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Thursday, November 19, 2009

Classical Music Sawara - Saptaka - Scales

Swara - The Note
is the musical name for the scientific term Frequency. It denotes the sound of a particular frequency. Since, all musical sounds (of a given pitch) actually constitute a combination of several frequencies, Pitch is more accurately, the predominant frequency of a sound. Given two sounds of two frequencies, the way we hear them comparatively has more to do with the ratio of their frequencies, rather than their difference i.e. we deal with geometric progressions. For eg. two frequencies which are exact multiples of two (i.e. ration of 2:1) have the highest consonance i.e. they make a pleasing sound together. Infact, in any natural sound, we not only get a fundamental (or dominant) frequency, say x, but also frequencies which are integer multiples, 2x, 3x, 4x etc, usually with decreasing intensity. This series of integer multiple frequencies are called overtones or harmonics (from which words like harmony and harmonium come). Also, the predominant frequency and its second hormonic (i.e. 2x ) are said to be an octave away from each other. Similarly a ratio of 10 is called a decade. The human audible range is usually given as 20Hz to 20,000 Hz, though with age it becomes difficult to hear the highest frequencies. Here, 20Hz and 40Hz are an octave away. 20Hz and 200Hz have a difference of a decade. 20Hz to 200Hz is called the bass decade (or just bass, pronounced base), 200Hz to 2000Hz is the middle decade and 2KHz to 20KHz is the upper decade. The middle decade is the most important part of the spectrum as for as human audibility goes.

Saptaka - The Octave
A note is a sound of a definite pitch. It is also sometimes called a tone. The Indian name for a Note is Swara (or svara). An Octave refers to a range of notes, with the highest one being two times in frequency compared to the first. Traditionally, in both Indian and western music, music is thought to be made of seven notes. The note after that, i.e. the eighth note would be double in frequency compared to the first. That is the reason for the name Octave (Okt- the root meaning eight in Indo-European). The classical name for the range of frequencies forming one octave is Saptak i.e. made up of seven notes. Other names for saptak are Sthayi (sanskrit) or Mandala (dravidian).

Just a note and its overtones can't be used to create music. We need a series of notes. The series of notes have to be such that when used with each other, a pleasant experience results. This series of notes is called a scale, or more correctly, a series of notes differing in pitch according to a specific scheme is called a scale. Traditionally, the Octave is made up of, or divided into, 7 basic notes, the Indian name being
SaptaSwara (sapta - seven, swara-note). They are denoted as Sa, Re, Ga, Ma, Pa, Dha, Ni. After Ni comes Sa', this time, double in pitch compared to the first Sa i.e. an octave higher. Usually three octaves are recognized. The middle octave, most used, is called Madhya Saptaka. The lower octave is called Mandra Saptaka and the higher one Taara Saptaka. So, for eg., if the basic reference note Sa is at 240Hz, Mandra saptak would be 120-240 Hz, Madhya Saptak would be 240-480 Hz and Taara Saptak would be 480-960 Hz.

# Name Called Symbol
1 Shadja Sa S
2 Rishabha Re/Ri R
2 Gandhara Ga G
4 Madhyama Ma m
5 Panchama Pa P
6 Dhaivatha Dha D
7 Nishada Ni N

From ancient times, musicians have noticed that music sounds coherent / sweet / natural only when the notes used are in to each other. The smaller the numbers used in the ratio, better the coherence and naturalness. The ratio 2:1 of course, yields an octave. The use of other simple ratios 3:2, 4:3, 5:3, 5:4, 9:8, 15:8 yield the other natural notes making up the seven pure notes of an octave, which can be expressed in terms of ratios as - 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8. But, most musical systems of the world use more than seven notes to an octave. Western music, for example, uses twelve. The classical octave , from ancient times, has been divided into 22 notes, called Shruthi (microtones), since it was thought that 22 distinguishable notes exist in an octave. But, the 22 shruthis have been approximated to 12 notes by musicologists. The other 5 notes out of 12 notes are also got by the use of ratio of natural numbers.

Depending on the exact scheme used to develop the scale, different scales can be made. Indian scales use notes derived using natural number ratios. In western music too, natural scales were used until a couple of centuries ago, when they were replaced by first well tempered and then equally tempered scales.
Ancient Scales and Modal Shift
In ancient times, different scales were created using a process we call modal shift (Grama Moorchana or GrahaBheda). If we start with the basic, 7 notes, we get a scale of natural notes (used in today's Bilawal / DhiraShankarabharanam). Then, if the reference pitch Sa (tonic) is shifted by one note, we get a completely different scale (Kafi/Karaharipriya). This happens because the relative pitch of various notes in the Just tempered scale are not the same (unlike in equal tempered, where a shift gives exactly the same scale). One more shift gives us yet another scale (Bhairavi/HanumaTodi). We can do this shifting a couple of more times to get a total of 6 different scales.

This method of generating different scales can be practically used, when a fixed fret instrument (Vina) is played. In ancient literature we continue to see references about this kind of Moorchana till the time of Sarangadeva. Practice of getting different scales by changing the pitch of notes (other than Sa and Pa) developed later.

