The yoga of sound and music that creates harmony and bliss in mind, body and emotions




The following is the text for a lecture held at Maharishi Vedic University, Vlodrop, Holland, in January 2011



GANDHARVA VEDA,
THE NATURE OF MUSIC
AND EQUAL TEMPER

By Ketil Helmersberg


CONTENT
The essence of music
Intervals of Just Intonation
- The nature of sound
- Intervals of the harmonic series
- Building natural scales - scales of Just Intonation
- Problems related to tuning in Just Intonation
The twelve-tone equal temperament system
Limitations of the twelve-tone equal temperament
- The influence of sound intervals on the mind
- Consonance and dissonance
Beating
- Examples of consonance and dissonance
Just intonation in combination with equal temperament Conclusion


Maharishi talked about Gandharva Veda as the music of nature, of natural law, that can create bliss and harmony and aid the development of consciousness. He also said that there were elements of Gandharva Veda in all types of music, but seemed to be of the opinion that traditional, Indian classical music was what was most in tune with the laws of nature - especially when the different Ragas, which are the tonal framework for this music, are played at their appropriate time.

However, in the latter 100 years or so, Indian music has been influenced by a foreign tonal system, different from its own original. This foreign system is the western tonal system, which has influenced Indian music mainly by Indian musicians adopting western instruments. This is especially the case with the harmonium, which has become very popular and widespread in India. Many scholars of Indian music consider this influence to be strongly distorting and polluting, lessening the original purity and strength of Indian music. Because we believe it important to raise awareness of this issue, we will try to explain what this is about, and what implications it has for Gandharva Veda.

The case is that unlike original Indian music, the modern western tonal system is not in accordance with natural tuning, and consequently neither the instruments based on this system. It's a question of intervals between notes. While the intervals used in the original Indian music is based on what is called natural harmonics, the intervals used in today's western music is tempered, which means that they are artificially maid. We will in the following describe these two tonal systems, their cons and pros, and make a comparison between them. We will start by describing the basic elements of music.

THE ESSENCE OF MUSIC

The essence of music can be said to be relationship of sounds. If we look up the word "music" in the Oxford dictionary, it gives the following definition:


Definition of music according to Oxford dictionary

Vocal or instrumental sounds (or both) combined in such a way as
to produce beauty of form, harmony, and expression of emotion




The sounds of music are both in traditional western and Indian music ordered in scales, which thus is the basis for the musical expression. Most commonly a scale has 7 notes, while it sometimes can have less and sometimes more. One of the notes of the scale is called the key note, the basic note of the scale. If you can liken it to a family, the key note is like the mother, while the rest of the notes are like the children, sometimes playing between themselves, but always with the mother in the background and always returning to the mother. Hence, the most important relationship between the notes of a scale is between the key note, the mother, and the other notes.

The key note is the starting note from which we build a scale. The different notes of a scale are often named by its position from the key note. Hence, the second note of a scale is called the second, the fourth note, the fourth, the fifth note, the fifth and so on.

INTERVALS OF JUST INTONATION

THE NATURE OF SOUND

When we strike a string on a guitar, it makes a sound. The sound comes about by the vibrating string making the adjacent molecules of air vibrate. The vibration spreads in all directions in space, like ripples in a pond. When it reaches our ear membranes, it makes them vibrate, and we perceive a continuous sound of a definite pitch.

The air vibrate in the same speed as the string, and the pitch of the sound is determined by the speed of this vibration, which is called the frequency. The frequency is measured in Hz, which is the number of vibrations pr. second. The frequency of the vibrating string is determined by its length, thickness and tightness.

Disregarding possible amplification devices, the amplitude of the vibration determines the volume. While the frequency, and thus the pitch of the sound, stays the same as long as the string is vibrating, the amplitude gradually diminishes, making the sound steadily fade away.

INTERVALS OF THE HARMONIC SERIES

An interval is the difference in pitch between two sounds. When we listen to the sound of a guitar string, we do not only hear one sound, but many sounds of different pitch. We actually hear a compound of different sounds. The intervals between these sounds are not at random, but very precise and orderly. What we hear is one sound that is most prominent, the basic sound, but in addition so called overtones.

