An introduction to Inversions

I spoke of intervals in the previous article. I will explain inversions in this article. A firm understanding of these concepts will help in grasping harmonic principles much better.

Take the interval from C to E. This is a major third interval. 

What about the interval from E to C? Applying the principles of the previous lesson tells us that it is a minor sixth. 

The second interval is an inversion of the first. 

Consider F to Bb - a Perfect Fourth.

And Bb to F - This is a Perfect Fifth.

To help in doing a quick math of this, the way it works is as follows:

An interval and its corresponding inverted interval always add to 9. (Inversion of 3rd is a 6th, inversion of 4th is a 5th), A major interval inverted becomes a minor interval. A perfect interval remains perfect, and a diminished interval inverted becomes an augmented interval - as simple as that. 

Take a quick exercise to figure if you got it right. What is the inverted interval of

1) Major Second

2) Major Sixth

3) Augmented Fourth

4) Perfect Fifth

While the harmonic relevance of this is not immediately relevant, get it in clearly. Trust me, it is important.
This has been one short article - I wrote it while I was waiting for a flight in the airport on Free Internet :)

In the next article, I step into chords. Did I hear you say "At last!!". Patience, my friend.

Article 4: Understanding Intervals : Stepping into Harmony

Understanding intervals is fundamental to understanding harmony. I have already touched on intervals in Articles 2 and 3 and have spoken about three kinds of intervals

1. Major Second
2. Minor Second
3. Augmented Second

If you notice there are two parts to the naming of the interval 

"Major" and "Second"

"Minor" and "Second"

"Augmented" and "Second"

All intervals will have this kind of a naming convention - the first refers to the quality of the interval, and the second to the quantity. The interval from Sa to Ga for instance, is a third. Depending on whether it is Ga1, Ga2 or Ga3, the quality of the interval changes.

Sa - Ga2 - Minor Third

Sa - Ga3 - Major Third

I missed Sa- Ga1 there for a reason. This introduces the fourth 'quality' called Diminished. Sa - Ga1 is called a diminished third. A question that pops up obviously is "Hey, Sa - Ga1 is the same as Sa - Ri2. Wouldnt that be a major second?". This answer is much easier for a carnatic musician to understand than for a beginner in western music. Just like the same note is referred to as Ga1 or Ri2 depending on the context in the raag, the interval is also referred to as either major second or diminished third depending on context. That said, diminished third is an interval that will rarely come up. (but it serves to illustrate this point nevertheless).

Lets flip over now to western notation and examine this concept of intervals. The common intervals that pop up in study are as below. I have illustrated in the key of C, but as I have mentioned before, there is nothing special about the key of C.

C- Db : Minor Second (1 semitone)  => Sa - Ri1

C - D : Major Second  (2 semitones)  => Sa - Ri2

C - D# Augmented Second (3 semitones) => Sa - Ri3

C - Eb : Minor Third (3 semitones) => Sa - Ga2

C - E : Major Third (4 semitones) => Sa - Ga3

C - Ab : Minor Sixth (8 semitones) => Sa - Dha1

C - A : Major Sixth (9 semitones) => Sa - Dha 2

C - Bb : Minor 7th (10 semitones) => Sa - Ni2

C - B : Major 7th (11 semitones) => Sa - Ni3

You will notice that I have omitted Sa - Ma and Sa - Pa. This is because the naming of these intervals is different. They are referred to as perfect intervals. 

Sa - Ma1 : Perfect Fourth

Sa - Pa : Perfect Fifth.

Corresponding to Ma2, obviously we have an interval which is

Sa - Ma2 : Augmentted Fourth

So the interval between the Sa and the Ma of Kalyani would be called an Augmented Fourth or a Tritone. 

Let us extrapolate this a little bit more. Move to Mayamalavagowla raagam.

Let us analyze the intervals between different pairs of notes to get a firm understanding of the interval concept. The Raag notes are of course

Sa - Ri1 - Ga3 - Ma1 - Pa - Dha1 - Ni3 - Sa

Let me use the note D as Sa, in order to illustrate this (I am assuming you will now be able to figure out the notes of D major based on previous lessons)

Sa - Ri : Minor Second (D - Eb)

Sa - Ga : Major Third (D - F#)

Sa - Ma : Perfect Fourth (D - G)

Sa - Pa : Perfect Fifth (D - A)

Sa - Dha : Minor Sixth (D- Bb)

Sa - Ni : Major Seventh (D - C#)

Sa : Sa : Octave (D - D)

Now let us analyze intervals between random notes (not starting only with Sa as we have done so far)

Ri - Ga : Augmented Second (Eb - F#)

Dha - Ni : Augmented Second (Bb - C#)

Ri - Ma : Major Third (Eb - G)

Ga - Dha : Diminished Fourth (F# - Bb)

Dha - Sa : Major Third. (Bb - D)

I am hoping your are getting my drift here. Intervals, as I have mentioned before is the distance between any two notes, and the above set of examples clearly illustrates the names corresponding to the intervals in various contexts. What does this have to do with harmony? Everything. If you dont understand intervals inside out, you are going to be completely lost when we talk chords and chord progressions and passing notes and suspensions. I am going to spend one more article talking about intervals, so we are not done yet :)

Article 3: The minor scale and a bit more

I dropped the last article off with a plan of talking about Keeravani and how other ragas can be mapped onto the western scale. Here goes.

