Complementary Ragas using Discrete Fuzzy Set Inversion (March, 2006)
Introduction
A Complementary-raga (C-Raga) is the opposite (or Complement) of a raga. A raga and its C-raga create the perception of maximum contrast in terms of their notes. The concept of C-raga explores the relationships between different pairs of complementary ragas and the effect they create. The C-Raga concept combines elements of Set Theory, Discrete Fuzzy Sets and Classical Raga Theory.
This experiment is very different from the Raga Morphing using Hamming Distance experiment, where transitions are smooth, with the smallest possible deviation. In the raga-morphing experiment, an important observation was made that the mood or the feel of ragas separated by a small hamming distance was often not small. The Complementary raga concept is a direct extension of this insight, to observe the effect on the mood of two ragas separated by the maximum possible hamming distance.
We conclude this paper by establishing the fact that the differences between the primary moods (the common or representative mood) of a raga and its C-Raga are largely different (in most cases). This is the first step in a series of Ragamatics papers exploring the relationship between ragas and communicating emotions.
Applications
1. Complementary Ragas allow one to generate maximum contrast between two musical pieces.
2. The Raga Morphing using Hamming Distance experiment can be performed by choosing any Raga, calculating the C-raga and then Morphing from Raga to C-Raga.
Formalization
A Raga is a set of notes, arranged in ascending order.
Set theoretically, a raga
,
where m is the number of notes in the raga. Each
is a member of the universal set of notes,
. Thus,
.
= { Sa, Re,
Re, Ga,
Ga, Ma,
Ma, Pa,
Dha, Dha,
Ni, Ni }
(Nomenclature: Underlined notes are komal, overlined notes are teevra, all others are shuddha)
A first-cut definition of Complementary-Raga can be:
The above definition makes use of the set theory nomenclature of representing the set S's Complement by S (read as S 'bar').
However, a quick observation reveals that this definition is incorrect, as it yields ragas without a Shadaj (Sa)! Without doubt, Sa should not be removed from the set of notes. Similarly, in ragas containing Pancham (Pa), there exist times when it becomes essential for the C-Raga to contain Pa to be aesthetically pleasing. We now introduce the notion of a discrete fuzzy set inversion by adding the optional subtraction of { Pa }.
A second-cut at the definition goes like this:
where,
This definition of C-raga introduces fuzziness by adding the optional subtraction of { Pa }. The operator <optional> represents binary fuzziness: Pa is either chosen, or not.
Notice that this optional Pancham ( Pa ) allows a one-to-many relationship between a Raga and it's Complementary-Raga. The reader can choose whether to include Pancham or not.
Let us consider a few examples to make sure that this definition gives us legitimate, aesthetically constructed ragas. Color Coding used is Subtracted Swar, Remaining Swar, Fuzzy Swar. Let us try some examples to test our C-Raga definition.
Complement (Basant-Mukhari) or Basant-Mukhari
=
- { Sa, Re,
Ga, Ma,
Pa, Dha,
Ni }
= {
Re,
Re,
Ga,
Ga,
Ma,
Ma, <optional>Pa,
Dha,
Dha,
Ni,
Ni
}
- {
Sa, Re,
Ga,
Ma, <optional>Pa,
Dha,
Ni,
}
= { Sa, Re, Ga, Ma, <optional>Pa, Dha, Ni }
= Madhuwanti
Let us try one more example:
Complement (Malkauns) or Malkauns
=
- { Sa, Ga,
Ma, Dha,
Ni }
= {
Re,
Re,
Ga,
Ga,
Ma,
Ma,
Pa,
Dha,
Dha,
Ni,
Ni }
- {
Sa, Ga,
Ma,
Dha,
Ni,
}
= { Sa, Re, Re, Ga, Ma, Pa, Dha, Ni }
= Undefined Raga (Such a raga is not defined in North Indian classical music)
We observe, that the presence of the two Rishabs (Re) make this raga unaesthetic.
Let us introduce some more discrete fuzziness in the Set-subtraction
to make the C-raga more palatable. If there exists a note in two forms (suddha
- komal or suddha - teevra), then one or more forms can used to form the
C-raga. Again, the reader can chose which combination(s) of notes to use.
Here, we cannot use the <optional> operator, as there are more than two
choices (3 to be precise), none of which is an empty choice
.
The choices are { Suddha }, { Komal (or Teevra)
}, { Suddha , Komal (or Teevra) }. For brevity,
this can be represented as
,
,
.
Continuing with our example, we chose one Re at a time and include Pa. Thus Malkauns can be written as
= { Sa, Re, Ga, Ma, Pa, Dha, Ni } or { Sa, Re, Ga, Ma,Pa, Dha, Ni }
= Puriya Kalyaan or Yaman
The final-version of the formal definition of Complementary-Raga
is as follows:
Let us try a final example to test this definition of C-Raga.
Complement (Bhoop) or Bhoop
=
- { Sa, Re,
Ga, Pa,
Dha }
= {
Re,
Re,
Ga,
Ga,
Ma,
Ma, <optional>Pa,
Dha,
Dha,
Ni,
Ni
}
- {
Sa, Re,
Ga,
Pa,
Dha
}
= { Sa, Re, Ga, Ma, Ma, <optional>Pa, Dha, Ni, Ni }
= 18 combinations
The 18 combinations come from the fact that there are
3 possible ways of using the Madhyam ( { Ma }, {
Ma }, { Ma, Ma }
) followed by 2 possible ways of using the Pancham (
{ Pa },
),
followed by 3 possible ways of using Nishad ( {
Ni }, { Ni }, { Ni,
Ni }) leading to 3 x 2 x 3, or 18 C-ragas in all.
