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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.
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
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
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.
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.
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 wth new posts, please subscribe.
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.
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.
The chromatic scale
The chromatic scale is made up of every one of the 12 musical note pitches, each 1 semitone apart from the next. It is the parent of all modes (…except in cultures with more than 12 note pitches per octave…).
Apart from some largely academic genres such as serial music, chromatic scales are mainly used as musical effects, such as the rapid chromatic passages used to build tension and drama in film music.
As a key, the chromatic scale has no inherent tonality: there’s no way to tell the root note from any other note. That’s not to say that you can’t nominate a root note, just that it requires a lot of effort to make that root note felt in a piece.
I don’t mean that it has no musical character, either. It does, but it’s a very nebulous one…
Modes
A mode is the pattern of intervals that determines which of the 12 different note pitches within an octave are used to produce a scale/key. Apart from the chromatic scale and the “whole tone scale” (6 notes within an octave, with 2 semitones between each note and the next), modes have an irregular pattern to their interval structure. As long as the root note is highlighted from time to time this irregularity allows a mode’s unique character to permeate the character of the music.
Diatonic Modes
The traditional Western modes, such as ionian (major) and aeolian (minor), are made up of one- and two-semitone intervals. Such modes are called diatonic modes; modes whose scales only have two different size intervals between their notes.
Just to be confusing, they’re also called heptatonic modes, meaning that they have 7 different notes within an octave. The traditional modes are all made up of 7 different notes, where the eighth note is the octave of the first.
A mode doesn’t innately have these limits. For example, the harmonic minor (discussed later in this course) still has 7 notes but one of its intervals is 3 semitones, that between the 6th and 7th notes of the scale. Other modes have a different number of notes per octave, such as pentatonic (5-note) modes and the blues mode. Some modes even use one or more different notes ascending and descending. The melodic minor works like that, as do some Indian modes.
However, the most common modes in most Western genres are the traditional Western modes which evolved out of the Renaissance era.
Although often associated with period music and traditional folk music, some modes, such as the myxolydian mode, are commonly used in a variety of other genres. Jazz goes even further by using relative modes as an approach to improvising around extended chords.
What Are The Traditional Modes?
Let’s have a look at the natural notes for two octaves. Each note can be the root note of a diatonic mode (yeah, I wouldn’t worry about the “heptatonic” bit, it seldom crops up in conversation…).
The traditional Western modes
Look at the notes for an octave, starting on each note in turn. Each root note produces a different pattern: a different mode.
We can compare their interval structure by lining them up underneath each other.
Each different mode has a unique quality that greatly influences the overall character of music played in that mode.
You can hear this, even by just playing a scale.
You can also teach yourself a new mode just by reading and playing the notes of the scale.
If you play by ear, follow its pattern of intervals on your instrument by counting semitones.
Below is a scale of each mode starting on A, so we can compare their character..
A aeolian (natural minor)
A locrian
A ionian (major)
A dorian
A phrygian
A lydian
A myxolydian
The character of each mode is easy to identify in a melody. Have a listen to this short, simple melody played in each of the modes…
Simple melody in A aeolian (natural minor)
Simple melody in A locrian
Simple melody in A ionian (major)
Simple melody in A dorian
Simple melody in A phrygian
Simple melody in A lydian
Simple melody in A myxolydian
Modes And Key Signatures
The way in which key signatures are used for modes other than major or minor depends on genre and school. There are two approaches: like major/like minor or the mode’s actual key signature.
Like Major or Like Minor
Classical musicians are brought up on a strict diet of major and minor (as well as two variations of the minor, melodic minor and harmonic minor). A classically trained player is only going to look for two possible root notes when interpreting a key signature. Music in a less familiar mode would be hard to interpret on first reading; the root note would seem to conflict with the key signature.
The most important notes of a scale are the root note and the note 7 semitones above the root note, called a perfect 5th (don’t worry about the interval name, we’ll look at interval names soon). The perfect 5th helps stabilise the root note.
There is one other important note, the 3rd note. One reason the 3rd is important is because it’s the most significant difference between major and minor. In the major mode the 3rd note is 4 semitones above the root note and in the minor it’s 3 semitones above the root note.
We can categorise the other modes as being “like major” or “like minor”, based on the 3rd note.
Below is a list of the modes staring on A, grouped in “like major” or “like minor”. The note in the other modes that differs from the major or minor is highlighted.
Note that the locrian mode is not in either list! The locrian mode wasn’t used in Western music, or even named, until relatively recently because it lacks the essential ingredient of a perfect 5th. By not having a note 7 semitones above the root note, music written in this mode is elusive. We naturally listen for a root note but we either can’t find it or we’re misled by other possible root notes that do have a perfect 5th but don’t hang around long enough to feel convincing.
