The mathematical beauty of Carnatic Music: Prelude
Music is mystical. Physically, there is no 'sound' out there in the real world. There are only vibrations. Our ears coupled with our brain helps make 'meaning' out of the vibrations that exist out here in the real world so that we can eat and avoid being eaten. However, evolution never produces 'perfect' machines. A vibration that has a frequency of 440hz is different from a vibration of 441hz. However, the average human ear, for possible reasons of optimisation can only differentiate between sounds that are atleast in the ratio twelfth root of two.
"Music" in the game of survival I believe, music is an accidental side-effect of how our psycho-acoustics work. Frequencies of the sounds in a baby's cry, if you noticed, is all in highly odd and complex fractions. The ratio of frequencies from laughter are in simple fractions (typicaly 4/3 or 5/3) whereas the ratios of sounds in a baby's cry are more complex (the ratio being the magical number 'twelfth root of two'). Rules of survival say, "Baby crying means bad... " and "Baby laughing is good". We have a 'natural aversion' to sounds that are in complex frequencies and 'natural desire' towards sounds that are in simple frequencies. Our brains are equipped to not only perform fourier analysis on frequencies but also make 'meaning' out of it such that it will help in survival. Seeing one's baby growing up happily is a reward for the parent. It makes the parent 'seek' more of the reward and therefore keep the baby happy. Also sounds that are in very simple whole number ratios like 2,4,8,16,etc.,. seem the 'same' to us.
Now that we've established that our brains tend to 'like' simple fractions and 'dislike' complex fractions, let's see how we can 'reward' our brains by producing sounds.
Mathematics of Music
Take any sound, say something that is at the frequency of '440Hz'. This note by itself doesn't produce any effect on our psyche. However, make another sound along with it and the brain automatically starts making meaning out of the sounds. Now produce another note of the frequency 587Hz. This combination of '440' and '587' sounds very 'pleasant' to us. Why? because these two frequencies are of the ratio 1.33 (which is a 'nice looking' fraction -- 4/3). Now that we know what sounds good (the ratio of '4/3'), produce another frequency starting from 587 using this ratio. We get 783. Do this process iteratively until you arrive at a whole number multiple of '440'. sort the frequencies in asending order and there you go! The set of frequencies so obtained correspond to the various notes that humans must have first discovered -- the major scale. For more details on how this is done, refer to Circle of Fifths.
There are a total of 12 notes: C C# D D# E F F# G G# A A# B (... and then C C# D D#, etc.,. again for the subsequent 2,4,8,etc.,. multiples). People who study music are taught that these are the 12 notes in an Octave. But did you know why there are 12 notes?
As mentioned earlier, since the average ear can only differentiate between sound frequencies that are atleast separated by the ratio twelfth root of two, all the notes mentioned here are at a ratio of twelfth root of two with its immediate previous and next note. That is to say, f(C#) / f(C) == twelfth root of 2 == f(D) / f(C#). Now, its just mathematical gymnastics to find out why there are only twelve notes in what we perceive as an 'octave'. The keyword being "12th root"... keep multiplying 12th root 12 times and what you get is '2'. sounds that are at the ratio of 2, 4, etc.,. are perceived as 'same' and therefore, we have reached the next 'octave' where the same set of notes begin again. Thus, a given set of frequencies where we see 'different' notes can only be divided into 12 parts due to the 12th-root-of-2 limitation imposed by our hearing system.
As soon as man realized that he could 'hack' the taste buds by cooking food there also sprang several rules on what is the best way to cook. From recipes to spices to what not. The various musical forms that were created were also nothing but such 'rules' which when followed will help you achieve the desired effect on our psyche.
What the indians call 'krodha rasa', the greeks call 'phrygian mode'. Humans have observed the effect various combination of notes have on the psyche. However, the means to achieving the effects seem vastly different at the outlook. But I believe every system of music is made up of very similar fundamental rules. There is definitely 'circle of fifths' in Indian classical music as there is 'melody' in western music.
The system of music that I'm very interested in -- Carnatic Music, is extremely intellectually challenging and I hope to demonstrate (in my subsequent posts) a few aspects of Carnatic music that I know about.
