Why lower notes appear to have higher bandwidth?

[jackoo]

Hi there!

I'm not sure if this is the best forum section, but here it goes:

When I play a sine patch on a synth and look at some frequency analyzer plugin (e.g. SPAN) I can clearly see graphically that - the higher the note, the more pure the sine appears to be. Higher frequency sine waves appear to have increasingly narrower spectrum spread.

While going down do A1 (110 Hz) i see a frequency 'blob' from 75 Hz to 144 Hz (measuring at -72 dB).

I thought a sine is a single / pure frequency. So is the sound produced by the synth wrong, or is this a limitation on the algorithm of the spectrum analyzer?

Also I noticed something very strange today. I seem to have some trouble recognizing the intervals on bass notes. For instance, if I play A1 (110 Hz) and B1 (123.5 Hz) I hear something closer to a minor triad rather than a whole tone. Is this normal or are my ears going to fail soon?
It seems my ears are wrong, because when I play in octaves, a dissonance clearly goes away when the notes are correct.

[DaveClark]

Hi jackoo,

Musical notes are not typically eigenfrequencies of a discrete Fourier analysis. So whenever you do a DFT (FFT) the pure sine wave you mentioned gets analyzed or resolved into multiple component sine waves (neglecting phase for the moment here), none of which are typically the frequency of the original wave. You are also probably looking at a plot with a log-frequency x-axis. That will make it appear that the higher frequencies are not analyzed or resolved into multiple frequencies, but typically they must be, just like the lower ones. Exaclty what happens for any particular note depends on the sample rate and the block size. Depending on how big of a block size you are using, you may also see effects from the different windowing functions. One should always keep in mind that any particular digital Fourier analysis is going to show the results of a particular basis set of functions.

Regards,
Dave Clark

[jackoo]

Thank you for your reply.
How could I miss the log X-axis....

I'm a little embarrassed

On the other hand, does anyone know if it is normal for human hearing to have less precision in pitch detection at lower frequencies (100-ish Hz)?

[Chaotikmind]

The human hear is an resonator, on the whole length of the cochlea you have hair like structure , the ciliated cells, i'm not sure if the density varies , but if that's not the case we should have higher resolution in the low frequency,
I think that's the case, since when we loose hearing in the higher frequency first with time.

[JCJR]

There has been much psychoacoustic research. One good seed search phrase for pitch/frequency discrimination is "critical bandwidth".
https://en.wikipedia.org/wiki/Auditory_ ... _bandwidth

There are several newer alternate measures to critical bandwidth. Just browsing articles and following links can be informative.

Re cochlea, last I heard unless scientific opinion has changed (which happens rather routinely), the wide funnel opening detects highs and the narrow endpoint detects lows. I cant recall anymore but possibly the cilia are all about the same but each cilia acoustic location in the conical tube determines one of the "first layers" of frequency selectivity. As with vision there are multiple layers of neural acoustic processing.

[cthonophonic]

Here is one study that shows the ability to distinguish pitches drops off dramatically at lower frequencies:

https://www.sciencedirect.com/topics/en ... difference

Look at Table 3.4, where you can see that for a 30 or 60 Hz tone, the minimal detectable change in pitch is somewhere around a full semitone (slightly better for louder signals, worse for quieter ones), while it improves to about 3-5 cents for higher frequencies.

I seem to recall (from a class some decades ago) that this is not only a physiological/psychoacoustic phenomenon, but also the result of an effect akin to the Uncertainty Principle. The shorter the duration or the lower the pitch of a vibration, the less possible it becomes to specify a frequency mathematically. When the period of a waveform is longer (i.e. the frequency is lower), there may simply not be enough information in the signal to distinguish a tone. Naturally, this applies to tones of limited duration, not long sustained drones, but since most musical notes are finite, this would seem to apply to music.

[jackoo]

Thank you, very interesting read!
So my ears may not be damaged if I recognize a minor triad instead of a semitone, during a rather fast sequence of low frequency notes (near 100 Hz range).

But this sometimes gives me the impression that the bass instrument is not tuned properly.

It might also be the result of my brain making me think I heard what I would expect to hear (I am used to hearing that note succession having a minor triad instead of a semitone).

[JCJR]

Pitch perception gets weirder at high spl, sound pressure level. Loud stuff is typically perceived as flatter, lower pitch. One more reason to monitor at conservative spl.

Long ago I would particularly notice if I accidentally let rhodes piano get too loud on stage monitors or phones. With the rhodes "pure tone" attack then decay at high volume almost sounded like whammy bar pitch bend.

[garrinm]

Expanding on what Dave Clark mentioned, I'd also point you to this page to understand why a pure tone doesn't show up as a single peak in an FFT plot:

https://en.wikipedia.org/wiki/Window_function

Mathematically, a Fourier Transform is defined for an infinite signal (has no beginning or end). When you play a pure sine tone that starts and then ends, mathematically that is equivalent to a signal that was 0, then becomes a sine wave, then goes back to 0. And that's what the Fourier Transform will see. The trouble is that a signal that goes from 0 to a pure sine tone actually uses frequencies other than the pure tone to create that transition (sort of explained here https://en.wikipedia.org/wiki/Discrete_ ... _transform).