Disregarding possible limitations of the physical medium of the string, the overtones are exact multiplied frequencies of the basic sound. If for instance the main sound is 200 Hz, the first overtone, which is the second sound, will be 400 Hz, twice the basic frequency. The second overtone, or the third sound, will be 600 Hz, three times the basic frequency. The third overtone, or the fourth sound, will be 800 Hz, four times the basic frequency. The fourth overtone, or the fifth sound, will be 1000Hz, five times the basic frequency, and so on. We can illustrate this by using a scale of frequencies in Hz:



A sound and its overtones



This sequence of sounds is called the harmonic series and represents a collection of natural intervals. They embody the sound intervals of nature, also called intervals of Just Intonation. In the most prominent cultures of the world's history, one built one's musical scales on such intervals. Also in Europe - all through the middle ages and in the renaissance - there was a general agreement that intervals of Just Intonation should be the basis for making music.

The intervals of Just Intonation can be expressed as ratios of the sound numbers, which are one number divided by the other. We can illustrate this in the following way:



Mathematics of Just Intonation



The interval ratio between two sounds in the harmonic series is the number of the last sound divided by the number of the first sound. By multiplying the number of the first sound with the ratio one gets the number of the last sound. The following demonstrates this:

The first interval we have is from sound 1 to sound 2, which in this case is from 200 Hz to 400 Hz. This is also called an octave. The interval ratio for this interval is 2/1. By multiplying 1 with the interval ratio, we get 2. Likewise, by multiplying 200 Hz with the interval ratio we get 400 Hz, the frequency of the 2. sound.

The second interval we have is from sound 2 to sound 3, which in this case is from 400 Hz to 600 Hz. The interval ratio for this interval is 3/2. By multiplying 2 with the interval ratio, we get 3. Likewise, by multiplying 400 Hz with the interval ratio we get 600 Hz, the frequency of the 3. sound.

The third interval we have is from sound 3 to sound 4, which in this case is from 600 Hz to 800 Hz. The interval ratio for this interval is 4/3. By multiplying 3 with the interval ratio, we get 4. Likewise, by multiplying 600 Hz with the interval ratio we get 800 Hz, the frequency of the 4. sound.

But we also have intervals comprising more than one interval, like for instance from sound 4 to 7, which in this case is from 800 Hz to 1400 Hz. The interval ratio for this interval is 7/4. By multiplying 4 with the interval ratio, we get 7. Likewise, by multiplying 800 Hz with the interval ratio we get 1400 Hz, the frequency of the 7. sound, and so on.

BUILDING NATURAL SCALES - SCALES OF JUST INTONATION

What we see is that there is a mathematical principal involved. All the interval ratios are expressed by whole numbers. So what characterizes natural intervals, or intervals of Just Intonation, is that they can be expressed as whole number ratios. This was also the discovery of the old Greeks, like Pythagoras, the old Indians, the Chinese and many other cultures of the world history, who all thought that musical scales should be based on such intervals.

Hence, one can choose a key note and build a natural scale, or a scale in Just Intonation, by adding intervals of whole number ratios. Which intervals to choose for different scales is a science by itself. There is, however, one basic guideline: The interval ratios of smaller numbers are more harmonious, or consonant, than those of larger numbers. As the numbers of the ratios become larger, the intervals become less consonant and more dissonant.

By applying whole number ratios, one can make a collection of intervals within an octave, from the smallest interval to gradually larger. These intervals then constitute a series of notes. From these notes or intervals, one then can select the ones to be used in different scales. In Indian musical theory, the largest number of notes in such a collection is 66, because this is considered to be the smallest degree of differentiation of sound one is able to perceive. Hence, the original Indian classical music, such as the genre of Dhrupad, is the only known form of music that systematically utilizes all possible intervals of naturally harmonic notes. Such a series of notes is also called shrutis or microtones.

