Harmonization of a Melody in Brief

Context

Suppose you have a melody and you don’t have chords for it. Let’s say you have written a tune starting with melody, perhaps using your voice. You would like to play the tune with chords so you need to figure out appropriate chords. What chords would be most appropriate? How should you do this?

If you know your instrument and you have a good musical sense you might do it through intuition combined with trial and error. However there is a systematic process for doing this. Even if you have good musical intuition, knowing the process may show you overlooked possibilities. I’ll give an outline of this method below.

The formal study of musical harmony is difficult, and not readily available to many. Although a person could presumably profit from such a study, learning many more aspects of how to harmonize; it is probably overkill for someone wanting to do a decent job of harmonizing a tune, but is not going to write complex pieces of music. In addition, it pretty much requires that a person is a fluent reader of western notation, which in itself is a difficult thing to master.

However, I will attempt to give a brief but useful method in this document. It does assume that you have some knowledge of music theory. Not a lot of knowledge, mind you, but some basics. If you don’t have understanding of these you will be lost trying to understand the document.

Prerequisites to Understanding

The basic things you need to understand are:

  • melody
  • musical notes
  • intervals
  • keys, scales, modes
  • degrees of the scale
  • tonics, sub-dominants, dominants, …
  • harmonization of a scale
  • triadic chord types
  • tertian chord construction
  • harmonization of a scale
  • primary triadic chord types
  • secondary triadic chord types
  • seventh chords
  • metre, beats
  • strong and weak beats in a measure
  • musical progression of chords
  • chromatic, diatonic and pentatonic scales
  • major and minor scales
  • intervals
  • scale tonic
  • chord roots
  • chord formulae

This knowledge is available from many sources; in paper books, electronic books and on the internet through numerous web pages and videos. Unfortunately, a lot of it is explained using musical notation, which is a study in itself. Until you know notation, it is hard to make sense of the explanations which use it. In any case, music theory is difficult enough regardless of notation. It is probably best learned with a keyboard in front of you.  At one time I stuck little letter names on my keyboard for each note. This works, although eventually a keyboard player would need to go beyond that. Even a cheap low end or second hand keyboard is a good investment for this purpose, if you do not have one.

First steps

First, let’s assume you have a melody. You need to determine:

  • the notes of the melody
  • the key, scale and the mode of the melody
  • the notes in that key and their degrees in the scale
  • the tonic note of the melody
  • the notes that comprise the triadic chords in that key
  • any notes in the melody that depart from the notes of the key
  • any modulations to a new key
  • the metre of the tune
  • where the strong beats are in that metre
  • the association between beats and the melody

This is probably sufficient information to start with.

Essential to harmonization is creating a piece of music that is enjoyable for you and, it is to be hoped, enjoyable for others.

Two Key Considerations

In harmonization there are two key technical considerations. Firstly: support for a melody note by a chord which contains it. Secondly: the flow or progression of chords through the piece. The context of chords surrounding a given chord and the movement from one chord to another is fundamental.

For the first point, if my melody note for a piece in the key of C major is A, I can put the F major, A minor, or D minor chords under it. The F consists of the notes F, A, C. The A minor consists of A, C, E. The D minor consists of D, F, A. Each contains the note A, so may be used.

So the first rule is that a melody note can rest on top of a chord quite nicely if that chord contains the melody note. The triadic chords contain three notes each and there are seven of them. There are three primary chords which are called major chords, and four secondary musical chords which are minor but one of them is also diminished.

For the second point, if I end up with the chords C major, A minor, D minor, G major, C major in progression, I have a conventional but strong sequence of chords. Any chord can move to any other within a given key, but some combinations will sound better; more interesting, and more pleasing. This is all a matter of taste of course, based on previous experience, with maybe some underlying physics and neuropsychology.

Scale Notes and Chords

The notes of the scale are numbered one through seven. Each of these notes in the scale is associated with a numerical value and a chord. This is shown in the following table for triadic and seventh chords, with examples for the key of C Major.

