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Adding Unison Tones

Don Gilmore

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Gentlemen:

I'm working on a device which involves shifting a harmonic partial on top of another one and I need to know how to determine the acoustic sum of the resulting two unison sine waves. I realize that the sum of two waves in phase has twice the amplitude and, if 180 deg. out-of-phase, interfere for an amplitude of zero, but I'm talking about actual volume in air. When two unison tones sound simultaneously the result is louder, but not twice as loud. I would think that it would be the RMS sum of the two separate RMS values of the waves, but does wave phase also come into play? It seems to me that it would, yet that would indicate that two given tones would have a random combined volume, depending on their phase difference.

Here is what I'm trying to do. I have two separate sine waves that are close together. One is at the correct frequency and one needs to be corrected and added to the other. Lets say the correct one is 100 Hz and the other one is 101 Hz. Rather than try to correct (shift frequency) and remix the bad tone, I'd like to determine its volume and increase the correct one by this amount to simulate the equivalent.

So, for example, if the 100-Hz tone is at 70 dB and the 101-Hz tone is at 62 dB, I want the equivalent of what the total volume would be if both tones were at 100 Hz. Then I would amplify the 100-Hz tone to this value and discard the 101-Hz tone in the output.

Incidentally, this is an analog circuit, not DSP. Thanks for any replies.

Don
 

RayDunzl

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When two unison tones sound simultaneously the result is louder, but not twice as loud.

6dB increase or double the voltage, when the two tones are in-phase.. Generally, 10dB is considered "twice as loud", a subjective impression.
 
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Don Gilmore

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Looking at your link (thank you), in the appendix at the end where it uses complex math, it would seem that my assumption of RMS is correct (neglecting the phase difference). Mathematically, it appears that the phase difference does affect the total volume, but you would think that every once in a while two tones would cancel or nearly cancel each other out in the real world, yet I can cite no examples. Hmmm...

Don
 

RayDunzl

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you would think that every once in a while two tones would cancel or nearly cancel each other out in the real world, yet I can cite no examples. Hmmm...

If you just want to look, use Audacity...

100 + 101Hz example: They cancel once per second, if the same amplitude

1574443771307.png


If combining different amplitudes, they don't fully cancel:

1574444005781.png
 
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Don Gilmore

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Well, I don't mean the two differing tones; I mean the two unison tones. The two different tones will obviously have a beat frequency of 1 Hz, like your graph shows.

Unison tones will not waver. They will remain steady in amplitude, but that amplitude will depend on the phase difference, which would be 2A for 0 deg. (double amplitude) and 0A for 180 deg. (cancelled, or silent).

Don
 

RayDunzl

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So, for example, if the 100-Hz tone is at 70 dB and the 101-Hz tone is at 62 dB, I want the equivalent of what the total volume would be if both tones were at 100 Hz. Then I would amplify the 100-Hz tone to this value and discard the 101-Hz tone in the output.

If they are the same phase:

http://www.sengpielaudio.com/calculator-coherentsources.htm

1574448254730.png


If they are not the same phase, use trig to calculate the maximum value of the resulting sine and convert to dB.
 

NTK

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Well, I don't mean the two differing tones; I mean the two unison tones. The two different tones will obviously have a beat frequency of 1 Hz, like your graph shows.

Unison tones will not waver. They will remain steady in amplitude, but that amplitude will depend on the phase difference, which would be 2A for 0 deg. (double amplitude) and 0A for 180 deg. (cancelled, or silent).

Don
The summation of two sinusoidal signals of the same frequency is related by the equation:

snip.JPG
 
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Don Gilmore

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According to Ray's site, the sum of the volumes is simply the logarithmic sum of the dB of the RMS of each wave. OK, this helps.

Apparently the phase difference gets lost in the shuffle.

Don
 

RayDunzl

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According to Ray's site, the sum of the volumes is simply the logarithmic sum of the dB of the RMS of each wave. OK, this helps.

Apparently the phase difference gets lost in the shuffle.

No, it doesn't.

Read above the calculation:

"Adding the level of two steady correlated (in phase) sources"

It adds "coherent" sources of different amplitudes.


http://www.sengpielaudio.com/calculator-coherentsources.htm

---

Incoherent can be added, but assumes the two waves are very different:

http://www.sengpielaudio.com/calculator-leveladding.htm
 
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Don Gilmore

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Well then, how do you add them in reality? Two real sources are almost never going to be exactly in phase. The phase will be random. I've never noticed any appreciable difference in the total volume between two tones played together several times in a row. The phase of each burst must have a different, random phase relationship, yet they all appear to have the same volume. Is there an averaging taking place?

Don
 

Blumlein 88

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So you are wanting to make two closely spaced tones of different frequency into one tone of higher volume to simulate the two tones? Can you describe in more detail how or when you wish to do this?

Tones within a few hertz of each other will cause obviously audible beat tones. Just try creating two tones in Audacity. 200 hz and 201 hz. You'll hear the beating at 1 hz. Even if you reduce one of the tones by 10 db relative to the other you get obvious beating. So I'm wondering what problem you are attempting to solve here?
 

RayDunzl

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Well then, how do you add them in reality? Two real sources are almost never going to be exactly in phase. The phase will be random. I've never noticed any appreciable difference in the total volume between two tones played together several times in a row. The phase of each burst must have a different, random phase relationship, yet they all appear to have the same volume. Is there an averaging taking place?

The sum of two tones of the same volume will add together with a result between +3 and +6 dB.

+3 for perfectly "incoherent" matching, +6 for perfectly "coherent". Differing levels of coherency will fall between and vary between those two extremes.

Example:

Create a noise track of -6dB level.
Copy it.
Add them together - coherent sum
Slide the copy of the noise track a little in time. It no longer perfectly matches the original track in timing/phase/instant amplitude/etc.
Add them together - incoherent sum

1574536863935.png


Displayed as the decibel value for each wave (not averaged, use your eyeball for that)

1574537024722.png
 
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Don Gilmore

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I have two input sine tones: one is correct and one is incorrect. I want to correct the bad one and mix it with the good one, as if they were both good all along. They come from two separate, independent channels. The output is a unison or single frequency; there are no beats.

I could carefully measure the frequency of the bad tone, shift it to correct it and then mix it with the good one, but this is a complicated circuit. If I can determine how loud the two tones would sound when played together, I can adjust an amplifier to boost the good wave to that level on the fly and not have to shift or mix anything.

Don
 
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