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B5. Inversions Of Intervals

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 make a comment or ask a question.

What’s An Inversion?

To invert an interval is to turn it upside down: to take the lower note and move it up an octave, or take the higher note and move it down an octave. The order of the notes is reversed.

Interval + Inversion = Octave

A Special Relationship

Octaves have the unique property of being musically interchangeable.
Going up or down by an octave is like going up or down a flight of steps in an apartment building. Each octave is a higher or lower version of the next.

This means that there is a strong musical relationship between an interval and it’s inversion.

Melody And Harmony

Take a melody and a harmony, for instance.

A typical close harmony more-or-less follows the melody, adjusting here or there to fit the key (and the chords, but more on that in a future post). Usually, the harmony is sung higher than the melody.

The above example is played at the pitch of a female or young child’s voice. Now imagine that the harmony is sung by a lower voice such as that of an adult male.
Typically, adult male voices are typically about an octave lower than female or children’s voices. This would put the harmony an octave lower: lower than the melody.

As you can hear, the notes fit together just as well as the original. Sure, you can tell that the harmony is lower than it was, but they fit together just as well – as they should, because the note names are still the same.

Inversions And Interval Names

Theory is meant to reflect reality. If there’s an audible relationship between an interval and its inversion then their interval names should reflect this relationship.

Fortunately, they do.

Semitones And Letters

An interval plus its inversion equals an octave.

Consider the following example:

The first interval, A-E, is 7 semitones. Its inversion, E-A, is 5 semitones. This adds up to 12. An octave is 12 semitones, so that’s what we expect.

It doesn’t matter which way round we view this. Just as E-A is the inversion of A-E, so, too, the other way: A-E is the inversion of E-A. They are inversions of each other.

Now let’s look at their interval names. A-E is a perfect 5th and E-A is a perfect 4th. 5+4=9, yet the total is an octave, which is 8 letters.

The simple reason for this is because when we count an octave as two intervals, we count the middle note twice; once as the upper note of the first interval and again as the lower note of its inversion.

An interval plus its inversion equals 12 semitones
Also, an interval plus its inversion equals 9 letters

Let’s look at a few more, this time on G:

From these, we can add another observation:

The inversion of a major interval is a minor interval
The inversion of a perfect interval is a perfect interval
The inversion of an augmented interval is a diminished interval
Also:
The larger the interval, the smaller its inversion

NOTE: There is nothing inherent in determining which is the interval and which is the inversion. They are interchangeable. They are inversions of each other.

How To Invert An Interval Name

To invert an interval is easy: we reverse the pitch order of the two notes. We can name the new interval from scratch using an interval ruler as shown in B2. Intervals 2: Augmented And Diminished Intervals, but there is a quicker way.

If we know the name of the original interval we can invert the interval name by using our observations of how the names of interval and inversion are related.

In general, the names invert as follows:

For example:

G-A is a major 2nd. What is its inversion?

The degrees add up to 9
9 – 2 = 7
A-G is a 7th
As for the quality, major goes with minor
A-G is a minor 7th

Eb-G# is an augmented 3rd. What is its inversion?

The degrees add up to 9
9 – 3 = 6
G#-Eb is a 6th
As for the quality, augmented goes with diminished
G#-Eb is a diminished 6th

F#-C# is a Perfect 5th. What is its inversion?

The degrees add up to 9
9 – 5 = 4
C#-F# is a 4th
As for the quality, perfect goes with perfect
C#-F# is a perfect 4th

Try These…

a) Name the following intervals
b) Invert the following intervals
c) Name the inversion by inverting the interval name

G-Bb
G-D
D-B
D-G#
E-Db
F-Gb

Example: C-E

a) C-E is a major 3rd
b) To invert an interval, reverse the pitch order of the notes. C-E becomes E-C
c) To invert the interval name, major becomes minor and 3rd becomes 6th. E-C is a minor 6th

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NEXT LESSON: B6. How To Name Intervals The Quick Way

PART 2 CONTENTS: Basic Music Theory Course Contents

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Answers To Try These…

a) G-Bb is a minor 3rd
b) Invert G-Bb to get Bb-G
c) The inversion of a minor 3rd is a major 6th

a) G-D is a perfect 5th
b) Invert G-D to get D-G
c) The inversion of a perfect 5th is a perfect 4th

a) D-B is a major 6th
b) Invert D-B to get B-D
c) The inversion of a major 6th is a minor 3rd

a) D-G# is an augmented 4th
b) Invert D-G# to get G#-D
c) The inversion of an augmented 4th is a diminished 5th

a) E-Db is a diminished 7th
b) Invert E-Db to get Db-E
c) The inversion of a diminished 7th is an augmented 2nd

a) F-Gb is a minor 2nd
b) Invert F-Gb to get Gb-F
c) The inversion of a minor 2nd is a major 7th

B4. Accidentals Etiquette: How And When To Use Accidentals

Note: Students who solely use text notation can skip this lesson. Text notation seldom uses key signatures so sharps and flats are written after every instance of a note. In text notation, accidentals are indistinguishable from notes in the key which have a sharp or flat.

