Intervals, octaves, the Chromatic scale
and transposition


This lesson is on octaves, intervals, the chromatic scale, and transposition.

Recall, that we use 7 letters (A,B,C,D,E,F,G) and accidentals (#,b,etc.) to name notes (last lesson).

Recall also, that an octave is the closest distance between two notes with the same name. (last lesson)

Recall that in the western tuning system we are using (twelve-tone system), that there are twelve distinct tones (which have distint names), and their octaves.

The set of all these notes taken together is commonly refered to as the chromatic scale.

Recall that we refer to notes in the chromatic scale in a relative way by using numbers and accidentals. Recall that the number "1" refers to the root note (letter name) given to a particular scale or chord.

Recall that in standard tuning, we have developed an understanding of where these 12 chromatic notes are in a relative way in 5 different positions that correspond with 5 open major chords. This process has built up in the previous lessons (major chords,pentatonic major scale,minor chords,pentatonic minor scale,diminished chords,major scale) and so far we've learned relative positions of the following notes:
1,2,b3,3,4,b5,5,6,b7,7
Filling in the two remaining notes b2 and b6, we have:

"E-shape" (root note on the 6th string)

|b7-|-7-|-1-|b2-|-2-|b3-|-3-|
|-4-|b5-|-5-|b6-|-6-|b7-|-7-|
|b2-|-2-|b3-|-3-|-4-|b5-|-5-|
|b6-|-6-|b7-|-7-|-1-|b2-|-2-|
|b3-|-3-|-4-|b5-|-5-|b6-|-6-|
|b7-|-7-|-1-|b2-|-2-|b3-|-3-|

"D-shape" (root note on the 4th string)

|b2-|-2-|b3-|-3-|-4-|
|b6-|-6-|b7-|-7-|-1-|
|-3-|-4-|b5-|-5-|b6-|
|-7-|-1-|b2-|-2-|b3-|
|b5-|-5-|b6-|-6-|b7-|
|b2-|-2-|b3-|-3-|-4-|

"C-shape" (root note on the 5th string)

|-3-|-4-|b5-|-5-|b6-|
|-7-|-1-|b2-|-2-|b3-|
|-5-|b6-|-6-|b7-|-7-|
|-2-|b3-|-3-|-4-|b5-|
|-6-|b7-|-7-|-1-|b2-|
|-3-|-4-|b5-|-5-|b6-|

"A-shape" (root note on the 5th string)

|-4-|b5-|-5-|b6-|-6-|b7-|-7-|
|-1-|b2-|-2-|b3-|-3-|-4-|b5-|
|b6-|-6-|b7-|-7-|-1-|b2-|-2-|
|b3-|-3-|-4-|b5-|-5-|b6-|-6-|
|b7-|-7-|-1-|b2-|-2-|b3-|-3-|
|-4-|b5-|-5-|b6-|-6-|b7-|-7-|

"G-shape" (root note on the 6th string)

|b6-|-6-|b7-|-7-|-1-|b2-|
|b3-|-3-|-4-|b5-|-5-|b6-|
|-7-|-1-|b2-|-2-|b3-|-3-|
|b5-|-5-|b6-|-6-|b7-|-7-|
|b2-|-2-|b3-|-3-|-4-|b5-|
|b6-|-6-|b7-|-7-|-1-|b2-|

Recall that two notes with the same name are octaves of each other. Also within a root-relative context, two notes with the same number name are octaves of each other.
So, looking at the above chromatic positions, we can find patterns for octaves on the fretboard.
First, consider the "E-shape" which has the 1 on the 6th,4th, and 1rst strings. Look at how the b2,2,7,etc. on the 6th,4th,1rst strings follow the same pattern. This is the "E-shape" octave pattern. There is an octave between any note on the 6th string, and a note on the 4th string two frets higher than the original note. There is an octave between any note on the 6th string , and a note on the 4th string two frets higher than the original note. There is an octave between any note on the 4th string, and any note on the 1rst string two frets lower than the original note. Also there is a distance of two octaves between any note on the 6th string and a note on the 1rst string in the same fret. Visually, here is the octave patterns for the "E-shape":

