History of writing music

                For a very long time I used to wonder how such a beautiful system of music writing came into existence. So this thought has been running in my mind for quite some time which pushed me to get back into the habit of reading again.

                The first thing I always tell my students is “Music is a language”. Language is the mode of communication and it has 2 forms SPOKEN & WRITTEN. So does music, instead of spoken form it is going to be singing/playing form. Music is a language specially dealing with the emotions. It is a form of human communication. Majority of music form was oral, passed on from one person to another by repetition and memory. Now before getting into the article first my disclaimer; this article contains my own guess work along with the history. 

The proof for the earliest form of written music belonging to 2000 B.C. was found in the form of tablet in NIPPUR, in Iraq. It was a fragmented instruction to perform music. The music in that tablet was composed in harmonies of thirds using a diatonic scale. A Tablet from the period 1250 B.C. shows more developed form of notation.

                Ancient Egyptians’ devised symbols for pitch & tone of melody. Around 6^th century B.C. to 4^th century A.D. the Ancient Greece had started writing notation that can represent pitch as well as harmony to a limited extent. As early as 3^rd century B.C. the Chinese had a sophisticated system of notation.

                Using alphabets to name notes of the scale dates to ancient Greece and possibly earlier. This system was established by around 500 B.C. The alphabets were given to whole notes of the diatonic scale and inflections of semitones and quarter tones are represented by rotation of symbols. Two different systems were used to write down notations for vocals & instruments.

Byzantine Music:

                This was followed after the Ancient Greece in terms music notation writing and it is almost similar to western music notation. The main difference between the European writing that followed this period is notes are not absolute, they just indicate the pitch change and the musicians need to deduce it correctly. The micro tonal changes can be given. This music originated near the eastern Greece and it is pre Islamic. Byzantine chants are close to Arabic and Turkish music which also make use of micro tonal differences. Later a form of notation using signs called neumes was developed for writing plain songs.  The graphic signs would show the rise and fall of the pitch. However the notes are not absolute, hence there was no precise idea of pitch or also rhythm. This system was intended to remind the singers, the relative melody that had already been taught to them.

Early Europe:

                As early as 7^th century A.D. it was believed impossible to write or notate music. In the middle of 9^th century this started changing. The first problem that was addressed was the one that was faced in the Byzantine music method. A single horizontal line was drawn to adopt the pitch absolutely.  Later another line was added the first one representing C and the later one F drawn in different colors. The earliest notation of this type is from about 850 A.D. This method is still in use for plain chants. Over a period of time 2 more lines were brought below representing E & A.

Guido of Arezzo further improved the system of notation by placing the notes vertically on the horizontal lines. He also placed the notes C & F on the appropriate lines at the beginning which is now replaced by the clef signs. For other type of music, staffs with different number of lines at various times for various instruments were used. This was probably the first comprehensive method where a whole melody can be notated. However, still there was no method to notate rhythm.

German music theorist FRANCO OF COLOGNE suggested individual notes could have rhythm represented by the shape of the note. He codified further and rationalized the system by introducing notes with different time values.

At the start of the modern music staff notation, 11 lines theory was floated which would accommodate maximum notes in order with the middle line (6^th line) representing the Middle C. The main issue with this method was reading the notes as we play. Readability became very difficult. The modern five line staff must have been deduced from this method. It was first adopted in France and started being followed globally by 16^th century.

to be continued...

Intervals 2

As I begin the second part, thanks to all for the appreciation and the discussions that followed appreciation... Coming back to the subject…. I left it at Middle C history and how the frequencies of Cs are double their previous…. And the last para goes as below:

There are other interesting features about this set of frequencies. Three times the frequency of C4 is almost equal to the frequency of G5! Five times the frequency of C4 is close to the frequency of E6! Or to put in simple terms, the ratio of frequencies of a G note to that of C note, in the same octave is 3:2, and that of notes E to C is 5:4. From here it will mostly be a reverse working to find out the answer to the question why 12 notes. Since it is a reverse working I am also going to start with my conclusion on this subject.

Our brain stores every detail in patterns and every rule that has been defined in the history owes to this biology of our brain. When I read all these articles the first thing that occurred to my mind was THE MATRIX movie. It seems like everything is mechanical and a code has already been written for the same and I want to get myself unplugged and find out some new form of music in arithmetic progression.

I am sure most of us have attended “paatu class” during some time of our schooling. Esp for singing during navarathri festival or some bhajanais. Of course some of us end up taking it as a career. I am no exception to this paatu class and singing for sundal during navarathri festival, but there was always one question which I never had answer; at least a scientific one until now.

