Stability Characteristics of Musical Contexts (October, 2006)

Introduction

The fundamental emotions can be classified into two broad classes:

• Stable emotions - Calmness, Happiness, Love, Peace, Courage, Kindness, Desire, etc
• Unstable emotions - Anger, Sadness, Melancholy, Enmity, Anxiety, Fear, Unkindness, Reluctance, etc.

Stable emotions have a feel of completeness, of satisfaction associated with them, while unstable emotions have a feel of incompleteness, of dissatisfaction. These emotions lie on a spectrum of stability, varying in degree of stability and intensity. Every musical context, as perceived by a listener can be said to lie on a spectrum, from highly stable to highly unstable. This paper proposes a mathematical model to understand this phenomenon. We use the concept of potential wells, attractor fields and energy contained in a field to model and formalize the stability characteristics of musical contexts.

With the model of stability formalized and understood, one can apply the concept practically to create accurate amounts of instability (or stability) in the listener. This instability along with the other contextual elements of music can be used effectively to induce a semantically rich emotion in the listener. Basic techniques to create the desired amounts of instability in a context are discussed in the paper.

This paper proposes concepts like Field of Stability, Force of Stabilization, etc. It conjectures that one can generate many (if not all) of the above emotions in a semantics-enriched context (Vocabulary, Inflection) by controlling the magnitude of instability (Conjecture 1).

Applications

1. The mathematical model proposed in the paper allows us to understand the principles underlying the stability/instability of a musical context, and thus its mood/emotional characteristics.

2. Techniques for creating accurate amounts of stability in a musical context are derived from the model. This allows a performer to generate varying emotional responses in the audience, either within the rules of a single raga, or by using an effective choice of ragas.

3. The model sheds light on fundamental phenomena observed in Ragas, e.g. why Ragas with the same (or slightly different) set of notes have significantly different stability/emotional characteristics.

Formalization

Along the swar-line (see Delta Fingerprinting of Ragas),Delta-fingerprinting-ragas.htm there exist potential wells corresponding to certain swars. Potential wells are local minima of potential energy. These wells can be visualized as powerful attractive fields at specific points along the swar-line that exert a force of attraction in the swar-space-time.

At the point of local minima, the field's force is at its maximum. Such points are called Stationary points, as the first derivative (slope) is zero. We conjecture that the field's strength decreases as the inverse square of the distance from it.

We define 3 swars per octave to form potential wells, namely Shadja (Sa), Shuddha Madhyam (Ma) and Pancham (Pa). From a listener's perspective, sustaining these swars (Nyaas) creates a feel of highest stability or completion or satisfaction. Classical Raga theory recognizes the importance and stability of the combinations of "Sa-Ma (Shadja Madhyam Bhaav) and Sa-Pa (Shadja Pancham Bhaav). We conjecture that the reason for their perceived stability is the simple ratios of the frequency of Ma : Sa (4:3) and Pa : Sa (3:2).

We thus call these Swars as Globally Stable Swars, the field these swars create as the Field of Stability and the force exerted by the field on a note as the Force of Stabilization.

Force of Stabilization

Let be the Force of Stabilization as felt by a note n on the swar-line. Force of Stabilization is the force felt by a note to converge to a globally stable swar .

{ Sa, Ma, Pa } over all octaves

n { } - { Sa, Ma, Pa } over all octaves

The neighborhood of swar in which a stationary note n feels a force to converge to , is known as the basin of attraction of swar . See Graph 1 for a visual representation of the forces exerted by stable notes on the swar line.

The magnitude of the force for an arbitrary note

(1)

where,

n is an arbitrary note not belonging to the set of globally stable swars,

is a globally stable swar to the left of note n,

is a globally stable swar to the left of note n,

is the strength of the force field generated by stable swar as felt by note n ( lies to the left of n).

is the strength of the force field generated by stable swar as felt by note n ( lies to the right of n).

Thus, is the summation of the strengths of fields exerted by all the globally stable swars to the left of note n and is the summation of the strengths of fields exerted by all the globally stable swars to the right of note n.