Keeravani's notes translated into intervals are
Sa - Ri2 - Ga2 - Ma1 - Pa - Dha1 - Ni3 - Sa
    IIM   IIm     IIM     IIM  IIm    IIA    IIm

There is a new interval here that I have called IIA. This refers to what is called an Augmented second interval. This is equal to 3 semitones. As a quick refresher, IIM is also called a tone, and IIm is also called as a semitone.

So the interval-lic pattern for the Keeravani raag is
IIM - IIm - IIM - IIM - IIm - IIA - IIm.
Starting with any note on the scale, if we build using this interval pattern, the Keeravani raag will result.

Starting with C for example, this would be
C - D - Eb - F - G - Ab - B - C

Starting with D, this would look as follows:
D - E - F - G - A - Bb - C# - D

This scale pattern is called the minor scale in western music.
So when a piece of music is written in C Major, it comprises the following notes
C - D - E - F - G - A - B - C

When a piece of music is written in C minor, it comprises the following notes
C - D - Eb - F - G - Ab - B - C

With this background, it becomes obvious that any raag (without the gamakam aspect) can be fitted onto the standard piano. Take the root note (Sa) and build the intervals on top of it.

To illustrate, Kalyani, starting on the C note as Sa, would be as follows:
C - D - E - F# - G - A - B - C
The F# is the equivalent of Ma2. Note that it is 3 tones away from C (Sa). This is referred to as a tritone in western music.

The fundamental way in which western music differs from carnatic is that carnatic music is largely horizontal - melody driven. Western music has two dimensions - horizontal and vertical (melody and harmony).

The basic elements of harmony will be the contents of the next article.

Article 2 - western music for the carnatic aficionado

Abstract
In this article, I will look at the way notes are represented in Western and how it maps into carnatic. At the end of this article, you should know how to take the sankarabharanam raaga and map it to any major scale. Along the way I will introduce the concept of intervals.

Warning
I am intentionally not spoonfeeding in this series of articles, since I assume you understand a branch of music fairly well, and can adapt using your way of thinking about and understanding music.

And we begin...
There are 12 notes in western music corresponding to the following
Sa - Ri1 - Ri 2 - Ga2 - Ga3 - Ma - Ma2 - Pa - Dha 1 - Dha 2 - Ni 2 - Ni3
Assuming for the moment, that the Sa corresponds to C, the corresponding equivalents would be
C - C# - D - D# - E - F - F# - G - G# - A - A# - B

The # is read as a sharp. So C# is C sharp.

Lets now pick and map the notes of Sankarabharanam
Sa - Ri2 - Ga 3 - Ma - Pa - Dha 2 - Ni 2 - Sa
From the above grid, these map to
C - D - E - F - G - A - B - C

To progress any further, it is important to introduce intervals. The interval is the difference between 2 notes. There are two intervals we must understand to be able to comprehend how scales work. These intervals are
- Minor second
- Major second

Examples of minor second intervals are

• Sa to Ri1
• Ga3 to Ma
• Pa to Dha1

Basically the distance between any two notes that are right next to each other is a minor second.

In the western equivalent, the same interval examples (assuming Sa as C) would become

• C to C#
• E to F
• G to G#

A major second is 2 minor seconds. Examples of major second intervals would be

• Sa to Ri2
• Ri2 to Ga3
• Dha 2 to Ni 3
  

The same examples translated to western music (assuming Sa as C) would be

• C to D
• D to E
• A to B

Note now that the concept of intervals is absolute. The interval between a C and a C# is ALWAYS a minor second interval. This is regardless of whether the Sa maps to a C or to a D or to anything else. Because C and C# are fixed frequencies (or multiples thereof in the case of octaves), the interval (which is essentially a ratio of pitches) remains constant.

The notations IIm and IIM are used to denote the minor 2nd and major 2nd intervals. Let us now take Sankarabharanam and understand how it maps to the interval pattern above.

Sa - Ri2 - Ga2 - Ma - Pa - Dha 2 - Ni 3 - Sa
IIM IIM IIm IIM IIM IIM IIm

This interval scheme gives us a clue on how we can start Sa on any note and successfully create the rest of the major scale successfully (remember that the major scale in western equates to sankarabharanam in carnatic).

