Draft:Beat (acoustics)/Helmholtz tables

Harmonic Matching
Here we verify that the equation for Hemholtz (amplitude) beats among harmonics of the two fundamental frequencies is correct. The two tables shown below list all the harmoincs of $$f_p$$ and $$f_q$$. The frequency of $$f_p$$ has been increased by $$1$$Hz.


 * Helmholtz beating is ordinary amplitude beating between higher harmonics of signals with two fundamental frequencies,
 * $$f_p=pf_0$$ and $$f_q=qf_0.$$


 * We use a pre-superscript, $$i\in \{1,2,3,...\}$$, to denote beats betweem the various harmonics, assuming that all harmonics exist:
 * $${}^i\!f_B= i\cdot\left| q\Delta f_p - p\Delta f_q \right|$$

Example 1: $$p=3 \text{ and } q=2 \text{ with } i=1. $$

The second harmoinc of $301$Hz is $602$Hz

The third harmonic of $200$Hz is $600$Hz

The (amplitude) beat frequency is:

$${}^2\!f_B=i\cdot q\cdot \Delta f_p=1\cdot 2\cdot 1= 2\text{ Hz}$$

Example 2: $$p=5 \text{ and } q=3 \text{ with } i=2. $$

The third harmoinc of $501$Hz is $1503$Hz

The fifth harmonic of $300$Hz is $1500$Hz

The (amplitude) beat frequency is:

$${}^2\!f_B=i\cdot q\cdot \Delta f_p=2\cdot 3\cdot 1= 6\text{ Hz}$$

Example 3: $$p=5 \text{ and } q=3 \text{ with } i=3. $$

The nineth harmoinc of $501$Hz is $4509$Hz

The fifteenth harmonic of $300$Hz is $4500$Hz

The (amplitude) beat frequency is:

$${}^2\!f_B=i\cdot q\cdot \Delta f_p=3\cdot 3\cdot 1= 9\text{ Hz}$$

Rank by consonance
Is this table a copyvio?


 * This table is taken from Lots & Stone:
 * Shapira Lots, Inbal, and Lewi Stone. "Perception of musical consonance and dissonance: an outcome of neural synchronization." Journal of the Royal Society Interface 5.29 (2008): 1429-1434. link


 * Lots & Stone references pages 183 and 195 of Helmholtz:
 * Hermann, L. F. "Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music." Trans. Alexander J. Ellis (New York: Dover, 1954) 7 (1954).


 * The fourth column lists ΔΩ, which the width of the stability interval discussed in Lots & Stone.