What Is An Accidental?

The term accidental has two meanings.

An accidental is a general term for a sign affecting the pitch of a note, such as a sharp, flat or natural sign.
Within a key, an accidental is the term for a note that has been sharpened or flattened, so that it’s no longer part of the key. Sharps or flats that are part of the key signature are not considered to be accidentals.
The following refers to how accidentals are used within the context of a key signature.

Unlike key signatures, accidentals only last within a bar and they are only applicable to the same octave: if the same note is used more than once in a bar, at the same octave, the accidental is only written for the first one.

The exception to this is that accidentals aren’t used in the middle of a tie, even if the tie crosses a barline. This is because tied notes are considered to be a single, longer note.

An accidental only lasts till the next barline.
An accidental only applies to a single note pitch. The same note at another octave requires it’s own accidental.
An accidental lasts the full length of a note, including tied notes. No accidental is used within a tie, even if it crosses a barline.

Courtesy Accidentals

An accidental lasts until the end of a bar. In the next bar, the note automatically reverts back to the key signature. As a reminder, this can be indicated by a courtesy accidental.

An example of this is the descending 7th and 6th notes in the melodic minor examples from the previous lesson, B3. Melodic And Harmonic Minor.

Courtesy accidentals, also known as cautionary accidentals, are sometimes written in parentheses () to indicate that each is only a reminder that a note is restored to the key signature.

Although not strictly necessary, it’s common practice to include courtesy accidentals. Whether or not you use parentheses is a matter of personal choice.

Example

The example below is in G melodic minor, requiring E natural and F# as accidentals when the melody ascends.

Accidental is only used at the beginning of a tied note, even if it crosses a bar.
Courtesy accidental because there is an F# in the previous bar, even though, as the end of a tied note, it is not written (see point 1).
Accidental is used for the first instance of each octave of a note within a bar.
Accidental is only used for the first instance of a note within a bar.
Courtesy accidental even though it’s in the key signature, because it was sharpened in the previous bar.

Ties And Slurs

A tie is a curved line that joins 2 notes of the same pitch to produce one longer note. It is placed adjacent to the notehead, opposite the stem.
A slur is a curved line that joins 2 (or more) notes of different pitches to indicate legato; full-length notes that are not articulated separately within the slur. It is placed adjacent to the notehead, opposite the stem.

Ties and slurs look the same. The only difference between a tie and a slur over 2 notes is the pitch.

Note: if the tied note is within a slur, the tie is always written closest to the notehead.

What if we wanted a slur to join a sharpened note at the end of a bar with its un-sharpened version at the start of the next bar? How can we distinguish this from a sharpened note tied over the barline?

In this case, the slurred note would receive a courtesy accidental, whereas there’s never an accidental within a tie.

Try These…

In the short melodies below, every note that’s sharpened is written with an accidental. Cross out any accidentals that shouldn’t be written and add any courtesy accidentals (or if you prefer, rewrite the exercises with the correct use of accidentals).

For example,

The answers, at the end of this post, show courtesy accidentals in parentheses. Parentheses are optional.

NEXT LESSON: B5. Inversions Of Intervals

PART 2 CONTENTS: Basic Music Theory Course Contents

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Answers to Try These…

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

B3. Melodic And Harmonic Minor

A Potted History Of Modes And Scales

Early Western music was based on modes. Religious music favoured those modes which had a solemn quality appropriate for worship. Secular music favoured modes suited to dance music and ballad singing.

The earliest recorded Western music, back in the 9th century, is Medieval plainsong; religious chants which were sung in unison. In Medieval times, secular music was somewhat more innovative, first by accompanying a melody with a drone and percussion, and later by layering several melodies on top of each other to form pieces called motets. The resulting “harmonies” were quite different to what we consider harmony to sound like nowadays.

During the Renaissance era, from the 14th century to the beginning of the 17th century, everything was done on a large scale, and music was no exception. Famous composers were commissioned by the Church to create ever more grandiose works, to be performed in the great cathedrals of Europe. By now, the combined melodies were crafted to produce rich harmonies. Renaissance music was still based on the Medieval modes but notes were organised harmonically as well as melodically, producing a more sophisticated sound.

By the 17th century we had entered the Baroque period, where music was sponsored as much by the royal court as the Church. Secular music flourished in high society and the sombre tone of some of the modes became unfashionable. The Ionian mode, most frowned upon by religious society for its frivolous nature, became the prominent mode of the time, to the point where it became known as the major scale.

The major scale is different to most of the other modes in that there is only 1 semitone between the 7th note and the octave, whereas in most other modes it is 2 semitones. From the Baroque perspective, this provides a sense of resolution that most other Western modes lack.

Harmony as we know it had developed, with melodies now being accompanied by chords. Along with this came the sense of key. It became possible for a piece to start in one key and visit another key.

Baroque music has many of the characteristics we can recognise in popular music of the 50’s and 60’s such as simple rhythmic structures, chord progressions, parallel (close) harmonies, and in particular the popular use of the major scale.

As popular as the major scale was, there was still a need to express the darker musical emotions. The Aeolian mode provided this to some extent and became known as the minor scale. However, it was still considered too reminiscent of the starker sounds of earlier times.

