This paper was prepared by myself & Rohan Jain, my friend , who is my class mate.

INTRODUCTION.

How do math and music relate to each other?

          For many people, mathematics is an enigma. Characterized by the impression of the numbers and calculations taught at school, it is often accompanied by feeling s of rejection and disinterest, and it is believed to be strictly rational, abstract, cold and soulless.

          Music, on the other hand, has something to do with emotion, with feelings, and with life. It is present in all daily routines. Everyone has sung a song, pressed a key on a piano, blown into a flute, and therefore made music. It is something people can interact with; it is a way of expression and a part of everyone’s existence.

          The motivation for investigating the connections between these two apparent opposites therefore is not very obvious, and it is unclear tins what aspects of both topics such a relationship could be seeked. Moreover, if one accepts some mathematical aspects in music such as rhythm and pitch, it is more difficult to imagine any musicality in mathematics. The count ability and the strong order of mathematics do not seem to coincide with an artistic pattern. However, there’r different aspects, which indicate this sort of relationship.  This assignment will show that mathematics and music do not form such strong opposites as they are commonly considered to be, but that; there are connections and simulations between them, which may explain why some musicians like mathematics and why mathematicians generally love music. At the fag end a simulation using a “C” program takes into account the underlying element of music in a particular mathematical function, revealing a brief coagulation of the two utterly different aspects of science.

TONE AND TUNING: THE PYTHAGOREAN PERCEPTION OF MUSIC.

          In the time of the ancient Greeks, mathematics and music were strongly connected. Music was considered as a tricky mathematical discipline, handling with number relationships, ratios and proportions. In the quadrivium,(the curriculum of Pythagorean school) music was placed on the same level as arithmetic; geometry and astronomy). This interpretation totally neglected the creative aspects of musical performance. Music was the science of sound and harmony.

Figure 1

Quadrivium

          The basic notions in this context were those of consonance and dissonance. People had noticed very early that two different notes do not always sound pleasant (consonant) when played together. Moreover, the ancient Greeks discovered that a note with a given frequency sounded pleasant only when other notes whose frequencies were integer multiples of the first could be properly combined with it. If, for example, a note of the frequency 220 Hz was given, the notes of frequencies 440 Hz, 660 Hz, 880Hz, 1100 Hz and so on sounded best when played together with the first.

          Furthermore, examinations of different sounds showed that these integer multiples the base frequency always appear in a weak intensity along with the basic note, when played. If a string whose length defines a frequency of 220 Hz is vibrating, the generated sound also contains components of the frequencies 440 Hz, 660 Hz, 880Hz, 1100 Hz and so on. Whereas the listeners perceive mainly the basic note, the intensities of these so-called overtones define the character of an instrument. It is primarily due to this phenomenon that a violin and a trumpet do not sound similar even if they play the same note.

          The most important frequency ratio is 1:2, which is called an octave in the western system of music notation. Two different notes in such a relation are often considered as principally the same (and are therefore given the same name), only varying in their pitch but not in their character. The Greeks saw in the octave a ‘cyclic identity’. The following ratios build the musical fifth (2:3), fourth (3:4), major third (4:5) and minor third (5:6), which all have their importance in the creation of chords.

          The difference between a fifth and a fourth was defined as a ‘whole’ tone, which results in ratio of 8:9. These ratios correspond not only to the sounding frequencies but also to the relative string lengths, which made it easy to find consonant notes starting from a base frequency. Shortening a string to two thirds of its length creates the musical interval of a fifth for example.

          All these studies of ‘harmonic’ ratios and proportions were the essence of music during Pythagorean times. This perception, however, lost its importance at the end of the middle ages, when more complex music was developed. Despite the ‘perfect’ ratios, there occurred new dissonances when particular chords, different keys or a greater scale of notes were used.

          The explanation for this phenomenon was the incommensurability of thirds, fifths and octaves when defined by integer ratios. By adding several intervals of these types to a base note, one never reaches an octave of the base note again. In other words, an octave (1:2) cannot be subdivided into a finite number of equal intervals of this Pythagorean type (x: x+1 | x being an integer). Adding whole tones defined by the ratio 9:8 to a base note with the frequency f, for example, never creates a new note with the frequency 2f, 3f, 4f or similar.

         Considering these characteristics of the Pythagorean intervals, the need of another tuning system developed. Several attempts were made, but only one has survived until nowadays: the system of dividing an octave into twelve equal (‘even-tempered’) semitones introduced by Johann Sebastian Bach founding on the ratio 1:2 for octaves, all the other Pythagorean intervals were slightly tempered (adjusted) in order to fit into this new pattern.

Mathematical music: Fibonacci numbers and the golden rule in musical compositions.

