Intervals, scale degrees and more

[Richard_close2u]

Prompted by a discussion in the theory topic on the Cycle of 5ths. The Cycle Of 5ths

Understanding the basics of intervals relies on knowing that they are viewed from the perspective of semitones (= half steps = two notes one fret apart from each other on a guitar). Starting from any note and thinking of it as in first position, there are thirteen intervals, each in their own position, at their own distance, from the start and each with their own name – some have multiple names as a result of enharmonic equivalent notes. Conventionally, intervals are considered in an ascending manner, from a starting note to a note higher in pitch. Descending intervals do exist and do need to be understood also.

1] (Perfect) Unison - Distance: 0 semitones

2] Minor Second - Distance: 1 semitone

3] Major Second (or Diminished Third) - Distance: 2 semitones

4] Minor Third (or Augmented Second) - Distance: 3 semitones

5] Major Third (or Diminished Fourth) - Distance: 4 semitones

6] Perfect Fourth (or Augmented Third) - Distance: 5 semitones

7] Diminished Fifth (or Augmented Fourth) - Distance: 6 semitones

8] Perfect Fifth (or Diminished Sixth) - Distance: 7 semitones

9] Minor Sixth (or Augmented Fifth) - Distance: 8 semitones

10] Major Sixth (or Diminished Seventh) - Distance: 9 semitones

11] Minor Seventh (or Augmented Sixth) - Distance: 10 semitones

12] Major Seventh - Distance: 11 semitones

13] (Perfect) Octave - Distance: 12 semitones

I have written the main names in bold according to those identified by Justin as being those most commonly used in this lesson Chromatic Intervals Worksheet | JustinGuitar.com

[Richard_close2u]

It is worth mentioning that the naming system relies on assigning each of the intervals an ordinal number (second, third, fourth, fifth, sixth, seventh) relative to the first or starting note. The numbering stems from the fact of having seven alphabetical letters in use for musical notes. A – B – C – D – E – F - G. Any interval describing the distance between A and C must be some type of third since making A the first letter automatically fixes C as the third letter.

Each interval is also assigned a quality.

The unison, fourth, fifth and octave are perfect (we can think of them as sounding ‘perfect’ with the first note - they are consonant with it). The second, third, sixth and seventh can be major.

Perfect intervals and major intervals in total make up all seven intervals contained within a major scale.

• Unison (= root)

• Major second

• Major third

• Perfect fourth

• Perfect fifth

• Major sixth

• Major seventh

• Octave (= root)

Any interval whose distance from the start note is one semitone more than these perfect and major intervals is described as augmented.

Minor is the assigned quality for intervals whose distance is one semitone less than major intervals. For major and minor intervals, a simplistic view would be to think of them as ‘big’ versions / ‘small’ versions of the same lettered note.

Diminished is the named quality given to intervals one semitone smaller than both perfect and minor intervals.

As clear as mud or as clear as day?

[Richard_close2u]

I will now shorten interval names from full words to their numeric equivalents, for example a sixth will be written as 6th.

We are going to focus solely on two notes and their respective intervals, keys and diatonic chords – namely the note C, the C major scale and the diatonic chords in the key of C major plus the note G, the G major scale and the diatonic chords in the key of G major.

It is important to recognise, as shown above, that a perfect 4th is defined as the distance of five semitones and a perfect 5th is defined as the distance of seven semitones.

We will see two diagrams each showing twelve notes (some enharmonic equivalents have not been written for ease). Upward movement, from the bottom to the top, represents notes getting higher in pitch and vice versa. The central column (containing only C or only G) shows the start note. Cells to the right depict intervals ascending from the start note and cells to the left depict descending intervals.

From C to G (ascending and descending)

Here we see the note C at the lower and upper extremes. Imagine moving along just one guitar string. This lowest note C could be the one we find on the first fret of the B string. Then note G would be at fret 8 and the topmost C would be at fret 13. The two C notes are one octave apart.

• Moving up in pitch from the lower C to G is a distance of seven semitones (= 7 frets). The interval between them is a Perfect 5th.
• Moving down to G from the higher C is a distance of five semitones (= 5 frets) meaning it is an interval of a Perfect 4th.

From G to C (ascending and descending)

Here we see the note G at the lower and upper extremes. Imagine moving along just one string once more. The lowest note G could be at fret 3 of the E string then we would find the note C at fret 8 and the higher note G at fret 15.

• Moving up in pitch from the lower G to the C is a distance of five semitones (= 5 frets) meaning it is an interval of a Perfect 4th.
• Moving down to C from the higher G is a distance of seven semitones (= 7 frets). Therefore, the interval between them is a Perfect 5th.

The charts contain a lot of information. Two simplified charts may help.

[Richard_close2u]

Within the double-sided charts is something vitally important in understanding ascending and descending intervals.

• C up to G is a perfect 5th.

• C down to G is a perfect 4th.

• Both are perfect and the values add to 9.

• It is similar with G up to C and G down to C.

Let us take a small digression and look at a few other intervals for a moment, to see if there are any patterns to guide us.

• C up to D is a major 2nd and C down to D is a minor 7th. (major / minor and 2 + 7 = 9)

• G up to Eb is a minor 6th and G down to Eb is a major 3rd. (minor / major and 6 + 3 = 9)

• C up to B is a major 7th and C down to B is a minor 2nd. (major / minor and 2 + 7 = 9)

• G up to Db is a diminished 5th and G down to C# is an augmented 4th (diminished / augmented and 5 + 4 = 9)

• The ordinal values always sum to 9.