We conjecture that the strength of the field is defined by the following equation:

(2),
Conjecture (2)

is the maximum magnitude of the strength of the field created by globally stable swar ,

Observe that the strength of the field in equation (2) falls off as the square of the distance, thus clearly following the inverse square law.

Thus, the resultant force of stabilization acting on a note n is the difference between forces generated by globally stable swars acting on it from opposite sides.

The magnitude of the force for a Globally Stable Swar

Let us try to derive the magnitude of the force of stabilization as experienced by a globally stable swar . Let be a small perturbation in either to the left or to the right. Thus we have,

(3)

A note that results from a small perturbation in to the left (or right) will experience a force of stabilization of magnitude that tends to in the limit as the amount of the perturbation approaches zero. Thus the magnitude of the resultant force of stabilization experienced by a globally stable swar is negligible.

This mathematical construction strengthens the intuitive understanding of the stability of globally stable swars. A practical manifestation of the force of stabilization acting on a stable swar is that a slight perturbation (out of tune, besur) creates extreme instability (and sometimes even pain) for most listeners!

The direction of the force

(4)

The resultant direction of the attraction force of a given note is towards that of the stronger force. If both left and right stabilization forces are equally strong, then the force acting on the note is zero.

Quantifying Stability of Musical Contexts

Let us quantify the notion of Stability and Instability of musical contexts.

if , ( is the threshold of stability), then the note n is said to be locally stable relative to the context.

(5)

if , then the note n is said to be locally unstable relative to the context

(6)

Using (5) and (6), let us define a Boolean function is_locally_stable(n) that returns 1 if the note n is locally stable, and zero otherwise. Note that is a constant that needs to be found using experimental techniques. For this paper, has been defined based on the author's experience.

Let us now define the magnitude of stability (and instability) of different contexts.

Case 1: Context = single note n

(7),
Conjecture (3)

(8),
Conjecture (4)

where,

is the perceived stability due to the volume of note n

is the perceived stability contributed by the sustenance time of note n

(9)

(10)

We can define proportionality constants and for both the above equations to determine the contribution of volume and sustenance time towards the total perceived stability of the note n.

The stability equation for a note says that for a locally stable swar (is_locally_stable(n) evaluates to 1), the perceived stability is directly proportional to the stability contributed by the volume of the note as well as its sustenance time. As an example, the Gandhaar (Ga) of Yaman can be said to be locally stable. A longer pause and/or a reasonably good volume on Ga will have the feel of more stability, as compared to a lower volume and/or a shorter pause. It also says that the stability is inversely proportional to the magnitude of the resultant attraction force exerted by a nearby globally stable swars. For example, the Gandhaar (Ga) in Yaman experiences 2 forces (Sa from the left, and Pa from the right). The magnitudes of these opposing forces are nearly equal, the force exerted by Pa being slightly stronger. This resultant force lies below the threshold of stability () thus making Ga a locally stable swar. The smaller this force, the higher the perceived stability, and hence the inverse proportionality.

For a locally unstable swar (is_locally_stable(n) evaluates to 0), the perceived instability is directly proportional to the magnitude of the resultant force of attraction exerted by nearby globally stable swars. For example the Teevra Madhyam (Ma) in Yaman experiences a very strong force from Pancham (Pa). It requires a very high amount of counterforce to balance the effect of Pa and hence the direct proportionality. As a result, a high amount of energy is required to sustain an unstable note (as the attraction force from a very close globally stable swar tries to move the swar towards itself). The higher the amount of energy expended, the higher the instability. Notice that a lower volume on an unstable note will make it seem even more unstable; thus the inverse proportionality coming from the Volume.

Case 2: Context = single phrase p

(11),
Conjecture (5)

Here, and are constants of proportionality; tending to be close to zero and tending to be close to 1.

Note that:

(12)

The stability equation for a phrase is directly proportional to the weighted sum of the individual notes that belong to the phrase. More precisely, we conjecture that the stability of a phrase depends largely on the stability of its last note and the other notes contribute only in a small proportion. This is because the last note is followed by some amount of silence thus creating a "lasting" effect.