Scale Degree1234567
Example NoteCDEFGAB
Chord TypeMajorMinorMinorMajorMajorMinorDiminished
ChordI Majii minii minIV MajV Majvi minvi dim
ExampleCDminEminFGAmiBo
Chord TypeMajor 7Minor 7Minor 7Major 7Dominant 7Minor 7min7 b5
ChordI Majii minii minIV MajV Majvi minvi min7 b5
ExampleCMaj7Dmin7Emin7FMaj7G7Ami7B min7 b5

Frequency of Chord Changes

When harmonizing a melody you can probably start by only harmonizing on the beats. You could decide to harmonize all beats but perhaps you could start by restricting chord changes to the strong beats. You may go farther and restrict that just to the first beat. It’ll be up to you decide what sounds best.

You can ignore melody notes that are not on the beat. If you tried to harmonize every note, things would get very messy, and quite often unmusical.

Note that in double and triple metre, the first beat is the strong one. In quadruple metre, the first and third beats are the strong ones. In sextuple metre, the first and third beats are the strongest. In all metres, the first beat is the strongest.

The Notes in a Chord

Below is a chart showing melody notes and triads in the key of C major. This understanding should become engrained for the chords in the keys you want to work in for easier harmonization. It is like learning the multiplication tables, but to my mind harder. I still have to think hard about it for any key but C.

Melody Note C D EF G A B
Triads             
C MajPrimaryC   E  G    
D minSecondary  D  F   A  
E minSecondary    E  G   B
F MajPrimaryC    F   A  
G MajPrimary  D    G   B
A minSecondaryC   E    A  
B dimSecondary  D  F     B
Sevenths             
C Maj 7PrimaryC   E  G   B
D min 7SecondaryC D  F   A  
E min 7Secondary  D E  G   B
F Maj 7PrimaryC   EF   A  
G Dom 7Primary  D  F G   B
A min 7SecondaryC   E  G A  
B min7 b5Secondary  D  F   A B

Matching Chords to Melody Notes

For any given melody note, you will have a choice of three chords to use if using all the triads in the key. It is your decision as to which to use. Clearly this decision can be changed later if you don’t like it. Some sequences of chords are going to be more musically pleasing than others. You get to choose.

To simplify things you can just use the primary chords, that is, the major chords, to do your harmonization. This will give you a harmonized melody. It may be very basic and maybe not interesting to you, but it’ll work.

If you choose to harmonize with all seven triadic chords you may be close to a finished product. This can be tweaked to your taste.

Notes to Chords in C Major

Below is an example, done without using notation, changing chords on the strong beats. The melody is childish, but good enough for my purposes. The possible triads are shown:

Beat1234123412341234
MelodyCDEDABCAFDDBC   
Chord                
1C Em Am C F Dm C   
2Am C F Am Dm Bdim Am   
3F Am Dm F Bdim G F   

You can see that there are numerous ways to navigate the chords in this grid. Which do you choose? Find one that sounds good, and use it. It would take some time to try them all by the way; there are 2,187 possible paths, if I have calculated correctly.

Example of Harmonization with Primary Chords

Here is a possibility using only primary chords, though not the only possible one using primary chords:

Beat1234123412341234
MelodyCDEDABCAFDDBC   
Chord                
1C C F C F G C   

Example of Harmonization with Secondary Chords

Here is another, using only secondary chords, again only one possibility:

Beat1234123412341234
MelodyCDEDABCAFDDBC   
Chord                
1Am Em Dm Am Bdim Dm Am   

Note: A diminished chord is not particularly inspiring, but if you put a B diminished over a G bass note, it becomes a G Dominant 7 chord, which is far more useful, especially if followed by a C Major.

Example of Harmonization with All Triads

Here is a possibility using a mix of primary and secondary chords:

Beat1234123412341234
MelodyCDEDABCAFDDBC   
Chord                
1C Em Am F Dm G C   


Using Seventh Chords

There are other possibilities for harmonization. One is to use seventh chords instead of triadic chords. These contain four notes. You can if you wish take that extra note in a seventh chord and use that as the basis for matching with the melody note. This may give you results you like. If not, don’t do it that way.

