Principles of tuning are essential for understanding the principles of harmony and composition in music. Complete scale comprises all possible swarasthanas. Different modes can be created based on the given scale.
The mode starts from the basic note (tonic), covers the whole scale in given number of steps (intervals). The sequence of the notes may also vary.
Tonic, notes, intervals, sequence of notes in the mode create a special flavor and character of the mode.
Combinations of notes are of two kinds: vertical (chords) or horizontal (melodic sequences).
Intervals between the notes are the measure of harmony (consonance) between given notes.
The mode could be imagined like a palette, harmonious combination of different colors, which evokes certain associations - winter or summer, morning or evening, tranquility or fury.
Fundamental frequency is any frequency from the continuum which defines the tonic (basic note of an octave).
Integer multiples of fundamental frequency are called harmonics, partial tones (partials) or overtones (at this, fundamental is the first harmonic of itself).
The term overtone is used to refer to any resonant frequency above the fundamental frequency. Thus, the fundamental frequency is the "first harmonic", the first overtone is the second harmonic or the second partial, and so on.
If one note has a fundamental frequency F, then its harmonics will have the frequencies 2F, 3F, 4F, 5F, 6F (this sequence is called "harmonic series").
|F||1||1st (odd) harmonic||fundamental|
|2F||2||2nd (even) harmonic||1st (odd) overtone|
|3F||3||3rd (odd) harmonic||2nd (even) overtone|
|4F||4||4th (even) harmonic||3rd (odd) overtone|
|5F||5||5th (odd) harmonic||4th (even) overtone|
|6F||6||6th (even) harmonic||4th (odd) overtone|
Overtones whose frequency is not an integer multiple of the fundamental are called inharmonic and are often perceived as unpleasant.
Human perception of pitch is logarithmic in nature (like most human senses) so a ratio of frequencies is perceived as a fixed size step in the pitch scale.
The ratio of the frequency of one note to the frequency of another note is the physical basis of an interval in music. In numerical terms an interval is the ratio between two pitches.
An harmonic interval is an interval that occurs between two members of an harmonic series. For example in the series 1F, 2F, 3F, 4F, 5F, and 6F the intervals between two adjacent harmonics are 1/1, 2/1, 3/2, 4/3, 5/4, and 6/5.
The harmonics of harmonics are themselves harmonics. Thus, any two notes related by an harmonic interval will be at least to some extent consonant.
Relative strength of the harmonics typically decreases with harmonic number. Thus, the lower numbered harmonics have the greatest influence on consonance.
The closest possible consonance is unison, 1 / 1, the consonance between two identical notes.
The octave, 2 / 1, is the next most consonant.
The perfect fifth, 3 / 2, and perfect fourth, 4 / 3, come next. These are the only intervals classified as perfect concords.
A musical scale constructed on such consonant intervals should itself be consonant.
Accordingly, 4F, 8F, 16F (i.e. fourth, eighth, sixteenth harmonics are 2, 3, 4 octaves from the fundamental).
Perfect fifth and perfect fourth taken together give octave: 3/2 x 4/3 = 2 / 1
A minor third and major third taken together give perfect fifth: 6/5 x 5/4 = 3/2.
Swara is the name of particular note. There are seven swaras - Shadja, Rishabha, Gandhara, Madhyama, Panchama, Daivata, Nishada.
Swarasthana is position of the note, i.e. one note can take different positions within the certain intervals, which are calculated based on the principle of harmony.
Sa is the basic note, it defines the pitch and a fundamental frequency to which all other swaras are tuned to.
Pa is 3/2 of fundamental frequency or "perfect fifth" of Sa.
Ma is 4/3 of fundamental frequency or "perfect fourth" of Sa.
There are two ways how to calculate the ratios of the other swarasthanas.
Ancient Greek philosopher Pythagoras (580 - 500 B.C.) devised a system of tuning based solely upon the interval of perfect fifth, the next most consonant interval after unison and the octave. One way in which he may have done so was by constructing a series of both descending and ascending fifths from the same starting point.
From a starting point 1/1 the ascending fifths are produced by successively multiplying by 3/2 to give 3/2, 9/4, 27/16 and so on to (3/2)^n, where n is an integer, from 1 to eternity.
The descending fifths are produced by successive multiplication by 2/3 to give 2/3, 4/9, 16/27 and so on to (2/3)^n, where n is an integer, from 1 to eternity.
According to this method, five "major" swarasthanas are derived from sa (1/1) by multiplication of 3/2 (i.e. to define such frequencies which stay the same "perfect" interval of 3/2 from each other). This cycle is called "cycle of fifths".