For example, a consider the phrase from Raga Yaman; p = { Ni, Dha, Ni, Re, Ma }. Here the phrase ends with an unstable swar (Ma). The lasting effect of this phrase is that of instability. The remaining phrase contributes to the resulting feel, but the largest effect is that of the last note, making the phrase as a whole, unstable. In contrast, the phrase p = { Ni, Re, Ma, Re, Ma, Ga } ends in a locally stable swar (Ga) and thus has an overall stabilizing feel, even when it contains a high proportion of Teevra Madhyam (Ma), a highly unstable swar.

Mathematically, it is possible that the effect of the last note can be overshadowed by the preceding notes if:

However we feel that in practice, this will be very rare.

Case 3: Context = set of phrases

Let us consider the set of phrases

(13)

The stability of a set of phrases is simply the average of the stability of its individual phrases. A more complex model can be proposed that takes the relative sizes of gaps of silence, instantaneous and average tempo, smoothened stability spikes, stability characteristics of percussion and melody accompaniment, frequency equalization characteristics (warm tones (bass boost; high cut) are relatively stable as compared to bright tones (high boost)), reverb characteristics (dry reverb relatively unstable as compared to wet reverb), amplitude characteristics (normalized/dynamically compressed amplitude has higher perceived stability than un-normalized audio), etc into consideration. However, for the sake of analysis and practice, this model is sufficiently powerful.

A graphical representation of Potential wells along the swar line

Graph 1 is a graphical representation of the potential wells formed on the swar line due to the presence of globally stable swars. The depth of the well is proportional to the strength of the field formed by the well.

The four sub graphs depict the presence of different combinations of globally stable swar in a given context.

Observations and Results

Fundamental Observations

• The shape of the curve depicting the field strength as a function of the distance, is that of a "bowl", or "basin" and is a characteristic of the inverse square law.
• The notes closest to the globally stable swars experience the highest force, and it falls of rapidly for notes that are farther away.
• The value of the Madhya Shadja (Sa) is the greatest, thus making it the most stable of the stable swars. The value of Taar Shadja (upper Sa) is high, but slightly lower than that of the Madhya Shadja. This is because even though it is stable, it is one octave higher, and thus has a slightly lower perceived stability than it's lower-frequency counterpart.
• The value of Pancham (Pa) is lower than either Madhya or Taar Shadja, but greater than that of Madhyam (Ma). We believe this is because Pancham lies at the exact center of the octave frequency spectrum and thus is perceived as more stable than Madhyam which splits the spectrum in a ratio of 1 : 3.
• Komal Rishabh (Re) and Shuddha Nishad (Ni) will always be unstable due to the the presence of the strong attractor fields at Shadja (Sa).

Derived Observations with Experimental Evidence

Graph 1.1

Teevra Madhyam (Ma) becomes relatively less unstable due to opposing forces exerted by Shuddha Madhyam (Ma) and Pancham (Pa).

A great example of this phenomenon is the feel of Ma in Purvi is relatively less unstable as compared to Ma in Puriya Dhanashri. Purvi contains Shuddha Madhyam (Ma) that Puriya Dhanashri lacks. Other examples include Basant vs. Puriya Dhanashri, Yaman Kalyan vs. Yaman, etc.

Graph 1.2

The presence of a strong attractor field at Shuddha Madhyam (Ma) causes the feel of Shuddha Gandhaar (Ga) and Teevra Madhyam (Ma) to become unstable, while stabilizing both the Dhaivats (Dha and Dha) and Komal Gandhaar (Ga).

The first example is Bhinna Shadja. In Bhinna Shadja, the Ga has an unstable feel to it, as compared to, say the Ga of Yaman. On the other hand, its Dha feels more satisfying, than say the Dha, in Nand. (In Shuddha Basant (Haveli Basant), in addition to the feel of Ga and Dha, the presence of Ma adds to its instability.)

The second example is raga Lalit. Lalit's Ga and Ma have unstable feels associated with them, in contrast with the Ga and Ma of Din-ki-Puriya. Also, the komal Dhaivat (Dha) in Lalit is not as unstable as say the Dha in Bhairav, in which the presence of Pa increases the instability of Dha.