After having used harmonization with triadic chords you can also convert the triadic chords to seventh chords. Quite often the fifth chord in the scale is converted to a dominant seventh, particularly if it’s going to be followed by the first chord in the scale. You can experiment.

Here is an example, where on the 3rd measure, 3rd beat I use an E minor7 chord, since it has a stronger transition to the A minor chord. In the 3rd measure, 1st beat I use a B minor 7 flat 5 chord, otherwise known as a half-diminished chord. It is a more interesting chord than the B diminished.

Beat1234123412341234
MelodyCDEDABCAFDDBC   
Chord                
1Am Em Dm Am Bmi 7b5 Em7 Am   

Going Further

There are many other varieties of chords that you can use. Once you have a fundamental harmonization done with triads you should look at other types of chords. These would include suspended 2nd and suspended 4th chords, temporary dominant chords, add chords, altered chords, poly chords, extended chords, and chords borrowed from closely related keys.

Notes Outside of the Key

I mentioned above that sometimes you find notes that are in the melody but not in the key. If these are on a beat, particularly on a strong beat, you can use these notes to make a chord by combining them with notes from the scale. This will probably sound fine although you might have to experiment to get the right chord.

Note that neither D Major nor B flat minor 7 flat 5 are in the original key, and the F# melody note is not in the original key either. This brief change in the note and the chord is called a change of tonicity.

Beat1&2&3&4&1&2&3&4&
MelodyEGEGEGEGF#AF#AF#AF#A
Chord                
1C   Am   D   Bm7   

Modulation to Another Key

A piece might actually exist in more than one key. One section may be in a first key and another in a second key. Treat these as two melodies and harmonize separately.

Bass Lines

You can add more interest to your progression by changing the bass line. The first method is through the inversion of chords, that is, taking a note that is not the root of the chord and making it the lowest note. The second method is to put a note in the bass of the chord that is not from that chord. This changes the chord type, and it may give it an entirely different name in another context. The third method is to add bass notes between the chords, to create what is often called a bass run. These can be notes from the scale of the piece, or chromatic notes. You might add chords to some weaker beats in order to get a more interesting bass line.

In Conclusion

This is pretty much a guerrilla treatment of harmony. It will get you started if you understand it, and will probably take you quite a distance. If you want to really see a deeper level of discussion, you will have to make a more formal study of harmony.

Music Structure and Navigation

Introduction

This document discusses the structural elements found in most popular songs, and the notational devices used to specify the sequencing of a piece, or how to navigate using the sheet music as a map.

Songs in notation format are available over the internet and in song books, but music notation is not something most people are familiar with. Of those familiar with it, there are only some who become fluent in reading it. However, it is possible to learn the sequencing of the parts of the song without really being able to read notation well. If you understand the methods used to document navigation through a song, you can grasp how it is structured, making it easier to learn. You can also make simple maps of the structure as cheat sheets for learning and playing the piece.

Standard Popular Song Structural Terms

There are a number of sections common in popular songs that any aspiring musician or singer should know. These include:

  1. Introduction (or intro)
  2. Verse
  3. Chorus
  4. Bridge
  5. Instrumental Break
  6. Coda (or outro)

In addition, you may encounter additional sections such as solo, pre-chorus and post-chorus. I suppose you could have a pre and post for any of the other sections. I don’t know that I have seen this, but you could keep your eyes open for such things. Not every song has all of these sections of course. Sometimes, these sections are just identified with letters, but more often, the names above are used.

The order of the sections is variable. The intro and the coda provide starting and ending sections, although it is not necessary that a piece has either. After that, the order can be highly variable. A typical form would be:

  1. Intro
  2. Verse
  3. Chorus
  4. Verse
  5. Chorus
  6. Coda

However, any of these elements can be mixed and matched to the taste of the writer.

The melody for the verse is usually the same for all verses, but the lyrics are typically different. The melody for the chorus is usually the same for all choruses, and the lyrics are usually more or less the same for each chorus.