Before we plunge into that, a short exercise is needed to ensure you get the western intervals down in your head. Figure out the intervals between the following sets of notes. Answers right at the end of this article
SELF CHECK 1
1) C - D
2) E - F
3) G - A
4) D - D#
5) E - F#
6) B - C
7) G - G#

If you couldnt get all of these right, go back to the beginning of this article, and get the mapping of carnatic to western clearly in your head.

The interval of IIm is also called a semi-tone(S), and the interval of a IIM is also referred to as a tone (T).

The intervals for a sankarabharanam therefore can also be represented as

Sa - Ri2 - Ga2 - Ma - Pa - Dha 2 - Ni 3 - Sa
T T S T T T S

The sankarabharanam scale is therefore built on the following interval sequence
TTSTTTS (Tone - Tone - SemiTone - Tone - Tone - Tone - Semitone)

Now this forms the basis of creating a major scale starting on any note.
Lets start with a C. Built on this interval pattern, we have the following notes emerging
C - D - E - F - G - A - B - C
This, then is the C major scale.

Start with a G and the following emerges
G - A - B - C - D - E - F# - G
This constitutes the G major scale.

Start with an F, and we get the following
F - G - A- Bb - C - D - E - F

I know I sneaked in a notation there that you probably were a little lost on - Bb. The b is read asa a flat. So a Bb would be a B flat. It is exactly the same as an A#. Just like you have an overlap between Dha3 and Ni2, so too there are overlaps in western music. However, while you cannot refer to the Bb in the F major scale as an A# even though the note is the same, and the sound it makes is the same. This is because in a scale, each of the 7 notes must have a representation (and certain other reasons which we wont get into now). Since A already has a representation, calling Bb as an A# would lead to two representations for A, and none for B, which is not ok.

If you have got this far, you should be able to figure out this next set of exercises to test your own understanding.

SELF CHECK 2
Work out the notes of the following major scales
1) D major
2) A major
3) Bb major
4) Eb major

Answers are at the bottom.

In the next lesson, we will explore how the Keeravani maps on to the western scale, and proceed for there to how we can get just about any raag mapped out.

ANSWERS TO SELF CHECK 1
1) C - D :: IIM
2) E - F :: IIm
3) G - A :: IIM
4) D - D# :: IIm
5) E - F# :: IIM
6) B - C :: IIm
7) G - G# :: IIm

ANSWERS TO SELF CHECK 2
1) D major :: D - E - F# - G - A - B - C# - D
2) A major :: A - B - C# - D - E - F# - G# - A
3) Bb major :: Bb - C - D - Eb - F - G - A - Bb
4) Eb major :: Eb - F - G - Ab - Bb - C - D - Eb

Article 1: First steps into western music theory - For the carnatic aficionado

I am hoping this is the beginning of a series of articles on the rudiments of western music theory. It assumes the reader understands carnatic music. This series starts as a result of a conversation with Harish Ganapathy, who figures on my top 3 list of carnatic singers. (I will post a link into a few of his songs shortly :)

I am not an expert at all on Carnatic music. My knowledge of it barely scratches the surface. But I believe I know the basics enough to take the carnatic expert through the basics of western music, and from there you (the carnatic music expert) are on your own.

The 7 basic notes of music remain intact, of course. Different singers pick Sa's at different pitches convenient to them. Basically they are picking a Sa at a frequency that is comfortable to them given the range of their voice. [For the violin player it makes no difference whatsoever, and so they can tune their Sa to whatever is the singer's comfort].

In western music, the concept of Sa is a bit more nebulous. Western music, as most of you probably know has the 7 notes A, B, C, D, E, F and G. Each of these notes maps to a particular fundamental frequency. So one carnatic singer's Sa could map the the note C. And another singer's Sa could map to the note D. Lets say these singers are singing Sankarabharanam. The first singer would then be singing the scale of C Major, and the second singer would be singing the scale of D Major.

Which brings us right over to the concept of scale. The scale concept should not really be confused with the Raga concept of carnatic music. There are only 2 well known and often used scale modes in western music - the major and the minor. These correspond, (for sake of understanding) to Sankarabharanam and Keeravani. Any piece of music in western music is written in either a minor scale or in a major scale.

The Sa of the scale determines what that scale is called. If the Sa is E, then the scale is E major or E minor. If the Sa is G, then the scale is G major or G minor. Within the same performance, singers and instruments effortlessly move between different scales for differnt songs. Even during a single song, the Sa can change from E to G. That means that a song in E major can quite possibly move to G major. (E and G are examples to illustrate. They are not necessarily good movements!). Even within the same piece of music, you can move from G major to E minor. These are all just movements of notes, and western music is extremely flexible in this respect. There is hardly a major piece of music in western classical music which begins and continues right through till the end on the same scale.

Compare this to Carnatic, where a song starting in Sankarabharanam in a Sa mapping to E, will always stay fixed between the defined 7 notes until the end of the song (unless it is a ragamalika or something, which is comparatively much rarer in carnatic music, anyway).