To keep all the notes in the same octave, some of them need to be transposed down by a number of octaves (to find the same proportion in the octave starting from fundamental 1/1 and to doubled fundamental 2/1). The table below shows the resultant series of fifths sorted by ascending order of pitch.
|Series of Fifths|
|Swara-sthana||Original fifth||Octaves to drop||Transposed fifth|
|Sorted Series of Fifths|
|Swara-sthana||Sorted fifth||Interval to next|
|sa||1/1||9 / 8|
The intervals of the scale are of two distinct sizes, 9 / 8 and 256 / 243. The logarithm of the interval 9/8 (0.117783) is very roughly twice the logarithm of the interval 256/243 (0.052116) so the interval 9/8 sounds like a step twice the size of the step 256/243. These are analogs of tone and semitone steps. At the other side, the interval 9/8 have meaning of four srutis, and interval 256 / 243 has meaning of one struti (as will be shown below).
Five "minor" swarasthanas are derived from sa using the cycle of fourths. The method is the same, starting from fundamental (sa) we define swaras which are 4/3 from each other, then transpose resulting values to the "basic" octave (from 1 to 2).
|Series of Fourths|
|Swara-sthana||Original fourth||Octaves to drop||Transposed fourth|
|Sorted Series of Fourths|
|Swara-sthana||Sorted fourth||Interval to next|
|Swara-sthana||Ratio||Interval to next|
|*2187/2048 = (9/8)/(256/243)|
The intervals of the scale are of two distinct sizes, 256/243 and 2187/2048 = (9/8)/(256/243).
Swarasthanas thus are twelve in number. At this, Swaras themselves are seven - sa, ri, ga, ma, pa, da, ni, sa.
Sa is basic (fundamental) thus one in number. Pa is perfect fifth, also one in number. Other five swaras have two varieties ("major", denoted by capital letter and "minor", denoted by small letter).
All twelve swarasthanas can be calculated based on the cycle of fifths, repeated twelve times.
|Series of Fifths|
|Swara-sthana||Original fifth||Octaves to drop||Transposed fifth|
Accordingly, if we repeat cycle of fourth twelve times, we will get another set of twelve swarasthanas, which ratios a different from the ratios calculated based on the cycle of fifths.
|Series of Fourths|
|Swara-sthana||Original fourth||Octaves to drop||Transposed fourth|
The difference between two sets of swarasthanas is demonstrated on the picture below. The swarasthanas calculated based on the cycle of fifths are shown as red points, and swarasthanas received from the cycle of fourths - as blue points.
If we keep only one swarasthana for sa and pa, and four swarasthanas for each of Nyasa swaras, i.e. ri, ga, ma, da, ni, we will reduce from 24 calculated values to the number of 22, mentioned in Natya Sastra.
Scales derived from the first and the second harmonics (3/2 and 4/3) are mutually symmetrical (3/2 multiplied by 4/3 gives 2). We can do ascending cycle of fifths (starting from 1/1, multiplying each time by 3/2 and transposing values to basic octave (1/1 to 2/1).
We also can do descending cycle of fifths, starting from upper Sa (2/1), dividing each time by 3/2, and transposing resulting values to basic octave.
We also can derive ascending and descending cycles of fourths:
|Cycle of fifths||Cycle of fourths|
|Swara-sthana||Ascending x3/2||Descending x2/3||Swara-sthana||Ascending x4/3||Descending x3/4|
Table above shows that ascending cycle of fifths and descending cycle of fourths give the same values of swaras. The same, descending cycle of fifths and ascending cycle of fourths gives the same values. Those two cycles are reciprocal to each other. They are very convenient in case of tuning musical instruments.
Thus, we can use only one cycle of fifths, ascending and descending, to get 24 different swarasthanas.
As we see from the pictures above, resulting sequences of swaras in cycles of fifths and fourths, follow each other, but corresponding swaras differ from each other for a small interval: swarasthanas of the cycle of fourths (blue points) are a bit lower then swarasthanas of the cycle of fourths (red points).
The value of this interval can be estimated using ratios of the final swara, sa, in both cycles (which is different from 2/1 ratio). The reason is, we use geometrical progression (not arithmetical), i.e. to move one step forward we multiply, not add. The upper limit of such geometrical progression is 2/1. Progression will climb up to this limit eternally, never achieving it. Thus, it seems not clever to go on more then for 12 steps in such progression.