In Malkauns, Dha and Ga have a relatively stable feel (as compared to Sampoorna Malkauns) than expected due to the small magnitude of the resultant force acting on them.

Graph 1.3

The presence of a strong attractor field at Pancham (Pa) causes the feel of Teevra Madhyam (Ma) and Komal Dhaivat (Dha) to become unstable, while stabilizing Shuddha Gandhaar (Ga) and Shuddha Dhaivat (Dha).

An example is Bhoop (or Deskaar), where Ga and Dha are locally stable notes.

In Puriya Dhanashri, Ga is locally stable but Ma and Dha are unstable. In contrast, Dha in Puriya Kalyan is stable.

Graph 1.4

The absence of strong attractor fields at Pancham (Pa) or Shuddha Madhyam (Ma) causes the feel of the notes belonging to the interval from Shuddha Gandhaar (Ga) through Komal Dhaivat (Dha) to become relatively stable while the remaining notes ( Re, Re, Ga, Dha, Ni, Ni ) to become relatively unstable.

For example, Din-ki-Puriya's Ma and Dha are relatively more stable as compared to those of Puriya Dhanashri. In Gujri Todi, the Ma and Dha are more stable as their counterparts in Miyaan-ki-Todi.

Fitting real world data to the mathematical model

Case Study 1. | Instability ( Marwa ) | > | Instability ( Puriya ) |

The feel of Marwa is more unstable than that of Puriya even though both these ragas have the same set of notes. The set of transitions and nuances for these ragas are significantly different, but this reason alone is not enough to explain why these specific differences affect the feel of the two ragas so much. This phenomenon can be explained by this model as follows:

Based on the model, in both Marwa and Puriya, Re, Dha and Ni are unstable swars while Ma and Ga are locally stable. Looking at graph 2 below, we can estimate the relative magnitudes of force of stabilization, and thus the magnitude of instability of these individual notes.

Graph 2 shows the relative magnitudes of the forces of stabilization felt by various notes in the Marwa/Puriya Swar-Set. Notice that the threshold of stability lies between the magnitudes of Ga and Dha.

| Instability (Sa) | < | Instability (Ma) | < | Instability (Ga) | < | Instability (Dha) | < | Instability (Ni) | < | Instability (Re) |

In Puriya, Ga and Ni are prominent, Sa is used frequently while Re, Ma and Dha are downplayed quite a bit. In Marwa, Re, Dha are very prominent, Ma is used frequently while Ga, Ni and Sa are significantly downplayed. Due to the combined effect of (Nyaas) and the magnitude of on instability , the magnitude of instability generated by Marwa is far greater than that generated by Puriya. (Keep in mind that although the feel of classic Puriya is not "stable", it is definitely more stable than that of Marwa!)

Case Study 2. Stability characteristics of Ragas with the same set of notes can be significantly different

This is a generalization of the above observation. Different Ragas with the same set of notes are distinguished by differing rules of Transition and the set of Nuances. The transitions along with the Nuances influence two important functions, and , that in turn influence the perceived stability of a raga. The techniques to manipulate these functions from a performance point of view are discussed later in this paper.

We believe that the Vaadi and Samavaadi swaras for a given raga is an emergent property of the positions of the fields of its globally stable swaras as well as the proportions and Nuances that define that Raga (Conjecture 6).

Examples

• Sohoni, Marwa and Puriya,
• Miyaan-ki-Todi and Multani,
• Bhairavi, Bilaskhani Todi and Komal Rishabh Aasawari Todi,
• Yaman and Suddha Kalyan,
• Adana, Darbari Kanada, Jaunpuri and Sampoorna Malkauns,
• Gawati, Bageshree, Bhimpalaas, Nayaki Kanada and Kafi,
• Puriya Dhanashri and Shree,
• Basant and Poorvi,
• Miyaan Malhaar and Bahar,
• Bhatiyar and Anand Bhairav,
• Yaman Kalyan, Maru Bihag, Shyam Kalyan and Bihag,
• Bhairav and Kaalingada,
• Megh Malhaar and Madmad Sarang,
• etc.