WHY DO WE SING SA----PA----SA at the beginning of every class?

It is as simple as this. As mentioned above Pa will be of the ratio 3:2 in comparison to the Sa. Hence therefore making it exactly the middle note of the scale and hence stabilizing the vocal tone.

When you pluck a string on a guitar, it vibrates back and forth.  This causes mechanical energy to travel through the air, in waves.  The number of times per second these waves hit our ear is called the ‘frequency’.  This is measured in Hertz (Hz).  The more waves per second the higher the pitch.  For instance, the A note below middle C is at 220 Hz.  Middle C is at about 262 Hz.

Do we know what frequency means? The rate of repetition of cycles of periodic quantity such as sound wave is called frequency.

Now, to understand the concept of why 12 notes, again as I did earlier I am taking C Major Scale as example.

The above picture shows the sine waveform of C (red) & F# (green).  Just on seeing the same we can say that these 2 sine waves doesn't match. The 2 wave pattern start together and in a given time (t) C’s wave pattern red is 8 lambda and in the same time Green wave is 11. And I do not like these numbers 11 & 8 as a pair for a simple reason I cannot simplify this ratio more. Yeah I know I am using a new term lambda without explaining what it is. I think this time instead of going tangent from the subject again I will reserve certain topics for next part of the intervals topic and for now lambda  is just like meters is a measurement used to measure waveforms. One trough to the next trough is considered 1 lambda. Now let us see how C & G sine waves look like:

These patterns if carefully observed we see the red and the green trough merge in equal intervals.

2 lambda of red = 3 lambda of green. So this proves our case of 3:2 between C & G (SA & PA) in the old method of pictogram.

Is this is the secret for creating pleasing sounding note combinations: Frequencies that match up at regular intervals.

Now let’s look at the ratios of the notes in the C Major key in relation to C:

C – 1 D – 9/8 E – 5/4 F – 4/3 G – 3/2 A – 5/3 B – 17/9

To tell you the truth, these are approximate ratios.  Remember when I said the ratio of E to C is about 5/4ths?  The actual ratio is not 1.25 (5/4ths) but 1.2599.  Why isn’t this ratio perfect?  That’s a good question.  When the 12-note ‘western-style’ scale was created, they wanted not only the ratios to be in tune, but they also wanted the notes to go up in equal sized jumps.  Since they couldn’t have both at the same time, they settled on a compromise.  Here are the actual frequencies for the notes in the C Major

Key:

Note	Perfect Ratio to C	Actual Ratio to C	Ratio off by	Frequency in Hz
Middle C				261.6
D	9/8 or 1.125	1.1224	0.0026	293.7
E	5/4 or 1.25	1.2599	0.0099	329.6
F	4/3 or 1.333…	1.3348	0.0015	349.2
G	3/2 or 1.5	1.4983	0.0017	392.0
A	5/3 or 1.666…	1.6818	0.0152	440.0
B	17/9 or 1.888…	1.8877	0.0003	493.9

You can see that the ratios are not perfect, but pretty close. The biggest difference is in the C to A ratio. If the ratio was perfect, the frequency of the A above middle C would be 436.04 Hz, which is off from 'equal temperament' by about 3.96 Hz.

Now let’s look at a chord, to find out why its notes sound good together.  Here are the frequencies of the notes in the C Major chord (starting at middle C):

            C – 261.6 Hz             E – 329.6 Hz             G – 392.0 Hz

 The ratio of E to C is about 5/4ths.  This means that every 5^th wave of the E matches up with every 4^th wave of the C.  The ratio of G to E is about 5/4ths as well.  The ratio of G to C is about 3/2.  Since every note’s frequency matches up well with every other note’s frequencies (at regular intervals) they all sound good together!

[When we press all the white keys in an octave the sound  that we hear can be described as just noise. However when we play chords like added 9^th, 11^th, 13^th (i.e. C-E-G-B-D-F-A) it sounds awesome. I am sure most of you would have and yeah as you have rightly thought the reason is wavelength of these notes and their ratio from the tonic C]

So it all comes down to number of times a note strikes... that is number of troughs and the crests created... and the mathematical fluency between them.

But then comes a big question; there are just 7 notes that I have mentioned here with this ratio but aren’t we looking to find out how 12 notes? Yeah, exactly that’s when I got hit with the concept of Consonance and Dissonances and that is precisely the topic as I have told above I am going to deal with in my next blog. I know I have promised to write on black and white keys in our piano but this a supplement to the current blog. So for this article let us find out the secret behind the 12 notes.

