Global Music Theory

Author: Brandon Alsup

  • What is Cantus Firmus Counterpoint? – how it improves any creative musician

    Yes, learning counterpoint is an old way composers have learned the basics of composition but I believe it is a skill all composers, songwriters, and producers should have in their tool belt.

    Cantus firmus literally means “fixed melody” which just means the main melodic line that all other voices will be based on. Think of it as the first layer in a multilayer music composition.

    In this article I’ll explain what the cantus firmus is and how it works but more importantly how learning the principles of writing a cantus firmus, and counterpoint, will improve your creative skills.

    Quick overview of cantus firmus

    This practice comes from at least as early as the 13th century and was primarily used in sacred chant music. Over the centuries of it’s usage, the cantus firmus has moved from the top voice to the lowest voice to finally settle in a middle voice called the “tenor” (tenor means “to hold”, as in holding the music together).

    I’d like to pause and just note that the idea of the cantus firmus as a way to organize, build, shape, and teach music has so far lasted over 700 years! That fact alone should make anyone creating musician take notice (I also strongly believe that other disciplines might be able to use some of the principles found in the cantus firmus in their approach to their own fields, but that is a topic for another article).

    As a pedagogical method, counterpoint (the cantus firmus) can stand tall because pretty much every successful and famous classical composer learned through this method at one point or another. It’s incredible that counterpoint has been a through line of success across so many geographies, individuals, times, and musical tastes. All that’s to say, this is great stuff to learn if you are interested in creating music – whether that is writing pop songs or string quartets!

    Lastly, cantus firmus forms the basis for Western music’s polyphonic writing. Other cultures also have polyphonic music that is built in entirely different ways – yet again, another topic for another article.

    Some examples of cantus firmus

    Before explaining all the “rules” that go into a cantus firmus, I’d like to point out that these rules were created after the fact. Most music theory explains music that is already written – it doesn’t necessarily predict. Music theory typically takes a look at music and tries to explain why the music works. Same goes for the rules of counterpoint.

    The music is always first. Rules and theories come after.

    – Common Knowledge

    Also, keep in mind cantus firmus and counterpoint began as vocal music. With that out of the way, let’s listen to a few examples.

    The first example is from the most famous of counterpoint books Gradus ad Parnassum by Joseph Fux.

    Another example, this time composed by me.

    You may note many things about these short lines of music, but one thing should stand out – it only has one rhythmic value!

    You should be able to hear the following characteristics of a cantus firmus:

    • it’s a melody within a fairly narrow range
    • it has virtually no rhythmic variety

    Music of the time certainly had more than one rhythmic value but rhythmic notation in Europe took a while to catch up – remember music first!

    Here is a nice two-part example from Pérotin, a French composer from the 1200s. Among many other features to notice, you may notice there is variety of rhythm.

    Why learn to write a cantus firmus?

    I already enumerated the reasons to learn counterpoint but more specifically, the cantus firmus can help in melodic writing. Many of the rules that shape a cantus firmus are still found in contemporary melodic writing. Of course, there are plenty of exceptions, I’m not saying all melodies follow the cantus firmus approach, but there are certain principles that have managed to persist. Aesthetically, what Western music deems to be a “good” melody, has some parallels with cantus firmus. This will become clearer as I show you the way a cantus firmus is composed.

    How to write a cantus firmus in 5 steps

    To compose a cantus firmus, the only prerequisite you need is to be able to place notes on a staff, know their pitch, and be aware of a few intervals. Understanding intervals is a critical skill and I highly recommend reading my interval article to get that skill under your belt.

    Step 1 – just try!

    Just try writing something that mimics the examples above. This is just to get you comfortable setting up the 8 -14 measures and getting notes on a page. I highly encourage you to always play back your work, whether that is on an instrument or by singing.

    Step 2 – the beginning and the end

    Whatever note you started on is the note you must end on. So, it’s a bit like fill in the blank. Setup your measures, place your first note, and then place your final note.

    Step 1 – the beginning and the end of our example

    Step 3 – plan your climax

    Climax in this type of writing just means the highest note. There should be only one highest pitch. Past the half-way mark of your cantus firmus (if you have 10 measures than from the 6th measure onward) and before the 3rd to last measure, make that your highest note. This is now the climax of your cantus firmus.

    For example, if you have an eight-measure melody, you can place the climax in measure 5.

    Example two, if you have an eleven-measure melody, you can place the climax in measures 6, 7, or 8.

