Beat (acoustics)/Helmholtz tables

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Harmonic Matching[edit | edit source]

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 and . The frequency of has been increased by Hz.

  • Helmholtz beating is ordinary amplitude beating between higher harmonics of signals with two fundamental frequencies,
and
  • We use a pre-superscript, , to denote beats betweem the various harmonics, assuming that all harmonics exist:

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:

The second harmoinc of 301Hz is 602Hz

The third harmonic of 200Hz is 600Hz

The (amplitude) beat frequency is:


Example 2:

The third harmoinc of 501Hz is 1503Hz

The fifth harmonic of 300Hz is 1500Hz

The (amplitude) beat frequency is:



Example 3:

The nineth harmoinc of 501Hz is 4509Hz

The fifteenth harmonic of 300Hz is 4500Hz

The (amplitude) beat frequency is:



Rank by consonance[edit | edit source]

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.