          The questions of tone and tuning are one aspect in which mathematical thoughts enter the world of music. However, music –at least in a modern perception – does not only consist of notes and harmony. More important are the changes of notes in relation to time i.e. the aspect of rhythm and melody. Here again mathematical concepts are omnipresent. Not only is the symbolic musical notation and all aspects very mathematical but also particular arithmetic and geometric reflection s can be found\d in musical compositions, as will be seen in the following paragraphs:

          A very interesting aspect of mathematical concepts in musical compositions is the appearance of Fibonacci numbers and the theory of golden section. The former is the infinite sequence of integers named after Leonardo de Pisa, a medieval mathematician. Its first two numbers are both 1 whereas every new number of the sequence is formed by the addition of the two proceedings (1, 1, 2, 3, 5, 8, 13, 21, 34….). However the most important feature in this context is that the sequence of Fibonacci ratios converges to the constant limit, called the Golden Ratio, Golden proportion or section (0.61803398….)

           More common is the geometric interpretation of the golden section: A division of a line into two unequal parts is called a golden if the relation of the length of the whole line to the length of the bigger part is the same as the relation of the length of the bigger part to the length of the smaller part. Due to its consideration as well balanced beautiful and dynamic, the golden section has found various applications in arts specially in painting and photography, where important elements often divide a pictures length or width (or both) following the golden proportion, However, such a division is not necessarily undertaken consciously, but results from an impression of beauty and harmony. Diverse studies have discovered that this same concept is also very common in musical compositions. The golden section- expressed by Fibonacci ratios –is either used to generate rhythmic changes or to develop a melody line.

           It is probably less important to evaluate whether people consider mathematics when they apply or perceive a golden proportion than to notice that harmony and beauty-at least in this aspect-can be expressed by mathematical means. Fibonacci ratios in relation to the division of a composition, as well as integer ratios in relation to Pythagorean intervals are examples of the fact that harmony can sometimes be described by numbers (even integers) and therefore have a very mathematical aspect. This could be one way to introduce an additional idea; that beauty is inherent in mathematics.

OCTAVE:  In modern music middle C generally has a frequency of 260 Hz. So the C an octave above would have frequency of 520=2 X 260 Hz. And the C and octave lower would be half 130Hz. Another pleasing ratio is the perfect fifth. Notes a fifth apart have frequency ratio 3:2.If we consider the musical scales starting at C and going down half steps to the C and octave higher, there are 12 notes. In Indian terminology these are SA, RE, GA, MA, PA, DHA, NI, SA  (an octave higher) along with this we have 5 notes like RA, GU, DA, NA and   MI, whose freq is a little above MA.

NOTES OF AN OCTAVE

MATHS: THE UNACANNY SILHOUETTE OF MUSIC.

Of all the academic subjects, math is most closely connected to music. Music is all based on fractions and patterns. Where fractions are concerned, music focuses on divisions of time for the rhythm and space for dealing with intervals such as octaves or fifths." 

.  Counting - It's fundamental to playing music. One must count beats per measure and count how long to hold notes.

.  Patterns - Music is full of patterns-patterns of notes, chords, and key changes. Musicians learn to recognize these quickly. Patterns, and being able to invert them (known as counterpoint), help musicians form harmonies.

.  Geometry -- Music students use geometric shapes to help them remember the correct finger positions for notes or chords (more than one note played simultaneously)-for instance, guitar players' fingers often form triangle shapes on the next of the guitar.

.  Ratios & proportions/equivalent fractions - Reading music requires an understanding of ratios and proportions. For instance, a whole note needs to be played for twice as long as a half note, four times as long as a quarter note, and so forth. In addition, since the amount of time allotted to one beat in a given time signature is a mathematical constant, the durations of all the notes in that piece are all relative to one another and are played on the basis of that constant. Finally, different frameworks of time with which musicians work are based on an understanding of fractions and multiples-for example, understanding the rhythmic differences between 1/4 and 4/4 time signatures.

.  Sequences - Music and mathematics are also related through sequences, particularly intervals. Although a mathematical interval corresponds to the difference between two numbers, a musical interval corresponds to the ratio of the frequencies of the tones.

…..AND FROM THE INDIAN POINT OF VIEW …

 There is a single principle that underlines all musicomathematical relations: 

An arithmetic progression in music corresponds to geometric progressions in mathematics; that is, the relation 

between the two is logarithmic.

MUSIC: A TEMPLATE TO ACT AS A BLUEPRINT OF MATHEMATICAL FUNCTIONS.

          While establishing an amalgamation of math and music and contemplating on their relation from music to math point of view, it won’t be a redundancy to try to confer upon mathematical function an attribute of music. Intuitively, if such notion can be made to exist, then every such function should have a character signature of music. But to define such an abstract relation a fixed set of rules is quite necessary. Of course a platform for defining such relations could be a computer language where it can be implemented in an exhaustive way.

          Large components of computer music systems are commonly written in the C programming language. Such concerns are central to successful implementations of reactive performance-oriented computer music systems. By judicious use of new features of the recently established ISO Standard C, real-time computer music applications may be developed that are more efficient and reliable than typical C programs, easier to understand and write, and easier to optimize for a particular operating environment. New features of ISO C relevant to reactive music system programming are illustrated by a new programming style for musical applications that exploits unique strengths of C. Many years of the effort involved the introduction of entirely new features many of which directly address efficiency issues that have prevented C from use in reactive music software.