Keeping in mind the rule of octave i.e. the middle C to the C above it the frequency will exactly be double.

The previous list shows only the 7 notes in the C Major key, not all 12 notes in the octave.  Each note in the 12 note scale goes up an equal amount, that is, an equal amount exponentially speaking.

And here starts the mathematical part of the reverse working. The moment I say Math, I have got nothing much to do in this part irrespective of whether it is simple or complex. So obviously it was my friend “Dr.Mathematics” Sriram who explained it in his article physics of music and I understood the same from that article (so here starts the cut copy paste part). Since we are following the rule of octave and fitting a G.P. into it, it is evident that the common ratio should be something of the form 2^1/n, where n is number of notes. But, I want 3f/2 to be a note (to be more specific the middle note based on the sine wave form above and f is the frequency of the tonic note which is C in our case) which means I should choose “n” such that

3/2 = 2^a/n

where “a” would represent the number of terms after which 3/2 would occur in the G.P. Calling a/n as x, we have

3/2 = 2^x

If you had not understood how the above equation came up, no worries!  Because even I did not at first. Only on doing the following steps I realized the logic behind it. Every note is assumed to be in GP, therefore if “a” is the 1^st note then a+1 will be the next until we reach a + (n-1) = n. So which would mean notes will be 2^1/n, 2^2/n… 2^n/n. And this is precisely the GP we have been mentioning all along. So as per this GP if I require the ratio of the 7^th note then the formula will be 2^7/12. And we are doing the reverse working to find out “n”

Taking logarithms on both sides, we get x = log(3/2)/log2 which is approximately, 0.5849 (OMG I am re learning how to use the Log tables, probably using it for the first time after my 12^th exam. So it is solid 8 years). Obviously x can never be a rational number, so we can never have integer solutions for n and a. So we choose integers n and a, which will yield a fairly good approximation for x. It so happens that choosing n=12 and a=7 gives x=0 .5833 or 2^x=1.4982, which is indeed a very good approximation to 3/2. If a similar exercise is carried out for the note 5f/4, then choosing n=12 and a=4 yields 2^x=1.2599, which is also quite close to 5/4. Thus, with n=12 we get fairly decent approximations of 3f/2 and 5f/4. This is the reason why there are twelve notes to an octave!

Take up the case of our own Carnatic music. The music which is taught for the beginners, Sarali Varisai, consists of just seven notes. To the beginner the other Ri’s and Ga’s and such are essentially non-existent. When you consider this, the number of notes is 7 and the placement of Pa is the fourth note after Sa. So for this case, n=7 and a=4, which gives x=0.5714 or 2^x=1.4859! Using a Geometric Progression in your system of notes just enables your system to have something called equal temperament! You can even do without it!!

Now I am thinking of creating music system with Arithmetic progression in place of Geometric Progression. The hurdle I will be facing is, I need to make people listen to the AP music system for at least a century to remove the GP swarams that got etched in our gene and make the brain to command wow! Really good! on the other system. The music system with AP should have 48 swarams for a simple reason, as a composer I will have lot of permutations and combinations of swarams that I can use and thereby I can keep creating different kinds of music at least for my life time. But for that I should live 100 years first.

On intervals part, concepts of consonances, dissonances, principle of super position, constructive interference, destructive interference (this particular physics theory is the most important part of sound proofing), Fourier law, equal temperament, etc. are very important.

So apparently it has taken about some 22 pages for me to explain 1/10^th of the things I want to. And respecting the complaints from all my friends that a 16 page article is too difficult to read, I am going to continue the remaining in my next parts. And people this time it is just a 4 page article.

Intervals

For some time I have totally forgotten that I started this blog and only when Divya posted an article in her blog after almost 2 years it struck me that I have a blog too. So the interest on posting some topic in my own blog resurfaced. What is the topic on which I am going to post was the big question. I had options starting from Ganguly not being picked up in IPL auction and Kochi suddenly deciding to take Ganguly in to 2G scam and how thamizh movies have become bad. At last I decided to post the current topic I am working on in music. Strictly speaking I started to work on Counterpoint theory and harmony. I had some doubts on basic stuff as I was working on it and ended up coming to the basic level of harmony, "INTERVALS" and from there I went to the core basic of music the NOTES or SWARAMS. And here goes the article….

This article includes compilation of information from various sources along with my views. Whatever I felt important for explaining the intervals concept has been included.

Basic Structure:

Interval: The distance in the pitch between any two notes is called as INTERVAL.

Though intervals can be classified in several ways the basic classification of intervals are Generic and Specific Intervals.

