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
|Notes||Frequency|| Ratio from
| Ratio from
|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.
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.
|Notes||Ratio from A||Notes of A-major scale||Ratio from previous note||Notes of C-major scale||Ratio from previous note|
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.
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:
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:
|Notes||Frequency equal temperament||Frequency Just Intonation||Frequency difference||Ratio from Previous note|
|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.
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.
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?
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:
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:
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.
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:
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.
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:
|Key note||Octave||Key note||Mistuned octave||Beating between sounds|
|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.
|Key note|| Fifth 3/2
in Just Intonation
|Key note||Fifth tempered||Beating|
|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.
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:
| Key note
| Fifth tempered
|Fifth 3/2 Just Intonation E||Beating between the fifths|
|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.
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.