To understand the role of harmonics, microtones and consonance in Indian Classical music, it is necessary to understand its underlying tuning systems. In this article, we reconstruct two significant tuning systems used in Indian Classical music:
In a couple of related articles, we cover how tuning systems influence musical scales and how this affects the tuning of a Tanpura to play these scales.
Before we begin reconstructing any tuning system, it is useful to familiarize ourselves with the four basic intervals of Dviguna (octave), Pancham (fifth), Madhyam (fourth) and Gandhar (major third). You can play and check these intervals using the keyboard below (just click or tap a key to play). Here Sa denotes the fundamental and Ga, ma, Pa and SA denote the Gandhar, Madhyam, Pancham and Dviguna respectively. Note that Sa is taken to be the pitch C, but the other notes do not match up with today's standard 12 tone equally tempered scale.
Note: This article features several audio samples which are an integral part of the narrative. For ease of understanding, these samples have been presented in the form of a musical keyboard which many would recognize. Simply click or tap a key to play. Please try and use a pair of headphones or good quality speakers to listen to the samples with maximum clarity.
As you can hear from Demo 1, Pancham, Madhyam and Gandhar are pleasant sounding intervals. For the musically inclined, they may be easy to recognise by ear. It is also possible to determine their frequency ratios in relation to the fundamental.
One way in which this can be done is by comparing lengths. Firstly, we can observe that the pitch of a string is inversely proportional to its vibrating length. Secondly, taking a certain string length to represent the fundamental, we can observe that if we touch the string at a third of its length from one end and pluck the remaining two-thirds, it results in the string sounding the fifth. Similarly, touching it at a fourth of its length from one end results in the remaining three-fourths to sound the fourth, touching it at a fifth of its length results in the major third, and touching it at the midpoint results in either half to sound the octave.
So by inverting these string lengths, we can infer that with Sa = 1, we have SA = 2, Pa = 3/2, ma = 4/3 and Ga = 5/4. Try and remember these frequency ratios, as we will use them repeatedly.
The first scale we will examine is the Sama Gana scale which was used to sing passages from the Sama Veda. Folk music or informal music was always around in India. But Indian music started to get formalized as the chants of Rig Veda evolved into a more sing-song way of rendering passages from Sama Veda. Rig Vedic chants already used three notes called Svarita (neutral drone), Udaatta (higher note) and Anudaatta (lower note). This evolved further in Sama Gana.
Sama Gana is the act of singing the Sama Veda. According to Ustad Zia Fariduddin Dagar, a typical arrangement employed three groups of singers with one group singing at the pitch Sa, the second at ma and the third at Pa. Each group in turn used three notes, a center note, a low note and a high note, similar to those used in Rig Vedic chants. The center note was whichever pitch they were singing, i.e., Sa, ma or Pa. The high note was higher than the center note by a specific tone and the low note was lower than the center note by the same tone. Refer to Demo 2 below.
Observe how the different groups stop and start chanting at appropriate points during the rendition, so that overall it creates an impression of a musical scale. Also, please note that this rendition is roughly in the scale of G.
Knowing the Pa and ma intervals, it is possible to derive the ma-Pa interval. For the musically inclined, you may be able to recognise it by ear. The size of the ma-Pa interval can be calculated as 3/2 ÷ 4/3 = 9/8. This ma-Pa interval was used as the tone to derive the high and low notes from the center note. Note that the ma-Pa interval is different from the whole tone used in today's standard 12 tone equally tempered scale. The whole tone is 200 cents wide, while the ma-Pa interval is a bit wider at 203.9 cents.
Using this, the Sama Gana scale as depicted in Table 1 can be derived. The method of derivation is given in the remarks column. The Sama Gana scale is discussed and documented in a number of references (for example, see Lakshanagrandhas in Music by Dr. Bhagyalekshmy).
ma-Pa Interval | 1.125 | 9/8 | |
Note | Ratio | Symbolic Ratio | Remarks |
Sa | 1 | 1 | |
Re | 1.125 | 9/8 | ma-Pa Gap above Sa |
ga | 1.185185185 | 32/27 | ma-Pa Gap below ma |
ma | 1.333333333 | 4/3 | Fourth |
Pa | 1.5 | 3/2 | Fifth |
Dha | 1.6875 | 27/16 | ma-Pa Gap above Pa |
ni | 1.777777778 | 16/9 | ma-Pa Gap below SA |
SA | 2 | 2 | Octave |
You can play and check how the Sama Gana scale sounds using Demo 3 below.