           The popularity of the C programming language and its derivatives in signal processing applications is increasing. A broad collection of numerical algorithms with implementations in C, FORTRAN and Pascal has been available for some years. Efforts are underway to develop an ANSI standard for Digital Signal Processing extensions to C. C is expressive enough to allow the careful programmer to obtain the numerical performance potential of a processor so important for many signal-processing applications.

          While there are numerous differences between these various programs when it comes to the details, software languages for sound synthesis all share one important similarity. Basically, they all follow a "modular" approach to sound synthesis. Each module (or "unit generator") is either creating or modifying a stream of numbers (an audio signal) and any number and type of modules may be "patched" together in a network. These modular networks are referred to as computer instruments.

A “C” SIMULATION OF MUSIC IN MATHS.

          Now any mathematical function has a typical range of values that can be easily expressed within a limit by multiplying with typical whole numbers or fractions. For example if a typical scale of three octaves is considered, each of 12 notes, then a range of 36 notes are obtained. Any type of music is based on these three octaves. Thus associating SA with 1,RE with 2,  RA with 3 and so on, a musical scale can be defined. Basically if three functions are considered: -

1. Logarithmic Function.

2. Trigonometric Functions.

3. Polynomial Functions.

          It seems that their structure resembled musical scores, so as an experiment let’s see what they sound like when the following rules are defined to convert the values of the function to a range of 1 to 36. 

Considering the polynomial function first:

           Its positive values may range from 1 to 32767(the limit for integer value in ‘C’). By dividing it into three ranges: i.e 1 to 36, 37 to 1296, &1297 to 32767.Any value of the function in the first region can be directly processed to get the corresponding note. A value in the second range can be divided by 36(a whole number) to get the value again in the first range. For the third range the values can be divided by 910(another whole number) to get the values within the first range and the subsequent sound output. A question arises as to why the negative values of the function are to be neglected when they can also add to the music DNA of the polynomial. Well they could be converted to a positive one by multiplying with –1 and given the same treatment as to their positive counterparts.

A logarithmic function can have the highest value of 10.39.(log (32767) ) which when multiplied with 3151 gives the range of values from 1 to 32767, which can then be treated the same way as that of polynomial.

A totally different treatment lies in store for the trigonometric functions:

           Since the trigonometric functions are periodic so let’s define a base when they obtain a zero value. Let this base be 18. Any negative values will be treated in the range 1 to 18 and positive values within 18 to 36.

SINE wave:

          Within the range 0 to 360 degrees, a sine function gradually rises to a value 1.00 from 0.0 and falls from 1. 00 to 0.00 in the range 90 to 180degrees.so the rise may be simulated as a rise from 18 to 36 then fall from 36 to 18 then go down from 18 to 1 and then again rise from 1 to 18.hence the cycle gets completed. Based on the same lines we can define the following tables for the remaining tables: - 

2. COSINE TABLE: -

3. TAN TABLE: -

          In a very similar way we can also simulate the remaining three ratios, so that they represent a particular note pattern.

          Essentially, the musical notes correspond to the depth of the proof tree as the proof verifier constructs the proof. One note is produced for each proof step as formulas are constructed (fast higher notes) and matched to other theorems (sustained lower notes, which correspond to the proof steps).

          Is it "music"? I guess that's for you to decide. It is richly structured, with underlying themes that on the one hand seem to repeat but on the other hand are interestingly unpredictable, teasing your mind as the piece progresses.

CONCLUSION.

          All these aspects of mathematical patterns in sound, harmony and composition do not convincingly explain the outstanding affinity of mathematicians for music. Being a mathematician does not mean discovering numbers everywhere and enjoying only issues with strong mathematical connotations. The essential relation is therefore presumed to be found on another level.

           It is noticeable that the above-mentioned affinity isn’t reciprocated. Musicians do not usually show the same mathematics and mathematicians for music. One therefore must suppose that the decisive aspect cannot lie in arithmetic, the part of mathematics people sometimes consider to be in fact the whole subject. It is probably more the area of mathematical thinking, mind setting and problem solving which creates these connections. The sensations in solving the mathematical problem seemed to be similar to those appearing when performing the musical work.

          Interesting evidence for this idea has been presented by Henle who compared the history of music with the history of mathematics based on the following three arguments:

1. Mathematics has many of the characteristics of an art

2. Viewed as an art, it is possible to identify artistic periods in mathematics: renaissance, baroque, classical and romantic.

3. These periods coincide nicely and share many characteristics with the corresponding musical epochs, but are significantly different from those of painting and literature.

          Following the idea of the “C” implementation, it may be used in defense systems to encode useful bits of information in terms of musical sequences by defining a fixed set of rules to do so.

         Whatever, links between music and mathematics exist, both of them are obviously still very different disciplines, and one should not try to impose one on the other. It would be wrong to attempt explaining all the shapes of music by mathematical means as well as there would be no sense of studying mathematics only from musicological point of view.

         However, it would be enriching if this relationships were introduced into mathematical education in order to release mathematics from is often too serious connotations. It is important to show people that mathematics, in one way, is as much as art as it is a science. This probably would alter its common perception, and people would understand better its essence and its universality. This task, however, will certainly not be completed by the end of this century.