= { Sa, Re, Ga, Ma, Pa, Dha, Ni } (Bhairavi / Bilaskhani Tod, etc) or,
= { Sa, Re, Ga, Ma, Pa, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Pa, Dha, Ni, Ni } or,
= { Sa, Re, Ga, Ma, Dha, Ni} or,
= { Sa, Re, Ga, Ma, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Dha, Ni, Ni } or,
= { Sa, Re, Ga, Ma, Pa, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Pa, Dha, Ni } (Miyaan-Ki-todi) or,
= { Sa, Re, Ga, Ma, Pa, Dha, Ni, Ni } or,
= { Sa, Re, Ga, Ma, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Dha, Ni } (Gujri Todi) or,
= { Sa, Re, Ga, Ma, Dha, Ni, Ni } or,
= { Sa, Re, Ga, Ma, Ma, Pa, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Ma, Pa, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Ma, Pa, Dha, Ni, Ni } or,
= { Sa, Re, Ga, Ma, Ma, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Ma, Dha, Ni } or,
= { Sa, Re, Ga, Ma, Ma, Dha, Ni, Ni }.
Notice that most of these ragas are not aesthetic, or well known. Some of these ragas are not a part of the North Indian repertoire, but may be found in Carnatic Music.
We can now chose any subset of the generated combinations as our Complementary-Raga set. Typically, the subset is limited to the known ragas.
Properties of Complementary Ragas
1. The C-raga transformation is a one-to-many transformation.
It is one to many as there can exist more than one C-raga for one raga. This is due to the fuzziness introduced in the set inversion. Furthermore, we have considered Ragas to be just a set of notes in ascending order. This definition can map multiple ragas to one set of notes. In our Bhoop example, we can map Bhoop to 18 possible complementary ragas (or more, if there exist more than one ragas that can be mapped onto that set of notes). One possible C-raga for Bhoop is Bhairavi, which maps to the same set of notes as Bilaskhani Todi, or Komal Rishabh Aasawari Todi.
2. The transformation is asymmetric if
and
are not empty sets, i.e
In the above example, Bhoop can be mapped to Bhairavi as one possible C-raga. However, Bhairavi is Yaman alone. This asymmetry is due to entropy introduced by the fuzzy-set-inversion.
3. The transformation is symmetric IFF
is empty, i.e.
For example, Puriya Dhanashri will be Kafi, and Kafi will be Puriya Dhanashri as there is no fuzziness considered in any notes, and Pa is a member of the resultant set.
4. If two ragas have a small hamming distance, their C-ragas will have a small hamming distance too.
In fact, many ragas with differences in the Pancham note
map to the same Complementary raga
(E.g. Puriya Dhanashri =
Din-ki-Puriya = Kafi)
Notable Examples
Raga
Complementary Raga(s)
Basant Mukhari
Madhuwanti
Bhairavi
{Yaman, Suddha Kalyan}
Sampoorna Makauns
Puriya Kalyan
Malkauns
{Yaman, Suddha Kalyan}, Puriya Kalyan
Puriya Dhanashri
{Kafi, Bageshree, Nayaki Kanada, Bhimpalas, Gavati}
{Marawa, Puriya, Sohoni}
{Kafi, Bageshree, Nayaki Kanada, Bhimpalas, Gavati}
Basant Bahar
or { Sa }, or { Sa, Pa }
Bhoop
{Bhairavi, Bilaskhani Todi, Komal Rishabh Aaswari Todi}, Miyaan-ki-todi, Gujri-Todi
Bhoopal Todi
Suddha Scale, Yaman, Khamaj, {Maru-Bihag Bihag, Yaman Kalyan, Shyam Kalyan}, Kedar
Lalit
(Shivaranjani - Ga))
Bhairav
Hemavati
Salagwarali
Nat-Bhairav
Miyaan-ki-todi
Rageshree
Madhukauns
Bhairav, Nat-Bhairav, {Anand-Bhairav, Bhatiyaar}
Gorakh Kalyan
Puriya Dhanashri, Din-ki-Puriya, Gujri-Todi, Miyaan-ki-Todi
Megh Malhaar
Puriya Dhanashri, Din-ki-Puriya, Gujri-Todi, Miyaan-ki-Todi, Puriya Kalyan
Mood or Emotional differences in Complementary Ragas
In the above examples of C-Ragas, the feel of the Raga and its C-Raga is significantly different due to the stark difference of notes. This "feel" is strong enough to translate into contrasting moods (e.g. Maarwa / Baageshri, Bhoop / Gujri-Todi, Miyaa-ki-Todi / Rageshree, Megh Malhaar / Puriya Dhanashri, Gorakh Kalyan / Gujri Todi, Madhukauns / Bhairav, etc). Empirically, the set of ragas where the mood translates into a contrasting mood is quite large.
In other cases however, the feel does not translate into a specific mood and thus the mood of the C-Raga is independent of that of the original Raga. (e.g. Malkauns / Yaman, Bhairavi / Yaman, etc). Empirically, the set of ragas where the mood does not translate into a contrasting mood is very small.
Notice, that the (Raga, C-Raga) pair demonstrate complementary stability characteristics in a majority of cases. We can classify the above examples of C-raga pairs as stable (S) / unstable (U) ragas. (e.g. Maarwa (U) / Baageshri (S), Bhoop (S) / Gujri-Todi (U), Miyaa-ki-Todi (U) / Rageshree (S), Megh Malhaar (S) / Puriya Dhanashri (U), Gorakh Kalyan (S) / Gujri Todi (U), Madhukauns (S) / Bhairav (U), etc).
We can thus conclude that for most ragas, their C-ragas create a large contrast in the primary moods. By primary mood, we mean the mood of the raga that is considered to be the representative or common mood.
Later papers explore techniques of generating moods other than the primary mood within a raga.
Future Work