Beat (acoustics)/Helmholtz tables

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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:

${}^{i}\!f_{B}=i\cdot \left|q\Delta f_{p}-p\Delta f_{q}\right|$

---

f₀ = 100

p=3     q=2

Δf_p = 1

fp	fq	fB
301	200	1
	400
602	600	2
903	800
	1000
1204	1200	4
1505	1400
	1600
1806	1800	6
2107	2000
	2200
2408	2400	8
2709	2600
	2800
3010	3000	10

f₀ = 100

p=5     q=3

Δf_p = 1

fp	fq	fB
501	300	1
1002	600
	900
	1200
1503	1500	3
2004	1800
2505	2100
	2400
	2700
3006	3000	6
3507	3300
4008	3600
	3900
	4200
4509	4500	9
5010	4800
5511	5100
	5400
	5700
6012	6000	12

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

The second harmoinc of 301Hz is 602Hz

The third harmonic of 200Hz is 600Hz

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 501Hz is 1503Hz

The fifth harmonic of 300Hz is 1500Hz

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 501Hz is 4509Hz

The fifteenth harmonic of 300Hz is 4500Hz

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

Interval ranking

quality	name	ratio	ΔΩ
absolute	unison	1/1	.075
absolute	octave	2/1	.023
perfect	fifth	3/2	.022
perfect	fourth	4/3	.012
medial	M 6th	5/3	.010
medial	M 3rd	5/4	.010
imperfect	m 3rd	6/5	.010
imperfect	m 6th	8/5	.007
dissonance	M 2nd	8/9	.006
dissonance	M 7th	8/15	.005
dissonance	m 7th	9/16
dissonance	m 2nd	15/16
dissonance	TT	32/45

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.