    If you like, you can take the total number of measures and multiply by 0.7 and that should tell you pretty well where you climax should be placed. This is just a guide not an absolute! There are always exceptions.

    Number of measuresX 0.7Climax
    85.65
    96.36
    1077
    117.77
    128.48
    139.19
    149.89/10

    The placement of the climax is one of those traits that has continued until this day. Many pieces, pop or otherwise, place their climax near the end of the song, much like the placement we find in cantus firmus.

    Step 3 – place the climax in our example

    Step 4 – steps and leaps (skips)

    There are two types of motion from one note to another – steps and leaps. A step means the note is either one line or space higher or one line or space lower. A leap is anything more than an interval of a 2nd. An example helps here:

    Steps and Leaps (skips)

    Leaps are treated differently than steps. In general, if your melody leaps up by a minor sixth (8 half steps) or octave it must fall back down by a step.

    If your melody leaps down by an octave, it must move back up by step. This is one of those melodic principles that has persisted until today. Many pop melodies follow the “rule” that after leaping up or down the melody needs to move in the opposite direction by step.

    Here is a cantus firmus from Joseph Fux that shows a couple of leaps in the same direction are sometimes resolved by stepping in the opposite direction. In general, you do not want your melody continuing up or down for too long – breaking up the direction of the melody after it has moved in the same direction for three-four notes is a good rule of thumb.

    Example Cantus Firmus from Joseph Fux

    There are some intervals you must not create when you leap. The following are “forbidden”:

    • tritones (6 half steps)
    • minor 7th (10 half steps)
    • major 7th (11 half steps)
    • any interval greater than an octave (12 half steps)
    Forbidden intervals

    You should already have your first, last, and climax measures filled in, now use steps and leaps to fill in the rest of the measures to connect the notes.

    Step 4 of our example

    Step 5 – don’t outline a tritone

    Be careful to make sure your melody does not outline a tritone (augmented 4th/diminished 5th).

    hidden tritone

    Completed cantus firmus

    You have now completed your first cantus firmus! Here are the steps right next to each other:

    Draw measures then place the beginning and ending notes.
    Place the highest note past the half-way point.
    Fill in the rest of the measures with steps and leaps watch out for forbidden intervals.

    Now you can play your cantus firmus!

    I’ve written a short book for composers that explores how music is organized and the roles it can play across the globe. The book is called Formative Forces in Sound. If you are interested, it’s available on Amazon here for $0.99 www.amazon.com/formativeforcesinsound

  • How to use a diminished chord

    How to use a diminished chord

    You know your major and minor chords and have heard of this diminished chord but are unsure how to use it in your music. I break down what a diminished triad is and provide a few examples on how it can function in music.

    Typically, a diminished chord is used to create tension and a sense of destabilization in the music. Diminished chords are dissonant and “want” to be resolved to a consonance. In a major key they are the seven chord (vii-dim) and in minor keys they are the two chord (ii-dim).

    Continue on to learn more about how this chord is built and how it functions in tonal music. In order for this article to not get too long, I’ll not be going over fully diminished, half diminished, or diminished intervals in this article.

    I’ve written a short book for composers that explores how music is organized and the roles it can play across the globe. The book is called Formative Forces in Sound. If you are interested, it’s available on Amazon here for $0.99 www.amazon.com/formativeforcesinsound

    What Is a Diminished Triad?

    A diminished triad is a three-note chord built from a root, a minor third, and a diminished fifth. Or put another way, a diminished triad is build by stacking two minor thirds on top of each other. Its formula is:

    1 – b3 – b5

    For example, a B diminished triad contains:

    B – D – F

    How a diminished triad is built

    The diminished triad is comprised of two minor thirds stack on top of each other (if you do not know what a minor third is, then check out my interval article) or a minor third and a diminished fifth. For example a B, D, F. It is called diminished because of that diminished fifth. In a major or minor chord the fifth would be a perfect fifth, but in the diminished chord it has been lowered by 1 half-step.

    Diminished triad notation

    There are a few different notations for diminished triads to be aware of.

    TypeNotationExample
    Roman Numeralsroman numeral followed by a superscript “o”. Any roman numeral followed by the superscript “o” is diminished not just seven or two chords.viio
    Chord SymbolThis notation is typically found in jazz or lead sheet style notation.VII dim
    Bdim
    IntegerThis notation is typically used in Set Theory. A system of looking at music as numbers rather than dots on a staff.036

    Function of diminished triads

    The vii diminished chord in a major key

    A vii-dim triad typically functions as a dominant. This means it is wanting to resolve to the tonic. Broadly speaking, tonal music has two poles pushing and pulling the music along: the tonic (the “I” chord of the home key) and the dominant (chords that provide tension and feel like they want to return to the stability of the tonic).