Octaves "E-shape"(root note on the 6th string)

|-1-|---|---|
|---|---|---|
|---|---|---|
|---|---|-1-|
|---|---|---|
|-1-|---|---|

Next, consider the "D-shape" chromatic position. Using the same reasoning as above, we see that there is an octave between a note on the 4th string and a note on the 2nd string 3 frets higher than the original fret. Visually, the "D-shape" octave pattern:

Octaves "D-shape" (root note on the 4th string)

|---|---|---|---|
|---|---|---|-1-|
|---|---|---|---|
|-1-|---|---|---|
|---|---|---|---|
|---|---|---|---|

Next, the "C-shape" chromatic postion. There is an octave between any note on the 5th string and a note on the 2nd string 2 frets lower than the original note:

Octaves "C-shape" (root note on the 5th string)

|---|---|---|
|-1-|---|---|
|---|---|---|
|---|---|---|
|---|---|-1-|
|---|---|---|

Then, the "A-shape" chromatic position. There is an octave between any note on the 5th string and a note on the 3rd string 2 frets higher. Visually, the "A-shape" octave pattern:

Octaves "A-shape" (root note on the 5th string)

|---|---|---|
|---|---|---|
|---|---|-1-|
|---|---|---|
|-1-|---|---|
|---|---|---|

And finally, the "G-shape" chromatic position. There is an octave between a note on the 6th string, and a note on the 3rd string 3 frets lower. There is an octave between a note on the 3rd string and a note on the 1rst string 3 frets higher. And also there is a distance of 2 octaves between any note on the 6th string and a note on the 1rst string in the same fret. Visually, the "G-shape" octave pattern:

Octaves "G-shape" (root note on the 6th string)

|---|---|---|-1-|
|---|---|---|---|
|-1-|---|---|---|
|---|---|---|---|
|---|---|---|---|
|---|---|---|-1-|

We've mentioned before that these shapes connect together (wrap around) the neck in a particular way. That is the "E-shape" connects with the "D-shape", which is connected to the "C-shape", which is connected to the "A-shape", which is connected to the "G-shape" which is connected to the "E-shape", etc. Visually it looks like:

|1|-|-|-|-|-|-|-|-|-|-|-|1|
|-|-|-|-|-|1|-|-|-|-|-|-|-|
|-|-|-|-|-|-|-|-|-|1|-|-|-|
|-|-|1|-|-|-|-|-|-|-|-|-|-|
|-|-|-|-|-|-|-|1|-|-|-|-|-|
|1|-|-|-|-|-|-|-|-|-|-|-|1|

Looking at the picture above allows us to see a few more patterns for octaves. Three more pop out in my mind as particularly useful.
1.) Notice that any note on a string has an octave 12 frets higher (or lower) on the same string.
2.) Based on the tuning of a perfect 4th between two adjacent strings (6 to 5, 5 to 4, 4 to 3, 2 to 1), there is an octave between a note on the lower string and a note on the next higher string 7 frets higher.
3.) Based on the tuning of a major 3rd between two adjacent strings (3 to 2), there is an octave between a note on the lower string and a note on the higher string 8 frets higher.

Many runs are based on the above patterns. Other runs are created using "lead patterns" which mix parts of the shapes together to get longer patterns on the fretboard. Looking above you could create a "lead pattern by using the octave (of the scale) from 6th to 4th string (E-shape), and then from 4th string to 2nd string (D-shape), and then from 2nd string to 1rst string, 7 frets higher.
Another one could be created by combining the octave from 5th to 3rd strings (A-shape) with the octave from 3rd to 1rst string (G-shape).

One more thing you could do with octaves (besides playing melody lines in octaves) is to learn the names of the notes on the fretboard using octaves. So for example, using the "E-shape" octave pattern, if you know that the 5th fret of the 6th string is an "A" note, then the 7th fret of the 4th string is an "A", and the 5th fret of the 1rst string is an "A". You could use the tuning (EADGBE) notes as reference if necessary.

