By sound we normally mean the sensation produced in our ears. Physically sound is produced by the vibration of objects like the vocal chords, stretched string of the Veena, the air column of the flute, the reed in the harmonium or simply the quick vibration of any object when it is struck. The vibrations have to reach the ear through a medium, usually the air. The air molecules surrounding the object producing the sound vibrate and the vibrations spread out in the form of 'waves' (by which we mean a disturbance moving through a medium while the medium itself undergoes only a to and fro movement at every point). When the wave reaches the ear, the ear drum vibrates in a manner similar to the original object and the vibrations are conveyed to the inner ear where, fine hair follicle like objects vibrate and convey the sensation to the brain through nerves. When objects vibrate at frequencies higher than or lower than what we can hear they are referred to as ultrasonic and subsonic sound. Some animals (dogs especially) are sensitive to high frequency sounds.

Three essential qualities of any sound- in particular musical sound are pitch, loudness and quality. These are the sensations produced in the person hearing the sound. They have their physical equivalents in frequency, intensity and harmonic content which are quantities that can be measured.

Frequency is the number of times the original object (string or air column) or the eardrum vibrates (makes a complete to and fro motion) every second. The higher the frequency the higher is the pitch. In music the pitch is usually referred to by the note name like sa, ri, C,D etc. but when discussing as a physical entity the frequency number is taken as referring to the pitch. Pitch is actually the sensation felt by the listener. Extraneous factors can make the pitch of two sounds of the same frequency appear different. Human ears are sensitive to the frequency range from 30 cycles per sec to 15000 cycles per second - the sensitivity to higher frequency falling at old age which explains the difficulty in distinguishing words although the sound is heard as vowels and consonants are identified by frequencies of the higher harmonics. Most of the sounds we hear are usually well below 3000 cycles but the presence of higher frequency harmonics affects the quality and clarity in speech (see later). To appreciate how sounds of different frequencies feel, you can set the frequency below and hear it. (Youngsters may be able to hear 12000 cycles but as one grows older sensitivity to highe frequencies becomes less. Over 80 one may not be able to hear above 5000!)

Loudness is a term well understood. Its physical equivalent is the intensity of the sound which can be measured by the energy it contains or more simply by the amplitude of the vibration (ex. the extent to which the Veena string is dis- placed from its normal position). When sound is recorded digitally into a .wav or similar file the extent to which the numbers deviate from the zero (no sound) position decides the loudness. Below are the graphs drawn from part of a digital data of a violin sound at 2 levels of loudness. You can click on them to hear the sounds. (The graphs shown cover only about 1.3% of the duration of the total sound you hear.)

Quality is a term more difficult to define. We speak of dull tone, rich tone, good voice or the good timbre of a musical instrument. We also recognize consonants and vowels though spoken in different pitches.

When the string of a musical instrument vibrates it does not make a simple oscillation with both ends fixed and the center making maximum movement. Actually the string can also vibrate in many different modes. It can vibrate as if it consists of 2, 3, 4 or higher integral number of segments with the points at midway, one third and two-thirds or every quarter of the length not moving at all. These points are called Nodes. You can see these movements in the animation below.

The frequency of a vibrating string depends on the weight per unit length , how tightly it is stretched (tension) and how long the vibrating part is. Other things being equal,in the case of an 'ideal strng', the frequency is inversely proportional to the vibrating length i.e. if the length is half, the frequency is double. So when a string vibrates in 2 segments it produces double the frequency, in 3 segments, three times the frequency and so on. (In Piano the frequencies of the upper harmonics are slightly higher then integral multiples and this leads to what is called 'Octave Stretch' with slightly sharpened upper notes)

The relative strengths of these different modes can also vary widely and we have hundreds of possible combinations, which explains the wide variety tones from different musical instruments. What is true of ideal strings is also true of air columns, vocal chords and almost any object producing sound. In general a flute or whistle like sound has less harmonics while stringed instruments are rich in harmonics. In the case of vibrating membranes (on drums) the harmonics may not be exact multiples of the basic frequency. In fact one of the important requirements of sustained musical note is that its harmonics should be 'true' or whole multiples of the fundamental. If you have an audio device (or software) with 'graphic equalizer' you can hear how the tone changes when the harmonics are adjusted. Speech also requires higher harmonics for clarity which is often the reason for very elderly persons' inability to follow the speech even though the sound is heard. The terms 'treble' and 'bass' imply higher and lower harmonics respectively. Absence of harmonics makes tone rather dull while very high harmonics make them sound 'metallic'.