The chorus is usually a more emotive section, and is the one most likely to be remembered. Since this is not a treatise on song writing, I will not discuss this farther.

Navigation

To traverse a piece of sheet music, you need to understand the common notations for navigation found in popular music. These are also used by all music which uses the standard notation system. Since this is not a document on the intricacies of notation, I will restrict the discussion to the devices used in navigation. These come in the following varieties:

  1. Play sequentially straight through with no side trips, which is the default.
  2. Go back to the start and repeat once up to the closing repeat sign, then continue on through.
  3. Repeat some contiguous section between open and close repeat signs, some specified number of times, with one repetition being the default.
  4. Repeat with alternative sections at the end of each repetition.
  5. Go back to the start and play until you hit the finish marker. This is called da capo al fine or go back and play from the start to the finish.
  6. Go to the sign and play until you hit the finish marker. This is called dal segno al fine or go back and play from the sign to the finish.
  7. Go to the start and play until you hit the go to coda sign, then jump to the coda. This is called da capo al coda or go back and play from the start to the coda.
  8. Go to the sign and play until you hit the go to coda sign, then jump to the coda. This is called dal segno al coda or go back and play from the sign to the coda.

These are shown below as flowcharts, and as musical notation. The melodies are silly, but serve as examples.

Deciphering the Structure

With knowledge of the navigational devices used, and the terminology for sections in popular songs, you can decipher and document the structure, as a sort of map to aid in understanding. You must identify the navigational notation, and put labels onto the sections that are revealed. Although you can just use letters, it is better if you can analyse the piece using the common names for sections, such as verse and chorus. The most common terms were presented above.

Any convoluted piece can be flattened out so that everything becomes a linear sequence. Sections that are repeated can be duplicated. However, you don’t want to create notation for the whole piece, only reveal the structure. I recommend that you create a description as a map of the flattened structure. So, if you have a piece that is like this in structure:

You can map it as follows:

  1. intro
  2. chorus
  3. verse 1
  4. chorus
  5. verse 2
  6. chorus
  7. verse 3
  8. coda

Note that the repetition has been removed. This obviously is not a real piece, but it should be sufficient to see how the process works. I have labelled the sections according to what I thought they represented, and then unrolled everything.

This process can be done for any of the navigational indicators. When a piece stretches over a number of manuscript pages, it is hard to understand the mapping, and requires a lot or careful looking at the notation, and a lot of page turning. Once a map is produced, the structure becomes much clearer.

Although some musicians think it is an advantage to have fewer pages with more complex notation, I think that a linear presentation should be easier to follow for a lot of people. Since printed linear song sheets are seldom an option, making a map accomplishes some of the same objectives for the student and the player. When I was teaching students a new piece, I would always create a map for their benefit, and for my benefit.

Here is another more complex example. We have the following piece, where the actual music has not been documented, but the structure is shown using musical notation. I don’t show all of the bars, the notes and the rests for the piece. In a real song sheet, a piece might extend over a number of pages, and it would take some work to find all of the important symbols.

Unrolling the notation, we get this linear structure:

  1. Intro
  2. [A] Verse 1
  3. [B] Chorus & Ending 1
  4. [A] Verse 1
  5. [B] Chorus & Ending 2
  6. [C] Solo
  7. [D] Bridge
  8. [A] Verse 3
  9. [B] Chorus
  10. Coda, Solo twice & Tag

This reveals the essence of the sections and navigation for the piece.