Oh yeah, one more thing, if you'll learn the octave pattern, that's shown by placing the shapes together, you can use that to draw your own fretboard maps in less than 30 seconds. You'll need a blank fretboard to draw on and either an idea of what you're chord/scale looks like or an understanding of one of the chromatic positions. Place your scale/chord on the fretboard where you want it, and then draw octaves up and down the fretboard from there. With practice, this can be done very quickly. You could then use the fretboard pattern to make your own patterns or create your own voicings, etc. (those with an eye for a quick buck could create your own pattern, and then try to sell it in book or on-line format as the "secret" that unlocks the fretboard. ROTFLMAO ;P )

Moving on to other intervals...
Recall, that the western music system we are using is analyzed against the major scale.
Recall also that an interval is the distance between any two notes.
Recall that in our system there are 12 notes that are distinct, and all other notes that are within the system (that are in tune) are an octave or multiple octave of one of these notes.

Now for historical reasons, all of the intervals in the major scale are either considered "perfect" or "major".
The perfect intervals are Perfect unison, perfect 4th, perfect 5th, and perfect octave. (A unison is when you play two of the same note. Think relative tuning for an everyday example)
The major intervals are major 2nd, major 3rd, major 6th, major 7th.

Note: the perfect intervals (unison, 4th, 5th, 8ve) are never major, and the major intervals (2nd, 3rd, 6th, 7th) are never perfect. ( you would never call an interval a perfect 3rd, or a major octave, etc. It's not done).

These names (P5, etc.) are intervallic names that are given to a note to tell its distance from a relative note (the 1). Since there are 12 notes in our system, and we compare all the notes against the major scale, we have the following names for the twelve notes/intervals in reference to the 1 note (tonic, root note). Multiple names for a note are enharmonic equivalents (different names for the same note). They show that there may be more than one context in which to view an interval.

Distance in 1/2 notes name

0      Perfect unison/augmented 7th/diminished 2nd
1      minor 2nd/augmented unison
2      major 2nd/diminished 3rd
3      minor 3rd/augmented 2nd
4      major 3rd/diminished 4th
5      Perfect 4th/augmented 3rd
6      augmented 4th/diminished 5th
7      Perfect 5th/diminished 6th
8      minor 6th/augmented 5th
9      major 6th/diminished 7th
10     minor 7th/augmented 6th
11     major seventh/diminished 8ve
12     Perfect 8ve/augmented 7th

Because these terms can be cumbersome, I shall use abbreviations in either letter/number combinations or number/accidental combinations. Below is the same chart as above but abbreviated.

0      PU/aug7/dim2; 1/#7/bb2
1      m2/augU     ; b2/#1
2      M2/dim3     ; 2/bb3
3      m3/aug2     ; b3/#2
4      M3/dim4     ; 3/b4
5      P4/aug3     ; 4/#3
6      aug4/dim5   ; #4/b5
7      P5/dim6     ; 5/bb6
8      m6/aug5     ; b6/#5
9      M6/dim7     ; 6/bb7
10     m7/aug6     ; b7/#6
11     M7/dim8     ; 7/b8
12     P8/aug7     ; 8/#7

Careful examination of the names will show that there is a half-step-wise progression through the names. that is to say you can go by half-steps through dim4-P4-aug4 or dim3-m3-M3-aug3, etc. (major and minor never show up in a perfect interval, perfects never show up in a major/minor type interval.) we do not use names below diminished or above augmented.

Compound intervals
From time to time, usually involving chords or other harmonic structures, you will see intervals with names above 7. For instance the octave 8, is the same note as 1, but an octave higher. 8 = 1+octave, 9 = 2+octave, m9 = m2+octave, M9 = M2+octave, etc. most common compound intervals are: 9,10,11,13.

Interval inversions
If I go fropm the note C up to the note A and find the distance, it is the equivalent of 9 half-steps. We call this a M6. However, if I go from C down to the note A and find the distance, it is equivalent to 3 half-steps distance between them. We call this a m3. And here we have two different intervals created by the same two notes.. we call this behavior/phenomena an interval inversion. when you move one of the notes in an interval an octave from its original position (and on the other side of the note that it is in context with), then you are said to have inverted the interval. Some interesting properties happen in interval inversion. They are the following:

Perfect intervals invert to become perfect intervals
Major intervals invert to become minor intervals
Minor intervals invert to become major intervals
Diminished intervals invert to become augmented intervals
Augmented intervals invert to become diminished intervals
2nd intervals invert to become 7th intervals
3rd intervals invert to become 6th intervals
4th intervals invert to become 5th intervals
5th intervals invert to become 4th intervals
6th intervals invert to become 3rd intervals
7th intervals invert to become 2nd intervals
Octave intervals invert to become unison intervals
Unison intervals invert to become octave intervals
always.