So the graph you're seeing is showing you the frequency of your pure tone, plus the frequencies needed to get the signal to start and end.

[jackoo]

Hi, thanks for the reply.
At some point in life (think 20 years ago) I understood rather well the concept of windowing.

While I understand that you have to use windowing because of the math you described involving starts and stops, your last statement has to be a little incomplete.

If it were purely like that (if your statement were 100% correct) , I would expect when starting a note to see some bandwidth as you described, then the spectrum would go to a pure frequency because now I'm holding down the key and there is no start and stop. And when I release the key, I would see an increasing bandwidth again.

But it's ok. I'll revisit my Fourier theory!

[BertKoor]

I thought it was quite simple: if you'd FFT with 4096 bins a signal from 20 to 20.000 Hz, then each bin is 19.980 / 4096 = 4.88 Hz. That is the precision you can measure: roughly 5 Hz. The difference in pitch between 100 Hz and 105 Hz (85 cts) is of course much bigger than the difference between 10.000 and 10.005 Hz (0.87 cts)

[Oden]

Fourier transform decomposes a signal of certain length into equally spaced sinusoids (or complex exponentials). We get S(t)=Sum_n a* sinusoid(n*t), where a represents the amplitude.

So if the windows is one second we get sinusoids of length one second, half a second, third a second, fourth a second and so on, in other words 1 hz, 2 hz, 3 hz... And the same in general.

This actually is equally spaced, but our preception is logarithmic and so the plots are usually scaled so that roughly the distance between 100 and 1000 and 1000 and 10000 is the same on the plot. But now the bin at 100 hz looks 10 bigger than at 1000.

FFT does not work like our ears, so that is a different animal. The objective of the auditory system is not to decompose signal so there is no requirement for any spacing or in fact any anything. It does roughly appear that the hearing works like many band pass filters set up in parallel, which is why FFT works (displays what we would expect) fine for short signals.

[DaveClark]

... then the spectrum would go to a pure frequency ...

Hi jackoo,

The played note itself will ideally go to a pure frequency, but even so, that frequency will not generally be an eigenfrequency of the analysis. The other "bins" may have values that are very small and difficult to see on a plot or graph, but they will not generally be zero. (Not only that, but the windowing of every block, including any abrupt cut made by rectangular windowing, introduces other frequencies in the same manner that Garrin mentioned.)

It is a common mistake to think that something that almost matches the eigenfrequencies of the analysis will be put into the closest "bin." That's why I object to the term "bin" and always try to remember put it in quotes. A musical note is generally not like a potato that gets put into one bin (without quotes) or another.

If you did manage to perform an analysis that used a musical note of a scale as an eigenfrequency of the analysis, all the other notes of the scale would not be because they are not harmonics of each other. Only that note and its higher octaves would be in single "bins."

Regards,
Dave Clark

[Oden]

Expanding on what Dave Clark mentioned, I'd also point you to this page to understand why a pure tone doesn't show up as a single peak in an FFT plot:

https://en.wikipedia.org/wiki/Window_function

Mathematically, a Fourier Transform is defined for an infinite signal (has no beginning or end). When you play a pure sine tone that starts and then ends, mathematically that is equivalent to a signal that was 0, then becomes a sine wave, then goes back to 0. And that's what the Fourier Transform will see. The trouble is that a signal that goes from 0 to a pure sine tone actually uses frequencies other than the pure tone to create that transition (sort of explained here https://en.wikipedia.org/wiki/Discrete_ ... _transform).

So the graph you're seeing is showing you the frequency of your pure tone, plus the frequencies needed to get the signal to start and end.

The analyzers that I have used do not even attempt to approximate Fourier transform, but instead compute the discrete equivalent of Fourier series. So what you actually get is a finite sinusoid for which the end points may not be the same. This is not a sinusoid of any integer size so it is a combination of multiple sinusoids.

Now it should be quite intuitive why windowing is applied to smooth the discontinuity. However, this is NOT why the bins appear larger at low frequencies. The same applies to any frequencies.

[Oden]

... then the spectrum would go to a pure frequency ...

Hi jackoo,

The played note itself will ideally go to a pure frequency, but even so, that frequency will not generally be an eigenfrequency of the analysis. The other "bins" may have values that are very small and difficult to see on a plot or graph, but they will not generally be zero. (Not only that, but the windowing of every block, including any abrupt cut made by rectangular windowing, introduces other frequencies in the same manner that Garrin mentioned.)

It is a common mistake to think that something that almost matches the eigenfrequencies of the analysis will be put into the closest "bin." That's why object to the term "bin" and always try to remember put it in quotes. A musical note is generally not like a potato that gets put into one bin (without quotes) or another.

If you did manage to perform an analysis that used a musical note of a scale as an eigenfrequency of the analysis, all the other notes of the scale would not be because they are not harmonics of each other. Only that note and its higher octaves would be in single "bins."

Regards,
Dave Clark

Índeed, and most instruments also have a drift and/or vibrato. Then the OP actually becomes very relevant. In the low frequencies the vibrato takes less bins than in the high frequencies. But this means the relative amplitude of the tone with vibrato spreads across more bins in the highs. Therefore on the FFT, tones with vibrato appear to have less high frequency content than in reality.

One big difference in how we hear and how FFT "sees".