In this scale, the intervals Re-ga and Dha-ni were regarded as Vivadi (dissonant). The concept of Vivadi notes is mentioned in Dattilam and Sangeet Ratnakar (see Chapter Sangita Ratnakara in Lakshanagrandhas in Music or Chapter Nada, Sruti and Svara in Sangeet Ratnakar, Part 1). The size of these intervals can be calculated to be 32/27 ÷ 9/8 = 256/243 = 1.053497942. This interval \(\frac{256}{243}\) was recognized and known as Purna Shruti. Note that the Purna Shruti is roughly comparable to the semi-tone used in today's standard 12 tone equally tempered scale. The standard semi-tone is 100 cents wide, while the Re-ga interval is a bit narrower at 90.2 cents. This scale may have been appropriate for chanting, but as a musical scale, it was not considered appropriate. In order to make it musical, the musical scholars of the day felt that the Re-ga and Dha-ni intervals needed to be slightly increased.
The Shadaj Gram scale is cited as a 7 note scale and also as a 22 note scale (which includes the original 7 notes). Let us reconstruct the 7 note scale first.
Note: Like the Shadaj Gram scale, an equally important scale called Madhyam Gram is also described by Bharat and by Sarang Dev. Later, we will note specifically how it differs from Shadaj Gram.
There are two ways in which Shadaj Gram can be understood. The first is to understand it as a derivative of the Sama Gana scale, to make it more musical by adjusting the Vivadi intervals of Re-ga and Dha-ni. The second is to understand it in terms of Shruti Intervals as specified by Bharat and Sarang Dev. Let us look at both explanations and see how they are mutually consistent.
First, we have to re-trace the line of reasoning that might have been employed by the musical scholars of the day to derive the Shadaj Gram scale from the Sama Gana scale. There are two options to fix the dissonant Re-ga and Dha-ni intervals:
The first option would result in losing the consonance between ma and ni (ni is the fourth of ma in Sama Gana scale), while the second option would result in losing the consonance between Re and Pa (Pa is the fourth of Re). We know that they chose the second option since Bharat and Sarang Dev have pointed out that Re and Pa are not consonant in Shadaj Gram.
Let us try and determine specifically by how much Re and Dha were reduced in pitch. Look at the notes from ma to SA in Sama Gana scale (Demo 3). Look at the relationship between ma and Dha and note that the interval ma-Dha is a bit wider than the Gandhar (major third) interval (Demo 1). The ma-Dha interval consists of the ma-Pa interval applied twice. You can try to hear Demo 1 and Demo 3 and see how two ma-Pa intervals combined (407.8 cents) are bigger than a Gandhar interval (386.3 cents). In contrast, in today's standard 12 tone equally tempered scale, two whole tones (i.e 2 x 200 cents) is exactly equal to a major third (400 cents).
In Shadaj Gram, the note Dha from Sama Gana scale was reduced in pitch to Dha♭ to maintain the Gandhar interval with respect to ma. The note Re from Sama Gana scale was reduced by a similar amount in pitch to Re♭ so that the Re♭-Dha♭ interval was maintained to be the Pancham.
Before we analyse this any further, hear for yourself what this scale sounds like using Demo 4 below. The keyboard may look a bit unusual, particularly the keys for Re♭ and Dha♭. But it has been depicted this way to contrast it from the Sama Gana scale in Demo 3 above.
Let us calculate the precise amount by which Re and Dha were reduced in pitch to obtain Re♭ and Dha♭ respectively. The ma-Dha interval consists of the ma-Pa interval applied twice, which is (9/8)^{2} = 81/64. Since the Gandhar interval is 5/4, the gap between the Gandhar and the ma-Dha interval can be calculated to be 81/64 ÷ 5/4 = 81/80. This interval \(\frac{81}{80}\) was recognized and known as Pramana Shruti.
Knowing the Pramana Shruti, the dissonance in the Sama Gana scale can be removed and the Shadaj Gram scale can be derived by tuning the Re and Dha down in pitch by an amount equal to the Pramana Shruti. Again, the Shadaj Gram scale itself is discussed and documented in a number of references (see Lakshanagrandhas in Music or Sangeet Ratnakar). Here is the scale (see Table 2).