    The chart below shows whether the chords function as tonic, dominant, or predominant in a major key.

    ChordPredominant FunctionTonic FunctionDominant Function
    Tonic (I)X
    Super Tonic (ii)X
    Mediant (III)X
    Subdominant (IV)X
    Dominant (V)X
    Submediant (vi)X
    Leading Tone (viio)X

    What determines the function of chords

    Without getting to deep at this point, it is context and how the composer treats the chords that determines function. In general triads that contain the leading tone (the seventh note in the major scale) will typically function as a dominant (but not the iii chord as you see in the example below). This is because our ears are trained to hear that leading tone as “wanting” to resolve to the tonic note.

    The example below shows three progressions. The first two have a tonic-dominant-tonic relationship. The last one does not. The iii chord does not function as a dominant primarily because it shares two pitches with the tonic triad and does not create enough tension to have a dominant sound.

    Function of a diminished chord in a minor key

    In a minor key the seventh chord is can take a few different qualities because there are three types of minor scales that could be used.

    Minor Scale TypeQuality of Seventh Chord
    Natural MinorMajor
    Harmonic MinorDiminished
    Melodic MinorDiminished or Major

    If the seventh chord has the diminished quality it will function as a dominant. It functions as a dominant because it has the leading tone in the chord. If the seventh chord has a major quality, it will then function as a predominant. Or better yet, the VII chord will function as a V chord in the relative major to get some modulation going!

    Diminished chords and substitutions

    If you followed the above discussion, then it is simple enough to follow the logic of which chords can swap in and out for each other. In tonal music it’s all about function when it comes to chord substitutions. Understanding what each chord is bringing to the music and where each chord is “leading” is essential. I believe it’s essential because it’s one of the way in which a composer can control two important aspects of tonal music: 1) variation (in particular through manipulation of harmony) and 2) expectations.

    Dominant substitutions

    The diminished seven chord can easily stand in for the five chord. Using the viio chord adds even more dissonance for a couple of reasons: 1) the diminished chord itself is more dissonant because it contains a diminished 5th rather than a perfect 5th and 2) the vii-dim shares no tones in common with the tonic (the V chord shares 1 pitch in common with the tonic making the V-I progression smoother than the viio -I)

    Diminished chords as a path to other chords

    i-vii-i: Example from Bach Chorale

    In the example above, the diminished seven chord is clearly a passing chord but shows how the pitches of the vii-dim chord still fulfils the function of the dominant. In the example the chord preceding the red box is a tonic (in 1st inversion) that moves to the vii-dim and then the chord after the red box is another tonic (in root position). This creates the expected movement of tonic-dominant-tonic.

    This step-wise motion from the tonic to the vii to the tonic is one of the main uses of the diminished seven chord.

    The diminished chords play a critical role in moving from one chord to another. Many time diminished chords are employed so that voice leading can be smooth. Look at the example below and notice the step-wise motion and maintenance of a common tone when moving between these chords.

    Typical Diminished Chord Voicing and Inversions

    Voicing refers to how the notes of the chord are presented – where on the staff is each note. Below are some typical voicings.

    The next consideration is which pitch will be the lowest (in the bass)? Determining which order (inversion) to place the notes will give the chord a different sound – even though you are using the same notes! Here are the three possible permutations of the three pitches that make up the B diminished triad.

    When you combine the voicing with the inversion you get the following options for how to present this chord.

    Typically, diminished chords are in the 2nd inversion. The main reason for this is that it softens the dissonance. You can test this on your own by playing an a C and B (one half step apart) and listen for the dissonance. Then move the B an octave higher and play C and the B again and now listen for the dissonance. You should be able to hear that it has been considerably reduced just by spacing the notes apart. If you want very intense dissonance, then bring the notes closer together. If you want a softer dissonance, spread the pitches apart.

    The key to using any chord – diminished included

    A final piece of advice when getting into chords and their functions and substitutions and the like: it’s all about voice leading. These “rules” about chords and where the “want” to go or how they function all stems from voice leading. Voice leading is the linear progression of each melodic line. Voice leading “rules” came from counterpoint. If you understand voice leading then you will understand chord progressions. Voice leading is the horizontal understanding and harmony (chords) are the vertical. Both have their place and use and when used in conjunction, you can write some great music! I like to use chord progressions as sign posts and then use voice leading to connect the sign posts.