C Major Scales and Zones

In a previous post, https://vorticity-martial-arts.com/similkam/2020/09/03/octave-and-unison-patterns/, I made several points:

  1. We can locate notes, their unison notes and their octave notes on the fret board by following a pattern.
  2. We can do this for any note.
  3. There is a common pattern for all octave and unison notes.
  4. This pattern can be shifted up and down the neck, and rotated at the ends to make the pattern for another note, following the chromatic scale.
  5. Each of these notes can be used in several contexts within a key, and for any degree of a scale.
  6. We can get an insight into the pattern if we regard the notes as the root of some chord or the tonic of some scale. In fact, the note could be any degree of any scale; the pattern would still hold.
  7. Pairs of octave notes above and below one and other can be regarded as delimiters for zones.
  8. These zones are equivalent to the CAGED sequence, so that:
ZoneCAGED
1C
2A
3G
4E
5D
Correspondence of Metafrets Zones and CAGED System

Below, I show the use of each zone to create a scale. These zones overlap slightly. I will use the C note as the tonic and show the C major scale for each zone. Note that I show partial scales below and above a complete octave. The root note of the C Major scale, the tonic note, is in yellow.

I used the freeware programs:

  1. Guitar and Bass, version 1.2.2 by Gabriel Fernandez found at https://www.gfsoftware.com/desktop/index.php to produce the diagrams, along with
  2. Greenshot version 1.2.10 build 6 for screen capture, found at https://github.com/greenshot/greenshot
C Major Scale Zone 1 (Nut)
C Major Scale Zone 1 (Nut) – CAGED C
C Major Scale Zone 2
C Major Scale Zone 2 – CAGED A
C Major Scale Zone 3
C Major Scale Zone 3 – CAGED G
C Major Scale Zone 4
C Major Scale Zone 4 – CAGED E
C Major Scale Zone 5
C Major Scale Zone 5 – -CAGED D
C Major Scale Zone 1 (Octave
C Major Scale Zone 1 (Octave) – CAGED C

Octave and Unison Patterns

This post assumes that you already have some knowledge of:
what a scale is, what an interval is, what an octave interval is, what a unison interval is, what a chord root is, how the ascending and descending chromatic scale works, and how each fret gives a new note on the chromatic scale.

These are explained elsewhere on my site.

What I am adding is how it all works on the fingerboard.

The white circles give the octaves and unison positions for the note C on the guitar fret board when the guitar is tuned in standard tuning.

In standard guitar tuning, the notes that are unison intervals are on adjacent strings, 4 frets up the neck if on the 2nd to 3rd strings, and 5 frets up the neck otherwise. The notes that are octave intervals will not be on adjacent strings. These notes can be the root notes for a scale or a chord. The root note for a scale is usually called the tonic note.

In the metafrets system, each root note pair for scale tonics, moving up the neck, gives a new zone, so zones are 1st, 2nd, 3rd, 4th, 5th, and octave 1st, and so on.

Guitar fret board in standard tuning showing the positions for the note C.

The pattern can move up the neck, giving different scales and different chords, following the chromatic scale notes.

Chromatic Circle
Chromatic Circle

Here is the pattern shifted up 2 frets, giving the note D. Note the D note at the nut. This indicates that the pattern has wrapped around, creating Zone 5 from D at the nut to D at the 3rd fret. Zone 1 starts at D at the 3rd fret. The zones overlap.

D Notes
D Notes – Unisons and Octaves

Arithmetic Inversion and Negative Harmony

January 22, 2020

Arithmetic Inversion

It is possible to translate the notes, scale, melodies and chords of a piece of music in one key into another key so that every note in the second key is an arithmetic inversion of the first. This gives a pairing of notes such that each note in the first key lies the same distance from a mid-point as does the corresponding note in the second key, but in the opposite direction. So for instance, assume the mid-point is the note G. The note C is 5 semi-tones above G. The note D is 5 semi-tones below G. See the table below:

NotesDD#EFF#GG#AA#BC
Intervals54321 12345

This procedure is followed for every scale note in the original scale, to produce an arithmetically inverted result, in some other key, or even in the same key.

Negative Harmony

This procedure is called negative harmony by some music theorists. It is being used to transform music, melodies and harmonies. I think it came to my attention first through an Internet teacher named Tommaso Zillio, but maybe it was Rick Beato. This information led to a video by musical prodigy Jacob Collier, and from there to a quick look at extracts of writings by Ernst Levy. What I saw in the latter was too difficult. Other sources gave simpler explanations, but I did not find any that really gave a full picture. There was always some hand-waving going on. In this document, I have tried giving a simpler picture of how it all works,  but explain what is really going on.