Three Types Of Minor

During the early Baroque period two variations of the minor scale were developed, called the melodic minor and the harmonic minor. As their names indicate, they were initially developed for use in melodies and harmony respectively.

To distinguish between these and the original minor, we often refer to the Aeolian mode as the natural minor.

Collectively, regardless of which variation is in use, we just use the term minor, based on the key signature.

Melodic Minor

Below are the scales of A major and A natural minor. By having both on the same root note, we can compare their character.

Listen to the the Aeolian scale, the natural minor. As a melody, it doesn’t arrive very strongly at the octave compared to the major, due to the 2-semitone interval from the 7th note to the octave. It’s a bit reminiscent of the modal sound of medieval times.

It turns out that, although parallel major and natural minor scales have 3 different notes, the most musically significant difference between their modes is the 3rd note. The 3rd is 4 semitones above the root note in the major scale and 3 semitones above the root note in the minor scale.

As long as we have the minor 3rd, the character feels like minor.
If we start to play a minor scale but finish like the major scale, we get the best of both worlds. We still have the essential character of the minor mode but a more “melodic” flow up to the octave at the end.

The problem is, when you play these notes descending, it sounds weird. It feels like we’re listening to the major scale until suddenly, right near the end, we hit the minor 3rd. It’s a bit of a shock!

The solution composers settled on is a hybrid.

On the descent, the natural minor flows quite well. The 2-semitone jump from the 7th to the octave that made it so ungainly (from the Baroque perspective) on the ascent isn’t an issue when descending. If anything, it helps the descent get started. Have a listen:

This leaves us with a new kind of mode, unlike any other mode we’ve seen so far; one that goes up one way and comes down another way. The 6th and 7th notes of the minor are sharpened while ascending but are returned to their key signature while descending.

Because the changed notes are temporary, they can’t be shown in the key signature. Instead, they are written as accidentals. If you’ve forgotten how accidentals work, see 9. Accidentals, Sharpen And Flatten.

Courtesy Accidentals

Note that on the descent, the 7th and 6th notes are written with natural signs, even though that’s part of the key signature. This isn’t strictly necessary because they’re not in the same bar as the sharpened notes, but it’s common practice to put them in when they’re in the following bar. These are called courtesy accidentals or cautionary accidentals. We will learn more about how accidentals are used in the next lesson.

How To Find The Notes Of The Melodic Minor

Start with the natural minor
When ascending, sharpen the 6th and 7th notes. Remember, a flat becomes a natural, a natural becomes a sharp and a sharp becomes a double-sharp
When descending, cancel the sharpened notes (a double-sharp becomes a sharp, a sharp becomes a natural and a natural becomes a flat)

Try These…

Write out the following melodic minor scales for 1 octave ascending and descending, in 4/4, in either the treble or bass clef
Use the rhythm of the example below
For each scale, write the key signature and use accidentals as needed

Example

G melodic minor
E melodic minor
Bb melodic minor
C# melodic minor

For bonus points, name the relative major of each scale…

Answers at the end of this post.

The Harmonic Minor

The harmonic minor was developed to make it possible to play a major chord on the 5th note of a minor scale. As we haven’t looked at chords yet in this course, I will leave the full explanation of the harmonic minor till a later lesson.

Unlike the melodic minor, the harmonic minor’s notes are consistent. Only the 7th note is sharpened, both while ascending and descending.

Because only the 7th note is sharpened, the harmonic minor has a 3-semitone interval between the 6th and 7th notes. As a melody, this gives it a rather exotic quality, as none of the standard Western modes have a 3-semitone interval between consecutive scale notes. Hence it is less-often used for melodies than the melodic minor.

Like the melodic minor, the sharpened note is not reflected in the key signature. Both melodic and harmonic minors are considered variations of the natural minor and in fact, often the same piece can incorporate both variations.

How To Find The Notes Of The Harmonic Minor

Start with the natural minor
Sharpen the 7th note, both when ascending and descending

Try These…

Write out the following harmonic minor scales for 1 octave ascending and descending, in 4/4, in either the treble or bass clef
Use the rhythm of the example below
For each scale, write the key signature and use accidentals as needed

Example

D harmonic minor
B harmonic minor
F harmonic minor
G# harmonic minor

For bonus points, name the relative major of each scale…

Answers at the end of this post.

Augmented 2nd

The harmonic minor is a good example of the need for alternate interval names.
As intervals, consecutive scale notes are called 2nds. When they are 1 semitone apart the interval is a minor 2nd and when 2 semitones apart the interval is a major 2nd.

The interval between the 6th and 7th notes of the harmonic minor is 3 semitones. Consecutive scale notes have consecutive letters so the interval must be called a 2nd but it is 1 semitone larger than a major 2nd. In other words, it’s an augmented 2nd.

We can’t call F – G# a minor 3rd because then G# would have to be renamed as Ab, implying that there is a possible scale note in between, F – G – Ab.
The difference between a minor 3rd and an augmented 2nd is quite noticeable when hearing the same pitches in different contexts.

