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:
1st note – Tonic (T)
2nd note – Super Tonic (St)
3rd note – Mediant (M)
4th note – Sub Dominant (Sd)
5th note – Dominant (D)
6th note – Sub Mediant (Sm)
7th note – Leading Note (L)
8th note – Octave (O)
To get a hold on this concept let us take C Major Scale.
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:
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.
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:
1st note – Tonic (T)
2nd note – Super Tonic (St)
3rd note – Mediant (M)
4th note – Sub Dominant (Sd)
5th note – Dominant (D)
6th note – Sub Mediant (Sm)
7th note – Leading Note (L)
8th 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 |
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:
· 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 3rd (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 13th (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.
• 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.)
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 21/12
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 21/12 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.
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.
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 6th and hence we can denote this interval as minor 6th. However, if we observe, the G# note is not part of the C major scale and it is an augmented 5th interval. Of course this is something basic which everyone knows. What is the harm in writing the same as minor 6th will be our next question. I found it to be easy to say minor 6th, but when the same is denoted as augmented 5th 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 21/12(thanks to wiki for this information). But why the hell should A4 be 440 Hz or should the common ratio be 21/12?? 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 21/12 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 21/12
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 21/12 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.