Whether or not this is really musically worthwhile is another matter. It is still something that I am exploring.
Chromatic Notes with Enharmonic Equivalents

There are 12 notes in the chromatic scale, with a semi-tone distance between notes. Some note pairs are enharmonic, that is, they are named differently but sound the same.

Chromatic Scale DegreeNotes
1C
2C#/Db
3D
4D#/Eb
5E
6F
7F#/Gb
8G
9G#/Ab
10A
11A#/Bb
12B
1C

Chromatic Circle

The notes of the chromatic scale can be organized as a circle. This shows the repeating nature of the names of the notes on different octaves. It is often more convenient to use the circle than a list of notes. The circle can be read in a clockwise direction for an ascending scale, and in a counter-clockwise direction for a descending scale. No note has a privileged position as a scale root note.


Counting Intervals in Semi-tones

We can use the chromatic circle to determine the interval size between any two notes in semi-tones. We can count clockwise (ascending) or counter-clockwise (descending) and in general get different results. It is a good practice to always specify whether your count is ascending or descending, although I think people default to ascending if the direction is not specified.

Example: Ascending Chromatic Intervals from note C in Semi-tones

TonicNoteInterval (semi-tones)
CC0
CC#/Db1
CD2
CD#/Eb3
CE4
CF5
CF#/Gb6
CG7
CG#/Ab8
CA9
CA#/Bb10
CB11
CC12


Diatonic Major Scale Degree Names

We can take a seven note subset of the chromatic scale, starting on a note called the tonic or scale root, and choose notes with a specific pattern of intervals between them. This pattern consists of full-tones (T) or semi-tones (S), and is called the major scale in the following interval formulation: T T S T T T S. This pattern can be rotated to get different scale root notes, seven rotations are possible. Each rotation is called a mode of the scale.

DegreeNameExample Using Notes of C Major
   
1TonicC
2SupertonicD
3MediantE
4SubdominantF
5DominantG
6SubmediantA
7Leading toneB
8OctaveC


Keys Sorted by Intervals of Ascending 5ths

We can create a scale on 15 different notes from the chromatic scale. Each is called a key. We can order the scales with different tonic notes according to the accidentals they contain, and also by going up 5 letter names and 7 semi-tones for successive roots. Several of the keys have different letter names, but are identical in pitch.

Ascending Sort OrderTonic of Major ScaleAccidental Count
1Cb7b
2Gb6b
3Db5b
4Ab4b
5Eb3b
6Bb2b
7F1b
8C 
9G1#
10D2#
11A3#
12E4#
13B5#
14F#6#
15C#7#

Circle of 4ths and 5ths

We frequently represent this series of keys as a circle. We follow the same pattern as in the preceding table, but since it is a circle, we can traverse in a clockwise or counter-clockwise direction.

General Inversion Rules for Any Key Pair

We now come to arithmetic inversion of notes from a key. We establish two keys, an original key and a resulting key. We apply the following rules to each note of the original scale to get the result. These rules are independent of the keys chosen. They produce a result in which the distance of paired notes around a mid-point is the same in size, but opposite in direction. With these rules, we don’t actually have to know the mid-point, which depends on the interval between the key tonics. Conceptually, we are creating the inversion around a mid-point, but the rules give the correct result without counting intervals. The mid-point depends on the interval between the two keys, just as though you were transposing one into the other. However, arithmetic inversion is not transposition. Not the intervals, TTSTTTS are mirrored across original and result. It has to be this way.

Rule NumberOriginal Diatonic Major Scale – Degrees AscendingInversion Result Diatonic Major Scale – Degrees Descending
113
222
331
447
556
665
774
883


The General Midpoint Formula

If you really want to know the mid-point, you can compute it as follows:

Ascending Mid-point with Respect to Original Tonic

= (Ascending Semi-tones Above Original Tonic + 4) / 2

There is actually another possible mid-point, at the exact opposite point if you have put it on a chromatic circle. The calculation is as follows:

Opposite mid-point = midpoint + 6 if original mid-point not greater than 6

else opposite mid-point = mid-point – 6

Rational for Mid-point Calculation

The original tonic to the resulting tonic is the interval for straight transposition.