In the European tradition, however, one has since long back used a tonal system with twelve intervals in the octave, constituting a series of notes called half tones. From these, one selects the notes for a scale. Originally, the twelve tones of western music were in accordance with Just Intonation.

While giving preference to the most consonant intervals, which have ratios of the smallest numbers, we can make a collection of twelve somewhat evenly spread out notes within an octave. Thus, we will have a series of half tones in Just Intonation. We can start with the standard A in western music, which has the frequency of 440 Hz.



Example of a series of intervals in Just Intonation

  Notes Frequency Ratio from
start note
A
Ratio from
previous note
1st A 440.0000 Hz      
2nd B b 469.3333 Hz 16/15 16/15 = 1.0667
2nd B 495.0000 Hz 9/8 135/128 = 1.0547
3rd C 528.0000 Hz 6/5 16/15 = 1.0667
3rd C# 550.0000 Hz 5/4 25/24 = 1.0417
4th D 586.6667 Hz 4/3 16/15 = 1.0667
  D# 616.0000 Hz 7/5 21/20 = 1.0500
5th E 660.0000 Hz 3/2 15/14 = 1.0714
6th F 704.0000 Hz 8/5 16/15 = 1.0667
6th F# 733.3333 Hz 5/3 25/24 = 1.0417
7th G 770.0000 Hz 7/4 21/20 = 1.0500
7th G# 825.0000 Hz 15/8 15/14 = 1.0714
Octave A 880.0000 Hz 2/1 16/15 = 1.0667


As we see from the table, one can also calculate the interval ratios between the half notes. To do this, one takes the interval ratio of a note and subtracts the interval ratio of the previous note. This is done by an interval ratio being multiplied by the inverse of the interval ratio to be subtracted. For instance, to find the interval ratio from Bb to B, one subtracts the interval ratio 16/15 from the interval ratio 9/8, which is done by the following multiplication: 9/8 x 15/16 = 135/128.

One can also add interval ratios. To do this, one multiplies one ratio with the other. For instance to add the ratio 16/15, which is the half note from A to B b , to the ratio 135/128, which is the half note from B b to B, one do the following: 16/15 x 135/128 = 9/8.

PROBLEMS RELATED TO TUNING IN JUST INTONATION

When we calculate the interval ratios between the half notes, we find that they are not equal. While the ratio from A to A# is 16/15, the ratio from A# to B is 135/128 etc. What this means in practice, is that to apply Just Intonation on a so called fixed-pitch instrument, like for instance an organ or a piano, one has to tune the instrument in accordance with one specific key note. If one should want to change the key note, which means starting the same scale from another pitch or frequency of sound, one most likely would have to retune the whole instrument, and to retune a piano is no small job.

This became a practical problem in western music when one started using fixed-pitch instruments, because one wanted to be able to frequently switch the key note. It also became a problem because one wanted to explore more complicated music with frequent modulations, which means transporting scales to different key notes in the middle of a composition. We can illustrate this problem by the following example:

The scheme above is based on the key note of A. So if we use this tuning with an A-major scale, we will see what will happen if we for instance try to change the key note to C. The A-major scale consists of the notes A - B - C# - D - E - F# - G#, while the C-major scale has the notes C - D - E - F - G - A - B.



Example of an A-major scale in Just Intonation changed to C

Notes Ratio from A Notes of A-major scale Ratio from previous note Notes of C-major scale Ratio from previous note
A   1st A        
B b 16/15            
B 9/8 2nd B 9/8      
C 6/5       1st C  
C# 5/4 3rd C# 10/9      
D 4/3 4th D 16/15 2nd D 10/9
D# 7/5            
E 3/2 5th E 9/8 3rd E 135/128
F 8/5       4th F 16/15
F# 5/3 6th F# 10/9      
G 7/4       5th G 35/32
G# 15/8 7th G# 9/8      
A 2/1 Octave A 16/15 6th A 8/7
B b 16/15            
B 9/8       7th B 9/8
C 6/5       Octave C 16/15


What we see is that the intervals between the notes of the two scales in many cases become different. For instance, the interval ratio between the first and second note of the A-major scale, which is from A to B, has the interval ratio 9/8, while the interval ratio between the first and second note of the C scale, which is from C to D, has the interval ratio 10/9. The interval ratio between the second and third note of the A-major scale is 10/9, while the interval ratio between the second and third note of the C scale, which is from D to E, has the interval ratio 135/128, and so on. Because the intervals of the notes of these two scales in many cases are different, they are actually two different scales. It is therefore not possible to change the key note of the major scale from A to C with this scheme of tuning.