Key Signature Shortcut

To this day, the melodic and harmonic minors are more popular in most genres than the natural minor. Since both involve the use of accidentals, either on the 6th and 7th note or just the 7th note, the frequent or early presence of an accidental in the part would likely indicate that the piece is in a minor key.
A key signature can equally represent a major and a minor key. Usually we look for clues in the first and last bars to find the root note, so we know whether the key signature represents the major or minor key for that key signature.
If there’s a recurring accidental, the piece is probably in the minor key of that key signature.

You can check this by seeing if the accidental is the 6th or 7th note of the key signature’s minor key.

For example, in the key signature of 1 sharp (F#), the key could be G major or E minor. If the part contains a C# or D# in the first few bars, the piece is most likely in E minor rather than G major.

Sing Along

Almost all voices and pitched instruments can sing or play one octave starting on C. Listen to all three minors on C and compare their character. Learn to sing and play all three.

Once you’ve learnt these, try singing or playing them in different keys (on different root notes).

NEXT LESSON: B4. Accidentals Etiquette: How And When To Use Accidentals

PART 2 CONTENTS: Basic Music Theory Course Contents

Answers to Try These…

B2. Intervals 2: Augmented And Diminished Intervals

Context

There are times when intervals of the same number of semitones require alternate interval names.

When taken out of context, this difference can’t be heard, and in fact, without the right context there’s no reason to use an unusual interval name. However, the same size interval can sound unrecognisably different in different contexts and requires different note names and interval names to reflect this.

Even if you don’t use note names in your practice, it’s worth becoming acquainted with augmented and diminished intervals. You can definitely feel the difference in the appropriate context.

Augmented And Diminished

In 16. Intervals 1: Major, Minor And Perfect Intervals there was one interval which was unnamed; the interval of 6 semitones. There isn’t a note 6 semitones above the root note in either the major or minor scale so we can’t call it a major, minor or perfect interval. Instead, we describe it as being 1 semitone larger than a perfect 4th or 1 semitone smaller than a perfect 5th.

Just as a note can be sharpened or flattened, an interval can be augmented or diminished.

Augmented means that the interval is 1 semitone larger than the corresponding interval in the major scale (major or perfect).
Diminished means that the interval is 1 semitone smaller than the corresponding interval in the minor scale (minor or perfect).

Let’s look at the interval ruler on A:

The note 6 semitones above A could either be called D# or Eb.

If the interval is A-D# we call it an augmented 4th; a perfect 4th plus 1 semitone.
If the interval is A-Eb we call it a diminished 5th; a perfect 5th minus 1 semitone.

The same applies for other intervals with unusual note names. Most notes have two or more possible names, resulting in different interval names.

For example:

A to C# is a major 3rd
A to C double sharp is an augmented 3rd, 1 semitone larger than A to C#
D is a perfect 4th above A
Db is a diminished 4th above A, 1 semitone smaller than D to A

NOTE: Remember to always count intervals from the lower note to the upper note. Use the major and minor scales of the lower note to find the upper note and name the interval.

Example:

Interval names are based on note names as well as size, so even though A-C# and A-Db are the same number of semitones apart, they can’t have the same interval name.

A-C#

C# is the 3rd note of A major, so
A-C# is a major 3rd

A-Db

There is no Db in A major or A minor. There is a D natural, though, the 4th note of both A major and A minor.
A-D is a perfect 4th.
Db is 1 semitone lower than D, so A-Db is 1 semitone smaller than A-D.
A-Db is a diminished 4th.

In general, if the upper note of an interval doesn’t fit either scale of the lower note, look for the nearest note in the scale with the same letter.

If the upper note is 1 semitone higher than the equivalent note in the major scale, the interval is augmented.
If the upper note is 1 semitone lower than the equivalent note in the minor scale, the interval is diminished.

NOTE: Augmented and diminished intervals can involve the occasional double-sharp or double-flat, depending on the lower note.

If the lower note is a sharp, an augmented interval will most likely require the upper note to be a double sharp. Similarly, if the lower note is a flat, a diminished interval would probably require the upper note to be a double flat.

Don’t be concerned. Just stick to the method:

sharpen = 1 semitone higher without changing the note name
flatten = 1 semitone lower without changing the note name

Interval Names Summary

2nds, 3rds, 6ths and 7ths have four possible qualities. From largest to smallest they are augmented, major, minor, diminished.
4ths, 5ths and 8ths have three possible qualities. From largest to smallest they are augmented, perfect, diminished.

It’s possible to have an augmented 1st but a diminished 1st is meaningless. There’s no such thing as a negative interval. Intervals are absolute…
8ths can be diminished or augmented but they should be called eighths, not diminished or augmented octaves. By definition, an octave is a perfect 8th.

Here’s a list of all intervals within an octave, with examples on C showing all the interval names including augmented and diminished intervals for each degree.

Here’s the same list of intervals shown as an interval ruler on C.

Just a reminder: the scales used to count intervals are built on the lower note of an interval and serve as a ruler to measure the name of the interval, in this case an interval whose lower note is C. For an interval with a different lower note we use scales on the new note to measure the interval.

