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.