Transposition
Consider the key of C (C major scale). The notes C,D,E,F,G,A,B,C = PU,M2,M3,P4,P5,M6,M7,P8 respectively. Let's now consider all twelve notes relative to C : C, C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab, A, A#/Bb, B, C = 1, #1/b2, 2, #2/b3, 3, 4, #4/b5, 5, #5/b6, 6, #6/b7, 7, 8(1) respectively.
Now let's look at another key, say the key of A. Key of A = A,B,C#,D,E,F#,G#,A = 1,2,3,4,5,6,7,8. All 12 notes in intervals (relative to the root note A) would be: A, A#/Bb, B, B#/C, C#, D, D#/Eb, E, E#/F, F#, Fx/G, G#, A = 1, #1/b2, 2, #2/b3, 3, 4, #4/b5, 5, #5/b6, 6, #6/b7, 7, 8(1).
So the b3 in the key of C is Eb, and in the key of A it is C.

Transposition is a method of changing keys. If we have a melody or something in one key and we want to write it into another key (maybe you're transposing it for another instrument like a horn), we can do so by changing each note to the new key keeping the same intervals for each note.

for example, if we had the notes in the key of C form a melody C-F#-G-E (1-#4-5-3) and we wanted to transpose it to the key of A, we would get A-D#-E-C# (1-#4-5-3). We can figure out how all 12 notes play out in the keys and create a transposition chart to use for easy reference. Here's one on my web page from my book. simianmoon.com/snglstringtheory/guitar/8theory3.html

Roman numeral system
recall that we use roman numerals to denote chords. Uppercase roman numerals denote major chords (e.g. I-IV-V), lowercase = minor chords (i,iv, etc.).
Now, in addition to this in classical theory, the following names are given to each relative note to express the functionality of the chord built off that note.

1 = Tonic
b2,2 = Supertonic
b3,3 = Mediant
4 = Sub-dominant
#4/b5 = Tritone
5 = Dominant
b6,6 = Sub-mediant
b7 = Sub-tonic
7 = Leading note
8 = Tonic

Without getting too deep into this, the tonic is the note/chord whose function is to set up the tonal system. The dominant note/chord has the function of being the most common chord (after the tonic) in the system. The sub-dominat note is the interval inversion of the dominant note with respect to the tonic. So the subdominant-tonic relation is a transposition of the tonic-dominant relation. (and the IV is the 3rd most commonly heard chord in western music, after the tonic and the dominant). The mediant is the note halfway between the the tonic and the dominant. The sub-mediant is the interval inversion of the mediant. The tritone is half-way between the tonic and its octave (the tonic). The supertonic is above the tonic, and the subtonic is below the tonic. the leading note when played in context (try playing it at the same time with the tonic below it) is said to create tension such that it leads your ear back to to the tonic (a half-step above it), hence the name leading note.

Consonnance vs: Dissonance
Consonnance - an interval is said to sound consonnant if it is pleasing to the ear.
Dissonance - an interval is said to sound dissonant if it is not (as) pleasing to the ear.

Of course, the above definitions would seem subjective and based on the nervous system of the listener, but traditionally the intervals are said to have the following characteristics (flavors, moods, etc.):

PU = open consonnance
m2 = sharp dissonance
M2 = mild dissonance
m3 = soft consonnance
M3 = soft consonnance
P4 = consonnace or dissonance
aug4/dim5 = neutral or restless
P5 = open consonnance
m6 = soft consonnance
M6 = soft consonnance
m7 = mild dissonance
M7 = sharp dissonance
P8 = open consonnance

Peace,
Christopher Roberts


How do I change all those numbers to letters (for notes, chords, etc.)? Here's a transposition chart simianmoon.com/snglstringtheory/guitar/8theory3.html

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Last updated December 31, 2002.
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