The interval ruler is no indication of the actual key of the piece! The actual key is determined by the key signature and the overall root note.

Why have two names for the same size interval?

Good question! Interval names are based on note names. Note names reflect a specific musical context. The same size interval can sound unrecognisably different in different contexts.

An interval name describes both the size of an interval and how many letters there are from the lower note to the upper note. This allows us to “reverse engineer” an interval name and arrive at the right note names as well as the right sound.

In the next lesson we will see a practical example of the use of an alternate interval name.

Why are there two note names for most notes?

There are actually more than two if you count double sharps and double flats…

The choice of note name depends on the context.

If a note belongs to the key of the piece, its name is determined by the key signature.
If a note doesn’t belong to the key, its name is based on which note in the key it replaces.

When reading a new piece, the reason for some note names may not be apparent. In the coming lessons we will encounter some examples where an unusual note name is required. You can see and hear two of these in Sleight Of Ear.

In the meantime, let’s assume that unusual note names are used for a reason, so interval names need to be able to reflect which note name is used.

How To Name An Interval

Write out the interval ruler; the major and *minor (phrygian) scales built on the lower note of the interval. One way to do this is to write out the major scale then flatten the 2nd, 3rd, 6th and 7th notes to get the *minor.
Look for the upper note of the interval in these scales.
If the upper note is in either scale or in or both scales, name it as we’ve already learnt, as a major, minor or perfect interval. You have the answer.
If it’s not in either scale, find the note of the same letter that’s closest in pitch to the upper note of the interval.
If the upper note is 1 semitone higher than the note of the same letter in the major scale, the interval is augmented.
If the upper note is 1 semitone lower than the note of the same letter in the *minor scale, the interval is diminished.

The degree is always the number of letters from the lower to the upper note, inclusive.

Try These…

Below is a blank interval ruler you can use as a template. For each of the following exercises, first write the scales of the lower note as per the template.

A. Name the following intervals, keeping the above method in mind:

A-G
A-Gb
C-C#
C-E
C-E#
Bb-Ab
Bb-Abb
D-A
D-Ab
D-G#

How To Name The Upper Note Of An Interval

Write out the interval ruler; the major and *minor (phrygian) scales built on the lower note of the interval.
For major, minor or perfect intervals, find the upper note by following where the degree and quality of the interval name intersect.
If the interval is augmented, sharpen the same letter note in the major scale.
If the interval is diminished, flatten the same letter note in the minor scale.

Try These…

B. Name the upper note of the following intervals. For the degree, count letters (including the starting note).

If the interval is augmented, sharpen the equivalent note in the major scale.
If the interval is diminished, flatten the equivalent note in the minor scale.

major 6th above G
augmented 6th above G
major 7th above G
augmented 7th above G
minor 3rd above E
diminished 3rd above E
perfect 5th above E
diminished 5th above E
minor 6th above C
diminished 6th above C

Answers at the end of this post.

Shortcuts

It may seem laborious to have to write out scales every time you want to name an interval.

If you know your keys well, you can do this in your head. In part, I have encouraged the learning of at least the key signatures of the major keys for this very reason. As we’ve seen, you can find the *minor by flattening the 2nd, 3rd, 6th and 7th notes. Key relationships also provide some shortcuts for remembering keys. Have a quick look at the relevant lessons from Part 1 if you’re not sure…

The good news is that there are a number of shortcuts to help us to name intervals without writing out scales. These will become apparent over the next few lessons.

NEXT LESSON: B3. Melodic And Harmonic Minor

PART 2 CONTENTS: Basic Music Theory Course Contents

Answers to Try These…

minor 7th
diminished 7th
augmented 1st
major 3rd
augmented 3rd
minor 7th
diminished 7th
perfect 5th
diminished 5th
augmented 4th

major 6th above G = E
augmented 6th above G = E#
major 7th above G = F#
augmented 7th above G = F##
minor 3rd above E = G
diminished 3rd above E = Gb
perfect 5th above E = E
diminished 5th above E = Eb
minor 6th above C = Ab
diminished 6th above C = Abb

B1. Syncopation Basics: Playing Off The Beat

A time signature has an inherent hierarchy of strong and weak points within the bar. Syncopation is the emphasis of weak parts of the time signature. This introduces a dynamic interaction between the time signature’s implied rhythm and the rhythm of the part; a sense of going “against the grain”.

The Back Beat

The simplest form of syncopation is to emphasise the weak beats instead of the strong beats. The classic example of this can be found in many popular music genres since the advent of rock & roll; the snare drum playing the “back beat”, emphasising beats 2 and 4 in a 4-beat bar.

A similar effect can be achieved on any instrument by accenting the weak beats. Below is an example of a quaver rhythm, first with accents on the strong beats, then on the weak beats.

Tap, clap or play along to the following rhythms:

Playing Off The Beat

The back beat is only syncopation in the broadest sense. The term syncopation typically refers to emphasising weak parts within the beat a rather than just the weak beats.

The most common example of syncopation within the beat is to emphasise the 2nd quaver of each beat, the “off-beats” or “and”s, instead of the beat itself.

