples in both written and audio form. That makes it super easy to just push play and listen along to what is happening in the music (since I find it much easier to Hear the beats than to read/count them on a page). And, that it's written out here means that I can go back and go over it again…