In the next article, I wll talk about the individual notes in western music and how they map to carnatic. That will give a clearer understanding of how the western music performer plays Keeravani starting from G, or Sankarabharanam starting from A, or extrapolating it further, how they play a Kalyani or a Shanmugapriya. The basics are the same, the fundamentals are the same, but the way of reading, the way of writing, and the way of understanding is a little different.

Two voice counterpoint - 2:1

With the appearance of passing notes, the detection of parallel 8ves and 5ths becomes a little more complex. Guiding rules are
- Open : Parallel with a passing note in one voice. This is bad.
- Symmetrical : Parallelism on strong beats is bad. On weak beats is possible
- Asymmetrical : Parallelism not in corresponding places in the mesaure - 8ve in strong beat followed by an 8ve on a weak beat. This is possible and good.

If one of the perfect 5ths is used as an unessential, then that is ok.

Two voice counterpoint - basics

The broad principles are
1) Each line must be good in itself.

2) The lines must be independent in terms of direction and rhythmic motion.
- Avoid parallel unisons, fifths and octaves
- Avoid moving to a unison, perfect fifth or octave in similar motion (direct unison, 5th or octave). Exception is movement to octave when the lower voice moves in a strong root progression.
- Avoid movement from a diminished 5th to a perfect 5th. The resolution of the diminished 5th here is incorrect.
- As a general rule, the same interval must not be used more than 3 or 4 times in succession.

3) They must have enough in common so they will fuse into a convincing whole when combined.
- Voices must not be too divergent in terms of rhythmic patterns.

4) Yhe lines must imply good harmonic succession
- Choice of chord tones is important. Given that all 3 chord tones cannot appear in 2 voice counterpoint, which chord tones should be chosen? Analysis will show that it is best to have the root and the 3rd chosen. (3rd and 5th will lead to a confusion of whether the chord is I or III. 1st and 5th will not make it clear what the color of the chord is - major or minor). That said, exceptions are possible. If the 5th is present in a voice, and tonic harmony is called for, then 3rd and 5th or 1st and 5th (as context dictates) can be used. In fact 5th in lower voice, and 3rd in upper voice, also can be used quite satisfactorily.
- Doubling: Avoid doubling 3rds of chords at accented beats. However, this is still ok, of both voices move stepwise in contrary motion.
- Choice of harmonic progression: Progressions that are not good in 4 part are not good here either.
- The second inversion chord is usable here in cadential, passing, embellishment or chord repetition. Since the lower voice in 2 voice counterpoint sounds like a real bass, the second inversion occurs whenever the 5th of a chord occurs in the lower voice. If the above conditions are not satisfied, it must not be used. Never jump to or from the bass of a second inversion, other than in IIb-Ic-V-1 or in chord repetition (Ib - I - Ic). IIc, IIIc and VIc are useless. If the principles here are stuck to, second inversions on two different scale degrees can never appear successively.
- Harmonic Rhythm : To keep a sense of motion, it is required that the bass is not retained from weak to strong beat. This is because the bass must provide a sense of motion forward at these points. Also, a changing bass with no change in harmony sounds static. However it is less objectionable than a non-changing bass.
- Cadences : Authentic cadences must imply one of the following progressions
Ib-IV-V-I
Ib-Ic-V-I
Ib-iib-(Ic)-V-I
The dominant chord must always be employed in root position. The Ic (if used) must always be on a strong beat.

5) The voices must be primarily consonant, dissonance being present, but lesser than consonance :
- The most usable intervals are 3rd, 6th and 8ve. Perfect 5th sounds bare, and if used, is quickly followed by the 3rd of the chord. Perfect 4th is extremely rare as an essential interval (top-heavy in sound, and also implies a second inversion possibly at inappropriate places). Diminished 5th or Aug 4th are commonly used, mostly as representations of the dominant 7th. 2nd and 7th are also not frequently used as essential intervals.

Mediant Triad

Complexity : Intermediate

This is the III chord. Its nature is minor in the major scale and augmented in the minor scale. (and major in the melodic minor context).

USAGE
- The chord may be preceded and followed by chords that have a note in common with it. V-III-VI is a useful progression.
- IIIa can also be used to harmonize the leading note in a descending scalic passage, preceded by I or VI and followed by IV.
- Descending scale passages can also be harmonized with a series of first inversions of which IIIb may be a component.
- An extremely important use of IIIb occurs when the melody ends in a mediant to tonic. The best harmonization here is IIIb-I for a perfect cadence (or IIIb-VI/IIIb-Ib for an intermediate cadence). Here IIIb is used as a substitute for Va.

Elementary Modulation

Modulation is passing from one key to another. There are two essentials
1) A cadence in the new key (usually some form of perfect)
2) The appearance of at least one note which belongs to the new key but not to the old.

