Category: Music Theory De-Mystified

A holistic approach to music theory. We investigate the principles behind the theory

Perfect 5ths & Perfect 4ths: An Octave Of Two Halves

An octave is 12 semitones. However, half an octave is 7 semitones – and the other half is 5 semitones!

How is this so? Surely half of 12 equals 6?

Frequencies

Each note pitch produces a repeating sound wave. Lower notes produce longer waves which repeat more slowly, whereas higher notes produce shorter waves which repeat more quickly. The speed at which a sound wave repeats is it’s frequency, measured in Hz (Hertz). 1 Hz = 1 wave cycle per second.

Composite Wave

When two (or more) notes are played together, their sound waves combine to form a composite wave. This wave also has a frequency. Playing two notes produces three!!!

The frequency of this combination wave is also a note. For example, below is an interval of a major 3rd.

Consonant intervals (intervals that sound musical) have frequencies which are closely related. The composite wave’s frequency is an octave of one of the two notes that make up the interval.

Dissonant (musically unpleasant) intervals such as a semitone or a tritone (augmented 4th/diminished 5th) have frequencies that are not closely related. As a result it takes many cycles of each note before they meet up to produce one cycle of the composite. The composite wave has a low frequency unrelated to either note which, if below our ability to detect pressure waves as continuous sound, can be felt as a disturbance known as beats or beating. 

For more on beats see The Secret To Tuning: How To Tune An Instrument To A Reference Note.

Octaves

When two notes are an octave apart, their sounds match so well together that we think of them more as being in different registers rather than as completely different notes. Notes which are whole octaves apart are considered to be different versions of the same note, to the extent that they share the same name.

When two notes are an octave apart, the upper note is 2x the frequency of the lower note. For example, if A = 440 Hz then the next A an octave higher is 880 Hz.

The composite wave is 440 Hz, the same as the lower note. 

Half An Octave

Half an octave is half-way between the frequencies of the two notes. In the above example, half an octave is half-way between 440 Hz and 880 Hz, which is 660 Hz.

660 Hz is E, 7 semitones above A 440 Hz.

Two Halves

  • A to E, the lower half of the octave, is 7 semitones
  • E to A, the upper half of the octave, is 5 semitones
  • A to E, the lower half, is a perfect 5th
  • E to A, the upper half, is a perfect 4th

If you’re wondering why a 5th plus a 4th is an 8th, please visit B5. Inversions Of Intervals.

Let’s look at the composite wave’s frequency of each half.

The interval between A 440 Hz and E 660 Hz has a frequency ratio of 3:2. That is, it tales 3 cycles of E and 2 cycles of A to form the composite wave. The composite’s frequency is 220 Hz, the A an octave below the played note A 440. This reinforces the lower note of the interval, making it stronger.

The interval between E 660 Hz and A 880 Hz has a frequency ratio of 4:3. The composite’s frequency is also 220 Hz, which is 2 octaves below the played note A 880. This reinforces the upper note of the interval, making it stronger.

In other words, the upper half of an octave, a perfect 4th, behaves upside down compared to the lower half, a perfect 5th.

  • In a perfect 5th, the lower note is stronger
  • In a perfect 4th, the upper note is stronger

Perfect 5ths and perfect 4ths are literally inversions of each other!

Half An Octave In Scales And Melodies

In a scale, the 5th note, the note half an octave above the root note, is called the dominant. The dominant has a double function:

  • The half-octave point is as far away from the root note as you can get
  • It is also a strong supporter of the root note, as seen by the composite wave examples

The dominant provides a polar opposite point allowing melodies to venture away from the root note and to return from.

This is easily demonstrated in the most simple melody of all, the scale. By splitting it in two, we can see that the first half of the scale leads away from the root note and towards the dominant and the second half of the scale leads from the dominant up to (the octave of) the root note.

In the example below I’ll use the major scale but it works equally well for the melodic minor.

Perfect 5ths And Perfect 4ths In Chords

The presence of a a perfect 5th or perfect 4th in a chord helps us to identify the root note. The root note will be the lower note of a perfect 5th/the upper note of a perfect 4th.

If a chord contains more than one perfect 5th (or perfect 4th), the chord has more than one possible root note and its interpretation is determined by the musical context.

For example, the notes A C E G could be seen as either

  • Am7
    an A minor chord; A C E, plus a minor 7th; G, or
  • C6
    a C major chord; C E G, plus a major 6th; A

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Intonation: Two Sides To Playing In Tune

Intonation is the ability to play in tune. When we compare one note to another, how do we decide whether they sound in tune?

If the two notes are in unison and played together, the answer is easy. They’re in tune when they match. When two unison notes played together are in tune, their frequencies match and they sound like one louder note.