•        Generic Intervals:  I am not going to give a definition for Generic Interval instead I am going to take a specific Scale and explain the same.

In a Scale the:

1^st note – Tonic (T)
2^nd note – Super Tonic (St)
3^rd note – Mediant (M)
4^th note – Sub Dominant (Sd)
5^th note – Dominant (D)
6^th note – Sub Mediant (Sm)
7^th note – Leading Note (L)
8^th note – Octave (O)

To get a hold on this concept let us take C Major Scale.

Note	Technical Name	Carnatic Equivalent	Generic Interval
C	Tonic	SA ‐Sadjamam	1st note – C to C
D	Super Tonic	RI – Rishabam	2nd note – C to D
E	Mediant	GA – Ghandharam	3rd note – C to E
F	Sub‐Dominant	MA – Madhiyamam	4th note – C to F
G	Dominant	PA – Panchamam	5th note – C to G
A	Sub‐Mediant	DHA – Dhaivadham	6th note – C to A
B	Leading note	NI – Nishadham	7th note – C to B
C	Octave	Upper SA	8th note – C to C

So to sum up if you start numbering all the notes in the scale in ascending order, then that would be the generic interval.

Now let us take it a step further to Specific intervals. Before I could give an explanation on specific interval, there are few basic concepts in music which needs to be explained.

Semitones: If I should explain this concept in a lay man‐ish way then it is nothing but a measurement like Hz, Kms, etc. between 2 notes. For instance let us take 2 notes C and F. The distance between these 2 notes are intervals and they are 5 semitones away from each other. The proper definition for Semitone goes as follows:

A semitone, also called a half step or half tone, is the smallest musical interval commonly used in Western tonal music. It is defined as the interval between two adjacent notes in a 12‐tone scale. This implies that its size is exactly or approximately equal to 100 cents, a twelfth of an octave.

Accidentals: There are basically 3 types of accidentals. Of course there are by products of every accidentals mentioned here. For almost 90% of the practical purposes we do not use them.

Sharp – The accidental used to raise the pitch of a note by 1 semitone.

Flat – The accidental used to reduce the pitch of a note by 1 semitone.

Natural – The accidental used to bring back the raised/reduced pitch back to normal.


Just in case the first sign on the left hand side of the note is the sharp sign followed by flat and natural.

I assume that people who read this know other basic concepts including octave, 7 white keys and 5 black keys in an octave, etc.

•        Specific Intervals: We now know what a generic interval is. Number the notes in a scale in ascending order and the number given to each note is the generic interval. Let us take an example and try to understand the concepts of generic and specific intervals in a better way. And once again I am going to take the same scale C Major to explain. Now that we know the concepts of accidental instead of the exact 7 notes of the scale, I am going to include the entire 12 notes in an octave.

Note	Generic Interval	Semi tones	Specific Interval
C	1st note – C to C	0	Unison
C#/Db	2nd note – C to D	1	Minor Second
D	2nd note – C to D	2	Major Second
D#/Eb	3rd note – C to E	3	Minor Third
E	3rd note – C to E	4	Major Third
F	4th note – C to F	5	Perfect Fourth
F#/Gb	4th note – C to F	6	*
G	5th note – C to G	7	Perfect Fifth
G#/Ab	6th note – C to A	8	Minor Sixth
A	6th note – C to A	9	Major Sixth
A#/Bb	7th note – C to B	10	Minor Seventh
B	7th note – C to B	11	Major Seventh
C	8th note – C to C	12	Octave

*you will get to know what this is in a few minutes.

So to put it in simple words, Specific intervals are those which are measured in half notes. Now immediately on seeing the above table the obvious question that came into my mind was why the flat sign was only considered to establish the generic interval for the black keys?

Actually speaking when I started learning piano somewhere during my fourth standard, I always used to wonder why they are confusing me with 2 names for every black key one with a sharp and one with a flat and my learning was pretty mechanical and trinity college exam oriented that I never really bothered to find out why it was so. As I tried to explore and find an answer to my question, I am just undergoing a very weird feeling for the first time in my life. Two mutually exclusive events seem to be happening at the same time. I really seem to know the answer for the question but at the same time I feel I do not know the answer. It is just like being in love without proposing to one another. You know the other person loves you certainly, yet you are not certain about it. So now I am trying to make the same into 2 parts. One part as I know part and the other as I do not know part. 

What I know part: There are 2 names to every black key. As we move right from a white key we call it sharp of that right key and when we move left from a white key it is flat of that white key. If I should say in a lay man‐ish way there are 2 names to every black key just for convenience.