Modulation may be gradual or abrupt.

Gradual Modulation
Gradual modulation passes through a pivot chord ( a chord that is common to both the scales).
- If the only pivot chord available is the root of the new key, then it should appear in a difference inversion from that used in the cadence.
- Pivot chords in second inversion should be avoided.
- If there is a chromatic alternation in a note as a result of modulation, then care must be taken that both these notes do not occur in the same part with just one note intervening. It gives an ugly effect.

Abrupt Modulation
This is also referred to as Transition. A chord present in the old key but not in the new, is abruptly followed by a chord that is present in the new key but not in the old. Both these chords will have a note in common. There will generally be chromatic movement in one of the parts.
- BE CAREFUL WITH DOWNWARD CHROMATIC MOVEMENTS: Chromatic movement up is common. Chromatic movement down is only good to the 7th of a dominant seventh or from major 3rd to minor 3rd of the same chord.
- AVOID FALSE RELATIONS. In dealing with transitions, false relations must be avoided. This is is the bad effect experienced when the a note in one part appears chromatically altered in another part.
- EXCEPTIONS TO FALSE RELATIONS: There are two cases where apparent false relation has no bad effect.
:: If the root of the second chord is the 3rd of the first chord. (eg. moving from C major to E major chord).
:: If the second of the two notes concerned is the 7th of a dominant 7th. (eg in modulating from C to F7, the E is one part of C and Eb in another part in the F7 is ok).

Escape Note

Complexity : 3/10 (Simple to Intermediate)

An escape note (or echapee) is a rare example of a non-harmonic tone that resolves by leap.

It is approached by step and then skips a 3rd in the opposite direction. It is, therefore, a non-harmonic tone that occurs between two consonant tones a diatonic step apart.

RULE
- They are never accented.

Accented Passing Note

An accented passing note is one which occurs on the strong part of the bar or beat. It is more strongly accented than the note to which it proceeds.

GENERAL
- Accented passing notes ascending need care. They should pass between the root and the 3rd or the 3rd and the 5th of the same chord.
- Two parts may take the same accented passing note, an octave apart.
- In a descending scalic passage, prefer accented passing notes to unaccented. Of course the strength of harmonic basis is a key factor.
- It is possible to have an accented passing note succeeding an unaccented passing note in a descending scalic passage.


RULES
- Rules of movement as they apply to unaccented passing notes also apply here.
- An accented passing note and its note of resolution may not be struck together unless the latter be in the bass.
- Accented passing notes in the bass, to or from the bass of a second inversion must be avoided.

IDENTIFICATION
- When a note leaps, it must invariably be treated as a harmony note. A note that moves by step may be a non-harmony note. This applies in any part. This can help at times in identifying the presence of an accented passing note.

Dominant Seventh

Complexity : 4/10 (intermediate)

In C scale, this is G-B-D-F.

The V7 chord is unstable and actively seeks resolution. The main motive for this instability is the diminished fifth interval between the 3rd and the 7th, which seeks resolution to a major 3rd or minor 3rd, with the 7th moving down, and the 3rd moving up.
A strong discord also exists between the root and the 7th and the 7th falling a step to its resolution makes the chords of resolution either I or VI.

GENERAL
- In root position, the fifth may be omitted.
- The inversions of the dominant seventh must be complete.

RESOLUTIONS
- Best :: V7-I, Good :: V7-VI, Acceptable :: V7-IVb
- The 3rd of the chord (being the leading tone of the scale) has a strong tendency to move to the tonic center (or root). This combined with the strength of root movement by 5th provides for a strong resolution tendency to I, and makes V7-I a powerful perfect cadence.
- Ornamental resolution is occasionally used. In this, the 7th moves either to the the 5th or the root of the chord before finally resolving. Resolution to IVc is also used as a decoration to the perfect cadence (the progression being V7-IVc-I).
- The inversions will resolve only to the tonic chord.
- The seventh may be transferred from one voice to another, resolving in the latter. In this case, the part which originally had the 7th must fall.

USAGE
- Broadly speaking, any chord which involved some form of V-I can be harmonized by some form of V7-I. In most cases, though, it is more fluent to introduce the seventh as a passing note.
- When the fourth note of a scale is repeated (or tied) weak to strong and falls a step, the strong note is to be harmonized as V7d.
- If a part leaps from the leading note up a diminished fifth and then falls a step, the leap is from the 3rd to 7th of a dominant seventh.
- If a part leaps up a 7th from the dominant and falls a step, the leap is from the root to the 7th of the dominant.
- Do not use V7 as the a substitute for V as the second chord of an imperfect cadence. The cadential effect is destroyed.

