Modelling relationship between Musical Intervals
Student name: Shreyansh Patel
Introduction ………………………………………………………………………………………. Pg.3
Sine Function of a Sound Wave …………………………………………………………. Pg.4
Consonance in Relation to the Octave ………………………………………………. Pg.6
Dissonance in Relation to Higher Harmonic Frequencies Within Diminished Fifth……………………………………………………………………… Pg.8
Conclusion ………………………………………………………………………………………… Pg.16
Bibliography ……………………………………………………………………………………… Pg.18
Background and Rationale
Music is a form of conveying sound through intervals and notes. The term ‘consonance’ represent pleasantness of the sound being conveyed through the notes and intervals, also referred to combination of notes which are in harmony with each other due to relation of their frequency. whereas the term ‘dissonance’ represent lack of harmony among musical notes. A slight change in notes being played at a time can transform listening experience from consonant to dissonant.
Sound waves of compression and rarefaction travel through waves of sound for human ear to perceive sound. Notes played or sung is apparent, when the human ear can decode the melody of the song. Different instruments sound different even if same chord is being played upon is due to the presence of higher harmonics. Higher harmonics are often indistinguishable pitches that are built for integer ratios above the single pitch we perceive. Fundamentals frequencies of two notes interact causing higher harmonic frequencies to interact.
Music plays a strong core of my life, specially singing and playing traditional instruments such as table, taal, dholak and sitar. I would always compare the tune and notes of each instrument and question what caused intervals to sound pleasant or unpleasant. Frequently through observations I would feel the sound change from fundamentals notes to higher frequencies as I play my instruments. Through applying mathematics to such model then only I realised that such music intervals are associated with quality of sound.
Through this paper investigating into perception of consonance and dissonance by taking frequencies of notes that are distance apart, as found in certain musical intervals, and representing them graphically to show the ways they interact with each other.
To deduce why the interval which in, western Music, is known as a diminished fifth, is so dissonant to the human ear, and why other intervals might be perceived as consonant.
Using mathematics, I will model a sound wave through sine function and figure the connection of the function’s shape and the pitch it I perceive. Also through the interval of an octave, I will represent the ways in which two sound waves of different frequencies interact with one another. Developing further, I will model an understanding of higher frequencies within the harmonic series of a note to explain the dissonance of the diminished fifth, identifying the frequencies within this series that interfere most severely with one another, and exploring this interference using a product-sum trigonometric identity.
Since function of a Sound Wave
Modelling sound graphically enables to explore the relationships between two intervals. An interval would be known as the similarity in two musical notes.
Sounds waves traveling through air are known as longitudinal waves which are found in oscillating regions of high and low air pressure within in relation to normal environment. Due to the increase in air pressure at a point in atmosphere, molecules naturally in such region travel along in the direction of the wave to restore normal air pressure; therefore, a force is exerted in the direction of the wave. Sounds waves are made up of repeated patterns known as periods enabling to be sketched on a graph. Taking the only movement of a sound wave at that fixed point to be shown on x-axis, and on y-axis, the changes in air pressure relative to atmospheric air pressure is to be showed. The graph will be constructed to represent the air pressure at this fixed point at a moment in time, where the horizontal axis (Y=0) is normal air pressure without the influence of sound. The basic form of such graph becomes a sine function, as seen below.
Y = Sinx4191001270000
The human ear distinguishes pitch – the degree of highness or lowness of a sound or musical note – by being sensitive to that frequency of a sound wave. The frequency, f, of a wave is the number of cycles (oscillations) that pass through a certain point in one second, and is measured in Hertz (Hz). The frequency of a wave is the reciprocal of its period, T, which is the time taken for a wave to go through one cycle, as indicated by the green arrow above. The period of the function sinx is 2Pi seconds, and resultantly the frequency is 1/2pi Hz (cycles/seconds). The amplitude – the absolute distance between the X axis and a peak of the function of sinX is 1, and represents the severity of the oscillating changes in air pressure caused by the sound wave and subsequently the volume perceived by the ear.
As the pitch of a note is related to the frequency of its sound wave and subsequently its period, it can be manipulated through sin function as period changes for to show different pitches. It can be done through manipulating x values of the function be placing a constant b in front of x, as shown below.
y = sin(bx)
The constant b dilates the function horizontally by a factor of its reciprocal, 1b. If b is increased, then the x values of the function of the will decrease. As the size of the period is dependent on the x values of the function. The period also decreases with the x values by the same factor, 1b, as shown below where 2? is the period of the unchanged function y = sinx, and T is the general symbol for the period of a wave.
T = 2?×1b=2?bFor example, if b was 2, then the period would half, becoming ? instead of 2?.