When they’re slightly different, two notes produce a difference frequency which we can hear as anything from a slow, swishing pulse to a more rapid, tremolo-like effect known as beating or beats. For more, including an audio example, please visit The Secret To Tuning: How To Tune An Instrument To A Reference Note.

If the notes are not in unison, how you play in tune depends on the context.

Playing In Tune

First, let me state the obvious. Fixed pitch instruments such as keyboards are unable to adjust their pitch on the fly. Playing in tune is entirely dependent on how well and how recently the instrument has been tuned. We’ll look at how they are tuned later in this post.

Two Types Of Intervals

An interval is the pitch difference between two notes. 

  • When two notes are played one after the other, as in a melody, it is called a melodic (horizontal) interval.
  • When two notes are played together, either on a polyphonic instrument or on two instruments, it is called a harmonic (vertical) interval.

Other than solo melodies, music consists of both melodic and harmonic intervals. Melodic intervals make up the flow of a part as you move through time and harmonic intervals are the parts interacting at any given moment, such as a melody note with a bass note or the notes of a chord with each other.

Generally, it’s easier to tune to a harmonic interval because, to varying degrees, we can hear the sound waves interact, much like when tuning to unison. 

Melodic intervals are easier to tune to if we can clearly recognise the key. One way to do this is to warm up on a scale in the same key as the passage you’re playing.

Two Types Of Intonation

Just Intonation

The earliest forms of music revolved around a single note; the root note or tonic. Other notes flowed towards or away from the tonic so the focus in tuning was on how well the other notes fit the root note.

Any harmonic interval has a “best fit”. Most intervals can be tuned to simple frequency ratios by listening to the difference frequency and adjusting the note so there is no beating effect. The stable sound makes chords extra strong and rich.

So why don’t we use just intonation all the time?

Just intonation is great while tuning to one root note, but over time Western music developed into a more outgoing direction. Rather than stay in one key all the time, melodies ventured from one key to another, exploring the relationships between different keys and chords.

The Problem With Just Intonation

With just intonation, the farther you move away from the original key, the more out of tune the notes sound. What’s more, the semitones aren’t even all the same size! 

The simple ratios of just intonation only work while the reference note is that of the key you’re in. To change keys, you need to change the tuning of the notes on the fly…

If you work out the frequency of every note using just intonation, by the time you get back to the starting note the frequency is significantly different!

Equal Temperament

The solution to playing in multiple keys is equal temperament. In equal temperament, the cumulative error over all keys is averaged out. All semitones are musically the same size. In equal temperament music can be played in any key and pieces can travel from key to key with every note sounding equally in tune.

No single interval is perfectly in tune as far as difference frequencies go but the amount is so small that we don’t hear it as beating. The only thing we might notice is a slight softening of the clearest intervals such as perfect 5th which, if anything, adds a sense of warmth.

Equal temperament is perfect for keyboards and other fixed pitch instruments and is widely accepted as the predominant tuning system in Western music.

Tuning On The Fly

Small consorts, in particular string quartets and small vocal ensembles, have a reputation for being able to play perfectly in tune.

This is achieved by combining the best of both worlds. Equal temperament is great for melodic intervals as the music progresses through time, while just intonation can be used to fine tune harmonic intervals to suit a chord at a given moment.

The ultimate guide is your ear. In any key, any note has a best fit. Be responsive to what you hear and you’ll find that best fit.

Feeling a bit nerdy? If you’d like to know a bit more about just intonation and equal temperament, please visit the Shop page for a FREE DOWNLOAD of my e-book, Intonation: From Pythagoras To Equal Temperament.

If you found this post helpful, please feel welcome to like, share or leave a comment. If you have any questions, leave them as a comment and I’ll respond as soon as I can. To stay up to date with new posts, please subscribe.

The Secret To Tuning: How To Tune An Instrument To A Reference Note

Have you ever had trouble tuning to another instrument? If so, read on…

In any ensemble, big or small, all the instruments need to be in tune with each other. If not, no matter how skilled the players, the music will not sound musical.

Who Tunes To What?

All the players need to agree on a pitch reference that everyone can tune to. This can be an external source, such as a tuner, or one of the instruments.

If your instrument is tuneable, you can tune a note on your instrument by matching it to the same note on a tuner or other instrument. We do this by listening for a disturbance called beating.

If you have a fixed pitch instrument such as a piano, your instrument becomes the reference and others tune to you.

Note: On fretted string instruments such as guitar or bass, once you’ve tuned one string to an external reference you can play a fretted note on that string as the reference for the next higher string and progressively tune the other strings.

How to Tune To A Reference Note: The Difference Frequency

When two (or more) notes are played together their sound waves combine. This combination wave has a frequency which is the difference between the frequencies of the two notes.

If the two notes are identical, there is no difference frequency. The notes lock together and sound like one louder note.

If your note is almost but not quite the same pitch as the reference note, the individual frequencies take many cycles before they match. This results in a very low (slow) difference frequency. The closer to unison, the lower the difference frequency.