Pramana Shruti | 1.0125 | 81/80 | |
Note | Symbolic Ratio | Remarks | |
Sa | 1 | 1 | |
Re♭ | 1.111111111 | 10/9 | 81/80 below Re |
ga | 1.185185185 | 32/27 | 9/8 below ma |
ma | 1.333333333 | 4/3 | Fourth |
Pa | 1.5 | 3/2 | Fifth |
Dha♭ | 1.666666667 | 5/3 | 81/80 below Dha |
ni | 1.777777778 | 16/9 | 9/8 below SA |
SA | 2 | 2 | Octave |
Bharat in Natyashastra and Sarang Dev in Sangeet Ratnakar describe an experiment using an Achal veena, an instrument with a fixed tuning, and a Chal veena, an instrument whose tuning is varied through the course of the experiment. Although the term veena is used in their literature, it is likely that these instruments were constructed in the form of a harp rather than like present day veenas.
These experiments require the Achal veena to be tuned according to the 7 note Shadaj gram scale. But here the Shadaj Gram scale is described in terms of intervals between notes. Each interval is described in terms of the number of Shrutis it spans. Here is the full description (Table 3) with the Sa beginning at the Shruti position of 4.
Shruti | String | Note | Shruti Interval |
4 | First | Sa | |
7 | Second | Re♭ | Sa-Re♭ spans 3 Shrutis |
9 | Third | ga | Re♭-ga spans 2 Shrutis |
13 | Fourth | ma | ga-ma spans 4 Shrutis |
17 | Fifth | Pa | ma-Pa spans 4 Shrutis |
20 | Sixth | Dha♭ | Pa-Dha♭ spans 3 Shrutis |
22 | Seventh | ni | Dha♭-ni spans 2 Shrutis |
4 | Eighth | SA | ni-SA spans 4 Shrutis |
Without understanding what these Shruti intervals meant, it is difficult to understand what really Shadaj Gram was. But it is possible to derive the precise meaning of these Shruti intervals.
Let us start with some observations from the above description of Shadaj Gram.
Using these observations, we can calculate the size of the 2, 3 and 4 Shruti intervals as follows. For ease of explanation, let us refer to these intervals as \(r_2\), \(r_3\) and \(r_4\), respectively.
Now, it is easy to verify mathematically that the Shadaj Gram scales defined by Table 2 and Table 3 are identical. For example, since \(r_3=\frac{10}{9}\), we know that Re♭ should be \(\frac{10}{9}\) which is consistent with Table 2. If you are keen, we encourage you to check the consistency of all the other notes in Shadaj Gram as well.
Note: The Madhyam Gram scale as described by Bharat and by Sarang Dev, has the note Pa at Shruti #16, making the ma-Pa gap 3 Shrutis and the Pa-Dha♭ gap 4 Shrutis. Knowing the values of \(r_2\), \(r_3\) and \(r_4\), you can determine the exact scale of Madhyam Gram too.
Let us perform the Bharat-Sarang Dev harp experiment and derive the full 22 shruti scale mathematically. The steps followed in the experiment are given below.
Note: Again, the Madhyam Gram scale as described by Bharat and by Sarang Dev, has the note Pa at Shruti #16, while the rest of the notes are taken to be the same as in Shadaj Gram. As observed above, this makes the notes Re♭ and Pa consonant with each other.
You can play and hear each of these notes in Demo 5 below. If you are interested in the actual values for each of the 22 shrutis, refer to Table 4 which summarizes the result of all these steps (scroll horizontally to see the whole table). Note that to play the Madhyam Gram scale, just play key #16 for Pa, instead of key #17.
Shruti | Note | Achal Veena | Round 1 | Round 2 | Round 3 | Round 4 | All 22 Ratios |
1 | 0.9 | 0.9 | |||||
2 | 0.9375 | 0.9375 | |||||
3 | 0.987654321 | 0.987654321 | 0.987654321 | ||||
4 | Sa | 1 | 1 | 1 | |||
5 | 1.041666667 | 1.041666667 | |||||
6 | 1.097393689 | 1.097393689 | |||||
7 | Re | 1.111111111 | 1.111111111 | 1.111111111 | |||
8 | 1.170553269 | 1.170553269 | |||||
9 | ga | 1.185185185 | 1.185185185 | 1.185185185 | |||
10 | 1.2 | 1.2 | |||||
11 | 1.25 | 1.25 | |||||
12 | 1.316872428 | 1.316872428 | |||||
13 | ma | 1.333333333 | 1.333333333 | 1.333333333 | |||
14 | 1.35 | 1.35 | |||||
15 | 1.40625 | 1.40625 | |||||
16 | 1.481481481 | 1.481481481 | 1.481481481 | ||||
17 | Pa | 1.5 | 1.5 | 1.5 | |||
18 | 1.5625 | 1.5625 | |||||
19 | 1.646090535 | 1.646090535 | |||||
20 | Dha | 1.666666667 | 1.666666667 | 1.666666667 | |||
21 | 1.755829904 | 1.755829904 | |||||
22 | ni | 1.777777778 | 1.777777778 | 1.777777778 | |||
1 | 1.8 | 1.8 | |||||
2 | 1.875 | 1.875 | |||||
3 | 1.975308642 | 1.975308642 | |||||
4 | SA | 2 | 2 |
The Shadaj Gram scale can also be represented by showing the relevant pitches in a circle (see Figure 3). The figure depicts how the Chal veena is depressed by the different shruti intervals in each round.