The following example of a bar of quavers uses accents, first to emphasise each beat, then to emphasise each off-beat quaver. Listen to the rhythms then tap, clap or play along:

Listen again and this time, only tap on the accented notes.

Syncopation adds excitement to a rhythm. Even in rhythms which are largely on the beat, the odd syncopated moment adds life to a part.

In the drum rhythm below, there is a brief syncopation is in the second half of bars 2 and 4.

Ways To Syncopate

Syncopation can be achieved in 2 ways:

by emphasising a note or notes on a weak part of the bar as above, with an accent.

by de-emphasising a strong part of the bar, in particular by not playing a note there at all. This can be because there’s a rest or because the previous note is still sounding.

Note that when clapping or tapping, there is no audible difference between these two bars.

Zooming In

Syncopation doesn’t just refer to emphasising the off-beat quavers. A more aggressive version would be to syncopate by a semiquaver.

A couple of the rhythms we learnt in 20. How To Read Rhythms 1 had semiquaver syncopation within the beat, by not playing a note on the “and”, the 2nd quaver. We can see now why these felt harder to learn than the others… Here’s an example of a bar with these two rhythms. Tap or play along:

Once you’ve experienced it, syncopation feels quite “natural”. In many popular genres, singers seldom sing exactly on the beat, even if that’s how the melody is written. Instead, they instinctively apply a degree of syncopation so the melody doesn’t sound too rigid. Rhythmic players rely on syncopation to add dynamics and drive.

Learning To Syncopate

Like many rhythms, syncopation is best learnt initially using a metronome. The secret to being able to syncopate is to feel the beat – to know where the beat (or strong beat) is, and then to know what relationship your note has to the beat.

Some musicians find it easy to tap the beat with their foot while playing. If this works for you, then by all means tap instead of using a metronome. However, many find it awkward to tap on the beat while playing off the beat, especially when first learning a new rhythm.

Foot Tapping Tip: In simple time, use the action of lifting your toes between taps to represent the half-beats; the “and”s.

If the rhythm seems tricky, remember to slow down the tempo and zoom in, as discussed in 20. How To Read Rhythms 1.

Ultimately, once you know a rhythm well enough to be able to feel it, you will no longer need the metronome. Metronomes can become quite annoying over time(!) so it’s worth weaning yourself off it as soon as you can feel the rhythm properly.

Mixed rhythms

Many parts, rhythmic as well as melodic, have a degree of variation in their rhythm, often achieved by brief syncopations in between overall on-beat rhythms.

Try These…

Play the movies below and tap the rhythms with your hand on a bench top, or if you prefer, clap. Listen carefully to the metronome click so you remain aware of the beat…

Once you’ve learnt each rhythm, play it to a metronome at 60 bpm without the movie. Gradually increase the tempo to 100 bpm or more. You can play along to the following movies of the rhythms at 100 bpm to see how you went.

Being able to tap the beats with your foot while playing is a useful skill. Practice tapping the beats with your foot, together with the metronome, while playing or tapping/clapping the above rhythms with your hands. As you settle in, stop the metronome and try it by yourself.

Notation Tip

Rhythms are usually notated so that it’s clear to see where the beats are. For shorter notes, this is indicated by beaming. For longer notes, the note is split into shorter notes and joined by a tie (see 21. Note Values 2: Ties).

When crotchets fall halfway between beats, on the “and”s, they can be written as crotchets: it’s such a common occurrence that most musicians, once they see a crotchet after a single quaver or quaver rest, are familiar with this shortcut.

However, crotchets which are a semiquaver off the beat must be split and tied to show where the beats are, otherwise the music is too hard to follow.

In the correct example above we can see that the next note starts just after each beat. The position of each beat is clearly shown by the beaming.

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.

NEXT LESSON: B2. Intervals 2: Augmented And Diminished Intervals

PART 2 CONTENTS: Basic Music Theory Course Contents

Top 10 Tips For How To Practise Efficiently

OK, so you have a piece you’d like to play but there are some tricky bits and it’s several pages long (or several minutes, if you’re learning by ear). What’s the most efficient way to not just learn it but master it?

Read on… but first, here’s a few general tips.

Practice Space

I believe that, whatever your practice space, be it a room or just a nook, when you enter that space all other thoughts should be left outside. If you need to finish practising at an exact time, set an alarm so you’re not constantly thinking of missing your appointment.

The enemy of successful practice is distraction. For children, that’s likely some fun activity they want to do. For adults, it’s often due to pressing concerns from the world outside practice. Either way, all distraction is best left outside your practice space.

What You Need

For learning from notated music, you will need:

a stand so you can see the music at playing height
a soft pencil (2B or 4B)
a rubber (an eraser)

If you have trouble playing at a steady tempo, you may also need a metronome.

If you learn by ear you will need:

a stand or flat surface
a music player (such as your phone)
either a pencil, rubber and a sheet of paper or a note taking app

Using Pencil And Rubber

For notated music, use the pencil to circle a difficult passage that needs further practice or a note that you’re inclined to misread.
At your next practice, the circle reminds you of a problem, which you can practice briefly before playing through the section you’re working on.
Once you’ve got the circled section up to the standard of the surrounding bars you can erase the circle so only relevant circles are visible at any one time.