Here is a diagram of 2 notes of nearly the same pitch (blue and green). Up/down represents amplitude. We hear amplitude as volume.

The difference frequency is produced by the overall shape of the composite waveform, shown here in yellow.

Look at the composite (yellow) wave: 

  • where both notes go up or down together, the composite wave is taller/louder
  • Where they oppose each other, the composite wave is shorter/quieter
  • If they oppose each other by the same amount, there is silence

Over one cycle, the composite wave gradually shifts from loud to silent (or near-silent) and back, creating a pulse.

Beating

We can hear the difference frequency as a pulse known as beating or beats. When both notes match, there is no pulse. When they’re almost in tune, there’s a slow, gentle pulse, maybe only 1 or 2 Hz (times per second).

As you tune your note away from the reference, the pulse becomes faster and the beating effect becomes more noticeable. This happens quite quickly, so make gradual adjustments!

Note: once the pulse is faster than 20 Hz we can’t hear the individual beats anymore. Instead, we start to hear two distinctly different notes. By now we’ve gone way too far…

Tuning the other way, as you get closer to matching the reference, the pulse slows down, then disappears when you’re in tune.

Example

Here is an example using 2 synthesiser notes, starting off in unison. After a couple of seconds, one note drifts flat, then back to unison, then sharp, before returning back to unison. I have chosen this sound because, like an organ, the notes don’t decay.

In the above example, the note is only just over a quarter of a semitone out at the furthest points. By then we can hear that the note sounds distinctly out of tune.

Sharp Or Flat? Listen For The Beats

Note that the beating effect is the same whether your note is slightly sharp or slightly flat. The beating only indicates how far you’re out: faster means further out, slower means closer.

Method

As you adjust your note, listen to the beating. Is it getting faster or slower?

  • If the beating becomes faster, you’re getting farther out of tune. Change the direction of your adjustment; tune the other way.
  • If the beating becomes slower, you’re getting closer to being in tune. Keep going. When the beating is slow, adjust more carefully.
  • When the beating is gone, the two notes will sound as one. Even with an electronic tuner, where the timbre of the reference is nothing like your instrument, the notes will feel like they’ve locked together.

If you’re not sure whether you’re sharp or flat, keep going in the same direction! Sooner or later it will either lock in because it’s in tune or the beating will have become fast and obvious enough to tell you that you’ve gone too far and you need to turn around.

Make slow, gradual adjustments as you listen.

Above all, don’t panic! If you make random adjustments you may end up tuning to a different note altogether, or you may end up going back and forth without ever reaching the note. 

Stay calm, listen for the beating and stick to the method outlined above.

Plucked Strings Take Note

Beating is more noticeable when the notes sustain well. On many acoustic plucked string instruments, notes don’t sustain for long before they fade away. 

One solution to this is to play repeated notes; each note long enough to hear the beating but not so long that they die down. For an acoustic guitar, once every 3 or 4 seconds should work. Instruments with a small body and nylon strings, such as the ukulele, have less sustain and you will need to play more frequent, louder notes in order to hear the beating clearly.

What Note Do We Tune To?

It doesn’t really matter what the note is, as long as everyone agrees. Different instruments find certain notes easier to play than others. Ideally, the note used for tuning is an easy note for all involved. 

A 440 – Concert Pitch

It’s possible for a keyboard or tuner to play different notes for different instruments to tune to. However, the tuning process is much quicker and more straightforward if everyone tunes to the same note. This is particularly true for large ensembles such as orchestras.

The closest to a universal tuning standard is A 440 Hz, or just A 440, also known as concert pitch.

Lower instruments can match A in a lower octave. Beating works when tuning notes at different octaves just as it does in unison.

Are Instruments Always Tuned To Concert Pitch?

The short answer is no.

Over the centuries the tuning reference has varied considerably, both above and below the current 440Hz. Some ensembles specialising in period music will use the appropriate pitch reference for the era.

More generally, apart from digital instruments, fixed pitch instruments don’t stay perfectly in tune forever and may end up sharp or flat overall. All other players will need to tune by ear to make sure they are in tune with that instrument.

As long as everyone tunes to a common reference, it doesn’t matter whether it’s in concert pitch or not.

Note: Beating can also be heard when other consonant intervals like a perfect 5th or perfect 4th are out of tune. You can tune to these intervals in the same way as tuning to unison.

If you found this post helpful, please feel welcome to like, share or leave a comment. If you have any questions, leave them as a comment and I’ll respond as soon as I can. To stay up to date with new posts, please subscribe.

(Guitar) String Theory 2: Why Do Frets Get Closer Together? 

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

All comments are welcome. If you enjoy my post, please give it a like and share it or subscribe to my blog.

Frets on a guitar are placed 1 semitone apart. The 12th fret produces a note one octave above the open (full-length) string.

The Relationship Between Pitch And Frequency

The frequency of a note is the speed at which a sound wave vibrates in order to produce a given pitch. The lower the frequency, the lower the pitch.

The common factor between the pitch of a note and its frequency is the octave. One octave equals 12 semitones, where each semitone sounds the same distance apart as the next, like centimetre or inch markings on a ruler. 

An octave is also the frequency ratio of 2:1. Every 12 semitones higher, the frequency doubles. We can look at the relationship between sound waves and what we hear by creating a graph with pitch on one axis and frequency on the other. It would look something like this:

The above frequencies are based on a guitar A string, A = 110Hz.

  • One octave higher = double the frequency.
  • Double the frequency = half the wavelength and thus half the string length.
  • One octave higher than the open (full-length) string is half the string length, half-way from the nut to the saddle.
  • The next octave higher is half of the remaining string length = 3/4 of the string away from the nut.

In other words, the first half of the string has 12 frets and the next quarter of the string also has 12 frets.

The effect of this relationship is that for every semitone higher in pitch, the frequency increases by a little bit more than the last semitone.

The Relationship Between Frequency And String Length

Frequency and wavelength are inversely related: as one goes up, the other goes down. As the frequency increases, the wavelength, and thus the string length, becomes smaller, a little less so for each semitone. 

Strings are effectively half a wave. Higher notes are produced by making the playing part of the string, and thus the wave length, shorter. For each semitone higher, the adjustment is a little less than the previous semitone. The frets mark these positions.

Why do we care? Maybe we don’t need to, but isn’t it nice to know why frets are laid out differently from piano keys?

(Guitar) String Theory 1: Strings and Octaves

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

All comments are welcome. If you enjoy my post, please give it a like and share it or subscribe to my blog.

A plucked guitar string is a good physical representation of half a sound wave. 

Sound waves, like ripples in a pond, are wave shaped pulses that travel and spread away from the source. Single frequencies have an evenly-curved shape called a sine wave. A complete wave, from the start to where it begins to repeat, is called a cycle.

One Wave Cycle

Unlike ripples in a pond, a string on a guitar (or any string instrument) is fixed and doesn’t travel. A vibrating string produces half a sine wave at a time, moving gradually upward then downward for each wave cycle. (The full sine wave is twice the length of the string.)

A Guitar (or other stringed instrument) String Is Half A Sine Wave

When you lightly touch the string above the 12th fret (half-way along its length) and pluck the string, we hear a pure sound called a harmonic. By not pressing all the way down, both halves of the string are free to vibrate: only the middle is blocked, allowing a complete sine wave of half the string length.

Guitar String With Octave Harmonic

The sound we hear is exactly one octave above the sound of the open (whole) string.

  • One octave higher = half the string length.
  • In other words, one octave higher = half the wavelength.

By the way, you can check the accuracy of a guitar’s intonation by comparing just touching the string at the 12th fret to pressing all the way down at (behind) the 12th fret. The pitch should sound the same.

In Why Are Octaves Special? we saw that one octave higher = double the frequency, so:

  • double the frequency = half the wavelength. As the frequency goes higher, the sound wave becomes shorter.

You can also place a finger lightly over the 5th fret, 1/4 of the string length, and hear a note 2 octaves above the open string, at 4x the frequency.

This is just another way of demonstrating the close relationship that exists between notes one or more octaves apart. The octave is fundamental to how music behaves. It is a universal musical phenomenon, independent of genre or culture.

Even though we don’t think of sound waves when playing or listening, I suspect that we are innately aware of them. We tend to think of bass notes as big and piccolo or tin whistle notes as little…

Bear with me- there’s a little more in the next post, (Guitar) String Theory 2: Why Do Frets Get Closer Together? 

How Can We Tell Which Key We’re In?

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

All comments are welcome. If you enjoy my post, please give it a like and share it or subscribe to my blog.

First I should say that there are two separate subjects here: the overall key and the key at a given point in the piece such as a visiting key. For the purpose of this post I’ll focus on overall key and assume a fairly simple melody but in principle, the same applies to sections and even individual phrases within the piece.

If you’re looking for a quick answer, scroll down to the summary.

When we play a scale, it’s easy to tell what the key is: it’s a combination of the root note and the mode. If we start and finish on C and play the major scale pattern of .2.2.1.2.2.2.1. (semitones), we’re in C major: C major is the major mode built on C.

But how can we hear/feel what key we’re in when the order of the notes varies, as in a melody?

If you’re reading notation, you could say, “look at the key signature”. This is true, but it’s only part of the answer. Within a key signature there are many possible tonalities. Even considering only the major and minor modes, you still need to find the right choice, to help interpret the music correctly. And if you’re listening or playing by ear, you need to be able to “feel” the key.

Look for the root note

To do this, we need to know what the root note is. For a given key signature (set of notes that make up a scale), the mode is determined by where we start the pattern i.e. the root note.

“The Spokes Of A Scale”

The best way to think of a scale is not as a strip of notes lying next to each other but as a series of spokes with the root note at the centre and the other notes around it. The double lines in the following diagram indicate the special bond between the root note and its octave (where the pattern repeats) and between the root note and the perfect 5th (more on that later in this post).

“The spokes of a scale”, diagram of the connections between the notes of a scale
“The Spokes of a Scale” over 3 octaves

(…I see it as a kind of spiral staircase extending up and down the octaves like storeys of an apartment block, where notes on the central column are octaves of the root note…)

Root Note Power

In a piece of music, it’s as much about the relationship of each note to the root note as it is from each note to the next. To be able to feel the key we need to be able to feel the root note.

There’s a good chance that the root note is first, or at least among the first few notes, and also at or near the end, but it’s not always the case. Fortunately there are many other ways it can be pointed out in a melody.

Longest, Strongest and Most 

In a melody, other than first and last, these are the three main ways we can highlight the root note.

Longest

Duration is power. Out of a series of different length notes, the longer notes are more prominent. If the root note is a long note it will stand out in the crowd.

Strongest

One way to emphasise the root note is to give it strength. There are two ways to do this:

  1. by playing that note louder than the others or giving the note an accent (a strong attack).
  2. by making the root note appear on the strongest parts of the bar. Time signatures have an implied hierarchy of strong and weak points- a default rhythm, if you like. Placing the root note on beat 1 gives it the most strength. In 3/4 and 4/4, beat 3 is also naturally strong. Similarly, on-beat quavers are naturally stronger than off-beat quavers.

Most 

Another way to reinforce the root note is to keep coming back to it. The more often we hear it compared to surrounding notes, the more we believe it.

Examples

Here are a few single-phrase “melodies” using just a single technique to highlight the root note. For the following examples, as you listen, try to hum the note that feels like the root note. Bear in mind that no single note will fit all the time. What we’re looking for is the note that fits most of the time. When you stop listening, which single note would you remember? Try a few if you’re not sure…

The examples are all in C major, so if the techniques I have described are effective, C should feel like the root note.

To make it a fair test, I have tried to make the (mini) melodies fairly random apart from the parameter we’re testing, so they’re not great. Real composers use a combination of these techniques when creating a melody.

Longest
Strongest
Most

Lowest & Highest, Direction 

Although not as significant overall, the lowest and highest notes of a passage within the melody will be naturally emphasised. I see this more as a sense of direction. When listening, we follow the direction of a scale-like series of notes, upward or downward, to its destination. The series directs us to the destination, giving that last note emphasis, before changing direction. 

A scale played ascending then descending is as good an example of this as any. Scales are essentially very simple melodies with no detours.

Lowest, highest, direction

Harmonic Reinforcement 

The perfect 5th, a great support act

The 5th note of a scale is almost as special as the root note itself and warrants a post of its own. I will say that it has both the capacity to blend well with the root note to support it (even if the notes are one after the other rather than played together) and to be a convenient destination for the melody to visit, a temporary root note of its own.

As a supporting note it is second to none. A 5th nearby will reinforce the presence of the root note by “pointing to it”. The 3rd note, especially the major 3rd, can also help in this way.

Some melodies place all 3 notes of the home key’s triad (chord) near each other, virtually acting as an arpeggiated chord, providing an even stronger emphasis.

Harmonic reinforcement from the perfect 5th

Accompaniment 

We haven’t talked about chords yet. Chords have an enormous say in what feels like home. Chords (at least the basic types) feature the three most important notes of a key, the 1st (the root note), the 3rd and the 5th. These notes blend so well together that they reinforce the chord’s root note. 

While the chord sounds (if it sounds for long enough), it’s hard not to feel that its root note is, temporarily, the root note of the piece.

The most prominent chord overall, especially towards the beginning and end of the piece, is generally that of the home key.

Other chords can also feel like home for a while if they sound for long enough, providing visiting points in the melody. This is one of the tools a composer can use to create music that has a journey, a sense of going places, rather than being stuck at home the whole time like a COVID lockdown.

The techniques described in this section are also used to establish the new key after a modulation (key change).

Even when no chords are played, the melody’s sense of direction and use of the above techniques can suggest some of these temporary keys. Chords can also be played melodically, as a series of notes called a triad. Many melodies are largely made up of scale-like passages and triads.

In A Nutshell

To answer the title question: look for the root note. The music makes sense when you can hear/feel how the other notes relate to the root note.

Look for a strong note or a strongly supported note early in the piece and towards the end. Feel the flow of the melody’s phrases- where the phrases start and end, which notes are emphasised. When a phrase is arriving home, you will hear it.

Once you know the root note, the mode will become evident, because you’ll interpret the other notes from the root note’s perspective. You can confirm this by playing the notes of the melody as a scale starting on the root note. If you have chosen the root note correctly, the tonality of the scale will match the overall tonality of the piece.

Much of the content is based on my upcoming music theory reference, Music Theory De-mystified, which is currently planned for release as an e-book by the end of 2022.

Please feel free to comment. I have a slightly unorthodox way of presenting music theory concepts but the concepts themselves are well established. If you like my posts, please subscribe so you can be informed of new posts.

A Melody Is A Journey

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

All comments are welcome. If you enjoy my post, please give it a like and share it or subscribe to my blog.

This is a key tenet in my approach to music theory. However, if you disagree, feel free to comment.

Most music has both a melody and chords. Even a melody by itself is usually built on chords, it’s just that we don’t hear them. We call this an implied chord progression (when no chords are written, we can deduce the likelihood of potential chords by the evidence provided in the melody such as phrase structure, the actual notes used and the use of accidentals).

A chord represents a key- at least basic major and minor chords and their common variants do. How effective they are at establishing their key depends on low long they’ve got. Yes, time. The longer the time spent on a chord, the more it feels like THE key. 

A melody is a journey. Typically it starts at home (in the home key) then travels to one or more visiting keys, represented by the main chords along the way, eventually arriving home again.

Just like a physical journey, the trip can be long or short, fast or slow, bumpy or smooth, visiting nearby or exotic places on the way. Everything that applies to a physical journey has its parallel in a musical journey.

There are three parts to any trip- the departure (including any prep such as packing), the travel and the arrival. Similarly, pieces (and the phrases within them) have a start, a middle and an end.

Time, Space and Culture Shock

A journey can be brief or extensive, or anything in between. The places you see can be familiar or exotic, near or far.

  • A trip to the local shop to get staples might be a 5-minute walk around the corner or up the street. You spend just long enough to do a common task in familiar surroundings and head back. This is the most basic journey; familiar and short.
  • You might drive across town to visit a close friend or family. Again, you are in familiar surroundings, yet you travelled beyond your immediate neighbourhood. This still feels like a small and safe journey yet you may spend hours on your visit and be surprised by the changed traffic or weather conditions coming home. A tiny bit more complex journey than the first example.
  • Maybe you’ve chosen to visit someone out of town or in another state. You might be invited to stay a few days. Longer distance means a little less familiarity: you don’t know the roads so well, where the post office is, the bed feels a bit different, etc. After a day or two, you start to get used to this. The longer you’re there, the more it feels like home.
  • While you’re there, you might take mini trips within the journey- go to the shop, the beach, maybe even camping out.
  • If you stay away long enough, when you come home it feels a bit strange at first. You almost turn the door key the wrong way, the colour of the wallpaper isn’t quite as you remembered it, you didn’t realise you were low on a few staples.
  • What if you set out on a grand adventure to visit strange and distant cultures? The journey is either massive, with strange and mysterious stops on the way, like an ocean journey, or super fast, almost like a blur, as in a flight. When you arrive, it’s almost alien. Everything’s different: the living conditions, the language, the food… Stay there for a while, however, and you gradually pick up a few basic words, learn a bit about the local neighbourhood and start to feel more settled.
  • Were you to stay in an exotic culture for long enough, it would start to feel like home, and your memories of your real home become less and less clear. Coming home after living there for years, home itself would feel like a very strange place at first. Stay somewhere long enough and you might even come home with a foreign accent!

All this can be mirrored in the way a piece of music progresses. The melody is the traveller, the main chords are the visiting points. Time is time.

The relationship between each chord and the home key (as well as between one chord and the next) is the relationship between home and the various places visited on our travels. As a (basic) chord represents a key, the main chords mark out the visiting keys in the journey.

Chord relationships are key relationships. A topic in itself, this is worthy of revisiting in at least one separate post. However, in general, keys (and chords) are related by how many notes they have in common. There are basically three types of key relationships:

The Cycle (or Circle) of 5ths

The cycle of 5ths is a sequence of all major and minor keys in increasing and decreasing key signature order, usually represented as a circle. Octaves are unspecified, as it’s just a list of keys. Adjacent keys in the cycle of 5ths have only one note different in their scales and both chords are made up of notes in the home key.

See my Beginner’s Tip for a graphic of the cycle of fifths, including relative majors/minors.

Adjacent keys in the cycle of 5ths are the closest companions. Many pieces only use 3 chords: that of the home key, previous key and the next key in the cycle, otherwise known as the Tonic, Subdominant and Dominant or I, IV and V. As we progress away from our neighbours, the keys sound less closely related and the chords a little more independent. Distant key relationships produce a startling or disorienting sensation in the listener.

Relative major and minor

All the common tonalities used in Western music have either a major 3rd or a minor 3rd from the root note. In this way, modes can be categorised as “like major” or “like minor” and be represented by a major or minor chord accordingly. It’s reasonable to talk in terms of major and minor chords, even if the piece is in another mode.

For every major scale, there is a minor scale with the same key signature (and vice versa). When the music changes between relative major and minor, the root note and tonality change but the notes all belong to the home key. As a chord progression, going from relative major to relative minor (and vice versa) feels more like taking a small step back rather than a significant change in key. Relative major/minor chords are often interchangeable in an accompaniment, depending on whether a more direct or a slightly indirect and more sophisticated effect is desired.

Major and minor on the same root note (parallel major and minor)

A major and a minor scale on the same root note have 3 notes that differ between them, so they only have 4 notes in common. In the cycle of 5ths that amounts to keys which are 3 steps apart, a relatively indirect relationship, yet they sound like they’re much more closely related. As it happens, only one of the three chord notes is different-the 3rd. The root note and 5th are both the same. The only thing that seems to change is the mood, the tonality.

Back to the present…

In short, closely related chords feel comfortable, almost predictable, as the melody arrives there – the friendly key next door…

Of course this is mitigated by the directness of the trip. We could potentially weave through a myriad of other keys before arriving next door, blindfolded and bedazzled, and it might then take us a while to realise where we are, but by and large, closely related keys can be freely visited in comfort.

More adventurous journeys use less direct key relationships or follow a cascading progression of keys in the cycle of 5ths to arrive in a new land.

When listening to a piece, try to feel not just the more rapid flow of the melody, but the deeper, underlying flow of the progression of keys through which the melody travels.

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Sleight of Ear: the effect of musical context on perception

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

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Musical context

Individual intervals and chords can be listened to by themselves, out of context, or within the context of a particular piece.

Any interval or chord has an effect; a character, based on how the notes interact. However, the context of the surrounding notes can produce “sleight of ear”. The interval or chord can appear to sound different than when played by itself or in another musical context.

Musical context is a combination of the overall key and mode and the development of the piece. Many pieces visit various keys along the way, resulting in a temporary key. As the music progresses through these visiting keys there is interplay between the home key and the visiting key and the listener’s viewpoint shifts.

Altered notes in either the melody or chords can also result in sleight of ear.

Sleight of ear example 1

Here are two examples of changing from an A major chord to an E major chord. The first example the melody feels like A is the home chord and we’re venturing out to E. In the second example, just one slightly different note in the melody suggests that E is home and we’re arriving home from a visiting key. This is especially noticeable when we hear the progression repeat itself.

Interestingly, the addition of D# in the melody implies the key of E major, and that’s how we hear it. To reflect this, the above example is written with the key signature of A major for the first example and E major for the second.

Sleight of ear example 2

The classic example of sleight of ear is the interval between the 6th and 7th notes of the harmonic minor, which is 3 semitones despite being consecutive scale notes (letters). This interval gives the scale an exotic quality reminiscent of Gypsy music.

From the 6th note to the 7th sounds like an unusually large step, a stretched out 2nd. It is called an augmented 2nd, reflecting how we hear it in the scale.

Normally, 3 semitones is a minor 3rd. When we hear this interval by itself we assume the first note to be the root note: it sounds like the first 3 notes of a minor scale with the 2nd note left out, or the start of a minor chord or minor triad.

The same size interval feels unrecognisably different in these two different contexts.

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Videos taken from Music Theory De-Mystified, due for release as an e-Book late 2022.

F flat Is a Note

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

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Most notes have at least two possible names. For example, F# is the same pitch as Gb. Even naturals have alternative names. E could be called Fb and F could be called E#. And that’s not to mention double sharps and double flats. G could be called Abb and so on.

Why so many choices? First, some background…

Modes

Major and minor keys are based on patterns of 2 and 1 semitone intervals between consecutive notes. We call such a pattern a mode. The series of notes generated by the mode is called a scale. Typical Western scales have 7 notes per octave, the eighth note being the octave of the first (hence the name “octave”).

The starting note of the scale is called the root note or tonic. The root note is easy to recognise when playing a scale because it is first and last. Melodies make the root note apparent by highlighting it in various ways so we can tell which mode we’re in when we listen to the music.

The choice of mode imparts an overall character to the music, called tonality.

Keys

A key is the combination of a mode and a root note. Keys allow us to choose the mode and the root note independently.

Let’s look at the major mode as an example. The original major, made up of only naturals, is C major. The name C major indicates that this key uses the major mode with C as the root note.

C major

Any other major key needs at least one sharp or flat. By starting the mode on a different root note we need some different notes in the key to preserve the pattern of intervals from note to note. The pattern of intervals defines the mode, in this case, major.

We can work out the notes needed for a chosen key by placing the new root note at the start of the pattern and counting the semitones from note to note. Let’s look at D major; the major mode starting on D.

D major

The note naming rule

There is one simple rule that determines the right choice of note name. In a standard Western mode such as major or minor, each scale note must have its own letter.

The letters indicate consecutive scale notes, just like they are written on a stave. A musical stave only has positions for notes as letters: sharps and flats are written as symbols beside the note.

When we work out the note names for a key, we start from the root note and count up. As we go, each following note must use the next letter as its name. In the example above, D major, the 3rd note is called F#. Gb is the wrong name because the third letter up from D is F, not G.

B#, Cb, E# and Fb

Remember BCEF? (see my beginner’s tip). This is the extreme end of BCEF. These notes look like they should never be used because they have equivalent pitches which are just naturals. B# = C, Cb = B, E# = F and Fb = E, so why use them? In truth their use isn’t all that common, but they do get used in certain keys.

For example, B# is used in C# major and Fb is used in Cb major.

This potentially begs the question, why use C# major as the name of a key when it could be called Db major? C# major has 7 sharps whereas Db major has only(?) 5 flats…

A valid question. I can’t answer it comprehensively in this post but there are three main reasons:

  • ease of playing/reading on a given instrument
  • movement within the piece from the home key to other keys
  • altered notes in the melody or chords

Easy keys

Players of some instruments such as guitar find sharps keys easier to read and play. Brass players, on the other hand, prefer flats keys. It depends on the base key and playing logic of the instrument.

Singers can be very specific about their choice of key for a particular song based on how the melody suits the different registers of the singer’s voice. This may force the rest of the ensemble to play in a key which is awkward to read, whichever name they choose.

For example, F# major has 6 sharps and Gb major has 6 flats. F# major has the note E# and Gb major has Cb.

Keys within a key

Typically a melody starts in the home key and goes on a journey. This journey takes it through various, usually related, keys, some of which are fleeting moments in the journey while others are visiting points; temporary homes. Campsites, if you like.

Visiting keys are named according to how closely related they are to the home key: in other words, how many notes they have in common. In general, if we start in sharps we continue in sharps, and the same for flats.

For example, in E major, a major key 2 semitones up would be called F# major, not Gb major. This is because F# is a note in the home key (E major) and Gb is not. In fact, none of the note names in Gb major are used in E major.

Which keys are related to which? That’s for another post.

Altered notes: weird note names in normal keys

Sometimes a melody or chord uses a note that doesn’t belong to the key. This could be as a variation or ornament, or the melody just might not be in a conventional mode.

We think of such a note as a replacement of the normal scale note or chord note. The context of the music determines which scale note has been replaced. To preserve the note naming rule, the new note is named with the same letter as the note it replaces.

  • if the altered note is a semitone higher than the scale note it is sharpened
  • if the altered note is a semitone lower than the scale note it is flattened

Sharpening or flattening allows the music notation to reflect which scale note is being altered, just as we would hear when playing and listening. However, depending on the key of the piece, this may require a double sharp or double flat.

NOTE: To avoid too much rambling I have only given a brief outline of the various topics raised in this post. I hope to cover some of these in future posts.

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Graphics taken from The Tiny Music Theory Book, a short, easy to read guide to the essentials of music theory and notation, available here.

Why Are Octaves Special?

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

All comments are welcome. If you enjoy my post, please give it a like and share it or subscribe to my blog.

Every musician discovers early on that octaves are special.

Notes which are one or more octaves apart have the same note name – that in itself means a lot. Furthermore, changing octaves feels more like changing voice or register than going to a different note.

Why is this so?

When we play a note, a sound wave is produced. Each pitch produces a wave which vibrates at a certain frequency: the higher the pitch, the higher (greater) the frequency.

Graph of a low pitch and a high pitch showing that higher pitches have a higher frequency and a shorter wavelength

The frequency is measured in cycles (vibrations) per second, called Hertz, Hz for short. You may have heard of A440, the frequency tuners are calibrated to. 440 means 440 Hz. A440 vibrates 440 times per second.

Playing a note an octave higher doubles the frequency: an octave above A 440 Hz is A 880 Hz. As the frequency gets higher, the length of the wave becomes shorter, so double the frequency is half the wave length.

When we play these two notes together, the higher note’s sound wave fits exactly twice inside the lower note’s sound wave. No other combination of two notes has such a direct relationship between their sound waves as an octave. This perfect fit is why the higher note of an octave sounds like it fits inside the lower note: because it literally does.

Graph showing 2 sine waves an octave apart
Graph showing the sound waves of two notes an octave apart such as A440 and A880. Twice the frequency = half the wavelength

Low and high octaves are large and small versions of each other. A musical part can be played at a different octave without introducing any new notes: it will still fit all chords and other parts equally well.

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*Graphics taken from Music Theory De-mystified, my upcoming music theory book, due to be released late 2022.