What I do not know part that I explored later on:

It has got something and everything to do with the concept of key signature. It seems to me like which came first egg or hen? key signature or the accidentals. For musicians this might seem little stupid because we very well know that accidentals came first. But the way the design goes really make me wonder. There is some big time basic math involved there. I have found out to an extent on this subject. But it needs to be a separate article. So I will explain the same later as to why there are 2 names to every black key in a scientific way and what is exactly convenient in it. However the primary question of why only the flat names are alone considered will be answered shortly.

Also I remember once talking to Sriram on why there were 5 black keys and 7 white keys and why are they arranged in such a manner. We thought of so many reasons but forgot something basic. We thought of reasons like how much our hand can stretch comfortably and so many other aspects. But finally today morning when I was playing one of my own pieces I realized something. If the black keys are not there then identifying where you are on a piano is not possible at all. It would completely be a blind run method then. Probably our other theories like how much our hand can stretch comfortably should have also played a part in determining the size of every key but not on the basic arrangement.

Other Classifications of Interval:

On the basis of Time:

·         Melodic Interval: When the notes are played subsequently one after another then the interval is known as melodic interval 
·         Harmonic Interval: When the notes are played simultaneously then the interval is harmonic interval.

On basis of Distance:

·         Simple Interval: The intervals are classified as simple interval if the distance between the notes is lesser than an octave then it is referred as simple interval. (e.g.) Minor 3^rd (3 semitones)
·         Compound Interval: The intervals are classified as the compound intervals if the distance the notes is more than octave then it is referred as compound interval. (e.g.) Major 13^th (14 semitones)

On the basis of the notes used intervals can be classified as follows: 

·         Diatonic intervals: If the interval consists of the notes that are in the scale the same is known as diatonic interval. E.g. Major, minor and perfect intervals
·         Chromatic Intervals: If the interval consists of the note(s) that are not in the scale then the same is known as chromatic interval. E.g. Diminished and Augmented intervals.

Again I am going to take the same table above with slight modification. I am just leaving the C Major scale alone.

Semi tones	Diatonic Interval	Short	Chromatic Interval	Short
0	Perfect Unison	P1	Diminished Second	d2
1	Minor Second	m2	Augmented Unison	A1
2	Major Second	M2	Diminished Third	d3
3	Minor Third	m3	Augmented Second	A2
4	Major Third	M3	Diminished Fourth	d4
5	Perfect Fourth	P4	Augmented Third	A3
6		‐	Diminished Fifth/ Augmented Fourth	d5/ A4
7	Perfect Fifth	P5	Diminished Sixth	d6
8	Minor Sixth	m6	Augmented Fifth	A5
9	Major Sixth	M6	Diminished Seventh	d7
10	Minor Seventh	m7	Augmented Sixth	A6
11	Major Seventh	M7	Diminished octave	d8
12	Perfect Octave	P8	Augmented Seventh	A7

So the above table clearly explains me my earlier question on why only the flat names were considered for generic intervals and the specific intervals given for the same. The above table considers the sharp name too for generic intervals and the specific intervals are given for the same. Now let us go back to our C Major Scale example for a better understanding:

Semi tones	Note	Diatonic Interval	Chromatic Interval
0	C	Perfect Unison
1	C#/Db		Augmented Unison
2	D	Major Second
3	D#/Eb		Augmented Second
4	E	Major Third
5	F	Perfect Fourth
6	F#/Gb		Diminished Fifth/ Augmented Fourth
7	G	Perfect Fifth
8	G#/Ab		Augmented Fifth
9	A	Major Sixth
10	A#/Bb		Augmented Sixth
11	B	Major Seventh
12	C	Perfect Octave

 

Similarly the intervals for each scale can be established. So every possible interval can be and should be named on this basis. Further if we closely observe we most likely would not use the chromatic interval names for all the Perfect Diatonic intervals like Perfect Unison, Fourth, Fifth and Octave, because irrespective of which scale we are going to take up, these perfect intervals will be present in almost all the scales. Then what are the unusual circumstances in which we would use chromatic intervals names. The first thing that comes to my mind right now is a fusion scale. A Carnatic Music Melakartha raga (or any other sampoorna ragam) being adapted in western music. However, still the perfect unison and the perfect octave will remain the same. I might be wrong, but I feel it is possible.