• Major and minor become inverted.

• Diminished and augmented become inverted.

[Richard_close2u]

As we saw previously, major scales comprise only these intervals:

• Unison (= root)

• Major second

• Major third

• Perfect fourth

• Perfect fifth

• Major sixth

• Major seventh

• Octave (= root)

The next step is to use just the perfect and major intervals (ascending from C and from G respectively) to create charts that will show the intervals, notes and scale degrees associated with their major scales.

[Richard_close2u]

• The intervals are perfect and major only

• The ascending semitone distances match the major scale formula (whole, whole, half, whole, whole, whole, half)

• G is the perfect 5th of C.

• C is the perfect 4th of G.

• The scale degrees are given as cardinal numbers (1, 2, 3 etc.) with no quality attached.

• Intervals and scale degrees have separation and important distinctions exist between them.

[Richard_close2u]

For both the C major scale and the G major scale, we can match up notes with scale degrees and arrive at their diatonic chords.
These are shown in the diagrams below with Roman numerals also shown.

…

[Richard_close2u]

Intervals and scale degrees (hence scales and diatonic chords) have many overlapping terms that are used in describing and discussing them. Making the subtle distinctions is not always an easy task and - most generally - is unimportant.

An analogy that comes to mind involves positive and negative numbers.

Everyone knows that on a warm day the temperature is positive, that our ordinary counting numbers are positive too.

[A] 2 plus 3 means positive 2 + positive 3 (= positive 5).

Everyone knows too that on a cold winter day the temperature is likely to be negative, below zero. It may be more abstract to think of counting with such values but of course it can be done, and calculations can be performed too.

[B] -2 plus -3 means negative 2 + negative 3 (= negative 5).

But how many of us actually call negative numbers by their proper, designated quality? I dare to suggest that the vast majority of people look at calculation [B] and say / think:

‘minus 2 plus minus 3 = minus 5’

Which, strictly speaking, is incorrect.
Negative is the actual correct term to use.
Minus is an operation performed on numbers, a verb, something you do.
It really is not at all right to call any number a minus number.
And it can lead to mathematical ugliness too. For instance:

[C] minus 2 minus minus 3 = 1
Yuck. Horrible.

But it really does not matter all that much if there is a shared understanding and if incorrect phraseogy does not lead to miscommunication. At least, not until it does matter.

So it is with intervals and scale degrees etc. Mixing up the language is generally no problem - until it is. And then a little care is required.

I hope this is useful.

Richard

[DarrellW]

OMG… talk about brain ache

[jkorvola]

Interesting, lots to take in .

Hopefully this is within the topic, but when transcribing or just playing for example a simple melody in the key of C going C → G → F , would it make sense to think this movement as individual intervals (5th up, 2nd down), or as scale degrees (1st, 5th, 4th), or maybe as both?

For me thinking this as scale degrees sounds a bit simpler, but it seems that in ear training (and thus in transcribing?) intervals are thought important.

[Richard_close2u]

Good question with no definitive answer.
If you know in advance that the music is in the key of C then either scale degrees or intervals describe what you need.
However, if you are transcribing an unknown piece whose key you are unsure of, only intervals would be appropriate. Once you have enough notes / chords figured out, enough pieces of the musical jigsaw, to deduce the key, then you could think in terms of either intervals or scale degrees.
Not all music is in a major key of course.
Also …
Caveat - non-diatonic music or key changes throw a spanner in the works too.

[brianlarsen]

@Richard_close2u

‘minus 2 plus minus 2 = minus 5’

[Richard_close2u]

Aha - you’ve spotted another of my many typos. Whoops. Thanks for making it through Brian and checking my work!

[roger_holland]

Hi Richard,
What a job … I can still remember something about this during the theory course … I did this a bit “too fast”, and it didn’t really land, in a way of speaking . …so for now I’m just going to play guitar
I’ll be back here later for a big rehearsal…
Greetings,.

[Jozsef]

So far I have thought about these “complementary” intervals as summing to 12 semitones, i.e. a perfect octave. But that’s probably being too reliant on the fretboard and counting the frets between notes.

There’s also a Wikipedia article on this topic for those who are interested.

Are there any plans to create a section on the website for your various supplementary material? They are a bit difficult to find here on the forum but would be quite handy to have somewhere close to the PMT modules.

[Richard_close2u]

@Jozsef

I have thought about these “complementary” intervals as summing to 12

You are correct. The list of 13 separate intervals does span 12 semitones. The additional interval arises from the fact of including a count from the start note to itself - the unison, a distance of zero.

@Jozsef

Are there any plans to create a section on the website for your various supplementary material?

It was discussed at one point with Justin - that content created by @LievenDV and myself might have a space on the main website.
We opted for this dedicated tips section in the community - for a whole host of reasons.

@Jozsef

They are a bit difficult to find here on the forum …

Oh, that is an interesting observation. We do want this area to have ‘visibility’.
I wonder if large numbers of people never visit here or are unaware of its existence.

[Jozsef]

I haven’t noticed that the posts are grouped under the “Tips from the Official Guides / Approved Teachers” category. This indeed makes it easier to find them here.