THE TWELVE-TONE EQUAL
TEMPERAMENT SYSTEM

Because of these problems related to tuning in Just Intonation, one started in Europe, sometimes during the Renaissance, to experiment with different types of so called tempered tuning, which means altering the intervals of Just Intonation so as to be able to change the key note of a scale without retuning. Many different systems of tempering were proposed through the years, but finally, in about 1850, the most simplistic system, called the twelve-tone equal temperament, became the standard and has remained so since in western music.

Equal temperament means equalizing the interval ratio between the twelve notes within the octave and fixing their frequencies. The frequency of the note A in the middle of the piano keyboard was for instance set to be 440 Hz. Hence, We can start from this frequency to calculate the equal temperament interval ratio:



Calculating the frequency ratio for
the twelve-tone equal temperament


R = The frequency ratio
Start frequency x R x R x R ... (twelve times) = Start frequency x 2
Start frequency x R 12 = Start frequency x 2
(440 Hz x R 12 = 440 Hz x 2 = 880 Hz)
1 x R12 = 2
(440 Hz x R12 = 880 Hz)
R = 12 √2 ≈ 1.0594630943593

R is an irrational number
can not be converted to a whole number ratio




One multiplies 440 Hz with the frequency ratio to get to the frequency of the next half note. Then one multiplies this new frequency with the same ratio to get to the next half note thereafter, and so on. This one does all together 12 times to reach the octave of A, the next A on the keyboard of a piano, which has twice the frequency of the previous A.

In the formula, one can replace 440 Hz with 1 and the octave with 2. One can then calculate the interval ratio to be 1.0594630943593, which is an irrational number, which means that it can not be converted to a whole number ratio, which again means that it is not an interval ratio in accordance with the natural harmonics.

So, to make it clear. We start with the note A of 440 Hz. We multiply this frequency with the frequency ratio 1.05946 and we get 466.1624 Hz, which is the frequency of next note on the keyboard of a piano, Bb. Then we take this last frequency and multiply with the same frequency ratio, and we get 493.8824 Hz, which is the next note thereafter on the keyboard of the piano, B, and so on. This is the twelve-tone equal temperament system.

This tonal system is a compromise solution, where one compromises the consonance, or the harmony, of the intervals with the possibility of playing a scale in any key without one scale sounding more dissonant than another. However, this also means that none of the intervals except the octave are in accordance with the natural harmonics of Just Intonation. So what does this imply? We can make a table that compare the previous twelve tones in Just Intonation with the twelve tones in equal temperament:



Twelve-tone equal temperament
compared to twelve tones of Just Intonation

  Notes Frequency equal temperament Frequency Just Intonation Frequency difference Ratio from Previous note
Equal Just
1st A 440.0000 Hz 440.0000 Hz 0.0000 Hz    
2nd B b 466.1624 Hz 469.3333 Hz -3.1709 Hz 1.05946 1.0667
2nd B 493.8824 Hz 495.0000 Hz -1.1176 Hz 1.05946 1.0547
3rd C 523.2524 Hz 528.0000 Hz -4.7476 Hz 1.05946 1.0667
3rd C# 550.0000 Hz 554.3648 Hz +4.3648 Hz 1.05946 1.0417
4th D 587.3296 Hz 586.6667 Hz +0.6629 Hz 1.05946 1.0667
  D# 622.2524 Hz 616.0000 Hz +6.2524 Hz 1.05946 1.0500
5th E 659.2564 Hz 660.0000 Hz -0.7436 Hz 1.05946 1.0714
6th F 698.4560 Hz 704.0000 Hz -5.5440 Hz 1.05946 1.0667
6th F# 739.9876 Hz 733.3333 Hz +6.6543 Hz 1.05946 1.0417
7th G 783.9920 Hz 770.0000 Hz +13.9920 Hz 1.05946 1.0500
7th G# 830.6100 Hz 825.0000 Hz +5.6100 Hz 1.05946 1.0714
Oct. A 880.0000 Hz 880.0000 Hz 0.0000 Hz 1.05946 1.0667