CADENCES
- V7-I is a perfect cadence. V7-VI is an interrupted cadence.
- V7b-1, V7c-I and V7d-Ib can be used as inverted perfect cadences. V7d-Ib is the most common, but all are good.
- In harmonizing a cadence involving V7-I, introduce the 7th as late as possible. Otherwise the cadence tends to feel unnaturally drawn out.

Unaccented Passing Notes

Complexity : 3/10 (simple)

A passing note is termed 'unaccented' when it is on a weaker beat or part of beat than the note that precedes it.
- Two passing notes may be used simultaneously in different parts, provided they move in parallel 3rds or 6ths. (called unessential 3rd or 6th). The movement MUST be parallel. Unessential 3rds or 6ths must NOT be quit in contrary motion.
- Two parts may take the same passing note simultaneously in contrary motion, at least an octave apart.
- Passing note and harmony note should be struck together only if the form a concord (3rd, 5th or 6th).
- Avoid passing notes that produce a minor 9th above the 3rd or 7th of the scale.
- Introducing passing notes increases the danger of introducing incorrect harmonic progressions (consecutive 5ths or 8s, exposed 8s etc). Parts that are incorrect without passing notes are incorrect with them. In addition, passing notes may also produce consecutive 5ths or 8s which also must be avoided.
- Never have more than 2 passing notes simultaneously. It creates congestion.

Second Inversion of Triads

Complexity : 2/10 (simple)

A chord is said to be in its second inversion, when the 5th from the root is in the bass.
- Two types need to be considered - cadential and passing.

Cadential Second Inversion
Only 2 are possible - Ic and IVc. They move to V and to I respectively on the same bass. The resolution must not be more strongly accented than the cadential second inversion.
These chords should NOT be approached by leap from the first inversion of another chord.
Uses of Ic
- This can be used as the first chord in a half-close, Ic-V.
- This can precede V in an interrupted or a perfect cadence. Of course, rhythm considerations set above must be met.
- Dominant of scale in S or B, lasting for 2 or more beats at a half or full close, must be harmonized as Ic-V, to avoid too long a halt.

Use of IVc Cadential
This may be used to decorate a perfect or a plagal cadence provided the final tonic bass lasts at least 3 beats. Its use is not very frequent.

Passing Second Inversion
Only IC and Vc are possible. The bass is approached and quit by step and the second inversion is always on the weak beat. Ic lies between IVa and IVb, and Vc will lie between Ia and Ib.

Stock treatments

Complexity : 6/10 (intermediate to high)

- If the bass has the first 3 notes of the scale, I-VIIb-Ib is good. If the 2nd is unaccented, I-Vc-Ib is a better option. (V7c is also fine). If the 2nd is accented though, Vc cannot be used and VIIb is better.
- 4th, 5th and 6th in the B or S can be harmonized effectively as IVa, Ic, IVb provided the 5th is on a weak beat.
- 5th, 4th and 3rd notes in the bass may be harmonized as Ic-V7d-Ib, provided the 4th is weaker than the 5th.
- When the B moves by a step, it is effective to make the S move by a 3rd is contrary motion if possible. (The reverse is also true). This is definitely possible in the case of step-wise motion from 7th to tonic (Vb-I), Tonic-2nd (VIb-II), 3rd to 4th (Ib-IV), 4th to 5th (IIb-V).

First Inversion of Triads

Complexity : 2/10 (simple)

When the 3rd from the root is in the bass, the chord is said to be in its first inversion.
- Doubling:
Major first inversions : The 3rd or 6th above the bass may be doubled. Avoid doubling the bass.
Minor first inversion: Any of the 3 notes may be doubled, but it is best to avoid doubling the bass here too, unless it is unavoidable.
Diminished first inversion: The bass may be freely doubled.

Cadences

Complexity : 2/10 (simple)

Cadences are points of rest - musical punctuations.

There are 4 types of cadences - perfect, imperfect, interrupted, plagal.

Perfect Cadence (full-close): V followed by I. This is the musical equivalent of a full-stop. It should be arranged rhythmically from weak to strong.
Inverted perfect cadences are also possible. V7b-I, V7c-I, V7d-Ib. The last is the most common, but all are good. These can be used for intermediate phrase ends.

Plagal Cadence : This is IV followed by I, and is another form of final cadence. Its rhythm is also weak to strong.

Interrupted (Deceptive) Cadence : This is V followed by VI. It is non-final. It is the musical equivalent of a comma. Musical rhythm may be strong to weak or weak to strong depending on context.

Imperfect Cadence (half-close) : Occurs in various forms, but the second chord must be V. Typically, the progressions in use are
Major : I-V, II-V, IV-V, VI-V, Ib-V, IIb-V, IVb-V
Minor : Same as above. II-V and IIb-V is not available here.
This is a musical semi-colon and is frequently used in the course of a sentence at the end of a subsidiary phrase. The rhythm may be weak to strong or strong to weak based on context.