As we can see from the table below, if we divide the value of swarasthana in cycle of fifths (swaras with indexes "2" ) by the value of corresponding swarasthana in cycle of fourths (swaras with index of "1"), we will get always the same value, which is equal to 531441/524288 = 3^12/2^19 = 1.013643265. This value is called "Pythagorean comma".
|Swara-sthana1||Original fifth||Transposed fifth||Decimal value||Swara-sthana2||Original fourth||Transposed fourth||Decimal value||Ratio of swara-sthana1 to swara-sthana2|
The picture below illustrates what happens if we shift cycle of fourths up (multiply) by the value of Pythagorean comma. Original values of swarasthanas are shown in gray color. After multiplication by Pythagorean comma, the whole set of swarasthanas "rises up", so red and blue plots become absolutely symmetrical and run through the same values of swarasthanas.
The next picture below illustrates that such shift is symmetrical. the cycle of fifths is shifted down by dividing the corresponding values by Pythagorean comma (original values of "red" swarasthanas are depicted in gray color). Again, the symmetry is achieved and both (red and blue) plots run through the same values of swarasthanas.
Above illustrated results prove the opinion of modern musicologists, expressed in the article (9) by Prof. Sambamoorthi as follows:
"A study of the notes obtained in the cycles of fifths and fourths enabled the ancient scholars to perceive the different musical intervals. They were already familiar with the chatussruti interval (9/8 or 204 cents), trisruti interval (10/9 or 182 cents) and the dvisruti interval (16/15 or 112 cents) in the sa grama. The ma grama helped them to appreciate the interval of a pramana sruti, 81/80 or 22 cents. When the notes of the cycles of fifths and fourths, worked up to the 12 the cycle in each case, were reduced to one octave and studied, it was found that there were 13 twins of notes, inclusive of the octave shadja, the notes constituting each twin being separated by the interval of a pramana sruti (comma 81/80 or 22 cents). It was also noticed that in each twin, the lower note belonged to the cycle of fourths and the higher note to the cycle of fifths. "
Besides the first two harmonics (giving rise to the cycles of fifths and fourths, described above), there are higher harmonics, which give "naturally" harmonious ratios, not covered by the cycles of fourths and fifths.
"Natural' values of ga and Ga are 6/5 and 5/4, and corresponding ratios of da and Da are 8/5 and 5/3.
The values derived for ga (32 / 27), Ga (81 / 64) and da (128 / 81), Da (27 / 16) using cycles of fifths and fourths differ from equivalent "natural" values, 6/5, 5/4, 8/5, 5/3, by the value of 81/80 (an interval known as the "syntonic comma" or comma of Didymus).
If we lower cycle of fifths (i.e. divide all values) by 81/80 and rise up cycle of fourths (i.e. multiply all values) by 81/80, we will get 24 alternative swarasthanas. The result of such transformation is shown on the picture below. We can see, that resulting sequence of swarasthanas includes the "natural" values of ga1 = 6/5, Ga2 = 5/4, da1 = 8/5, and Da2 = 5/3.
Again, if we discard the extra values of sa and pa (which cannot be doubled in an octave), we have 22 swarasthanas of two "natural" scales, which we got by shifting the first two gramas by 81/80.
In vocal and instrumental music we use only twelve swarasthanas. Cycles of fifths and fourths along with two natural set give us 44 variations in together. We have to understand the principles of selecting swarasthanas in order to create harmonious and balanced scale.
To simplify this task, let us consider the more precise values, i.e. twelve ratios received from the first seven six steps in cycle of fifths and the first five steps in cycle of fourths (twelve swarasthanas intogether).
One more set of twelve swarasthanas we get by shifting the first set by the value of 81/80.
If we keep only one swarasthana for sa, only one swarasthana for pa, and four swarasthanas for each of Nyasa swaras, i.e. ri, ga, ma, da, ni, we get 22 swarasthanas. Two extra values received for sa and pa should be discarded, as those swaras cannot be doubled in octave.
Let's explore the intervals between each swarasthana and thus consider the measure of harmony between them.
In the table below contains the result of analysis of intervals between 22 swarasthanas. The scale includes two swaras, sa and pa, which have only one swarasthana each, and five Nyasa swaras, ri, ga, ma, da, ni, which have four swarasthanas each, two "minor" sthanas and two "major" sthanas.