Before going for the next classification I have something to say. I have always wondered, actually for a very long time that, why do we use the terms augmented and diminished when there is a very easy way to mention the intervals as major and minor. Probably I never liked the words augmented and diminished and I don't know why? Actually speaking I just did not stop with the thinking part. Every time I wrote a score I always avoided using these terms and rather go with the terms major and minor. After about 3 years of recording now I realize my mistake in writing the score. Let me explain this-

If we write a score in the scale C Major, G# is the minor 6^th and hence we can denote this interval as minor 6^th. However, if we observe, the G# note is not part of the C major scale and it is an augmented 5^th interval. Of course this is something basic which everyone knows. What is the harm in writing the same as minor 6^th will be our next question. I found it to be easy to say minor 6^th, but when the same is denoted as augmented 5^th it conveys something else too. This term explains that the note does not belong to the scale however ,the same has been used for the feel. A very important factor called "FEEL" is getting conveyed here and that made me wonder how much the composers of the earlier years must have thought. WOW!

•        Minute intervals: Microtone difference. As a matter of fact till date I have not seen this particular interval practically anywhere as part of Western Classical Music. Though not very popular in western classical this particular interval is very much evident esp. in Carnatic Classical music. I would term this particular interval as the most important as it triggered my next question and made me go to the very basic to find out why do we like these specific frequencies which has been set as standards like A=440 Hz and other corresponding frequencies I hear on my piano. I am always a fan of round numbers in math. The audible range is 20 Hz to 20 kHz. So why don't we like some sound that is coming at 100 Hz or 1000 Hz, etc. The explanation goes as below:

For next few minutes we are going to go bit tangent from our original topic.

Before I go for that I should thank my friends Sriram and Shankar the sangarapondy whose articles helped me a lot in understanding the basic structures of mathematical concepts like Geometric progression, Trigonometry, etc. and also their application in music. Yes, I really felt bad that I didn't know these things are connected to music so much when I was in school. If I had known probably I would have studied Math much better than what I did ;). For all we know I might have become a mathematician :P. Actually the right thing to say would be I have done a CUT COPY PASTE from his article "Physics of Music". Physics of Music is the result of sangarapondy and Sriram's hard work. To be exact they did complete research in this part and I understood from them. Also I cannot forget my sound engineer Vijay who was there for me for all my stupid questions on sound engineering and at last my violinist Harini, who actually made me read a lot with the question on How to define a SA? Actually if I think back this question has a very important role in my life. In an urge to find an answer to this question I started reading lot of books and eventually ended up having so many basic doubts in music.)

The question is "what is the problem we have to solve, to understand what frequencies we 'like'?" I think this is a vastly neglected field. I am sure there must have been many people who would have done research on this and surely the twelve notes in western music were not hit upon in an arbitrary manner! But how many of us know the reasons for it?

I always thought the frequencies had something to do with our human ear. So obviously the initial interest was in finding out how our ear functions. This is where Sriram's article and my mom's BA music books helped me a lot. The ear consists of three parts namely the outer ear, middle ear and the inner ear. The outer ear consists of the ear lobe called the pinna and the auditory canal. I think the shape of the pinna is pretty much easy to explain ‐simply directing the sound waves into the middle ear like a funnel! Then the auditory canal is just a tube which connects the outer ear to the middle ear. The auditory canal ends in a tympanic membrane or what is commonly known as the ear drum. And though I was never very good at physics, I was very keen at it. Small information here, the length of the auditory canal seems to be 26 mm (thanks to wiki!).

Further examining the structure of the ear, the middle ear has 3 small bones called ossicles which are named malleus, incus, and stapes. The vibrations of the tympanic membrane are transmitted through these three bones to an oval window in the inner ear. The inner ear has the cochlear fluid and semi lunar canals – stuff that help in converting the mechanical vibrations of the ossicles to electrical impulses which are then transmitted by the auditory nerve to the brain. This is what we perceive as sound. While looking at the shapes of these ossicles, the bones actually have a very distinct appearance. Malleus looks like a hammer, the incus looks like an anvil and the stapes looks like a stirrup. Let's take a pause for a second. I think we really need to admire the biology here. The sound enters you ear. The mechanical wave gets converted as the electric impulses, search our hard disk for the pattern that is similar to those electric impulses and return back. What amazes me more is the time it takes, the speed at which all these processes happen. Ironically I just pressed CTRL+S before writing this sentence and my computer was blinking for almost 10 to 15 seconds. Why didn't I study medicine? Esp. the subject on neurology. It is really interesting. I wish I live for about some 200 or 300 years. And that reminds me that i have one other topic on which I wanted to blog. A blog on a movie "Man from Earth" where a man lives for unusually long time. So that once I complete music to an extent, I can take up Math and then medicine. It is really bad that we live just maximum for 75 years in the current condition and considering the way I live I think I have very less time to do all this. And in the middle of all this, people expect me to be a CA!!!!