As seen from the table, the difference in frequency between equal temperament Just Intonation might seem to be small. The supporters of the twelve-tone equal temperament system will therefore probably claim the this difference is not of great importance. They also might ask why the intervals of Just Intonation should be more preferable, even if they can be considered to be so called natural, which means in accordance with the natural harmonics.

To answer this question, we will first consider the limitations of the twelve-note equal temperament system and then its influence on the mind of the listener as compared to Just Intonation.


LIMITATIONS OF THE TWELVE-TONE
EQUAL TEMPERAMENT

The twelve-tone equal temperament system has great limitations for musical expression. While one in just Intonation have a large number of natural intervals available, one has in the twelve-tone equal temperament system only 12 fixed intervals to use.

As an illustration of this limitation, much of the world's folk music and contemporary music would actually not have existed if one only had to stick to the tempered system. This includes genres of music like Irish and English folk music, Negro Spirituals, Blues, Soul, many types of Jazz and Rock and Roll. The reason is that these genres of music rely heavily on intervals that simply are not available in the tempered system, as for instance the so called blue notes, which often are a lowered third, fifth or seventh of a scale, but not lowered as much as reaching the next half note in the equal temperament. These are notes that in many ways are the life-blood of these genres of music. Without them, they would loose their vitality and power of enchantment.

It is possible to create a kind of an illusion of a blue note on for instance a piano by playing very fast intervals of half notes, and thereby create a feeling of a blue note, which is situated somewhere between two half notes. Hence, some pianists can to a certain degree compensate the limitations of the tempered system by their technical ability. But this is definitely not the same as playing the blue note itself, which is not available on an equal tempered piano.

The limitations of the tempered system are even more apparent when it comes to Indian music. There are so many intervals in Indian classical music that are not available in the tempered system. The tonal framework for composition and improvisation in Indian classical music is called Raga, of which there is recorded to exist about 300, and each of them has their own specific scale. The difference between the scales of two Ragas, for instance a morning and evening Raga, can sometimes be only a microtone on some of the notes.

Furthermore, an important part of a Raga is to move certain notes away from their position after they have been sounded, so to slide between the microtones or from note to note in the scale, enhancing the beauty of the composition. This is certainly not possible on a piano or a harmonium, which therefore makes it impossible to play a Raga properly on them, even if they should be tuned in accordance with the natural harmonics.

Moreover, the limitation of the equal temperament is not only the reduced selection of intervals, but also that each note is fixed to a certain frequency. In traditional Indian music one never did that. Every frequency of sound has a particular influence, a particular quality or feel to it. If there weren't different feelings connected to different sound frequencies, there would, for instance, be no point of playing in different keys. By the fixity of the frequencies of the notes, the twelve tone equal temperament excludes many frequencies - obliterate them from nature's palette. If you liken the sound frequencies to the spectrum of colors, it is as if artists only had a small limited number of set colors to work with.

THE INFLUENCE OF SOUND INTERVALS ON THE MIND

Another very important consideration regarding intervals of sound is how they affect the mind of the listener. In the classical texts of Indian music, as also in the theories of the Greek philosopher Pythagoras, a key factor for music to have a positive effect is that it should be pleasing to the mind. Studies show that when people hear intervals of just intonation, they find them to be more pleasing, more beautiful than the equivalent intervals in equal temperament. People are actually often amazed that the intervals of equal temperament at all can be considered consonant, or harmonious, when hearing them after having heard the equivalent intervals in Just Intonation.