However, the match, according to rule 1 for inversion, is not tonic to tonic, but tonic to mediant, that is to degree 3. The rule says that the tonic of the original scale transforms into the mediant of the resulting scale. The mediant of the resulting scale after inversion is 4 semi-tones above the tonic of the resulting scale. Hence, we add 4 semi-tones to get the tonic to median match. Then, we divide by 2 to get the distance of the mid-point for inversion for that pair of keys. Notes are arithmetically inverted about that point, or equivalently, the point opposite that on the chromatic circle.

Example: Keys in Ascending Order of Distance from Key of C for Transposition

Below I calculate the mid-points for the key of C Major with respect to the 15 keys in the circle of 4th/5th. I show the distance from tonic to tonic in semi-tones for each key. I determine the mid-points using the general mid-point formula.

Tonic OriginalTonic of inverted resultSemi-tonesAscending Mid-point with Respect to COpposite Ascending Mid-point with Respect to C
CC#12.58.5
CDb12.58.5
CD239
CEb33.59.5
CE4410
CF54.510.5
CF#6511
CGb6511
CG75.511.5
CAb8612
CA96.50.5
CBb1071
CB117.51.5
CCb117.51.5
CC1282


Original Key to Resulting Key Tonic Intervals and Mid-points

This table gives the two mid-points for any interval between any two keys, without you having to do the calculation. It is based on the calculation formulae.

Ascending Semi-tones for TranspositionAscending Mid-point with Respect to Original TonicOpposite Ascending Mid-point with Respect to Original Tonic
12.58.5
239
33.59.5
4410
54.510.5
6511
75.511.5
8612
96.50.5
1071
117.51.5
1282


Example Ascending

Here is an example of a transformation from the key of C Major to the key of Eb Major, using the rules. The transposition distance is 3, counting clockwise around the chromatic circle, and by table lookup, we see that the Inversion Mid-points are 3.5 and 9.5

C MajorEb Major
AscendingDescending
CG
DF
EEb
FD
GC
ABb
BAb
CG


Inverting the Chords

We can invert melodies and chords both by inverting the notes. With chords, we have to remember that the root of a chord in the original becomes the 5th of the chord in the inversion, and the 5th of the chord becomes the root. Major chords become minor, minor chords become major, and diminished chords stay diminished. Here is an example of scale harmonization for C Major inverted to Eb Major. It is best to do this a note at a time.

C Major OriginalC Major Inversion Result
Chord DegreeChord NameChord NotesChord NameChord Notes
ICC, E, GC min/GG, Eb, C
II minD minD, F, ABb/FF, D, Bb
III minE minE, G, BAb/EbEb, C, Ab
IVFF, A, CG min/DD, Bb, G
VGG, B, DF min/CC, Ab, F
VI minA minA, C, EEb/BbBb, G, Eb
VII dimB dimB, D, FD dim/AbAb, F, D


Graphical Method

The chromatic circle can be used as a graphical method of doing this inversion. It demands that you have a mid-point (which is unlikely), or that you have one pair of original and inverted notes, along with a starting key. The rule set is easier to use, but the graphical approach can give you a visual appreciation of the technique.

Start with a chromatic circle, determine your keys, original and inverted, and use the rules to get the first pair. Draw a line from original to inverted note, and then make lines parallel to this for the rest of the pairs. Here is an example, going from C Major to A Major; start with E to A for instance:

Note that some pairs have nothing to do with either key. They are ignored. The mid-points are between quarter-tones and are joined with a line. Here is the result in tabular format:

Scale DegreeOriginal Key of C MajorScale DegreeInverted Result in A Major
1C3C#
2D2B
3E1A
4F7G#
5G6F#
6A5E
7B4D
8C3C#