Listen to note of a plucked string with some harmonics and listen the same note with much less harmonics -sounding 'dull'. Listen to note of freuency 524 with no harmonics and Listen to same note with high harmonics

Of the 3 attributes of sound, it is pitch that has been given maximum importance in musical theory . A musical note can be defined as a sound where the pitch is constant for some period (say at least one twentieth of a second - but generally much longer). When there is a certain relationship among the pitches of notes they sound pleasant when heard in succession (melody) or heard together (harmony). It is found that when two notes whose frequencies are in simple ratios are heard together or in succession they are pleasant. The simplest ratio of 2 (one note being twice in frequency of the other) gives maximum blending when heard together and pleasant when heard one after the other. Listen to two notes with frequencies in the ratio 1 : 2, first heard in succession and then sounded together. For this reason same symbols are used for notes whose frequencies are in the ratio 1:2, 1:4, 1:8 etc. with suitable indications. The interval between a note and another with twice its frequency is called an octave because in most musical systems 7 notes are used before reaching the 8th note with double the frequency. The next best blending is where one note is 1.5 times in frequency of the other note. Listen to such a pair of notes in the ratio 2: 3, in succession and together. 2:3 is the inverse of 3:2 and since a note of double frequency blends completely notes with frequencies in the ratio 3:4 also blend as well as the notes of the ratio 2:3. Also listen to notes which are not in simple ratio - not very pleasant.

When a note of frequency (say) 400 is sung or played on a musical instrument it also has the harmonics with frequencies 800, 1200, 1600 etc. When another note with 1.5 times the frequency of 400 i.e. 600 is played we also hear the harmonics of 1200, 1800, etc. The common figure 1200 in these two notes makes them blend well. The ratio 1:2 blends thoroughly because the second note is the 2nd harmonic of the first note. When the ratio becomes smaller (say) 9:8 the blending is less as we have to hear the 10th harmonic (9 times) the first and 9th harmonic (8 times) the second (for the base of 400, the second note will be 450 and the harmonics 9 X 400 = 3600 and 8 X 450 = 3600). Generally very high harmonics are weak. But we can find another note i.e. with frequency 600 (1.5 times 400) which blends with both since 600 = 400 X 1.5 and 600 = 450 X 4/3. Listen to the notes in the frequencies 400, 600, 450 and 600 again.

The ratio 1.5 or 3:2 plays an important part in musical theory. In Indian Music it is called Samvaadhithva and ancient Sanskrit and Tamil texts describe generation of scales using this ratio. This process is called the 'cycle of fifths'. If we take any note as sa then the note which is 1.5 times in frequency is pa (in western notation C and G). Pa is the 5th note in sa-ri-ga-ma-pa and hence the term cycle of fifths. We can calculate the frequencies of a number of musical notes by repeated multipli- cation by 1.5 and dividing by 2 whenever the frequency goes above twice the frequency of sa so that we get a note within the octave.