Ramamatya laid down a practical procedure for setting the frets on a Veena, and in the process dividing an octave into 12 notes. Venkatamakhin documented the 12 note divisions of the octave and calculated numerical ratios for each of the 12 notes. Taken together Ramamatya and Venkatamakhin created a tuning system for Indian Classical music which was both practically and theoretically well defined. The tuning system defined by Venkatamakhin and Ramamatya is equivalent to the Pythagorean Tuning system.
Ramamatya in his Swaramelakalanidhi has described 3 types of Veenas: Sudha Mela, Madhya Mela and Achyutharajendra Mela. For the purposes of our discussion here, we will consider the Sudha Mela Veena (for details, see Swaramelakalanidhi of Ramamatya by M.S. Ramaswami Aiyar).
Ramamatya's procedure starts with using a four string Veena with its notes tuned as Sa, Pa, Sa" and ma" from the lowest to the highest. Here, Sa denotes the fundamental, Pa the fifth and ma the fourth, as before. But, we denote notes in the upper octave with a ". Thus, Sa" denotes the octave and ma" the octave of the fourth. The tuning of the four strings is prescribed to be done by ear, and it is expected that a veena tuner can accurately set the octave, Pancham (fifth) and Madhyam (fourth) intervals.
Then the following procedure is prescribed:
Ramamatya's procedure is so clearly defined that it is used even today by veena makers to set the frets on the instruments they produce.
Starting with the fact that Pancham is \(\frac{3}{2}\) and Madhyam is \(\frac{4}{3}\), it is possible to derive the ratios for each note of the 12 notes which would result from Ramamatya's procedure. This documentation is credited to Venkatamakhin.
You may have noticed that ratios for Ma and re at the first and sixth fret are different. Venkatamakhin prescribes that Ma is taken as \(\frac{729}{512}\) (defined by the sixth fret) but that re is taken as \(\frac{256}{243}\) (defined by the first fret).
You can hear the Venkatamakhin-Ramamatya scale in Demo 6 below. The calculations for deriving the ratios for each note are summarized in Table 5.
Note | Ratio | Symbolic Ratio | Remarks |
Sa | 1 | 1 | |
re | 1.053497942 | 256/243 | ma-Pa Gap below ga |
Re | 1.125 | 9/8 | ma-Pa Gap above Sa |
ga | 1.185185185 | 32/27 | ma-Pa Gap below ma |
Ga | 1.265625 | 81/64 | ma-Pa Gap above Re |
ma | 1.333333333 | 4/3 | Fourth |
Ma | 1.423828125 | 729/512 | ma-Pa Gap above Ga |
Pa | 1.5 | 3/2 | Fifth |
dha | 1.580246914 | 128/81 | ma-Pa Gap below ni |
Dha | 1.6875 | 27/16 | ma-Pa Gap above Pa |
ni | 1.777777778 | 16/9 | ma-Pa Gap below SA |
Ni | 1.8984375 | 243/128 | ma-Pa Gap above Dha |
SA | 2 | 2 | Octave |
We have reconstructed the two significant tuning systems used in Indian Classical music. The Shadaj Gram and Madhyam Gram scales gave rise to musical structures called Jatis from which many modern day Indian Ragas have evolved. Similarly, the Venkatamakhin-Ramamatya scale gave rise to a classification system for Indian Ragas called Melakarta system which has again given rise to many modern day Indian Ragas. It is necessary to understand and define these tuning systems, to understand the role of harmonics, microtones and consonance in Indian Classical music and musical scales.
In a following article, we trace the evolution of Indian Classical music over two centuries to determine how these tuning systems have influenced the music of today.