In Indian music, Alankar or Alamkara means ornaments or adornments. In the context of Indian classical music, the application of an alankar is essentially to embellish or enhance the inherent beauty of the genre. Note transitions play other important roles in the domain of Indian Classical Music. Style, emotions, gharana characteristics, raga characteristics and even the personal characteristics might be embedded in these transitions. Therefore it is very important to objectively categorise these transitions to understand their cognitive implications. With the advent of modern measuring methodologies and statistical tools these tasks are relevant now-a-days.

Analysis and synthesis of transitions between musical notes are open-ended problems today. While much research has been done on the proper analysis and synthesis of musical timbres, sequence of notes, etc., less attention has been paid to what occurs between successively played notes.

A pitch contour describes a series of relative pitch transitions and an abstraction of a sequence of notes. It has been found to be more significant to listeners in determining melodic similarity, and it also includes rhythm information. In the rendition of Ragas in Indian Classical
Music, the used notes, not only their sequences but also the nature of transitions between notes are said to be relevant to characterize the Raga. Understanding the relationship between cognition (which creates appreciation) and the physical events triggering them is one of the important problems in music research.

How strange that music is deemed a phenomenon in need of scientific explanation. We don’t, in general, construct objective theories of how great paintings ‘work’, or great literature, dance or sculpture. We are interested in what is happening at a perceptual level when we experience these arts, but there is always a space in which we leave them to speak for themselves, beyond the reach of cold facts. Yet with music, scientific studies seem to be on the trail of an absolute, all-encompassing explanation that connects neurology with creativity, auditory physiology with acoustic physics. There seems to be a conviction that the composer Arnold Schoenberg was right when he cautioned: “One day the children’s children of our psychologists will have deciphered the language of music.”

This ’scientification’ of music is part of a very old tradition. In antiquity and the Middle Ages music was not an art in the modern sense; it was one of the four sciences of the syllabus called the liberal arts, alongside geometry, arithmetic and astronomy. Scholars studied music to learn about the natural harmony of the world, and performed music was often dismissed as frippery. The early sixth-century Roman philosopher Boethius ranked it as the least of his three classes of ‘music’, and agreed with Pythagoras that music should ideally be studied while “setting aside the judgement of the ears”.

The practice of music does have something of the mathematical about it. Some of the experiments in compositional symmetry, such as the palindromes and mirror reflections of Wolfgang Amadeus Mozart and Joseph Hadyn, are little more than the parlour tricks of an age that delighted in such amusements. But many other musical forms and theories have deeper, more formal organization, from the interwoven fugues of Johann Sebastian Bach to the quasi-mathematical laws of composition developed by Paul Hindemith.

In the final throes of Schoenberg’s twelve-note serialism in the 1960s, composers such as Pierre Boulez insisted on a mathematical rigidity that almost sucks their music dry of ex-pression and makes onerous demands of the listener’s ability to perceive ordered forms, and in some types of non-Western music, pattern and structure rather than emotion or tone-painting provide the foundations of composition. This is the case in polyrhythmic African drumming, for instance, and the shimmering soundscapes of Javanese gamelan.

Even musicians are uncertain of what kind of art it is they are engaged in, and what, if anything, can be said about it. ‘Is there meaning in music?’ asked US composer Aaron Copland. He felt there was, but admitted to being unable to articulate what that meaning is.

Almost the only thing we can say about music as a cultural phenomenon is that it seems to be universal. Music serves very diverse ends, sometimes with more apparent emphasis on the ritualistic than the hedonistic. Even when it is taken very seriously - in some Native American cultures a ceremony has to be started again if a single note is out of place - anthropologists have often struggled to understand how or to what extent cultures apply intellectual and aesthetic judgements. Sometimes music is a commodity for sale and exchange; elsewhere it is inseparable from dance.

Given this range of what music is and what functions it serves, how can we make sense of it as an acoustic, cognitive, cultural and aesthetic phenomenon? That need not be deemed an entirely hopeless task, but it is not one that science will accomplish alone.

Have you ever had a meeting, or a brain storming session, that involved a lot of coffee and enthusiasm, with everyone throwing out ideas at a breakneck pace, and quickly becoming convinced of their brilliance? I had just such a meeting one morning not too long ago. Everything moved really, really fast, and we were convinced that we’d hit upon a really good idea. Later that evening, everything about the idea that we’d come up with began to fall apart. The next morning, I received an email from one of the meeting’s participants with the subject heading, “Maybe this is why we thought it was such a good idea.” The email had no text, only an attached paper by Emily Pronin and Dan Wegner titled “Manic Thinking: Independent Effects of Thought Speed and Thought Content on Mood”.