Esthetic appeal was also Pythagoras' starting point. He discovered that the length of a string is equivalent to the difference in sound frequencies. If for instance the length of a string was twice the length of another with the same thickness and tightness, the interval between them would be an octave. By this, he discovered that the intervals of sound were the most beautiful when the difference in the length of the strings were in small whole number ratios. On the basis of this discovery, he was even able to use music to cure people from diseases.

Esthetic reasons were also the main argument against the equal temperament when it was introduced in Europe. Musical theorists of the time felt that equal temperament degraded the purity of each chord and the esthetic appeal of music. It is also interesting to note that none of the renowned western, classical composers wrote for equal temperament, including Bach, Mozart, Beethoven, Schubert, Schumann, Chopin, Liszt, Wagner, Brahms and Chaikovskii. Mozart is even quoted to have said that he would kill anyone that would play his music in equal temperament.

However, considering that the differences in frequency between the notes in equal temperament and the equivalent notes in Just Intonation are not very large, as seen in terms of percentages, why should there be such a difference in the pleasantness of hearing their intervals? Can it be just a question of imagination? Or some kind of a placebo effect?

CONSONANCE AND DISSONANCE

The answer to this is that the intervals of Just Intonation are more consonant, or harmonious, than the equivalent intervals in equal temperament, which also can be shown by modern scientific experiments. Consonance is a word derived from Latin: com, "with" + sonare "sound." If we look it up in the Wikipedia, it will be defined as the following:



Consonance

A harmony, chord or interval that are considered stable, as opposed to dissonance, which is considered unstable.



Dissonance is also a word derived from Latin: dis "apart" + sonare, "to sound." It defined by the modern musicologist Roger Kamien in the following way:



Dissonance

An unstable tone combination is a dissonance; its tension demands an onward motion to a stable chord. Thus dissonant chords are 'active'; traditionally they have been considered harsh and have expressed pain, grief, and conflict."



Both consonance and dissonance are important for musical expression, but it has a value that the consonant intervals are truly consonant. To show by modern scientific experiments that the intervals of Just Intonation are more consonant than the equivalent intervals in equal temperament, we have to go into a branch of physics called acoustics. This is a comprehensive science, because many features are involved in the relationship between sounds. We will therefore only look at what is considered to be the most important factors for consonance and dissonance. These are concurrence of overtones and a phenomena called beating.

BEATING

When the difference in frequency between two sounds is more than zero Hz and less than about 20 Hz, we will perceive them as one sound. The frequency of the combined sound that we hear, will be the average of the two sounds. The volume of the combined sound, however, will for certain reasons constantly vary, and this is what is called beating. It is a phenomena that is considered to be the principal cause of dissonance. The reason why beating occurs is because of the constantly changing relationship between the vibrations of the two sounds.

When we strike a string on a guitar, the air molecules surrounding it starts to vibrate back and forth. The vibration spreads in all directions of space. The volume of the sound is dependent on the amplitude of the vibration. When the frequency of two sounds are so close together that they are perceived as one sound, the amplitude of the combined sound is the summation of the amplitudes of the two sounds. As one of the two sounds vibrates slightly faster than the other, the relationship between their vibrations will continuously vary. At one point they will be synchronous, which means that they will swing back and forth simultaneously. Then they gradually will be less synchronous, which also means that the sum of the amplitudes gradually will be less, until they reach a point when they vibrate opposite each other. If they then have the same amplitude, the sum of their amplitudes will be zero, making no sound. If the amplitude of one of the two sounds is larger than the other, the amplitude of the combined sound will not be zero, but less. Then gradually the vibrations of the two sounds will move back to being synchronous, which also means that the amplitude of the combined sound gradually will increase, and so on.

