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Topic: Question to help understand "octave bands" (Read 3871 times) previous topic - next topic
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Question to help understand "octave bands"

Now from what I understand octaves are always twice as high as the previous frequency, so if you started on 32 hz, it would go up 64 hz, 128 hz and so on. What I thought this meant was that if you have a center frequency of 1000 hz, and a bandwidth of one octave, the high end of that band would be 1500 hz, and the low end would be 750 hz. Yet according to this chart that's not true (scroll down a little) https://courses.physics.illinois.edu/phys40...ctave_Bands.pdf

According to this the high end for 1000 hz is 1414.214 and the low end is 707.107.

Could someone help me understand?

Question to help understand "octave bands"

Reply #1 –
Quote from: peterh on
Could someone help me understand?


As the bandwidth gets higher, individual Hz are a smaller fractional difference.  So instead of defining it like you're thinking with the arithmetic mean (which would mean that relatively speaking the high limit was closer to the center than the low limit), they define it with the geometric mean.  This means that both the high and low limit are the same fractional bandwidth apart, and the overall fractional bandwidth is 70.7%.



Question to help understand "octave bands"

Reply #2 –
Wait so each of those bands in .71 of an octave wide? I thought they where each an octave wide. Or am I mixed up about the definition of fractional bandwidths.

Question to help understand "octave bands"

Reply #3 –
Imagine for a moment that you have bands that are two octaves apart. If you have one band centered at 1kHz, you would have another at 4kHz and 250Hz. The width of the 1kHz band would be from 500Hz to 2kHz. With these numbers it is easy to grasp the idea, each band is two octaves apart, and each band spans two octaves. With single octave bands, each band is an octave wide (2 * 707.107 = 1414.214). The square root of 1/2 is 0.707, that's where that number comes from. I'm not entirely sure what fractional bandwidth means exactly, but the lower and upper end of the octave band are each sqrt(1/2) octaves from the center, one is above, one is below, so with a sqrt(1/2) on the top and bottom of the fraction they cancel each other out and you end up with one octave as the resulting width.

Question to help understand "octave bands"

Reply #4 –
Quote from: peterh on
Wait so each of those bands in .71 of an octave wide? I thought they where each an octave wide. Or am I mixed up about the definition of fractional bandwidths.



Fractional bandwidth is bandwidth divided by center frequency.  Its also one over the Q factor.  Basically, its how wide a band is relative to its frequency.  In terms of filters, its the relative width that counts, not the absolute width in Hz. 

Thus each one of those bands is spaced one octave apart, has a fractional bandwidth of 1/sqrt(2) = .707 and a Q factor of sqrt(2) = 1.414.

Question to help understand "octave bands"

Reply #5 –
Thank you guys, extremely informative!

Just as an exercise so I understand the math right. Let's take the opposite of that problem (finding a center frequency given 2 end points). If I wanted to make a 2 octave band that stretched from 1414.214 (The bottom of the 2k band) to 5656.854 (the bottom of the 8k band), i'd make a 2 octave band with a center frequency of 2828.427, since an octave down would  be 1414.214 and an octave up would be 5656.854, right?

Question to help understand "octave bands"

Reply #6 –

To an engineer those numbers make perfect sense but they should be familiar to most. An octave has a 2:1 frequency spread. That 1414.214 is 1000 ( the center frequency ) times the square root of 2 and the 707 is 1000 divided by the square root of 2. The square root of 2 squared is 2 so 2 times 707 is 1414 Hz. He makes the math look 'mathematical' (difficult) but for 1 octave it's simply the square root of 2. Now that you know where the numbers come from you can calculate the spread for any center frequency that you wish. For 1/2 octave the number is the 4th root of 2. For 1/3 octave it's the 6th root of 2 and for the truly nerdy, the note to note (C - C# etc) the number is the 12th root of 2. Any scientific calculator can do this stuff.

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