Basic rules of progression

Complexity : 1/10 (simple)

Rules of melodic progression
- No two parts may move in parallel 8s or 5s. These faults are called consecutive octaves and consecutive fifths respectively.
- Parts should not move from unison to octave and vice versa.
- Parts should not move to octaves in contrary motion.
- S and B may not approach a 5th or 8ve in similar motion with a leap in the S. This fault is called Exposed Octave. Approaching an octave in similar motion with S moving by step is perfectly ok.
- Parts may not overlap between consecutive chords. Overlapping is undesirable since it endangers the identity of the parts. However, there is no harm in overlapping between two positions of the same chord.
- Perfect fifth followed by diminished fifth is ok between upper parts, but must be avoided with the bass.
- Avoid doubling the leading tone. It gives it undue prominence.
- Avoid ungainly leaps, too many consecutive leaps, sticking around one note for too long.
- When the melody moves by a third, it is effective to have the bass move by a step. This is good strong to weak, or weak to strong. Making the bass take the same notes as the soprano in the reverse order is also good, though not weak to strong.

Some guidelines on harmonic progression
Chords without inversion
- Chords whose roots are a 4th or a 5th apart typically give strong chord progressions.
- Roots falling a 3rd are good strong to weak, as well as weak to strong.
- Roots rising a 3rd are good strong to weak, but not weak to strong.
- II-I in root position is not a good progression.
- No cadence should be immediately preceded by its first or second chord. Otherwise the effect of the cadence is marred.
Chords in first inversion
- Avoid VIIb-V, VIIb-IVa or VIIb- Vb. VIIb is generally best followed by Ia or Ib, but IIb, Vb and VIb are also possible.
- Vb generally sounds best if the leading note in the bass moves to tonic. This means Vb-I, or Vb-VIb should sound good.
- In IIb-V, it is best to double the bass of the former.
- Change from root position to first inversion of the same chord is good strong to weak, not weak to strong.

Suspensions

Complexity : 4/10 (intermediate)

The suspension is a tone whose natural progression has been rhythmically delayed.

There are 3 factors in the management of a suspension
- Preparation - the sounding of the note to be suspended as a harmony note in the same part, in the first chord.
- Percussion - the actual sounding or striking of the discord
- Resolution - the progression of the discord to a concord, by falling one step.

Suspensions are named on basis of the intervals of the relevant notes above the bass. A 4-3 suspension means the suspended note is a 4th above the bass.

GENERAL
- The suspension may occur in any part, and any note of the triad may be suspended.
- The tied note is characteristic of the suspension as this is the means by which the note is delayed while the root changes. If tied, the second of the tied notes must not be longer than the first.
- The resolution of the suspension is usually to the scale step below, but upward resolution is possible (retardation). If the suspended note is a leading note or a chromatically raised note, it will have a natural resolution to the note above.
- The suspension does not usually resolve on a fraction of a beat, especially the second half of a first beat. The amount of time a tone is suspended is typically at least 1 whole beat.
- Ornamental resolutions are possible before the actual note of resolution arrives. These could be in the form of auxiliary, escape notes, cambiata, or anticipation, or there may be a note of harmony interpolated between the suspension and its resolution.
- The suspension of the 5th of a chord is really only effective in connection with the second inversion. This is because the 6th of the scale actually does not form a discord with the first inversion and the root position. (In the case of the second inversion, a discord of a 9th is produced).
- Suspensions in the bass need extra care. Only the root or the first inversion may be effectively suspended. In the early stages, it is best to suspend only the first inversion.
- Suspensions can occur in the root, 3rd and 5th of the dominant 7th.

RULES
- A suspension MUST be prepared by a harmony note.
- The note on resolution may not be struck against the suspension unless the latter be in the bass. There are NO exceptions to this rule. The note of resolution may NOT be struck against the note of suspension.
- Suspensions do not remove consecutives.

EFFECTIVE USAGE
- Cadences may often be effectively used to decorate a cadence.
- A long note where no change of chord is desirable may well have a suspension above or below it.
- A note repeated weak to strong and falling a step can be effectively harmonized using a suspension. In the case of the 4th of the scale repeated and falling a step to the 3rd, there are two possibilities (treat the repeated note as V7d or as a suspension. In the case of short value notes, the suspension is preferable).
- 4-3 suspensions are very effective. 9-8 and 7-6 suspensions are also good. 2-1 suspensions rarely sound good.
RESOLUTION
- A suspension may resolve into another position of the same chord or into another chord.

DOUBLE SUSPENSION
If two notes of a chord are both to fall a step into the next chord, moving in parallel 3rds or 6ths - a double suspension is possible.