Pronin and Wegner note that the psychiatric illness, mania, is associated with both increased thought speed and elevated mood, along with delusions of grandeur, and the feeling of heightened creativity and inspiration. However, the effect of thought speed has not been studied independent of clinical mania. To explore this relationship, they had college undergrads read out loud a series of emotion-inducing statements (58 in all) in at either a fast or slow pace. The statements, which have been used to manipulate mood for a few decades, start out emotionally neutral, and then become more and more emotionally positive or more and more emotionally negative. The idea is that reading a series of progressively more positive or negative statements will affect the reader’s mood accordingly. The letters in the statements were presented one at a time, either for 40ms per letter (fast thinking condition) or 170 ms per letter (slow thinking). A pilot study indicated that the 40ms per letter reading time was about twice the normal reading speed for college undergrads, with 170 ms being about half the normal speed. The time between statements also varied, with only 320 ms between statements in the fast condition, and 4 seconds between statements in the slow condition.

After reading all 58 statements, participants were asked to answer a series of questions designed to assess their mood, energy level, feelings of power, creativity and inspiration, and “grandiosity or inflated self-esteem,” along with their own perceptions of their speed of thought. Not surprisingly (and confirming that the manipulation was working), participants in the fast thought condition reported faster thought speeds than those in the slower condition.
Consistent with the hypothesis that faster thought speeds affected mood and mania-related feelings, participants in the fast thought condition reported being happier, had higher energy levels, experienced greater senses of power and creativity, and higher levels of grandiosity (though self-esteem did not differ between conditions). Furthermore, these effects were independent of the mood manipulation (positive or negative statements). So there you have it.
Thinking fast produces effects on mood and self-view similar to those of clinical mania. Of course, nothing in these results says that thinking faster doesn’t actually lead to more creativity and inspiration, but my colleague was probably right: the fact that we’d been thinking so fast during our meeting likely had something to do with the fact that we thought what turned out to be a bad idea was so good. I wonder how much of a role coffee played in this. While I don’t know of any research on the effect of caffeine on the speed of thought, it is a stimulant, and the fact that we were drinking it in large amounts (the meeting took place at a local coffee house) couldn’t have helped.
The implications are clear, then. It’s important, when you’re dealing with something important, to slow down now and then (and cut off the supply of coffee) in order to be able to objectively evaluate the ideas your producing. Otherwise, you might end up like we did, with hours of work that produce only bad ideas.

Reference:

Pronin, E., & Wegner, D. M. (2006). Manic thinking: Independent effects of thought speed and thought content on mood. Psychological Science, 17(9),807-813.

I was inspired by a research paper showing that we may represent positive numbers on a “mental number line.” In one experiment testing the mental number line hypothesis, participants who were asked to indicate whether a number was positive or negative did so faster for large numbers when the response key was on the right side of the keyboard, and small numbers when the response key was on the left side of the keyboard. Since the concept of a mental number line presumably involves representing small numbers on the left side of the mental number line, and large numbers on the right side, this result is consistent with the existence of such a line.

The paper I discovered while looking for the research paper isn’t about mathematics, though. Instead, it’s about music. The work, by Rusconi et al., was designed to show a connection between how we represent music with how we represent number. They note that across many languages (they list Chinese, English, French, German, Italian, Polish and Spanish), the words use to denote differences in pitch are spatial (e.g., high pitches and low pitches). The idea, then, is that like number, pitch may be represented on a mental line, with (when the terms for pitch are vertical terms) high pitches represented higher on the line than low pitches.

In their first experiment, Rusconi et al. first presented Chinese participants (most of whom spoke both Cantonese and English) who had no musical experience with a reference pitch (C4), followed by a target pitch that was either higher or lower than the reference pitch (E3, F3#, G3#, A3#, D4, E4, F4#, G4#). Participants were asked to indicate whether the pitch of the target was higher or lower than that of the target pitch. In one condition, the response keys, the spacebar and the 6 key, were above and below each other, and in a second condition, the response keys, Q and P, were across from each other on the same keyboard row. In both conditions, half of the participants were told to to higher pitches with one key (e.g., 6 or P) and lower pitches with the other (spacebar and Q), and the other half were told to respond in the opposite way. The prediction, then, is that when the response keys are on a vertical axis (spacebar and 6), people will respond to higher pitches faster when the response key is the upper one (6), and lower pitches when the response key is the lower one (spacebar). There should be no difference between the two response key configurations when the response keys are on the same horizontal axis (Q and P).

First, they found that bigger differences between the target and reference pitches resulted in faster response times than smaller pitches. No surprise there. They also found that participants were faster to respond to higher pitches with the P key (on the right), and lower pitches with the P key, though this difference wasn’t quite significant. I’m not exactly sure what to make of this difference. Consistent with their predictions, though, when the response keys were on the vertical axis, their responses were faster when the upper key (6) was used to respond to higher pitches, and the lower key (spacebar) was used to respond to lower pitches.