In melodic music the actual frequency is less important than the ratio between the frequencies. In Carnatic Music the musician is at liberty to chose any frequency as his tonic and then it becomes sa. In discussing frequencies of notes we use 'Relative Frequency' (r.f.) which is a ratio rather than the actual frequencies. Thus pa has the r.f of 3/2 or 1.5 which means that the actual frequency of pa will be 1.5 times the frequency of Sa. By applying cycle of fifths 5 times we can get the following frequencies: 1 (the tonic), 3/2 (pa), 9/8 which is 3/2 X 3/2 = 9/4 divided by 2, 27/16 (9/8 X 3/2) and 81/64 (27/16 X 3/2 /2). These are roughly the white notes sa, pa, ri2, da2 and ga2. Arranging in ascending order sa,ri2,ga2,pa,da2 and adding upper Sa the r.f's are 1, 9/8, 81/64, 3/2, 27/16 and 2. Listen to these notes (you can perhaps recognize Mohanam). Continuing, we get 243/128 (ni2) and taking 4/3 as ma1 we get all the white notes (not exactly, as on the keyboard which uses equally tempered scale - explained later). To avoid more unwieldy figures we can use 15/8 instead of 243/128; these are very close (differ only by 81/80) and go further to get more notes (notes of black keys ma2,ri1,etc.)

We can also calculate frequencies by multiplying by 4/3 instead of 1.5 (cycle of fourths as 4/3 represents the fourth note from sa). Trying the cycle of fourths we get the ratios 1, 4/3 (ma1), 16/9 (4/3 X 4/3 ni1), 32/27 (16/9 X 4/3 /2 ga1) and 128/81 (32/27 X 4/3 da1) or 1, 32/27, 4/3, 128/81, 16/9 and 2 in ascending order. Listen to these notes. (Can you recognize Hindholam ?).

The diagrams below will help identify the notes in the cycle of fifths and fourths. On a keyboard or fretted instrument there are 12 notes in the octave and pa is at the 7th position and ma1 is at the 5th position. So, to apply the cycle of fifths we count 7 notes up each time (counting all the notes - black and white keys) and if the result is above the upper Sa we come back by 12 notes to be within the octave. On a cyclic diagram this is automatically done. For cycle of fourths count 5 notes each time. (Colors of the lettering correspond to key colors.)

   

There is a catch in calculating frequencies by cycles using 3/2 and 4/3 as ratios because 3/2 multiplied 12 times is 129.746.. and dividing by 2 repeatedly to come back to the octave we land up at 2.027.. instead of 2 which should be the r.f. of upper Sa. In fact the pa arrived at by cycle of fourths is 1.4798 and not 1.5. Listen to natural sa- pa and the cycle of fourth sa-pa and the two pa's one after the other. Calculation of the frequencies by cycle of fifths and fourths gives two sets of values for the notes differing by the fraction 81/80 which has dogged musicians especially for orchestration involving extensive harmony. From these pairs of notes the following set of simple fractions are generally mentioned for 'natural notes':

sa

(ri1)

ri2

(ga1)

ga2

ma1

(ma2)

pa

(da1)

da2

(ni1)

ni2

Sa

1

16/15

9/8

6/5

5/4

4/3

45/32

3/2

8/5

5/3

16/9

15/8

2

Although 9/8 (ri2) and 3/2 (pa) have a relative ratio of 4/3, the ratio between da2 (5/3) and ri2 (9/8) is 40/27 and not 3/2. There is no way to have every pair of notes separated by 7 or 5 notes to have the perfect 3/2 or 4/3 ratio. As this problem caused difficulties in orchestration, western system has adopted a scheme in which every adjacent pair of notes has the same r.f of twelfth root of 2 - about 1.05946. This is called the 'equally tempered scale'. In this scheme ma and pa are 1.3348 and 1.4983 instead of 1.3333 and 1.5, a difference not usually distinguishable by the human ear (0.1 percent).

The idea of simple ratio for notes can be illustrated on stringed instruments. Most stringed instruments have a few strings only and notes are produced by damping a part of the string (on the finger board on a violin or on the frets in Veena or Guitar) and making the rest vibrate. By damping lengths which are simple fractions like 1/2, 1/3, 1/4 etc. we can get pleasant notes of r.f. 2,3/2,4/3 etc. The relative frequency is the reciprocal of the vibrating part fraction. The ratios 2 (Sa of upper octave), 3/2 (pa), 4/3 (ma1), 5/4 (ga2) and 6/5 (ga1) are all natural notes because of their simple ratios. 7/6 and 8/7 are however not used but 9/8 (ri2) and sometimes (10/9) are used. The sketch below shows how notes are generated by damping part of a 60 cm long string. In a fretted instrument there is some pressure applied to make string contact the fret and this increases the tension and actual positions would be slightly to the left.

