# 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 ${\displaystyle f_{p}}$ and ${\displaystyle f_{q}}$. The frequency of ${\displaystyle f_{p}}$ has been increased by ${\displaystyle 1}$Hz.

• Helmholtz beating is ordinary amplitude beating between higher harmonics of signals with two fundamental frequencies,
${\displaystyle f_{p}=pf_{0}}$ and ${\displaystyle f_{q}=qf_{0}.}$
• We use a pre-superscript, ${\displaystyle i\in \{1,2,3,...\}}$, to denote beats betweem the various harmonics, assuming that all harmonics exist:
${\displaystyle {}^{i}\!f_{B}=i\cdot \left|q\Delta f_{p}-p\Delta f_{q}\right|}$

f0 = 100 p=3     q=2 Δfp = 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 Δfp = 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: ${\displaystyle p=3{\text{ and }}q=2{\text{ with }}i=1.}$

The second harmoinc of 301Hz is 602Hz

The third harmonic of 200Hz is 600Hz

The (amplitude) beat frequency is:

${\displaystyle {}^{2}\!f_{B}=i\cdot q\cdot \Delta f_{p}=1\cdot 2\cdot 1=2{\text{ Hz}}}$

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

The third harmoinc of 501Hz is 1503Hz

The fifth harmonic of 300Hz is 1500Hz

The (amplitude) beat frequency is:

${\displaystyle {}^{2}\!f_{B}=i\cdot q\cdot \Delta f_{p}=2\cdot 3\cdot 1=6{\text{ Hz}}}$

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

The nineth harmoinc of 501Hz is 4509Hz

The fifteenth harmonic of 300Hz is 4500Hz

The (amplitude) beat frequency is:

${\displaystyle {}^{2}\!f_{B}=i\cdot q\cdot \Delta f_{p}=3\cdot 3\cdot 1=9{\text{ Hz}}}$

## 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.