beats & harmonics

[mattrix]

What are you hearing?
I'm totally confused, but hopefully you will see where I'm going wrong after reading this :).

According to some reference if you play a perfect interval, the combination of the harmonics of the two tones reinforce each other as if they were the harmonic serries of a new tone. eg 440 with 550, then all the harmonics are those of (440)/4=(550)/5=110. But you dont hear this 110hz tone but the lower of the original tones (440hz), right? (* see below)

If the intervals are not perfect then you will hear a tone which "beats". I'm not sure where "beats" come from as:

1.Some references claim that beats are produced in the air through addition, resulting in a tone half way between the two original tones, with an envelope equal to the difference of the two tones.

2.Others claim that beats are produced in the ear by replacing the original tones with two new tones through multiplication. The new tones being the sum and difference of the original tones.

I'm not sure which of these, or if both are correct, but either way they both result in "beats" equal to the difference.

This is fine if you have "pure" tones, but what if you have real tones that are rich in harmonics.

Each harmonic of one tone will produce beats with each harmonic of the other tone, resulting in a whole range of beats. And presumably the harmonics of a tone produce beats with the other harmonics of that tone, but these "beats" will all be harmonics of the tone anyway.

Can we hear these individual "beats", or can we do the same thing as in the first paragraph of this post??. ie find the "base" that would make all the harmonics, harmonics of this "base"?. Then all the beats would also be harmonics of this "base". ie find the highest common factor of the tones. eg 440 and 548 have a HCF of 4, so our "envelope" is 4 beats per second, right?

* From above then, would we hear a tone with an "envelope" of 110 hz?

But what is the tone inside the "envelope"? is it half way between the original fundamentals, or is it the sum of the original fundamentals or something else?.

matt

(ps separation of the notes may also be a factor here, but this should at least apply when they are close?)

[ttw]

Beats come from superposition of tones. It's the same thing as the addition formulae for Sin and Cos in trigonometry. For example, a tone at 440 cps and a tone at 441 cps will form a beat at 1 cps. The beats are actually there in the air.

How beats are perceived is another question though. Fast enough beats, for example beats between 440 cps and 220 cps will occur at 110 cps which is heard as a tone itself (although the combination will more likely be heard as an octave combination.) One can use two slightly mis-tuned organ pipes (and some baffling to suppress the high notes) to make a 64' stop out of a 32' and 16' stop.

Google "combination tones."

[mattrix]

Thanks TTW,
So when we hear beats we take the difference, but when we hear a tone we take half the difference? What about the other term using the sin/cos equations, shouldnt we hear (220 +440)/2 = 330?

[JonR]

Thanks TTW,
So when we hear beats we take the difference, but when we hear a tone we take half the difference? What about the other term using the sin/cos equations, shouldnt we hear (220 +440)/2 = 330?I'm not sure tts is right about hypothetical beats of 110 from a combination of 440 and 220. IMO they would be 220, the difference.
Tones of 440 and 330 would produce 110.
The point (I think) is that beats any faster than around 20 cps will be heard as a pitch, because around that frequency beats blur into a smooth tone.
The lowest A on a piano is 27.5 cps, tho some organs (AFAIK) go down to a C below that (16.35): such a pitch would be more felt than heard, IMO.

As you seem to be guessing, actual musical pitches (as opposed to synthesized single frequencies) are so complex that this kind of analysis gets pointlessly complicated. Eg, every note of a chord has its own harmonic series, and (in equal temperament) each one is out of tune with the harmonic series of the root anyway.
Eg, a pure ("just") A major triad in root position would be 440-550-660 (or multiples or fractions of those figures). With absolutely pure synthesized tones, with no harmonics of their own, I guess these 3 would produce a virtual - but audible - difference tone of 110 (A). This is why the major triad sounds so "right" as a harmonic unit - and also why A is the root, regardless of what order the chord tones go in. (A 2nd inversion triad of 330-440-550 still "points" to a root of 110, as they all represent harmonics of 110.)

However, in equal temperament, these frequencies are 440-554-659.2. We are so used to the complex sounds of real musical instruments (combining non-harmonic partials and inharmonicity as well as pure harmonics), that the discrepancies don't bother most people, in most situations.
But if we play that chord on a guitar with a lot of distortion, the overtones are enhanced, and the normally negligible clash between the 5th harmonics of the root (440 x 5) and the octave harmonics of the major 3rd ( (554 x 4) start to become apparent; which is why rock guitarists use power chords in preference to major triads. The discrepancy of the 5th remains negligible (2 cents from pure), while the equal tempered major 3rd (14 cents sharp) becomes a real issue.

However, I'm neither a mathematician nor an acoustics expert. Try some of the following:

http://en.wikipedia.org/wiki/Combination_tone
http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-hearing-difference.htm
http://www.patmissin.com/ffaq/q26.html
http://www.lifesci.sussex.ac.uk/home/Chris_Darwin/Perception/Lecture_Notes/Hearing4/hearing4.html#RTFToC2

[mattrix]

Thanks Jon,
That bit about the chords explains a lot. :)
I have heard that organs can be designed to sound as if they had longer pipes than they really have, but the reading suggests this effect is all in your head and hense subjective.
And somewhat more complicated than simply math.

[JonR]

Thanks Jon,
That bit about the chords explains a lot. :)
I have heard that organs can be designed to sound as if they had longer pipes than they really have, but the reading suggests this effect is all in your head and hense subjective.
And somewhat more complicated than simply math.Talking of "organ pipes", how about the human vocal kind!
Check this out - Huun Huur Tu and their Tuvan throat singing, in which a high flute-like tone is produced along with a deep throat tone:
http://www.youtube.com/watch?v=RxK4pQgVvfg&feature=related
(don't try this at home; unless your home is a yurt in Mongolia, perhaps...)

[AsturiasFan]

http://people.hofstra.edu/steven_r_costenoble/graf/graf.html (http://people.hofstra.edu/steven_r_costenoble/graf/graf.html)
I found the above site where you can graph functions like

(1) sin(110x) + sin(220x)
(2) sin(110x) + sin(111x)


My understanding is that (1) would be the graph for a note played with its octave, while (2) would be a note played with a nearby dissonant note.

Do you agree? If so, can you see beats from the graph itself? How do you tell the dissonance by looking at the combined graph? Anyone have any examples or web references?

[Actually I don't know if it takes a physicist to answer these questions, so if they go unanswered that will be ok]

[JonR]

http://people.hofstra.edu/steven_r_costenoble/graf/graf.html (http://people.hofstra.edu/steven_r_costenoble/graf/graf.html)
I found the above site where you can graph functions like

(1) sin(110x) + sin(220x)
(2) sin(110x) + sin(111x)


My understanding is that (1) would be the graph for a note played with its octave, while (2) would be a note played with a nearby dissonant note.

Do you agree? If so, can you see beats from the graph itself? How do you tell the dissonance by looking at the combined graph? Anyone have any examples or web references?

[Actually I don't know if it takes a physicist to answer these questions, so if they go unanswered that will be ok]I don't know how to work that graph site, but I can tell you you would hear beats at 1 Hz from combined pitches of 110 and 111.
Beats are due to the waves moving in and out of phase with each other. Once per second the positive values of those waves coincide, which doubles the amplitude and therefore the volume. In between those moments, positive peaks in one coincide with negative in the other, so they subtract. If the waves were pure sine waves of equal volume, the sound would cancel out entirely at those moments. With real musical sounds (with overtones and other imperfections) the sound just throbs between two different volume levels.
IOW, the two waves together form a 3rd wave with a frequency of 1 Hz, which modulates the other two.