Chromatic Passing Notes

Complexity : 6/10 (intermediate to advanced)

Chromatic notes that may be used to join up two notes that are a major 2nd apart of some greater distance than this.
If a passage begins with chromatic passing notes, it should continue with them until the harmony note is reached.
- A 4-note figure combining a chromatic passing note with an appoggiatura is sometimes met with. The first and last notes are harmony notes a major 2nd apart. (eg. D, D#, F, E)

Nota Cambiata

Complexity : 3/10 (simple to intermediate)

This is a four note figure. The fourth note is a 3rd below the first. In between come the diatonic passing note followed by the appoggiatura below the last note.
The shape of the figure and soundness of harmonic basis form the guides to identifying the nota cambiata.
This figure can move in parallel 3rds or 6ths. In such a case, the lower note must be either both diatonic or both chromatic.
Change of chord on the third note is also possible.

Changing Note

Complexity : 3/10 (simple)

The changing note figure consists of a double appoggiatura with the principal note added at the beginning.
The lower note follows the same rules as in the double appoggiatura.
Changing notes can be used simultaneously in parallel 3rds and 6ths. The lower note must be either both diatonic or both chromatic.
If harmonically appropriate, a change of chord may be introduced in the third note of the figure.

Transient modulations

Complexity : 5/10 (intermediate)

Modulations is not necessarily confined to phrase-ends or sequences. A cadence will be presented (modulation is not possible without one), but it will be less obvious.
- The original tonic must be firmly established before modulation is attempted.
- Transient modulations need inverted cadences
- Cadences may be masculine or feminine

Major scale

- Transient modulations to the supertonic minor are often possible and are quite acceptable when the central cadence is in the dominant.
- Passing modulation to the relative minor is also useful.
- Syncopation in the middle of a melody can provide a clue. There must be a chord change in the middle of a syncopated note, and a change of key is also effective.

Minor Scale
- If the central cadence is in the relative major, then a transient modulation to the subdominant is surprisingly, often possible.
- If the central cadence is in the dominant, the modulation to the subdominant or to the relative of the subdominant is undesirable.
- If the central cadence is in the dominant minor, passing modulation to the relative major is often a possibility.

Anticipatory notes

Complexity : 4 (intermediate)

If two harmony notes are a step apart, the second note may be anticipated in the same part, just before the beat.
This practice is most common before the cadence, though it may occur elsewhere. The note of anticipation is always a small value - quaver or semiquaver.
- Anticipations in the Tenor and Alto must be combined with one in the soprano. In this case, they either move in parallel 3rd and 6th or they approach octave or unison in contrary motion.
- Anticipations should never be used in the bass.
- Occasionally passing notes, suspensions can also be anticipated. Experience is the ultimate guide, but it is good to consider whether the 3 notes in question, would normally be harmonized with or without a change of chord.
- Anticipatory notes are usually not tied. They are ordinarily much shorter in time value than the principal tone.

Appoggiaturas

Complexity : 5/10 (intermediate)

These are simply unprepared suspensions or retardations and the rules governing suspensions and retardations govern appoggiaturas too.
- Accented discord (particularly at the beginning of a bar) is preferable.
- If a note is chromatically altered, it is to be treated as an appoggiatura unless modulation is implied. In such cases, chromatic alteration of accompanying parts can be used judiciously to enhance the effect.
- A chromatic appoggiatura may be approached by an augmented interval.
- Appoggiaturas in the bass are possible but can be ineffective and ungainly if used carelessly. Beginners must avoid this.
- The note of resolution should, as a rule, not be struck against the appoggiatura particularly if the appoggiatura is not in the upper voice. For chords in root position though, the note of resolution may be struck in the bass, provided the appoggiatura is far enough above to be clearly followed melodically.
- Appoggiaturas may obviously be struck below the principal note, too. A typical example is the leading tone appoggiatura to the tonic.


Unaccented Appoggiaturas
These come before the beat instead of on it. If above a principal note, they are diatonic. If below it, they are always at a distance of a semitone, and are therefore more often chromatic.
- Unaccented appoggiaturas of a major 7th or minor 9th are bad against the major 3rd of a chord. Against the root and 5th of a chord, these intervals are acceptable.
- The presence of an unaccented appoggiatura can be usually easily diagnosed if it is remembered that a note quitting by leap is a harmony note. Though this is a guideline, soundness of harmonic basis is the ultimate guide.

Double Appoggiaturas
Both the upper and lower appoggiaturas are taken consecutively, in either order, before the principal note, the figure beginning on the accent.
- The lower appoggiatura is normally a semitone below the principal note, but if the principal note be the major 3rd of a chord, the appoggiatura may be a whole tone below. Upper appoggiaturas are diatonic.
- If the lower appoggiatura is taken first, and is a semitone below the principal note, that principal note may be struck against the appoggiatura in another voice. But this is better avoided if the principal note is the 3rd of the chord.
- The indicator to the presence of a double appoggiatura is the curve of the figure and the use of chromaticism.
- The double appoggiatura can occur simultaneously in two parts, and can be combined with itself, moving in the opposite direction in some other part.