While I wanted to cover some carnatic related topics, I figured this has been a pretty long (hopefully a useful!) write up on my views of music and hence I stop here.
Up next: Griha Bedham: View points to ragas.
"Music" in the game of survival I believe, music is an accidental side-effect of how our psycho-acoustics work. Frequencies of the sounds in a baby's cry, if you noticed, is all in highly odd and complex fractions. The ratio of frequencies from laughter are in simple fractions (typicaly 4/3 or 5/3) whereas the ratios of sounds in a baby's cry are more complex (the ratio being the magical number 'twelfth root of two'). Rules of survival say, "Baby crying means bad... " and "Baby laughing is good". We have a 'natural aversion' to sounds that are in complex frequencies and 'natural desire' towards sounds that are in simple frequencies. Our brains are equipped to not only perform fourier analysis on frequencies but also make 'meaning' out of it such that it will help in survival. Seeing one's baby growing up happily is a reward for the parent. It makes the parent 'seek' more of the reward and therefore keep the baby happy. Also sounds that are in very simple whole number ratios like 2,4,8,16,etc.,. seem the 'same' to us.
Now that we've established that our brains tend to 'like' simple fractions and 'dislike' complex fractions, let's see how we can 'reward' our brains by producing sounds.
Mathematics of Music
Take any sound, say something that is at the frequency of '440Hz'. This note by itself doesn't produce any effect on our psyche. However, make another sound along with it and the brain automatically starts making meaning out of the sounds. Now produce another note of the frequency 587Hz. This combination of '440' and '587' sounds very 'pleasant' to us. Why? because these two frequencies are of the ratio 1.33 (which is a 'nice looking' fraction -- 4/3). Now that we know what sounds good (the ratio of '4/3'), produce another frequency starting from 587 using this ratio. We get 783. Do this process iteratively until you arrive at a whole number multiple of '440'. sort the frequencies in asending order and there you go! The set of frequencies so obtained correspond to the various notes that humans must have first discovered -- the major scale. For more details on how this is done, refer to Circle of Fifths.
There are a total of 12 notes: C C# D D# E F F# G G# A A# B (... and then C C# D D#, etc.,. again for the subsequent 2,4,8,etc.,. multiples). People who study music are taught that these are the 12 notes in an Octave. But did you know why there are 12 notes?
As mentioned earlier, since the average ear can only differentiate between sound frequencies that are atleast separated by the ratio twelfth root of two, all the notes mentioned here are at a ratio of twelfth root of two with its immediate previous and next note. That is to say, f(C#) / f(C) == twelfth root of 2 == f(D) / f(C#). Now, its just mathematical gymnastics to find out why there are only twelve notes in what we perceive as an 'octave'. The keyword being "12th root"... keep multiplying 12th root 12 times and what you get is '2'. sounds that are at the ratio of 2, 4, etc.,. are perceived as 'same' and therefore, we have reached the next 'octave' where the same set of notes begin again. Thus, a given set of frequencies where we see 'different' notes can only be divided into 12 parts due to the 12th-root-of-2 limitation imposed by our hearing system.
As soon as man realized that he could 'hack' the taste buds by cooking food there also sprang several rules on what is the best way to cook. From recipes to spices to what not. The various musical forms that were created were also nothing but such 'rules' which when followed will help you achieve the desired effect on our psyche.
What the indians call 'krodha rasa', the greeks call 'phrygian mode'. Humans have observed the effect various combination of notes have on the psyche. However, the means to achieving the effects seem vastly different at the outlook. But I believe every system of music is made up of very similar fundamental rules. There is definitely 'circle of fifths' in Indian classical music as there is 'melody' in western music.
The system of music that I'm very interested in -- Carnatic Music, is extremely intellectually challenging and I hope to demonstrate (in my subsequent posts) a few aspects of Carnatic music that I know about.
While I wanted to cover some carnatic related topics, I figured this has been a pretty long (hopefully a useful!) write up on my views of music and hence I stop here.
Up next: Griha Bedham: View points to ragas.