The sound that we hear is the result of the two sounds working against our ear membranes. When their vibrations are synchronous, they will push and pull the ear membranes simultaneously, and thus with twice the force as by one sound. Then when their vibrations are opposite each other, one sound will push the membranes, while the other will pull it, and thus they will block each other, making us perceive a reduced sound or no sound at all. This is what is called beating. It can be likened to listening to the radio while turning the volume rapidly up and down.The frequency of the beating is the difference in frequency between the two sounds. We can illustrate the phenomena by the following figure, which shows the sound vibrations as waves:



Example of beating



The two upper waves are the sounds, while the wave under is the combined sound that we hear. The two sounds have the same amplitude. The changing amplitude of the lower wave shows the change in volume of the combined sound. At point A the two sounds are somewhat synchronous, and the combined amplitude is at its largest. Then they become less synchronous and the combined amplitude becomes less. At B they vibrate opposite each other and the combined amplitude becomes zero, making no sound. Then they gradually move back to synchrony, while the combined amplitude gradually increases and reaches its maximum when the two waves again become synchronous, and so on.

When the difference in frequency increases between two sounds, while it's is still less than about 20 Hz, the frequency of the beating increases. When the difference between the sounds becomes larger than about 20 Hz, the beating is replaced by a general experience of roughness. When the difference reaches a point somewhere between a whole note and a minor third, the beating stops, and we hear two separate sounds.

EXAMPLES OF CONSONANCE AND DISSONANCE

The phenomena of beating denotes that just a slight variance in frequency have a strong impact on the degree of consonance or dissonance. We can illustrate this by a mistuned interval of an octave:



Example of beating for a slightly mistuned octave

    Key note Octave Key note Mistuned octave Beating between sounds
O
v
e
r
t
o
n
e
s
10. sound 4000 Hz 8000 Hz 4000 Hz 8020 Hz 10. + 5. = 10 Hz
9. sound 3600 Hz 7200 Hz 3600 Hz 7218 Hz  
8. sound 3200 Hz 6400 Hz 3200 Hz 6416 Hz 8. + 4. = 8 Hz
7. sound 2800 Hz 5600 Hz 2800 Hz 5614 Hz  
6. sound 2400 Hz 4800 Hz 2400 Hz 4812 Hz 6. + 3. = 6 Hz
5. sound 2000 Hz 4000 Hz 2000 Hz 4010 Hz  
4. sound 1600 Hz 3200 Hz 1600 Hz 3208 Hz 4. + 2. = 4 Hz
3. sound 1200 Hz 2400 Hz 1200 Hz 2406 Hz  
2. sound 800 Hz 1600 Hz 800 Hz 1604 Hz 2. + 1. = 2 Hz
  1. sound 400 Hz 800 Hz 400 Hz 802 Hz  


We see that between the key note and the pure octave, the overtones are in concurrence with each other on different levels, which means that there are no beating and that the two sounds have a very high degree of consonance. If we however mistune the octave by 2 Hz, we get beating between the overtones on many levels, between the 2nd and 1st sound, between the 4th and 2nd sound, between the 6th and 3rd sound etc. This will result in reduced consonance, or increased dissonance.

We can now compare the consonance of Just Intonation with that of equal temperament by using the fifth as an example. The fifth is considered to be a highly consonant interval.



The comparative consonance of a fifth
in Just Intonation and in equal temperament

    Key note Fifth 3/2
in Just Intonation
Key note Fifth tempered Beating
O
v
e
r
t
o
n
e
s
10. sound 4400 Hz 5940 Hz 4400 Hz 5931 Hz  
9. sound 3960 Hz 5400 Hz 3960 Hz 5391 Hz 9. + 6. = 6 Hz
8. sound 3520 Hz 5280 Hz 3520 Hz 5272 Hz  
7. sound 3080 Hz 4620 Hz 3080 Hz 4613 Hz  
6. sound 2640 Hz 3960 Hz 2640 Hz 3954 Hz 6. + 4. = 4 Hz
5. sound 2200 Hz 3300 Hz 2200 Hz 3295 Hz  
4. sound 1760 Hz 2640 Hz 1760 Hz 2636 Hz  
3. sound 1320 Hz 1800 Hz 1320 Hz 1977 Hz 3. + 2. = 2 Hz
2. sound 880 Hz 1320 Hz 880 Hz 1318 Hz  
  1. sound 440 Hz 660 Hz 440 Hz 659 Hz  

The tempered fifth is actually 659.2564 Hz.