Using A Music Player

If you prefer to learn by ear, you’ll need a way to break the piece up into manageable segments for learning, unless you’ve listened to the piece enough times to be familiar with the notes and timing.

Note down the start time of each phrase and treat each phrase as a segment. As you learn the segments, you can indicate your progress by checking off their start times.
Mark the start time of a difficult passage that needs further practice.
As above, at your next practice, the mark acts as a reminder of the problem section.
Cue the music player to the marked time so you can listen to the passage and practice it.
Once you’ve got the marked section up to the standard of the surrounding bars you can erase or delete the mark so only relevant marks are visible at any one time.

Ideally you want a music player that gives control over playback speed without altering the pitch (some phone-based music players work like this)… or a patient friend who knows how to play it and can slow it down for you…

If your music player doesn’t provide that option, one workaround is to play the track at half speed for learning. It will play back an octave lower so as you play along, all the notes you play will still fit the music.

Before You Start

Make sure your instrument is optimised for playing. If it’s a new instrument, make sure it is set up correctly.

For older instruments, replace any worn parts such as dull or rusty strings, cracked reeds, worn pads etc. if you can. Give your instrument some love! You want your instrument to work well so you can enjoy playing it and hear the reward of your progress.

If your instrument is tuneable, make sure it’s in tune. This may seem obvious but playing an instrument that’s in tune is like a free power-up. An out-of-tune instrument immediately depreciates your practice- you will have less positive feedback from the instrument as you play. When your instrument is in tune, it’s much easier to hear how well you’re playing and much more rewarding when you get it right.

The 10 Top Tips for how to practise a piece

1 Practise Regularly

The single most important tip for successful learning is to practise regularly, ideally at the same time each day, 5 to 6 days per week. Your mind and body will adapt to this schedule and you’ll both learn quicker and play better.

During your practice, pace yourself. Take regular short breaks. It’s hard to maintain intense concentration for very long before starting to lose focus.

2 Warm Up

Before trying something tricky, warm up on a few easy notes. Forget about the piece, just relax into something simple or something you’re really familiar with as you settle in to playing and become one with your instrument.

3 Play A Scale

Play the scale of the key of the piece. If you play chords rather than melodies, play the main chords of the piece.

For bonus points, play in a rhythm related to the piece, in the same time signature. By the time you start practising the piece you will already be attuned to the root note, tonality and rhythmic character of the piece. Another free power-up!

4 Little Wins

Set yourself achievable goals. You want to have success almost every practice.

Don’t expect to get through all of a long or difficult piece first time. Break it up into smaller segments that can be joined together. In a fast or difficult piece, that might only be a bar or two at a time.

I say almost because everyone can have an off day. My recommendation for an off day is to just play through something easy or a section you have already learnt. That might draw you in to practice mode. If not, just finish up for now. Coasting through a few easy bits or even skipping a practice altogether is preferable to a negative practice! After all, you can always come back to it later…

5 Start Slow

The first time you read (or listen to) each segment, it’s all about making sure you have interpreted the instructions correctly. You don’t want to discover, a week later, that the notes you have so diligently learned are not all the right ones.

Make sure you learn all the instructions from the outset, including timing, dynamics and articulation.

Slow the tempo down. If it’s tricky, slow it down a lot. An easy way to do that is to count the next note value down as a beat. In simple time such as 3/4 or 4/4 (and in compound time) this means counting quavers as beats.
Don’t change tempo while playing! Whichever tempo you choose, you must stick to it for the whole segment. If it’s too fast to play the whole segment, slow the tempo even further and start again. Also resist the temptation to speed up for the easy bits. If necessary, use a metronome to make sure you play the entire segment at a consistent tempo.
Initially it’s all about percentages. Wait until you’re well over 50% success rate before you increase the tempo. I personally believe that something is only ready for the next stage when you can play it 3 times in a row without mistakes.

6 Gradually Build Up The Tempo

As you become more confident that you’re playing the right notes in their correct timing, increase the tempo of the segment little by little. Beats are most easy to feel in the range of 60-120 bpm (1-2 beats per second).

For example, in simple time,

If you’re counting quavers at 60 bpm when first reading it, keep counting quavers while you gradually increase the tempo until you have reached about 120 bpm.
Now play at the same tempo but count half as often (crotchet = 60). This way you can switch to counting crotchets without having to change how fast you play.
Continue gradually increasing the tempo until you are playing the segment at or near its final tempo.

The method in Top Tips 5 & 6 is demonstrated clearly in 20. How To Read Rhythms 1.

Note: For pieces in compound time, we have to switch from quaver beats to dotted crotchet beats, so we should increase the tempo a bit further before switching. For example, quaver beats at 150 bpm result in dotted crotchet beats at 50 bpm.

7 Layers Of Practice

The mind needs diversity of focus and intensity. You can’t practise the same bar over and over for more than a few minutes without losing clarity. After a while, it starts getting worse rather than better and you risk having a negative practice.