So coming back to music, the whole thought process on why our ears like these frequencies seems to be wrong and I realized this after knowing the biology behind it. We like these frequencies because of the electric impulse pattern in our brain. We like what our brain likes. So as we take up a small concept on music like INTERVALS I ended up finding that there is so much of Math, Physics and Biology involved in it and not to mention a little bit of philosophy too. No wonder people say music is an ocean. ADMIRATION GAURANTEED! So then why do we like these frequencies? To answer this question, let us look up to the Standard Western notation where the frequency for each piano key note is fixed. The A above middle C is fixed to be 440 Hz, which is in general referred as A4. So once one note is fixed, the frequencies of all other notes can be found out by making use of the fact that the frequencies are all in a Geometric Progression (thanks to Shankar for this brilliant finding), with a common ratio 2^1/12(thanks to wiki for this information). But why the hell should A4 be 440 Hz or should the common ratio be 2^1/12?? Supposing we choose some other frequencies set and try to compose with that, will it sound as music to our ears? One notable thing about choosing A4 as 440 Hz is that middle C turns out to be very close to 2⁸ Hz – meaning all the C notes are close to integral powers of 2. (again a very important finding by Sangarapondy and Sriram). So does the 'magic' of music lie in the fact that the frequencies of C notes are close to that of integral powers of 2 or is it due to the common ratio 2^1/12 or is it a combination of both? And it is time to go a bit tangent from the concept which is already tangent from the original topic to find out the history on this A = 440hz which would give us a better picture to understand our current concept. And I think there is another subject that fascinates me now as I go on. HISTORY! Most likely the period that follows immediately after our lunch break and the period in which most likely we would be sleeping.

So once upon a time….

An English pitch pipe from 1720 plays the A above middle C at 380 Hz, while the organs played by Johann Sebastian Bach, Leipzig and Weimar were pitched at A = 480 Hz, a difference of around four semitones. In other words, the A produced by the 1720 pitch pipe would have been at the same frequency as the F on one of Bach's organs. Ok if I go on writing like an essay I think u will sleep so….

Period/Year	Frequency of A above middle C	Short note
1720	380 Hz	An English Pitch Pipe from this period plays A at this frequency
1720	480 Hz	A difference around 4 semitones. Composers like Johann Sebastian Bach, Leipzig and Weimar's Organs were pitched at this frequency. Hence A of an English pitch pipe will be F of Bach's organ
1740	422.5 Hz	A Tuning Fork associated with Handel was pitched at this frequency
1780	409 Hz	Almost a semitone lower.
1815	423.2 Hz	The advent of the orchestra brought pitch inflation to the fore.
1826	435 Hz	The same opera that gave the frequency as 423.2 Hz changed it. 1815 to 1859 was period characterized with rise in pitch.
1826 – 1859	451 Hz	At La Scala in Milan. Probably after Bach's period this was the most highest pitch of A.
1859	435 Hz	The most vocal opponents of the upward tendency in pitch were singers, who complained that it was putting a strain on their voices. Largely due to their protests, the French government passed a law on February 16, 1859 which set the A above middle C at this frequency. This was the first attempt to standardize pitch on such a scale, and was known as the diapason normal. It became quite a popular pitch standard outside France as well, and has also been known at various times as French pitch, continental pitch or international pitch
1859‐1900	430.54 Hz Middle C = 256 Hz	The diapason normal resulted in middle C being tuned at approximately this frequency. An alternative pitch standard known as philosophical or scientific pitch, fixed middle C at 256 Hz (that is, 28 Hz), which resulted in the A above it being approximately at this frequency. On a personal note this seems to be the most appropriate one for me because of the reason it is a mathematical idealism (the frequencies of all the Cs being power of two). However the same did not become popular.
Towards the end of 19th century	452 Hz	British attempts at standardization gave rise to this old philharmonic pitch standard.
1896	439 Hz	considerably "deflated" new philharmonic pitch
1939	440 Hz	An international conference recommended that the A above middle C be tuned at this frequency, now known as concert pitch. This standard was taken up by the International Organization for Standardization in 1955 and reaffirmed by them in 1975 as ISO 16. The difference between this and the diapason normal is due to confusion over the temperature at which the French standard should be measured. The initial standard was A = 439 Hz, but this was superseded by A = 440 Hz after complaints that 439 Hz was difficult to reproduce in a laboratory owing to 439 being a prime number.
Current	Despite such confusion, A = 440 Hz is the only official standard and is widely used around the world. Many orchestras in the United Kingdom adhere to this standard as concert pitch. In the United States some orchestras use A = 440 Hz, while others, such as New York Philharmonic and the Boston Symphony Orchestra, use A = 442 Hz. Nearly all modern symphony orchestras in Germany and Austria and many in other countries in continental Europe (such as Russia, Sweden and Spain) play with tune to A = 443 Hz. A= 442 Hz is also often used as tuning frequency in Europe, especially in Denmark, France, Hungary, Italy, Norway and Switzerland. Some orchestras like the Berliner Philharmoniker now use a slightly lower pitch (443 Hz) than their highest previous standard (445 Hz).
Current Special Cases	Many modern ensembles which specialize in the performance of Baroque music have agreed on a standard of A = 415 Hz. An exact equal‐tempered semitone lower than A = 440 would be 440/21/12 = 415.3047 Hz; this is rounded to the nearest integer. In principle this allows for playing along with modern fixed‐pitch instruments if their parts are transposed down a semitone. It is, however, common performance practice, especially in the German Baroque idiom, to tune certain works to Chorton, approximately a semitone higher than A‐440 (460–470 Hz) (e.g., Pre‐Leipzig period cantatas of Bach).