|Number of Sruti||Swara-sthana||Name (according to number of Sruti)||Name (by protecting deity)||Ratio (in natural scale)||Interval to previous swara-sthana||Previous swara||Interval to previous swara|
|2||Ekasruti Rishabham||Tivra||ri1||256/243||256/243||sa = 1/1||256/243|
|3||Dvisruti Rishabham||Kumudvati||ri2||16/15||81/80||sa = 1/1||16/15|
|4||Trisruti of Suddha Rishabham||Manda||Ri1||10/9||25/24||sa = 1/1||10/9|
|5||Chausruti Rishabham||Chandovati||Ri2||9/8||81/80||sa = 1/1||9/8|
|6||Suddha Gandharam||Dayavati||ga1||32/27||256/243||Ri2 = 9/8||256/243|
|7||Sadharana Gandharam||Ranjani||ga2||6/5||81/80||Ri2 = 9/8||16/15|
|8||Antara Gandharam||Raktika||Ga1||5/4||25/24||Ri2 = 9/8||10/9|
|9||Chutha Madhyama Gandharam||Raudri||Ga2||81/64||81/80||Ri2 = 9/8||9/8|
|10||Suddha Madhyamam||Krodha||ma1||4/3||256/243||Ga2= 81/64||256/243|
|11||Tivra Suddha Madhyamam||Vajrika||ma2||27/20||81/80||Ga2= 81/64||16/15|
|12||Tivrata or Prati Madhamam||Prasarini||Ma1||45/32||25/24||Ga2= 81/64||10/9|
|13||Tivrata or Chuta Panchama Madhamam||Priti||Ma2||729/512||81/80||Ga2= 81/64||9/8|
|14||Panchamam||Marjani||pa||3/2||256/243||Ma2 = 729/512||256/243|
|15||Ekasruti Daivatam||Ksiti||da1||128/81||256/243||pa = 3/2||256/243|
|16||Dvisruti Daivatam||Rakta||da2||8/5||81/80||pa = 3/2||16/15|
|17||Suddha Daivatam||Sandipani||Da1||5/3||25/24||pa = 3/2||10/9|
|18||Chatusruti Daivatam||Alapini||Da2||27/16||81/80||pa = 3/2||9/8|
|19||Suddha Nishadam||Madanti||ni1||16/9||256/243||Da2 = 27/16||256/243|
|20||Kaisiki Nishadam||Rohini||ni2||9/5||81/80||Da2 = 27/16||16/15|
|21||Kakali Nishadam||Ramya||Ni1||15/8||25/24||Da2 = 27/16||10/9|
|22||Chuta Shadja Nishadam||Ugra||Ni2||243/128||81/80||Da2 = 27/16||9/8|
|-||Shadja||Ksobini||Sa||2/1||256/243||Ni2 = 243/128||256/243|
Prof. Sambamoorthi in his article (9) describes the example of tuning two veenas, mentioned by Bharata Muni in Natya Sastra:
"In his Natya Sastra (4th cen BC) Bharata has suggested an interesting experiment to get a clear grasp of these three types of ekasruti intervals. These three types of ekasruti intervals are in the increasing order of magnitude respectively termed Pramana sruti, Nyuna sruti and Purna sruti intervals or the srutis of minimum, medium and maximum values."
As we see from the calculations above and Prof. sambamoorthi conclusions, three different values for one sruti intervals are as follows:
Fox Strangways  says about Bharatas experiment (explanation of sadja and madhyama gramas): "The only thing this establishes is that the srutis were not equal in size. They are of three sizes: (I) difference between the major and minor tone, pramana sruti, 22 cents; (2) the difference between the minor Tone and Semitone, 70 cents; and (3) the difference between the Semitone and Pramana sruti, 90 cents."
B. Breloer , also mentions three different srutis, i.e. srutis of 24, 66 and 90 Cents.
Omkarnath Thakur  also infers that "Bharatas system gives rise to three different srutis, which might be indicated by the following ratios: 1. Comma = 81/80, i.e. syntonic comma or comma of Didymos, which is named after the Greek mathematician and which indicates the difference between a major and minor tone; 2. Limma = 256/243, i.e. Pythagorian diatonic semitone; and 3. Minor semitone = 25/24, i.e. the smaller chromatic semitone."
Stone Powers in  states: "The theoretical ekasruti would represent barely discernible differences of pitch, such as the various commas, which could arise only as theoretical pitch differences between svaras in two different scales - for it is a fact that nowhere in any listed scale does an akasruti occur..."
Note: interesting fact is, that Muttusvaami Dikshitar refers to the 22 shrutis in his kriti "vamshavati shivayuvati" in the raga Vamshavati.
There are following important conclusions:
In case of ga, previous swara is Ri2 (9/8), thus:
The exact values of intervals between adjacent swarasthanas are not equal (i.e. this scale do not belong to the class of equal scales). There are four values, fixed between each swarasthana of each Nyasa swara. Multiplication of those four values gives us:
256/243 by 81/80 by 25/24 by 81/80 = 9/8. Thus, value of 9/8 is equal to four Srutis.