It’s much better to have a few different types of practice in the same practice session. Once the first segment, say bars 1 & 2, starts to become a little familiar, even while still at a fairly slow tempo, you can introduce the next segment, bars 3 & 4, reading it slowly and carefully.

The next day, bars 1 & 2 are starting to speed up a little, bars 3 & 4 are becoming solid at the slow tempo and you introduce bars 5 & 6. We practise most efficiently when our practice occurs at several different levels: beginning one segment, building up another and consolidating a third.

8 Practice The Joins

Once you have a couple of segments up to a fast enough tempo to feel the character of the music, practise joining them together. For longer segments, practise the transition from the last bar or so of the first segment to the first bar or so of the second. As you learn more segments, add them to those you’ve learnt.

This now becomes the consolidation part – developing a sense of how the segments flow as an entity, as a piece of music, even if they’re not quite at the final tempo.

9 Continuity Practice

At some point, even if it’s not perfect, you have to assume that you’ve learnt the piece. There may be some fluffs and glitches to iron out, but you’re now ready for the last stage, continuity practice. Here’s how:

Play at a tempo that you know you can manage. Unless it’s a slow piece, this should be in the proper time signature at 60 bpm or more.
If you make a mistake, stop, go back to the nearest convenient point before the mistake and take another run at it. If it still doesn’t work, that section or join needs separate practice next time. You can circle the area on the part in pencil so that you know where to find it next practice.
Whatever you do, don’t slow down or speed up. A piece should always be practised at a steady tempo (unless the tempo variation is part of the piece). If you feel you have to slow down in order to play the right notes, you’ve chosen a tempo that’s too fast. Find out how slow you need to go, then start the whole piece again at that slower tempo. If that feels too slow, practise the offending segment or join separately to get the tempo up a bit and come back to the whole piece later.
Make sure you stick to the tempo. Don’t hesitate or pause between segments unless it is written into the part, such as a fermata or comma. If you do find yourself pausing, practise from before the pause to after it.

10 The Finishing Touches

Once you can play the whole piece at or near its final tempo with only the occasional slip-up, it’s reward time. Try to enjoy the piece while playing, even if there are a few precarious moments (but never slow down for a tricky bit! At this stage, any dodgy bits should be practised separately until they fit the tempo of the rest of the piece).

Now you can apply what I think of as the cheap tricks: polishing the expression and dynamics: flair and pizzaz. When I say cheap, I mean cheap in cost: high value for relatively low effort. These finishing touches take the music to a whole new level!

This part is the most fun. You get to decide how much you want to push the accented notes, how sharply you cut off that staccato note, how boldly you want to swell and fade with the phrases and much more. Expression is like acting: visualise yourself as one of the greats playing to an arena of fans.

The rewards of expression are instantaneous- the playing immediately sounds more impressive. And it’s fun!

A big thank you to Jo Haythornthwaite design42@iinet.net.au for creating the illustrations for this post.

See It, Hear It: Dynamics

The contrast of loud and soft (quiet) is important in shaping a performance.

Dynamics is the musical term for how a musician varies loudness. When printed, dynamics are written below the stave with symbols and words in bold italics.

In a score, each part can have its own dynamics, whereas tempo/character markings apply to the whole score. As with character and tempo markings, symbols are based on Italian words.

The following list shows the most commonly used dynamics. The default dynamic, when nothing is written, is mf (mezzo forte), or moderately loud.

Dynamics Flyover

See It, Hear It: Tempo Markings

We have already looked at tempo: the ability to play in varying tempi, speed up or slow down while playing, or play freely, with totally flexible timing.

Overall tempo within a section of music is indicated by a metronome marking indicating the tempo in beats per minute (bpm). As we have seen in previous chapters, this is usually written as the note value representing 1 beat in the time signature followed by = and a metronome number.

Other terms and symbols are used to indicate changes in tempo within a piece. The list below is of the more common of these.

The dashed line following terms which indicate gradual tempo change extends over the number of bars or beats over which the change occurs. Tempo markings that describe an overall tempo are listed in Character Markings And Their Tempo Ranges.

Tempo Change Flyover

Character Markings And Their Tempo Ranges

These are usually descriptive words and can appear in a variety of languages including English. The Italian names are prevalent in Classical music, but in Jazz they’re usually English. Other genres often use the language of their origin.

Character markings appear above the start of the first stave on the first page, where metronome markings are also written, in Bold type.

We associate Italian character names with tempo, largely because metronomes attach tempo ranges to them, but it’s more about the character than the tempo. The ranges are quite wide, though, and it’s likely that, to achieve the appropriate character, the tempo will end up within the suggested range.

The list below lists the more common tempo markings. Other variations add –issimo to the end of the word to intensify the meaning (Larghissimo means extremely slow, Prestissimo means extremely fast), or –ino or –etto to soften the meaning (Allegretto is slightly slower than Allegro).

If the piece is to be played at a specific tempo, a metronome marking can be written after a character name.

*At the lowest tempi it becomes difficult to count the beats accurately without counting the ”ands”, the next shorter note value. For more on this, please visit 20. How To Read Rhythms 1.