T = 2?2=?As the frequency of a sine function is the reciprocal of its period, 1T, then as b increases and x values decrease, the frequency of the function will increase. An alternative explanation is that frequency is the number of cycles that pass through a particular point in one second, and so if those cycles take place more quickly, then the frequency will increase. As such, the frequency increases by a factor of b, as shown below, where 12? is the frequency of the unchanged function y = sinx, and f is the general symbol for the frequency of a wave.
In the same situation, b being 2, the frequency would double, becoming 1? instead of 12?.
f=12?×2=1?Rearranging so that the constant b becomes the subject, we find the way by which the frequency of a certain pitch may be directly related to its since function.
2?×f=b2?×2?2?f=bThe equation of a sine function with a modifiable period may be now be written with b replaced with 2?f, as seen below.
y=sin2?fxThis enables to solve any frequency of any pitch and find it’s sine function which models the alternating regions of high and low air pressure that make up its sound wave.
Consonance in relation to the Octave
The most basic interval is known as the octave. Two notes are to be an octave apart when the frequency of one note is double the frequency of the other, or the frequency ratio of the two notes is 1: 2. An example would be, the note A4, which is found near the middle of a western Piano, has a frequency of 440 Hz, and the note an octave above it, known as A5, is double that frequency. At 880 Hz.
A4: A5 = 1: 2 440:880 = 1: 2
The octave is considered the most consonant interval. The equation derived in section II, I decided to investigate the graphical representations of the pitches that make up an octave, so graphing the two notes A4 and A5 with frequencies of 440 Hz and 880 Hz respectively, as shown below.
Blue: y=sin (2? × 880x)Red: y=sin (2? × 440x)
The graph above shows two functions that intersect at x axis regularly and in repeating pattern. This regular, repeating intersection explains the consonance of the interval of an octave. If we apply these functions back to the sound waves they represent, where the horizontal axis is air pressure without the influence of sound, then these two sound waves would seem to have little contact with another as they regularly repeat and intersect points while moving through air with normal atmospheric air pressure.
Similarly, if a sound wave is uncertainly travelling through high and low air pressure in the context of normal atmospheric air pressure and modelling a single sine function with such context, then the addition of two sine functions could be used to model the net changes in air pressure of when two sound waves occur at the same time. Thus, through graphing two waves in form of functions represents the interference between both graphs. The above graph of two notes A4 and A5 is a better representation of the difference that occur when two notes are played together, enabling to construct deeper insights why music intervals are perceived as consonant to the human ear.
y=sin (2? × 880x) +sin (2? × 440x)
The above graph is a representation of these two functions being a sine function with a regularly repeating waveform, such waveform could explain the consonance of the sound to the human ear.
Dissonance in Relation to Higher Harmonic Frequencies Within Diminished Fifth
Frequencies of Higher Harmonics
The taal, my first instrument I learned from a very young age. I was intrigued by the amount of pitches I produced at the same speed throughout while playing the instrument. Taal is loud instrument, controlled through handles and grip. As both the sides of Taal is banged uninterrupted, this send vibrations through air to produce high pitches of sound. I could always produce several frequencies of sound at once, all of them follow same patterns in terms of their frequencies and subsequent interactions with one another. As I became more aware and advanced in playing Taal, the term ‘Harmonic Series’ and its significance to my instrument, coming to understand Taal consisting range of series as my ability to control upon particular pitch was constructed through the fundamental pitch.
Harmonic Series is the series of frequencies an instrument produces when a note is played. A basic frequency of a certain note, the first harmonic, is its fundamental frequency. As for an example, the note A4 is most often referred to as having a precise fundamental frequency of 440Hz as it is played through an instrument. As frequencies raise above the fundamental frequency known as overtones which consist of frequencies integer ratios above the fundamental. These higher frequencies are produced because of the way in which a column of air in a wind instrument may vibrate in different ways, each of which is known as a mode of oscillation. Much like a string on an instrument vibrates in various modes with having different wavelengths, so does the column of air inside a wind instrument. As the mode of oscillation increases, the wavelength and subsequently the period of the wave halves. As such, the frequency doubles. The relationship between the mode of oscillation, or the harmonic of the wave, and its frequency may be presented as seen below, where fn is the frequency of the nth harmonic of the wave, and f1is the first harmonic, or its fundamental frequency.
fn=nf1If the note A4 is used to work out frequencies of its first, second, and third harmonics as follows, where the fundamental frequency (first harmonic) of the note is 440Hz. Higher harmonics do exist through instruments but are not relevant.
The above frequencies present that the second harmonic of the note A4 contrasts with the frequency of note A5, at 880Hz, leading into an explanation as to why in the interval of an octave is perceived as being so consonant. Using the equation derived in part 2, these frequencies may be graphed together as sine functions on the same axes, as seen below.