Questions concerning combined/resultant/difference tones

Factual Questions
1

I’m working on a new composition and I want to employ the use of combined/resultant/difference tones. To be clear, my intent is not to actually try and create them audibly (it’s my understanding that this is difficult to do in practice and requires incredible precision on the part of the performer as well as the orchestrator). However, I want to understand how to properly calculate them and use the harmonic relationships to form my chord voicings.

Unfortunately, I’m having difficulty finding a reliable resource to use for this. I can find a chart of frequencies for all chromatic pitches within a well-tempered spectrum. However, I can’t find any such reference for just intonation. I understand that this is because the tones are reliant on the fundamental, but it would still be a very useful reference to me to see a chart based on C0 or G3, for instance.

I’ve read that adding the frequencies of two pitches will give me the resultant tone and subtracting them will yield the difference tone. Is this correct?

If I have three fundamental tones do I add all three or should I only work with two of the tones at once (for D-F-A, dealing with D-F, D-A, and F-A separately)? The latter appears to be how Tartini handled them, but I’ve also read that his math wasn’t entirely correct.

I guess I have more questions, but I’ll keep quiet for now since I’m not even sure if there’s someone in the teeming millions with the expertise on this subject.

2

I do not understand some of the terms you mention (e.g., I have not heard “resultant tone”).

Two tones with pitches sufficiently close together will be perceived as a single pitch with a frequency that is the average of the two tones, pulsating at a frequency (beats) that is the difference of the two tones. Once the tones get farther apart this effect isn’t noticeable. It seems to happen before you get a half-step apart so has limited practical use in music, except for tuning.

I don’t know if that is in the same vein as what you are trying to do musically, and I am not familiar with Tartini.

3

I don’t know if this will help or not, but in the world of electronics mixing two frequencies yields the sum of the two frequencies, the difference of the two frequencies and the two original frequencies. The process is called heterodyning.
There is some info here
https://secure.wikimedia.org/wikipedia/en/wiki/Heterodyne

4

Here is a chart of frequencies for a just intonation of the C diatonic scale anchored on A=440 Hz.



  Hz
C 1/1  264.00
D 9/8  297.00
E 5/4  330.00
F 4/3  352.00
G 3/2  396.00
A 5/3  440.00
B 15/8 495.00
C 2/1  528.00

I got the ratios from the Wikipedia article and calculated the frequencies. I believe there is not a single just intonation system, however.

5

People may argue about technical differences, but essentially this is what I’m talking about.

6

Neat. This is basically the equivalent of combination tone theory. It’s funny how the Wikipedia page says that this was discovered in 1901 even though Tartini knew about summation and difference tones in the 18th Century.

7

[quote=“CookingWithGas, post:4, topic:556513”]

Thank you! This has already been very useful to me.

Any other informative minds out there?

8

I know in organ construction, the effect of simultaneously sounding two notes an octave apart will produce a resultant an octave below the lower of the two. E.g., on a smaller installation, you can get the effect of a 16-foot pipe by having a stop that simultaneously plays the 8-foot and 4-foot registers of a given voice, or on larger ones, a 32-foot resultant can be achieved by simultaneous use of 16-foot and 8-foot registers. Better theoreticians than I will need to explain why this happens, though.

9

I just found this great reference: