Music Theory De-mystified BlogOctave

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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Interval Names And Their Size In Semitones

Interval names are based on counting scale notes (letters) and are always counted from the lower note to the higher note, even if the higher note is played first.

An interval name is made up of two parts, quality and degree.

Degree

Treat the lower note of the interval as the root note of a major scale.
Now look for a note in the scale with the same name as the upper note of the interval. The degree is the position of that note in the scale: 1st, 2nd, 3rd, 4th, 5th, 6th, 7th or 8th.

Quality

There are 5 qualities: major, minor, perfect, augmented and diminished, depending on the degree and the sign of the upper note (#, b etc.).

Major

The upper note is the 2nd, 3rd, 6th or 7th note of the major scale built on the lower note.

Minor

The upper note is 1 semitone lower than the 2nd, 3rd, 6th or 7th note of the major scale built on the lower note and has the same letter name.

A minor interval is 1 semitone smaller than the major interval of the same degree.

Perfect

The upper note is the 1st, 4th, 5th or 8th note of the major scale built on the lower note.

Perfect intervals are common to both major and minor scales.

Augmented (made larger)

The upper note is 1 semitone higher than the equivalent major or perfect interval (1 semitone higher than the same letter in the major scale).

An augmented interval is 1 semitone larger than the major or perfect interval of the same degree.

Diminished (made smaller)

The upper note is 1 semitone lower than the equivalent minor or perfect interval.

A diminished interval is 1 semitone smaller than the minor or perfect interval of the same degree.

Note:

A perfect 1st is called a unison.
There is no such thing as a diminished 1st: the smallest interval is 0 semitones.
A perfect 8th is called an octave (not a perfect octave).
A diminished 8th or augmented 8th is NOT called a diminished or augmented octave. An octave is, by definition, perfect.

Example: Intervals Whose Lower Note Is C

Examples

C-E is a major 3rd
C-E# is an augmented 3rd (1 semitone larger than a major 3rd)
C-Eb is a minor 3rd
C-Ebb is a diminished 3rd (1 semitone smaller than a minor 3rd)
C-G is a perfect 5th
C-G# is an augmented 5th (1 semitone larger than a perfect 5th)
C-Gb is a diminished 5th (1 semitone smaller than a perfect 4th)

Interval names are dependent on note names. if the upper note has two possible note names, each option will have a different interval name.

For example, C- G# and C-Ab both are 8 semitones apart.

C-G# is an augmented 5th (perfect 5th + 1 semitone)
C-Ab is a minor 6th (major 6th – 1 semitone)

List Of Interval Names And Sizes In Semitones

Example with C as the lower note.

NOTE: The scale used for working out an interval name is built on the lower lower note of the interval. It is no indication of the key of the piece.

For more on how to name intervals, please visit 16. Intervals 1: Major, Minor And Perfect Intervals and B2. Intervals 2: Augmented And Diminished Intervals

Learn how to count intervals by singing. Visit 18. Listen & Sing: Learn Major And Perfect Intervals By Singing and 19. Listen & Sing: Learn Minor Intervals By Singing

18. Listen & Sing: Learn Major And Perfect Intervals By Singing

This post is one of a 2-part series of free basic music theory lessons on my blog, musictheoryde-mystified.com. You can see the complete list here. Please feel welcome to make a comment or ask a question.

Learning To Sing Intervals

Interval names are based on scale notes.

If we can sing, hum or imagine the sound of a scale, we can teach ourselves the character and name of various intervals by ear. We can count how many scale notes there are from the lower note of the interval to the higher note.

The easiest scale to sing, at least in Western culture, is the major scale. If you can’t sing a major scale straight away, please have a look at 17. Listen And Sing: How To Sing The Major Scale before reading on.

Major scale intervals

In 16. Intervals 1: Major, Minor And Perfect Intervals we saw that intervals are always counted from the lower note to the higher note, regardless of the order in which they’re played. The lower note of the interval becomes the root note of a major scale. We count scale notes to find the higher note and name the interval.

Counting up from the root note, the major scale contains the major 2nd, major 3rd, perfect 4th, perfect 5th, major 6th, major 7th, and, of course, the octave.

Treat the root note of the scale as the lower note of an interval.
Now sing from the root note to the 2nd note. This is a major 2nd.
To sing a major 3rd, sing the first 3 scale notes in a row but sing the 2nd note quieter or shorter than the first and third notes (see below). After a few times, leave the second note out altogether.
Repeat this exercise from the root note to each of the other notes in the scale.

Tip: the most useful intervals to become really good at are the major 3rd, perfect 5th and the octave. They are the notes of a major triad, a sound which will feel familiar to the ear and provide a shortcut for larger intervals (more on triads in Part 2 of my course).

Try These…

Below are the intervals of C major. Most voices can find a comfortable way to sing a C in the lower part of their range. The note number/scale degree is indicated below the notes.

In the first line, sing along to the first bar, then sing the same notes again in the second bar while you hear the interval played together. Feel your voice hit the lower and higher notes of the interval at the start and end of the bar.
In the second line the in-between scale notes are left out. Again, keep singing the first bar while you hear the interval played together in the second bar.
Practice each interval long enough until you don’t need to listen to the example while you sing.

Major 2nd

Major 3rd

Perfect 4th

Perfect 5th

Major 6th

Major 7th

Octave (perfect 8th)

Once you build a little confidence, choose a slightly lower or higher note for your intervals.

The more you do exercises like these, the easier it will be to recognise the interval between two notes, whether you hear them as a melodic interval (consecutive notes) or as a harmonic interval (both notes sounding together).

How To Sing An Interval Above A Note

This is just like how we learnt the intervals starting on C

Choose a major or perfect interval by name, such as a perfect 4th.
Play a note towards the bottom of your range.
Sing that note, then sing a note that’s the chosen interval above it
If you need to, you can quietly sing the in-between scale notes like in the first exercise.

How To Name An Interval You’re Hearing

You can use the same method to name an interval that you hear.

First, identify both notes of the interval by singing them. They are a little harder to pick when played together.
Sing the lower note, then sing the notes of the major scale until you hear your note match the higher note, counting notes as you sing (the starting note counts as the first note).
2 notes is a 2nd, 3 notes is a 3rd, etc. The 2nd, 3rd, 6th and 7th are major intervals, the 4th and 5th are perfect. (Technically the octave is also perfect, we just don’t need to say so. An octave is just called an octave.)

Try These…

Below are audio files of a few harmonic intervals. Remember to sing both notes of each interval before singing (or thinking) scale notes. To make it a little easier, the two notes are quickly played as a melodic interval before hearing the two notes together.

Name each interval using the steps outlined above:

Answers at the bottom of this post.

NEXT LESSON: 19. Listen & Sing: Learn Minor Intervals By Singing

PART 1 CONTENTS: Basic Music Theory Course Contents

___

Answers to Try These…

major 2nd
major 3rd
major 6th
perfect 4th
major 7th
perfect 5th
octave

(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.

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.

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?

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.

Please feel welcome to post a comment or ask a question.

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

1. Note Names, Semitones and Octaves

If note names mean nothing to you, start here…

In my posts:

A PIECE is any musical work.
A PART is one instrument’s component of a piece.
An ENSEMBLE is any combination of instruments collaborating to perform a piece, be it one person singing and playing, a band, choir or orchestra.

Note names

Most musicians are familiar with the note names A to G. After G comes A again and the pattern continues repeating from the lowest pitches to the highest.

A B C D E F G A B C etc.

Over the audible pitch range there are many A’s, many B’s and so on.

From one A to the next is an octave, as is from any letter to the next instance of the same letter.

Octaves

Notes which are an octave (or several octaves) apart enjoy a special relationship. When played together, the higher note blends in to the lower note. If they’re perfectly in tune (that’s for a later post), the higher note blends in so well that it almost merges inside the lower note. Even when played one after the other, what we hear sounds more like a change in register (or voice) than a different note.

Try this on your instrument. If you can play two notes at once or play one and sing the other, the effect will be the clearest, but you can still tell by playing one after the other.

Now try combinations of two different notes, such as A and G or A and C. None feel as closely connected as when they’re an octave apart (or a unison; two notes of exactly the same pitch).

In musical terms, in an ensemble, any part can be played an octave higher or lower without clashing with the other parts. All chords or harmonies will still fit. It is because of this relationship that notes which are octaves apart can, and do, share the same note name.

Intervals

The difference in pitch between one note and another is called an interval. A to the next A, an octave, is an interval, A to G is an interval, F to C is an interval.

Intervals can be measured in octaves and semitones. Each octave is divided into 12 musically equal intervals called semitones. This gives us 12 different notes, the 13th being an octave. The semitone is the centimetre (or inch) of pitch.

On a piano, 1 semitone is the interval between consecutive keys, regardless of the key’s colour.
On a guitar, 1 semitone is the interval from one fret to the next (or from an open string to the first fret).

We started with the letters A to G, followed by A etc. that’s 7 letters, the 8th being the octave of the first (as it happens, octave means 8th). So how do 7 letters add up to 12 semitones?

Not all letters are 1 semitone apart: in fact, most are 2 semitones apart. This is how the letters are spaced:

A . B C . D . E F . G . A
2 1 2 2 1 2 2 = 12

This means that 5 of the 12 different notes (per octave), the ones represented here by dots, have no name.

On a piano keyboard, all the named notes are white keys. You can see when two white keys are 2 semitones apart because there is a black key to represent the so far un-named note between them.

Piano keyboard layout showing naturals for 1 octave

On a guitar, you can find the named notes by starting on an open string, then following the above pattern by skipping a fret for every 2-semitone interval. The dots above represent the frets you skip.

Guitar fingerboard layout, A string, showing naturals for 1 octave

The named notes are called naturals. The un-named notes can be described as being 1 semitone higher or 1 semitone lower than the nearest natural.

Sharps and flats

Any natural can be raised by 1 semitone by adding the sharp symbol, #.
Any natural can be lowered by 1 semitone by adding the flat symbol, b.

For instance, the note between A and B could be called A# (A plus 1 semitone) or Bb (B minus 1 semitone).

This may seem confusing: we’ve gone from having no names for some notes to having two names. Fear not. For now, either name will do. The most common note names in general terms are:

A Bb B C C# D Eb E F F# G G# or Ab

Once we look at the notes in the context of a piece of music, the choice of note names will matter but by then it will be quite obvious which names to use. The correct note names for a piece are based on its key, a subject for a future post.

The graphic below shows how any natural can be raised by 1 semitone by adding a sharp or lowered by 1 semitone by adding a flat, resulting in two possible note names for most notes. Notice that even some of the naturals have an alternate name, although their use is relatively uncommon in most keys.

In my next basic post we will look at how note pitches are written on a stave.

Try These…

How many semitones between the following pairs of notes? (count up from the first note until you reach the second note of the pair):

A to C
A to C#
A to E
A to G
Bb to F
B to F
C to A
C# to A
D to Bb

Answers at the end of this post.

This post is one of a growing series of free basic music theory lessons on my blog, musictheoryde-mystified.com. You can see the complete list here.

Please feel welcome to like, comment or to share this post. If you have any questions, pleased leave them as a comment and I will respond as soon as I can. If you enjoy my posts and would like to be kept up to date, please subscribe.

NEXT LESSON: 2. Notes on a Stave: Pitch

PART 1 CONTENTS: Basic Music Theory Course Contents

Answers to Try These…

A to C = 3 semitones
A to C# = 4 semitones
A to E = 7 semitones
A to G = 10 semitones
Bb to F = 7 semitones
B to F = 6 semitones
C to A = 9 semitones
C# to A = 8 semitones
D to Bb = 8 semitones