The amplitudes and time relationship of harmonic number

 When a musical note is played, apart from the fundamental frequency, there are overtones above are generated. On a classical string, the harmonic number and time are related. 

1.1

A as the amplitude of the frequency. k is the harmonic number. t is time dominant.

This also implies that a frequency takes time to vibrate, ringing a note. As time pass, the sound fades out. For any fundamental or overtone,

1.2
For the intensity of the fundamental, it provides about 50% of the power of a note. The next overtone provides about half of the previous etc. resembling as an exponential decay.

For a complex soundwave that we hear daily, a tone for an instrument consists of a fundamental wave and multiple overtones. For k is any individual waves, a note can be described as,
1.3

Using cosine function 
The cosine function can also be used for expressing a wave by adding the variable e to shift the phase to achieve the same waveform as what the sine function did.
1.4

Fourier Series
Since a wave can be expressed in sine or cosine terms, the Fourier series brings these two terms together and can express any periodic wave. w=2*pi/t, n = any positive integer
1.5a
1.5b
dt is the integral of the time domain, the area between the function (the waveform) and time axis.

Fourier Transform
An alternative and better way to express a waveform are using the imaginary number i = sqrt(-1). It can deal with a waveform which is not period or continuous.
1.6

When an imaginary number is used, a complex number is formed. There are a real part and an imaginary part. With Euler's identity, 1.6 can be expressed in sine and cosine functions, back to where we started at the beginning of this article. We can substitute .
1.7





Comments

Post a Comment

Popular posts from this blog

Post-spectralism influence on tuning

According to Helmholtz (2011), a musical note is due to the fundamental frequency and the quality of a musical sound is due to the presence of upper partials. For traditional western music, pitch perception is mainly fundamental, rather than the frequencies in the spectrum which Teodorescu-Ciocanea (2003) describes as a “note proper” tradition. The spectralism composers shifted the domain from absolute pitches, intervals and chords to the spectrum. Spectral music is a complex approach to timbre, Teodorescu-Ciocanea (2003) classified 9 tendencies and their possibilities in spectral music. To conclude the techniques, tuning the frequencies according to the harmonic series is the major approach; while working with noise, generated digitally and/or extended techniques on instruments, is a sort of timbral dissonance which dilutes the pitch perception of a musical sound. Teodorescu-Ciocanea describes inharmonicity as the combination of frequencies that are not integral multiples of the funda
Read more

Haba's field shifting - and the expansion of traditional functional harmony

Image
Hába characterizes the quarter-tone gamut as separated into two fields, one half comprising the twelve conventional pitches, and the other half, shifting all the twelve notes by a quarter-tone, the twelve quarter-tone pitches. In harmony, Passages occur in which chords composed purely of conventional pitches alternate with chords composed purely of quarter-tone pitches. Two fields are not always rejecting each other when he equally divided a perfect 4th (5 semitones)into 2 equal intervals, equal 2.5 for each, the 0.5 quartertones always step into the other field. Sometimes, this fourth interval can be further divided into 2.5+2+0.5 or other permutations. This kind of splitting a fourth can be traced to ancient greek tetrachord, a scale of four notes, bounded by the interval of a perfect fourth. A chord across both fields, the chord spell like a flatter dominant 7th chord. the 7th harmonic on harmonic series is quite flat (-31cents). This chord combines both fields sounds better accordi
Read more

Temperament, Tuning, and Timbre -- the underrated trinity in music

 The trinity here is not about the Christianity faith, but similar to the idea of the Christian doctrine of it defines God as being one God existing in three coequal, the core and fundamental definition, temperament, tuning, and timbre (TTT) are the basic building blocks of music, yet, underrated and even neglected in our music education system. TTT correlates with each other. A note is played, it carries out a series of frequencies, the harmonic series. The series multiples from the fundamental frequency with a series of integers (1f 2f 3f 4f 5f 6f 7f 8f 9f 10f 11f 12f 13f...). The loudness of individual particles, and different patterns, creates different timbres. View the series in a different way, the combination of the integers creates intervals. The combination of the beginning of the series sounds more harmonious and vice versa; simple ratios sound pure and vice versa. The pure perfect 5th interval is based on the ratio of 2:3 (2nd and 3rd harmonics at the same time), Pytheogoro
Read more