The process of cycle of fifths or fourths is actually visible on stringed instruments especially those with frets. Different notes are produced on the different strings at a given fret. The Veena has its first string tuned to sa and the next to lower pA with r.f 3/4 (3/2 / 2). Thus on every fret, the frequency of a note on the pA string is 3/4th of the note on sa string because the vibrating lengths are reduced by the same proportion in both cases. Against the pa fret of sa string we get ri2 on the pA string, against ri2 on sa string we get dA2 on pA string, against da2 of sa string we get ga2 on the pA string and so on, generating the same sequence of notes as in the cycle of fifths. In fact this is how the frets are placed. If pa fret is placed correctly others can be placed by alternately listening to notes produced on sa and pA strings. If we tune the second string to mA we will get the notes of the cycle of fourths.

As mentioned earlier the slight increase in tension when the string is pressed to the fret requires some adjustment. In the Guitar it is done by a slanting saddle next to be bridge.

The question as to the frequencies of notes used in Carnatic Music has been subject of much discussion. The pairs of frequencies obtained by cycle of fifths and cycle of fourths have been equated with the 22 sruthis of ancient texts - without theoretical justification or based on practical measurements with modern computers . It is difficult to speak of frequency when a note is constantly oscillated, often with unequal timings of rest at the ends. When ri1 of Saaveri is sung the voice remains most of the time in sa and reaches ri1 quickly and comes back giving the feeling of a very low ri1. It is possible that the presence or absence of other notes with sa-pa or sa-ma relationship in a raagam influences the pitch of a note and the manner in which it is held. However, some notes on the key- board (equally tempered scale) such as ga2 (r.f 1.26) appear somewhat higher for an ear accustomed to the natural ga2 (r.f 1.25 a difference of about 1% ) Listen to sa-ga equally tempered, sa-ga natural and the two ga's. The difference is noticeable when sounded one after other and in other cases a background properly tuned thambura sound may also help to distinguish the two notes

Given below are the frequencies of equally tempered scale and one possible set of natural notes for sa at 262 (261.62 to be accurate, which corresponds to 440 for for da2, standard used in western music). Notes of black keys are in brackets. Actually the notes of black keys sound much lower in Carnatic Music as they are usually oscillated from the lower note (except ma2 which is oscillated from pa and sounds higher).

In Music Pitch frequencies are related by ratios. Thus a note which is one octave higher has double the frequency of the lower note. A note having 3/2 times the frequency of another note is called the fifth (pa in carnatic music). Comparison of pitch values of fractions like 9/5 or 16/9 is difficult. The cyclic cent system coverts the ratios into additions by use of logarithms. As most music systems use 12 notes in an octave, the value of 1200 cents is assigned to a note having double the frequency of the tonic (sa and Tara Sa in Carnatic Music). In equally tempered scale each note will be (dividing 1200 by 12) 100 cents higher than the adjacent lower note . The cyclic cent of a ratio (relative frequency or rf) is given by log(rf)*1200/log(2). You will find that it is easier to appreciate the difference between 9/5 (1017.6) and 16/9 (996.1) and see which is higher. Some common note values are 'pa' (3/2)-702 cents. 'Suddha ma' (4/3) 498 cents. Adding them you get 1200 which is the value of Tara Sa. Others - Anthara Gaandhaaram 5/4 is 386, Chathusruthi Rishabham (9/8) is 204. Add 702(rf of pa) to 702 giving 1404. Subtract 1200 to come within middle octave giving 204 an this is Chathusruthi Rishabham - first step in Cycle of Fifths

In most papers or articles discussing frequencies/pitches in music the cent system is used rather than the r.f. itself. Below you can enter r.f and get cyclic cents

Enter Relative Frequency-as decimal fraction (like 1.25)
or in a/b style (like 5/4) and click OK