We see in this table that there are a large degree of concurrence between the overtones of the key note and the fifth in Just Intonation. By none of the sounds are there such a difference that beating can occur.

But if we look at the tempered fifth, none of the overtones are concurrent with the overtones of the key note, and we get beating between many of the overtones. We get beating between the 3rd and the 2nd sound, between the 6th and the 4th sound, between the 9th and the 6th sound etc. This means that the consonance is much weaker, or the dissonance stronger, compared to the equivalent interval in Just Intonation.

JUST INTONATION IN COMBINATION
WITH EQUAL TEMPERAMENT

Still, somebody might say that even though an instrument like the piano or the harmonium, which are tuned in equal temperament, have their limitations, and that their intervals are less pleasing, why should it not be okay to use them for accompaniment, for instance for a singer?

To answer this question, let us first consider the singing by itself. In the Indian classical tradition of music, as for instance in the genre of Dhrupad, one is trained by ear to sing in Just Intonation. Studies also show that people in general, singing alone or in a vocal group, naturally tend to sing in Just Intonation when not being accompanied by an equal tempered instrument. But what happens if you try to sing in Just Intonation when being accompanied by an instrument tuned in equal temperament? Let's say that you are singing a tune in the A-major scale based on the intervals of Just Intonation that we have shown above. And let's say that you sing a fifth, which in this case is the note E, while an A-major chord is being played on the instrument, having the notes A - C# - E. Let's look at the following table to see how this will work:



Just Intonation fifth in combination with equal tempered fifth

    Key note
A
Fifth tempered
E
Fifth 3/2 Just Intonation E Beating between the fifths
O
v
e
r
t
o
n
e
s
10. sound 3900 Hz 6590 Hz 6600 Hz 10 Hz
9. sound 3520 Hz 5931 Hz 5940 Hz 9 Hz
8. sound 3200 Hz 5272 Hz 5280 Hz 8 Hz
7. sound 3080 Hz 4613 Hz 4620 Hz 7 Hz
6. sound 2640 Hz 3954 Hz 3960 Hz 6 Hz
5. sound 2200 Hz 3295 Hz 3300 Hz 5 Hz
4. sound 1760 Hz 2636 Hz 2640 Hz 4 Hz
3. sound 1320 Hz 1977 Hz 1980 Hz 3 Hz
2. sound 880 Hz 1318 Hz 1320 Hz 2 Hz
  1. sound 440 Hz 659 Hz 660 Hz 1 Hz

The tempered fifth is actually 659.2564 Hz.



The piano or harmonium will play the key note and the tempered fifth, which is the notes A and E, while the singer will try to sing the note E in Just Intonation. If we then look at the difference in frequencies between these two Es, we see that not only are neither of their sounds concurrent with each other, but that we will get beating on all levels, both between the basic sounds and all the nearest overtones. This will probably create a very strong degree of dissonance, which most likely will force the singer to sing in equal temperament. Hence, to sing a Raga in accordance with the natural harmonics will probably not be possible when being accompanied by for instance an equal tempered harmonium.

What these examples taken from the science of acoustics show, is that very minute modifications of natural intervals, which in isolation might seem to be trifles, might have far reaching distorting consequences on many levels.

CONCLUSION

So to conclude. Western musicology has made a prison for its music. It has locked it out from a vast universe of sounds and potential musical expressions. In addition, it has marred the harmony and beauty of the sound intervals by distorting their natural relationship. It has even conditioned musicians to hear music that are in accordance with natural harmonics as out of tune!

By incorporating western musical instruments into Indian music, the original strength and purity of the music is distorted and polluted. One is bringing the music away from natural law, while it should do the opposite, bring us more in tune with natural law. We believe it therefore to be important that everybody interested in Gandharva Veda should become aware of this.