Classical Music Scale
First let us see how classical scale can be derived. Starting with Sa of 240Hz for the sake of easy arithmetic, using the ratios as stated above, the nominal frequencies of Re, Ga, Ma, Pa, Dha and Ni come to 270Hz, 300Hz, 320Hz, 360Hz, 405Hz and 450Hz, in the Just or Pure (shudha) scale used in Hindustani Music. There is a geometric progression of scales with a note being 1.5 times the fifth earlier note. So,Pa(360Hz)/Sa(240Hz) = Dha(405Hz)/Re(270Hz) = Ni(450Hz)/Ga(300Hz) = Sa'(480Hz)/Ma(320Hz) = 1.5

In the west, this is called the perfect fifths and attributed Pythagorous. There is considerable debate as to historically whether the Greeks got the scale from Indians or the other way round or they were independently developed. Additional notes are got by lowering the pitch of Re, Ga, Dha and Ni by one or two shruthis (microtones) to get Komal (flat) Reshab (R1), G1, D1 and N1. But Ma is slightly moved up to get Teevra Madhyam (M2). Thus we have 12 notes (Sa + 2 Ri + 2 Ga + 2 Ma + Pa + 2 Dha + 2 Ni) to an octave, just like in western music. The associated frequencies (nominal) are slightly different because an equal tempered scale is used in western music, instead of the pure scale.

Sa and Pa are denoted as S and P respectively. For the other notes, small letters are used for the notes of lower pitch (r, g, d, m, n) and capital letters for the note of the higher pitch (R, G, M, D, N). Thus small letters denote komal notes, except Shudha Madhyam. Capital letters denote Shudh notes except Teevra Madhyam

In actual practice, all classical notes except Sa and Pa, can move a microtone or two depending on the raaga (see Shruthi below). Sa, is thus called the reference note or tonic. Pa is the secondary reference.

Also, note that the absolute pitch of the basic reference note Sa is a variable in classical music. It is set by the main artist according to his voice or the instrument. In western music, a standard has been established where the note A (Dha) has the frequency of 440Hz (which makes C, 261.63Hz). Usually, male vocalists in classical use C(First White key) or C#(First Black key) as the reference note, Sa. The female vocalists use F#(Fourth black key) or G#(Fifth black key). Usually this reference is specified according to the harmonium keys, since harmonium is the accompaniment. Harmonium being a western import is tuned in equal temperament and thus is ill suited as an accompaniment, even though it is used because it is an economical alternative.

As noted above, all classical notes except Sa and Pa, can move a microtone or two depending on the raaga. This practice can be traced back to the ancient concept of Shruthi, where an octave is divided into 22 shruthis. Each of the 10 notes (i.e. excluding Sa and Pa) have two variants. For Eg. Komal Rishab (r) can be either the normal r or r+. The pitch difference between these two notes is small, but the use of one instead of the other, guarantees a simple ratio with the other notes used in that Raaga. That is the reason, why r is used in some Raagas and r+ is some others. It has been observed that the shruthis are such that, between two adjacent saptasvara notes there is a ratio of 256/243 and between shudh and komal notes the ratio is 25/24. It is this simple ratio that makes the music pleasing, natural and sweet. The 22 shruthis, their frequencies and equivalents are given in the table 3. An octave can be divided into 1200 cents. Scales are usually defined in cents so that for any base pitch, the scale can be calculated. The difference in cents between two frequencies, f1 and f2, is given as log(f1/f2) * 1200 * log(2) or approximately log(f1/f2) * 3986.3137.

Hundastani Scales
In Carnatic music, the scale is somewhat different, mainly in nomenclature. Carnatic also starts with 22 Shruthis for an octave. Even here the basic reference note Sa and the secondary reference Pa are fixed. Of the rest of 5 notes, 4 can be one of three Shruthis. In Hindustani, as given above, they can be only one of two. Ma though, can be only one two, in both Carnatic and Hindustani. But, not all the 3 pitches that can be Re or Ga are actually different. If we assign R1, R2 & R3 to the three possible Rishaba notes and G1, G2 & G3 to Gandhara notes, the actual pitch of R2 is same as that of G1 and R3 is same as that of G2. So, instead of 6 possible notes between Sa and Pa, we only have 4 (R1, R2 or G1, R3 or G2, G3). Similarly, we have four possible notes between Pa and Sa' of the next octave (D1, D2/N1, D3/N2, N3). So, even in Carnatic, we have 12 notes in the octave, even though they are referred to by 16 names.

Western Scales
The western music, now uses what is called the equal tempered scale, created around 19th century. Since there are 12 notes, and the octave is to be divided equally (in a logarithmic sense), each note is got by multiplying the previous note by twelfth root of 2 (which is about 1.059). Thus the actual frequencies used are slightly different from the pure scale used in classical. The keys of a harmonium, piano or an electronic keyboard are tuned to this equal tempered scale. Some notes of equal tempered scale are very close to those of pure scale that only a trained musician can tell them apart. Some other notes are close, but not close enough, so that casual listeners can also make out the difference. The sensitivity of average listeners is about 5 cents and trained musicians as low as 2 cents. Some of the keys of equal tempered scale differ from just scale by as much as 16 cents, and some like the fifth (G or Pa) by a mere 2 cents. Equal tempered scale is a trade off between pure notes and ability to use any key to start a mode without retaining.

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