Case Study 3. Stability characteristics of Ragas related by the Moorchana relationship can be significantly different

The Moorchana relationship (See delta fingerprinting properties) shifts the origin (Sa) for a Raga keeping the original raga's relative distances of notes constant. The question thus becomes, "if the relative distances between the notes of two ragas are the same, why are the feels of these ragas so different?". (Note that this is different than the case with two ragas with the same set of notes. With Moorchana, the distances between the notes remain constant, but the absolute notes change.)

The answer lies in the placement of the potential wells on the swar line. When Moorchana is applied to a Raga, the shift of origin also shifts the positions of the globally stable swar. This changes the distribution characteristics of forces and thus the stability characteristics. Another factor is the transformed raga (after Moorchana) can have rules of transition and nuances that can influence the stability characteristics by a great amount.

Examples: (note, the stability classification is relative and is based on the "average" feel of the raga. The degree of stability (or instability) is variable)

• Jog (stable), Gujri Todi (unstable),
• Jog (stable), Vrindawani Sarang (stable),
• Bhoop (stable), Malkauns (unstable),
• Bhoop (stable), Durga (Stable),
• Bhairavi (both stable and unstable), Yaman (both stable and unstable),
• Shivaranjani (unstable), Marwaa (unstable),
• Lalit (unstable), Miyaa-ki-todi (unstable),
• Ahir-Bhairav (unstable), Nat-Bhairav (unstable),
• Chandrakauns (unstable), Madhukauns (stable), etc.

Manipulating the stability characteristics of a musical context

We can derive the techniques that can control accurately the amount of instability in a musical context from the mathematical model proposed in this paper. Some of these techniques are known and are taught. Others are learnt through experience, although they may never be understood consciously. Yet others are new insights into the realm of stability.

We know from equation (1) that the force of stabilization for a given musical context is constant through the lifetime of the context and depends purely on the set of globally stable and other notes in the context. The variables in that equations are and . Let us look at some of the musical techniques that can manipulate the magnitudes of these two functions.

The proportions of the swars in the context will have an average effect on the stability of the context; higher proportions of unstable swar will create a predominantly unstable effect while higher proportions of stable swar will create a strong stable feel. As a corollary, the touch of a stable swar, local or global, will raise the stability of a swar by a small but noticeable amount. Another corollary is that increased Nyaas (sustenance time) on a stable swar will add a lot of stability to the context.

From equation (11) we see that the end note of a phrase is very important as its contribution towards the overall stability of the context is large. Thus we can control the end note and it's Nyaas to boost the magnitude of . It is important to remember to stay within the prescribed rules of the raga. This can be a challenging task with respect to amount of Nyaas as most often, rules of Nyaas are used to distinguish one Raga from another with the same set of notes. That is why for some ragas with stringent prescribed note proportions and rules of Nyaas, the range of stability may be quite constrained (examples: Marwa, Puriya, Puriya Dhanashree and Shree, etc). With other ragas with liberal rules of Nyaas, the range can be quite large often covering almost the whole spectrum (examples: Malkauns, Bhairavi, Yaman, etc)

Volume modulation can be used to great effect to augment stability characteristics thus effectively controlling the function . A healthy volume for a locally stable note makes it more stable, while a soft volume on it will make it appear less stable. Conversely, a healthy volume on a locally unstable note will make it feel more unstable while a softer touch will soften it a little.

Large jumps in notes creates a feeling of instability as the energy to transition between a larger frequency gap is correspondingly larger. Similarly, shorter transitions help stabilize the context. (Example, the context {Ma, Ga, Re, Sa} in Bageshree is relatively more stable than the context {Ma, Ga, Sa} in Malkauns). Tempo influences a context provided all other variables are held constant over that context. A higher tempo for a stable phrase tends to decrease it's perceived stability while a higher tempo for an unstable phrase tends to decrease its perceived instability. This follows directly from equations (7) and (8).

Future Work

• Empirically calculate the various constants of proportionality.
• Experimentally prove conjectures 1 through 6.
• Quantify the model of stability of musical contexts.
• Create and test a model of stability based on Nuances and Inflections in a musical context.
• Associate degrees of stability with other contextual elements to create a map of performance techniques with the created emotion and its intensity, etc.