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{{No footnotes|date=May 2011}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


'''Time–frequency analysis for music signals''' is one of the applications of [[time–frequency analysis]]. Musical sound can be more complicated than human vocal sound, occupying a wider band of frequency. Music signals are time-varying signals; while the classic Fourier transform is not sufficient to analyze them, time–frequency analysis is an efficient tool for such use.  Time–frequency analysis is extended from the classic Fourier approach.  [[Short-time Fourier transform]] (STFT), [[Gabor transform]] (GT) and [[Wigner distribution function]] (WDF) are famous time–frequency methods, useful for analyzing music signals such as notes played on a piano, a flute or a guitar.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Knowledge about music signal==
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Music is a type of sound that has some stable frequencies in a time period. Music can be produced by several methods. For example, the sound of a piano is produced by striking [[Strings (music)|strings]], and the sound of a violin is produced by [[Bow (music)|bowing]]. All musical sounds have their [[fundamental frequency]] and overtones. Fundamental frequency is the lowest frequency in harmonic series. In a periodic signal, the fundamental frequency is the inverse of the period length. Overtones are integer multiples of the fundamental frequency.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:{| class="wikitable" border="1"
<!--'''PNG''' (currently default in production)
|+Table. 1 the fundamental frequency and overtone
:<math forcemathmode="png">E=mc^2</math>
|-
!Frequency
!Order
!
!
|-
|''f'' = 440&nbsp;Hz
|''N'' = 1
|Fundamental frequency
|1st harmonic
|-
|f = 880&nbsp;Hz
|''N'' = 2
|1st overtone
|2nd harmonic
|-
|''f'' = 1320&nbsp;Hz
|''N'' = 3
|2nd overtone
|3rd harmonic
|-
|''f'' = 1760&nbsp;Hz
|''N'' = 4
|3rd overtone
|4th harmonic
|}


In [[musical theory]], pitch represents the perceived fundamental frequency of a sound. However the actual fundamental frequency may differ from the perceived fundamental frequency because of overtones.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Short-time Fourier transform==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


[[File:Chord.jpg|thumb|Fig.1 Waveform of the audio file "Chord.wav"{{where|date=October 2012}}]]
==Demos==


[[File:Garbor of Chord.png|thumb|Fig.2 Gabor transform of "Chord.wav"]]
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


[[File:Spectrogram of Chord.jpg|thumb|Fig. 3 Spectrogram of "Chord.wav"]]


===Continuous STFT===
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


Short-time Fourier transform is a basic type of time–frequency analysis. If there is a continuous signal ''x''(''t''), we can compute the short-time Fourier transform by
==Test pages ==
:<math> \mathbf{STFT} \left \{ x(t) \right \} \equiv X(t, f) = \int_{-\infty}^{\infty} x(\tau) w(t-\tau) e^{-j 2 \pi f \tau} \, d \tau </math>
where ''w''(''t'') is a [[window function]]. When the ''w''(''t'') is a rectangular function, the transform is called Rec-STFT. When the ''w''(''t'') is a Gaussian function, the transform is called [[Gabor transform]].


===Discrete STFT===
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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However, normally the musical signal we have is not a continuous signal. It is sampled in a sampling frequency. Therefore, we can’t use the formula to compute the Rec-short-time Fourier transform. We change the original form to
*[[Inputtypes|Inputtypes (private Wikis only)]]
:<math> X(n \, \Delta t,m \, \Delta f) = \sum_{p=n-Q}^{n+Q} x(p \, \Delta t) e^{-j 2 \pi p m \, \Delta t \, \Delta f} \, \Delta t</math>
*[[Url2Image|Url2Image (private Wikis only)]]
Let <math> t = n \, \Delta t </math>, <math>f = m \, \Delta f</math>, <math>\tau = p \, \Delta t </math> and <math> B = Q \, \Delta t</math>. There are some constraints of discrete short-time Fourier transform:
==Bug reporting==
*<math>\Delta t \, \Delta f = \frac{1}{N},</math> where ''N'' is an integer.
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
*<math>N \ge 2Q+1</math>
*<math>\Delta < \frac{1}{2f_\max}</math> , where <math>f_\max</math> is the highest frequency in the signal.
 
==STFT example==
 
Fig.1 shows the waveform of a piano music audio file with 44100&nbsp;Hz sampling frequency. And Fig.2 shows the result of short-time Fourier transform (we use Gabor transform here) of the audio file. We can see from the time–frequency plot, from ''t''&nbsp;=&nbsp;0 to 0.5 second, there is a chord with three notes, and the chord changed at ''t''&nbsp;=&nbsp;0.5, and then changed again at&nbsp;''t''&nbsp;=&nbsp;1. The fundamental frequency of each note in each chord is show in the time–frequency plot.
 
==Spectrogram==
 
Figure 3 shows the [[spectrogram]] of the audio file shows in Figure 1. Spectrogram is the square of STFT, time-varying spectral representation. The spectrogram of a signal ''s''(''t'') can be estimated by computing the squared [[magnitude (mathematics)|magnitude]] of the STFT of the signal ''s''(''t''), as shown below:
 
: <math> \mathbf {spectrogram} (t,f) = \left| \mathbf{STFT} (t,f) \right|^2 </math>
 
Although the spectrogram is profoundly useful, it still has one drawback. It displays frequencies on a uniform scale. However, musical scales are based on a logarithmic scale for frequencies. Therefore, we should describe the frequency in logarithmic scale related to human hearing.
 
==Wigner distribution function==
 
The [[Wigner distribution function]] can also be used to analyze music signal.  The advantage of Wigner distribution function is the high clarity. However, it needs high calculation and has cross-term problem, so it's more suitable to analyze signal without more than one frequency at the same time.
 
===Formula===
 
The Wigner distribution function <math>W_x(t,f)</math> is:
 
:<math> \mathbf W_x(t,f) = \int_ {-\infty}^\infty x(t+\tau/2)x^*(t-\tau/2) e^{-j2\pi\tau\,f} \,d \tau, </math>
 
where ''x''(''t'') is the signal, and ''x''*(''t'') is the conjugate of the signal.
 
==See also==
 
*[[Musical acoustics]]
*[[Harmonic pitch class profiles]] (HPCP)
 
==Sources==
 
* Joan Serra, Emilia Gomez, Perfecto Herrera, and [[Xavier Serra]], "Chroma Binary Similarity and Local Alignment Applied to Cover Song Identification," August, 2008
* William J. Pielemeier, Gregory H. Wakefield, and Mary H. Simoni, "Time–frequency Analysis of Musical Signals," September,1996
* Jeremy F. Alm and James S. Walker, "Time–Frequency Analysis of Musical Instruments," 2002
* Monika Dorfler, "What Time–Frequency Analysis Can Do To Music Signals," April,2004
* EnShuo Tsau, Namgook Cho and C.-C. Jay Kuo, "Fundamental Frequency Estimation For Music Signals with Modified [[Hilbert–Huang transform]]" IEEE International Conference on Multimedia and Expo, 2009.
 
{{DEFAULTSORT:Time-frequency analysis for music signals}}
[[Category:Musical analysis]]
[[Category:Time–frequency analysis|Musical signal]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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