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Reading over my class notes, I see that you can consider a spectrogram is being comprised of many "slices" (static spectra? Whatever that means) horizontally, making it so "time appears continuous".

This sounds very strange to me, and I can't get my brains around it. I don't think I understand what "slices" are or what it means for "time to seem continuous" in the spectrogram itself. Can someone help me come to terms with this?

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Everything about sound in the digital world presents discrete (not continuous) values which represent "what is happening in this (physically continuous) time span out there". It is impossible to compute amplitude at a given frequency and a single instant in time (for one, there is no such thing as an absolute "instant"). When you decompose a waveform into a graph of frequency and amplitude, you have to select a specific continuous chunk of the waveform, in order to ferret patterns of frequent amplitude changes, versus infrequent amplitude changes. A Fourier spectrum uses the concept of a "window" and throws away any information about when in time a given component occurred; instead, it gives you the amplitude at a given frequency, for the entire signal between time n and time n+j. This can be displayed, and gives you a very accurate representation of frequency and amplitude within that window. This is a spectral "slice" -- the frequency and amplitude values derived from the Fourier transform of the signal at some (small) interval.

A spectrogram is created by computing a frequency-and-amplitude display (which has no time component) for very many adjacent (partially overlapping) windows of time, and until we get 3-D view screens, amplitude information has to be degraded by translating it into scales of grey, so that you are effectively looking at a single spectrum edge-on.

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Good answers! A simpler way to look at it might be that frequencies only exist over time, never in an instant. So, if you analyze a waveform for frequencies, you have to decide over what period of time to analyze it. If you choose 10 milliseconds, you'll get an average strength for the various frequencies as they appear in those 10 milliseconds, but you don't know how strong each frequency component is for each millisecond. If you use 1 millisecond, you get much more accuracy timewise, but you will know fewer frequency components. Crazy to think that the human brain isolates frequencies in sound all the time, albeit not always accurately!

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There are three parts to this answer: why slice, how to slice, and ‘what does it mean for time to seem continuous’. You can also review first (wikipedia) if it’s easier.

First, we may want to slice because a spectrogram (e.g., in Praat) is a 3D figure on a 2D display. Humans aren’t great at reading 3D figures. If you slice that 3D figure (intensity by frequency by time), you get a 2D figure. For a spectral slice, that’s (intensity by frequency); slicing lets us set aside time for the moment, and focus on the intensity-frequency distribution.

Second, this kind of slice is made by ‘cutting’ the spectrogram at a particular point in time (i.e., on the x-axis). The metaphor may be clearer if you think of the spectrogram as a real 3D object, like little plastic mountain range. If you use a knife to cut a perfectly vertical slice of that mountain range, you can see the silhouette by looking at the end. That outline is the spectral slice: a simple plot of intensity (y-axis) vs. frequency (x-axis).

Third, why even talk about time being continuous or not? The answer is that a spectrogram really is a big stack of slices put together. Mathematically, that’s where you get it (Fourier transform). In order to convert a single complex sound into a spectral representation, you’re forced to cut time into discontinuous pieces and analyze each little slice. The length of this slice in time is the window length (Praat manual).

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