This is the history of western classical with respect to A = 440 Hz. If we take up Carnatic Classical there is no such concept of A=440 Hz. We tune the instruments in line with the Lead Singer or the Lead Instrumentalist's comfort. And that was when it struck me, I have known situations where my grandfather (Dr.T.K.Murthy) used to reduce the pitch of his mirdangam slightly lower, probably half a semitone when the singer has a sore throat.

This history and the Indian Classical music scenario clearly explain the magic is not in the frequency of the notes. So obviously if my first factor (A=440 he and all the C notes are close to the integral powers of 2) is not right, the combination of the first and second factor cannot be right. So it is the GP, the common ratio 2^1/12

If you remember, above this tabular column I mentioned "One notable thing about choosing A4 as 440 Hz is that middle C turns out to be very close to 2⁸ Hz – meaning all the C notes are close to integral powers of 2. So does the 'magic' of music lie in the fact that the frequencies of C notes are close to that of integral powers of 2 or is it due to the common ratio 2^1/12 or is it a combination of both?" This is precisely why I would have been much happier with the scientific pitch (A4 = 430.54 Hz) rather than the concert pitch (A4 = 440 Hz) for a simple reason that I like whole numbers better than the decimals. However, still it seems to be close to 2⁸, so let us continue our journey.

On observing the history carefully, the one thing that was common thro all these periods was 12 notes in an octave. Further we all know now that this geometric progression has got everything to do with our brain liking the sound we hear. Ultimately there is only one question left. Why was this particular GP chosen? To expand a little more, this GP confines the number of notes in an octave to 12. What if it is 13, 19 or 32? Since so much of math was involved and considering the fact that when I started all this I didn't even remember what Geometric Progression was, I had 2 options

Option 1: Learn Math first then come back to music.
Option 2: Use my friend Sriram's and Sangarapondy's article to the fullest possible to understand all these questions, so that I can get back to my own intervals concept ASAP.

Obviously as you have guessed I chose the second option.

We already know that all the C notes are close to integral powers of 2. As we closely observe the frequencies of all the Cs, they are actually 2 times of their previous. This seemed to me like a very important line from my friends' articles.

Middle C = 261.626 and the C next to middle C = 523.251 (take out your Calc and confirm it for yourself….. done… yup it is twice). Now the C before middle C is 130.813 (repeat the calc exercise… and that proves it is half of Middle C's frequency). The rule of the octave is followed – meaning, multiply the frequency of a note by 2 then you get the same note in the next octave. So C5 is two times C4, C6 (Soprano C) is four times C4 and so on. Similarly, C3 (Low C) is half of C4; C2 (Deep C) is one fourth of C4 and so on. So instantly I jumped up to a conclusion on completing this part. The concept of 12 notes need not be right as far as the rule of octave is followed there can be any number of swarams. Right or wrong? I would say partially because of what I understood after this.

There are other interesting features about this set of frequencies. Three times the frequency of C4 is almost equal to the frequency of G5! Five times the frequency of C4 is close to the frequency of E6! Or to put in simple terms, the ratio of frequencies of a G note to that of C note, in the same octave is 3:2, and that of notes E to C is 5:4.

I sign off at this point and promise to return ASAP as I finish writing the next part(s) on this particular concept.