It’s VERY hard to make the root note stick in the locrian mode. Modes without a perfect 5th need almost constant reinforcement of the root note in order to be musically stable. One way to achieve this is by having a drone accompaniment, where the root note persists throughout the piece.
Accidentals
For modes which are like major, we use the key signature of the parallel major mode; the major mode on the same root note.
For modes which are like minor, we use the key signature of the parallel minor mode; the minor mode with the same root note.
This requires the use of an accidental. Anyone who’s seen music written in the melodic or harmonic minor will be used to accidentals used in this way.
For example, let’s look at one of the more common modes, the myxolydian mode.
Using only naturals (key signature of 0 sharps/flats), the myxolydian mode starts on G. The actual key signature of G myxolydian is 0 sharps/flats.
The myxolydian mode is like major, so for a ”like major/like minor” key signature we would use the key signature of the major mode on G, G major, which is 1 sharp (F#).
To preserve the intervals of the myxolydian mode we need to flatten the 7th note, the note which differs from the major of the same root note, F# (remember, in the cycle of 5ths, the latest sharp is the 7th note of the scale…).
When we flatten F# we get F natural, so for the first F in every bar that has one, we write a natural sign as an accidental.
*NOTE: An accidental is only written for the first instance of a given note in each bar.
Try These…
Like Major, Like Minor
Write the scale of the following modes for 1 octave ascending using the key signature of its parallel major or minor (the major or minor key with the same root note) and an accidental where required. Base your decision on whether the 3rd note belongs to the parallel major or parallel minor.
NOTE: Accidentals aren’t written as part of the key signature. They must be written beside the first instance of that note in every bar where that note occurs.
A dorian
D lydian
C ionian (trick question… just seeing if you were paying attention)
C myxolydian
B phrygian
F# dorian
Bb myxolydian
Answers at the end of this post.
Actual Key Signature
Players of early music and traditional folk music, as well as a number of other genres, are quite familiar with the traditional modes. Typically, in such genres the actual key signature is used.
In the above example, G myxolydian is written in the key signature of 0 sharps/flats, rather than in the key of G major with F natural as an accidental. It’s cleaner and simpler to read (as long as you interpret the key signature correctly when reading).
When first reading a piece knowing only it’s key signature, you might wonder how to determine the root note of the piece, with so many modes to choose from. The approach is the same as when you’re just looking for major or minor. Look for an obvious note near the start and end of the piece.
Pentatonic modes (5 notes per octave) are quite popular in various genres. Having only 5 notes, we can think of them as a subset of the traditional heptatonic modes that we’ve already looked at.
The most well known of these is the minor pentatonic mode, notes 1, 3, 4, 5 and 7 of the natural minor/aeolian mode. In this instance, we would use the key signature of the “parent” minor key. Other Western pentatonic modes are also subsets of parent modes, so we would use the key signature of the “like major” or “like minor” mode of which it is a subset.
Here are the two most common Western pentatonic modes.
Not all pentatonic modes are related to a parent Western mode. Any combination of 5 notes within an octave can be used as a pentatonic mode.
The Blues Scale
The most common mode in blues is a hexatonic (6-note) mode based on the minor pentatonic, with an extra note added between the 4th and 5th notes. From a notation viewpoint, there is no room in the naming system for the extra note to have its own letter: instead, it is considered an alternative to the 4th or 5th note and should be named according to its use. Either option requires an accidental.
Other Modes
Most musical cultures around the world have a concept of modes, a system of choosing a set of notes within an octave based on a root note and the intervals between its scale notes. Some use different intonation (tuning) systems: such modes can sound unfamiliar to the Western ear. Many are based on the same intonation as Western music, 12-Tone Equal Temperament, yet may have different numbers of notes per octave or notes that differ when ascending or descending. If you wish to experiment, you can create your own mode for a new composition or improvisation.
Freedom From Modes
The Western modes were originally used as a rigid framework for determining which notes were used in a composition. Over time, as music developed towards the journey through various visiting keys that it largely is today, the use of accidentals became more and more common, to accommodate these temporary keys.
Accidentals are also used for embellishment (ornamentation). Ornaments are treated as effects: an ornament may well use notes outside the key, requiring an accidental. This may extend to chromatic passages involving several accidentals.
The contemporary view is that a key is based on the overall use of the notes of a major or minor key, with the option of sharpening/flattening notes or incorporating other notes as desired.
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 wth new posts, please subscribe.
