Category Archives: harmonic

The mathematics of harmonic means and the beat frequency

Two bits of math associated with the harmonic scale and the beat frequencies. See recent posts on music and math for where this came from.

1. Why is the word “harmonic” used for the “harmonic” series 1+1/2+1/3+1/4+…?
This is easy, the fractions in the series are exactly those proportions of a stretched string that yield the harmonics of the open string.
Why is the word “harmonic” used in the definition of the “harmonic” mean?
This looks obvious after the event, but I was stupidly surprised by it.
Let p and q be two frequencies, and a and b the corresponding periods.
Then a = 1/p and b = 1/q
The harmonic mean k of p and q is 2/(1/p + 1/q), which is 2/(a + b)
Call the period corresponding to this frequency h
Then h = 1/k = (a + b)/2
So the harmonic mean of two frequencies corresponds to the arithmetic mean of the two corresponding periods.
You can check to see that it works both ways !
So it’s all about sounds, music and harmony.

2. The formula for the beat frequency for two notes with frequencies f and g.
Beat frequency = |f – g|
But why ? This is TOO simple !!!
I found this by looking at the plots and making a table, then confirmed it by a quick visit to the internet, but no proof.
Here is one of the plots
We are adding two sine waves together, so back to school math and always having to work the sine and cosine formulae out from scratch I got to sin(A + B) + sin(A – B) = 2sin(A)cos(B).
After turning this into the sin(P) + sin(Q) form and getting nowhere I realised that the first version held the answer.
Take A + B for the higher frequency f and A – B for the lower one g, so we have A + B = f(2πt) and A – B = g(2πt),
Solving we get A = ((f+g)/2) x (2πt) and B = ((f-g)/2) x (2πt).

So the combined signal has an amplitude of 2, a fast wave from the ‘A sine term and modified by a slow wave from the (f-g)/2 cosine term.
The slow cosine wave has a frequency of half the difference between f and g, but in each cycle of the cosine there are two pulses perceived as volume change, so the frequency of these pulses is twice as big, which is the difference between f and g.

We have ended up with a sine wave of frequency (f+g)/2 modulated by a cosine wave of frequency (f-g)/2. In radio communication this is called AM, or amplitude modulation (as opposed to FM).

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MUSIC: tuning,harmonic, equal temper, beat frequency :MATH – part trois !

So, what is the difference in musical effect between the harmonic scale and the equal temper scale ?

Whenever two notes are played together the ear “hears” the two notes both separately and together. The “togetherness” is a consequence of the perception by the ear of a third note, usually quite faint, the beat note with associated “beat frequency”. The effect is very noticeable when two recorders are played, as the notes are very “pure”. A pure note is one which consists of vibrations at exactly one frequency, and this is described by a sine wave or sine function y = sin(2πkx) where x is time and k is frequency.

Below, and with many thanks to DESMOS, which made the job almost painless, are plots of the sum of a frequency 5 wave and a frequency 5+b/2 wave, for various decreasing values of b. (The first has b = 0 to show the sum of the frequency 5 wave with itself, giving a reference point of 2sin(5*2πx)


Now with frequencies 5 and 10

Now with frequencies 5 and 9beatpic3a

Now with frequencies 5 and 7, something going on herebeatpic4a

Now with frequencies 5 and 5.75, notice the appearance of the beat in the signalbeatpic5a

Now with frequencies 5 and 5.5beatpic6a

Now with frequencies 5 and 5.25beatpic7a

And finally with frequencies 5 and 5.05beatpic9a

The first thing to note is that the beat frequency is the difference between the two “added” frequencies. This can be seen by seeing the period of the beat in this last one as 20, which is a frequency of 1/20 or 0.05 (= 5.5 – 5). (The math for all of this will be in another post). See London police whistle

The second thing to note is that in the case of 5 and 5.75 the frequency ratio is 5.75/5 = 23/20, and this is greater than the relative frequency of the 9/8 whole tone (C to D), and a bit less than 6/5, one of the estimates for the interval C to Eflat. This one has quite a large effect on the perceived sound of the whole tone interval, and is one reason why it is difficult to hear the two notes separately.

Now we can see what this all means in the harmonic tuning system. Let us take the notes C and E, frequency ratios 1 and 5/4.

The difference is 5/4 – 1 = 1/4. What note, if any, is this?

We have seen that multiplying by 2 doubles the frequency and produces a note one octave higher, so dividing by 2 produces a note one octave lower. Do it again and we get a note two octaves lower.

So the beat frequency for the pair CE is the C two octaves down. I will write this as C,, and the C two octaves up will be C”.

This means that each (in this case) of the two notes C and E is a harmonic of the beat frequency.

Consequently the CE interval will appear to have more “body” than might be expected. Just try playing the C on its own and then with a quieter 2 C’s below added.

This can be done for all the intervals on the harmonic scale (results shown in the table below).

The main conclusion I have come to is that the reason a choir trained without a piano accompaniment has a fuller sound than one used to singing with a piano, or other musical “backing” is that the first type automatically tunes itself in the harmonic system, and consequently has the benefit of beat frequencies which are in tune with the notes being sung.

This is particularly noticeable with barbershop singing.

Let us compare equal temper with harmonic on the major third interval CB

Harmonic: interval is 1 to 5/4, beat frequency is 1/4 or 0.25 , nice !

Equal temper: interval is 1 to cube root of 2 (twelfth root of 2 raised to power of 4, 4 semitones from C to E), and this is 0.260. So when we look at the second harmonic of this we get 4*0.260 = 1.04, which is NOT C. It is a bit less that a semitone above, and creates a low volume buzz that causes the major third to feel harsh.

The fifth, C to G, is not as much affected as the two systems give almost the same frequency.

Table coming

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