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Accepting beats an not fighting them makes them more discrete in the end, and they also goes well with the piano tone, and the shape it takes.

The consonances I hear/tune are an addition of slow beats (within the 2 octaves framework for instance.

I think it's the same thing: sound waves ride on the atmospheric pressure (the DC) and you can't hear beat cancellation on the telephone either.

If you're not convinced; I convinced myself by looking at an example. Consider tuning P4 A3D4 1bps wide and P5 A3E4 1bps narrow (equal beating). Ignore IH as it plays no role in this.

The P4 beats come from partials 4 and 3 which are at 880 and 881Hz. The P5 beats come from partials 3 and 2 which are at 660 and 659Hz. If we play A3D4E4 simultaneously we should be able to demonstrate beat cancellation by synthesizing just the aforementioned partials (880,881,659,660).

Here I put audio files of the individual beats of the P4 and P5 and the combined beats with different relative phases (0, 90 and 180 degrees). You can hear that when they are in phase you hear a stronger 1bps beat, if they are out of phase (180) the beats interleave and we effectively hear 2bps, and at 90 it's more complicated.

No cancellation can occur at any phase difference because beats are periodic variations in loudness of a carrier wave and there is no such thing as negative loudness and hence no cancellation, unlike periodic variations in pressure (deviation from atmospheric) which can be positive or negative.

Kees

Create two sine waves. One at 440 and one at 665. A horribly wide fifth. No beats. It's weird.

Indeed beats can not physically cancel, unlike pressure waves. The reason is that pressure (or rather deviation from atmospheric pressure) can be positive and negative and can add up to 0, whereas beats are fluctuations in energy, and energy is always positive and no physical cancellation is possible.

If there is any advantage of "equal beating" it will have to be a human perception thing.

Kees

That may be the case.

I was only trying to say that currently, I'm using ET via Marpurg, which tunes some intervals pure (specifically, it's the whole-tone scale from F#3 to E4, i.e. six notes in total). Any short-cut that reduces the number of purely tuned intervals, is helpful to me - even though it would still be cumbersome or "tinkering" to a professional.

Have you something in mind to resume the temperament from the whole tone scale F3 G3 A3 B3 C#3 D#4 F4 ?

Kees:

Thanks for your response and your patience! I am just not getting it. What about noise cancelling headsets? http://en.wikipedia.org/wiki/Noise-cancelling_headphones

I tried this with audacity - both 665 and 661 plus 440. It sure sounds like beating to me as I can hear periodic variations in loudness. You definitely can have beats that are not due to partials near the same frequency - just do the summation of the two sines and you'll see periodic loudness variations in the 440+661 case.

[edit] just throw graph sin(4.4*pi*x)+sin(6.61*pi*x) and graph sin(4.4*pi*x)+sin(6.60*pi*x) into google, zoom out horizontally and you'll see the amplitude modulation in the 440+661Hz case.
Paul.

I don't hear beats at low volume, only when I crank it up nonlinear distortion, either in my ear or in my speakers, kicks in.

The amplitude modulation you see is a visual effect only. If you plot the energy over say 50ms windows you'll see it is constant.
Unlike when you have a true beat when it oscillates at the beat rate.

Kees

My guess is that our brain creates the resultant of 661 and 440, for example, as 221, which will produce a harmonic at 442, which will beat at 2bps.

I don't hear beats at low volume, only when I crank it up nonlinear distortion, either in my ear or in my speakers, kicks in.

The amplitude modulation you see is a visual effect only. If you plot the energy over say 50ms windows you'll see it is constant.
Unlike when you have a true beat when it oscillates at the beat rate.

Kees

If someone comes up with a better word than "canceling effect" to describe that phenomenon, I will certainly consider using that term, whatever it may be.

I can hear them clearly at low volume. Two sines at 440Hz and 441Hz also have constant energy over time in the simple summation case surely

You could check that by playing the notes separately to each ear with headphones. If you still hear beats they must be made in the brain.

Kees

I seem to recall some years ago that someone told me that the "canceling" effect that I was hearing was not the same as what happens with noise cancellation systems. That is why I always use the word, "effect" when I talk about it. If it isn't truly a cancelation, then it is something else which mimics that effect.

When I tune the EBVT or ET via Marpurg, I always check my equally beating intervals by playing them simultaneously. If the two intervals are not quite equal, the slow beat between them emerges. When the two intervals, whether slowly or rapidly beating are truly equal, the beating sound becomes fainter, certainly not louder. The entire cluster of tones, whether a consonant, mildly dissonant or sharply disonant sound becomes quieter, not louder than if either interval is played alone. If the two intervals have very slow beats, the beat seems to vanish entirely.

If someone comes up with a better word than "canceling effect" to describe that phenomenon, I will certainly consider using that term, whatever it may be. I have long known that this is a way to take the "noise" out of a piano. Now, I am being told it will make it twice as noisy but I know that simply is not true.