In their second experiment, a second group of musical novices (again, Cantonese as first language and English as second language) were presented with different tones (F3#, G3#, A3#, C4, E4, F4#, G4# and A4#) played by either wind instruments (French horn or tenor trombone) or percussion instruments (marimba or vibraphone). Their task was to indicate the instrument family (wind or percussion). Once again, the response keys were either on a vertical axis (spacebar and 6) or a horizontal one (Q and P), and as in the first experiment, the prediction was that participants would be faster to respond to high pitches with the upper key than the lower, and for low pitches, responses would be faster with the lower key than the upper. This time, they didn’t find a difference between the horizontal response keys (maybe the difference in the first experiment was a fluke?), but as in the first experiment, they did find the predicted difference for the horizontal keys: responses to high pitches was faster when the response key was the upper one (6), and they were faster for low pitches when the response key was the lower one (spacebar), despite the fact that the task didn’t involve making any distinctions between pitches.

Their third experiment was identical to the second, but this time their participants were musicians. Once again, they found that high pitches led to faster response times with the upper key than the lower one, and low pitches resulted in faster response times with the lower key. In fact, the difference between consistent keys (6 for high pitches and spacebar for low keys) and inconsistent keys (spacebar for high pitches and 6 for low pitches) was even greater for the musicians than it was for the novices in the second experiment. Furthermore, the musicians also responded to high pitches faster with the right key than the left, and lo pitch with the left key than right (OK, so maybe it wasn’t a fluke).

It seems, then, that to some extent, we do represent pitch on a mental line, with high pitches at the top and low pitches at the bottom, at least in languages where the terms for pitch are vertical terms. We may also represent pitch on a horizontal axis, too, as evidenced by the differences in response times for the horizontal keys in the first and third experiments. Rusconi et al. argue that the horizontal effect may be due to a “remapping” of the vertical dimension onto the horizontal dimension, which musical training somehow facilitates (the horizontal effect was strongest in the experiment with experts). I think that explanation can be translated as, “We haven’t the slightest idea why we got this result.” Perhaps future work will figure it out, though.

References:

Dehaene, S., Dupoux, E., & Mehler, J. (1990). Is numerical comparison digital: Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16, 626-641.

Rusconi, E., Kwan, B., Giordano, B.L., Umilta, C., & Butterworth, B. (2006). Spatial representation of pitch height: The SMARC effect. Cognition, 99(2), 113-129.

A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity.
The term was coined by Benoit Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken” or “fractured.” A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals - for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.

Musical research over the last century has become increasingly entwined with the scientific areas of acoustics, psychoacoustics, and electro acoustics, among others. During the last half century, the computer has become the central site of this research, including sound synthesis, digital signal processing and computer-assisted composition.

The application of fractal geometry to musical signals with respect to Indian musical instruments has not been widely experimented.

Tanpura is a multi-stringed instrument extensively used as a drone instrument and is an integral part of classical music in India. The instrument is plucked by finger and is used as an accompaniment with the vocal music. The special form of the bridge has a remarkable influence on the tone quality. When the adjustment of the position of contact of string on the oval shaped bridge is made carefully by trial, by using a cotton thread, the instrument is highly sonorous, giving a tone of fine musical quality. This is known as “jwari”. The average fundamental frequency is extremely steady and consistent. The amplitude however shows a regular long-term variation, a sort of waxing and waning, three to four in number during the course of a single plucking. It has been noticed that the harmonics of the tanpura strings’ sound exhibit a periodic waxing and waning. This period is shorter as one goes up along the higher harmonics. This is connected with the jwari of the tuning. These are, however, regular predictable nature of complexity variation, which may have functional values other than providing a rich melodious sound. The source of origin must be related to some sort of non-linearity associated with the strings, the slightly convex form of the lower bridge and the mode of attachment of strings and/or some sort of feed back mechanism which uses the total acoustic environment including the global resonance structure of the instrument.

The best known example of this process (non-linear dynamic systems) in the field of Indian musical instruments is Tanpura. The analysis of the Tanpura sound signals defy the conventional assumption based in linear models that complex behaviour results from complex factors. It is also characteristic of these models that their component elements act as cells or quanta, and that the global behaviour emerges over large numbers of iterations, usually such that a computer is required for the intensive calculations involved.

Fractal dimensions of time series data generally reveal the presence of non-linearity in the production mechanism. Tanpura signal is considered as repetitive quasi-stable geometric forms. Time series data is a quantitative record of variations of a particular quality over a period of time. One way of analysing it is to look for the geometric features for categorising data in terms of concept. The signals emitted by a Tanpura is characterised by varying complexity with undulations of intensity of different harmonics with different frequencies as well as multiple decay. All these suggest interplay of source at various point of time like attack time, quasi-static state and end decay. Study of fractal dimensions might be a technique to analyse this bevaviour. Non-linear dynamical modeling for source clearly indicates the relevance of non-deterministic approaches in understanding these signals.