# Tag Archives: frequency

## 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).

Filed under beat frequency, harmonic, mean, series

## MUSIC: tuning, scales, fractions, ratio, harmonics :MATH ! Part deux.

Recap from previous post. here is a readable table of the notes and relative frequencies for the scale of C:

Notice that the whole tone intervals are not all the same size. There are two distinct sizes, with frequency ratios 9/8 and 10/9. Only the two semitone intervals EF and BC’ are the same. Check the others ! Use the slightly not obvious relationship between ratios  P/Q = (P/R)/(Q/R)

This was the situation faced by J. S. Bach when he wanted to write music in any key. He figured that if two intervals had the same relative size then the two notes from the first, played together, would have the same musical “feel” as the two notes from the second interval. He wanted the major chord Bflat-D-F to sound like the major chord C-E-G, and so on. Not only that, but the scale devised so far is lacking four of the five “black” notes, and hitherto these had been fudged in. If one pursues the harmonic approach one finds that for example F-sharp and G-flat are different. See later for details.

So he figured that as there were 12 semitones in the full octave the frequency of each note would have to be a fixed multiple of the previous note, so multiplying 1 by this fixed number 12 times must get you to 2. In other words the magic number is the twelfth root of 2. Its value is 1.059463094 from my calculator.

The table below shows the comparison between the new Equal Temper scale and the harmonic scale.

In 1722 Bach composed twelve preludes and fugues for keyboard called Das Wohltemperierte Klavier

If the link doesn’t work here it is:

https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier

Part three will look at beat frequencies and reasons for preferring the harmonic system, though not for keyboard instruments.

Filed under fractions, math, music, musical scales, teaching

## MUSIC: tuning, scales, fractions, ratio, harmonics :MATH !

A spinet, from1704

Nobody knows when or how humans began to make music, but as soon as they could make things that could be used to play “nice” sounds (call them “notes”) they wanted to make musical instments, objects that could produce a number of different notes. The ocarina was an early one of these, and a strange collection of notes it makes. Eventually it was found that some sequences of notes of increasing pitch sounded “better” than others, and the result was a “scale”. The one I am considering here is the well known eight note scale, which runs from a start note to a finish note one octave higher. Found on a piano as the “white” notes, they are C D E F G A B and C’. As usual, it’s one thing knowing what you want, it’s often quite another to find a way of achieving it. In this case it is “string to the rescue”.

A stretched string can be made to vibrate by plucking it. The result is a “nice” sound. This sound will have a “pitch”, which is musical jargon for the frequency of the vibrations, which is physics/engineering jargon for the rate at which the string repeats its vibrations. This rate is expressed in repeats or cycles per second (or other unit of time).
For the record the pitch of the middle C on a piano is 256 (movements of the string up and back down
per second).

Now it was observed millenia ago that if a finger is placed on a string halfway along and the string is then plucked the vibration rate or pitch is doubled, and the sound is described as one octave higher than from the unfingered string. This new note is also known as the first harmonic of the original. The second harmonic is what you get if the finger is placed one third of the way along the string, and its pitch is three times the original. The third harmonic, with pitch four times the original, two octaves up, comes from a quarter of the original string length, and so on …….
Taking the second harmonic, and halving the frequency, or pitch, we get a new note which is between the original and the first harmonic, and either of these played at the same time as the new note gives a “pleasant” sound.
Doing this for the the fourth harmonic (one fifth of the original string length) we get another new note and the original plus the two new notes together produce what is known as the “major chord”.
On the piano, with C the original note, the new notes described above are the G and the E.
Now we can do some math !
If the pitch of C is 256 then the (harmonic) pitch of G is 256 times 3/2. Why? Because the pitch of a harmonic from a plucked string is inversely proportional to the fraction of the string used to make the harmonic.
First harmonic – half the string – pitch 2 times 256 (one octave up)
Second harmonic – one third of the string – pitch 3 times 256 (to get the G)
Third harmonic – one quarter of the string – pitch 4 times 256 (two octaves up)
Fourth harmonic – one fifth of the string – pitch 5 times 256 (to get the E)
and continuing..
Fifth harmonic – one sixth of the string – pitch 6 times 256
Sixth harmonic – one seventh of the string – pitch 5 times 256
The fifth gives a note one octave up from the second, and the sixth gives a note which we call B-flat,
with pitch 256 times 7 divided by 4

The result of all this is that we now have five notes in the scale:
C 256 … E 320 … G 384 … B-flat 448 … C’ 512
but it is easier to see what is going on if we just look at these as proportions of the “home” note C
Then we get
C 1 … E 5/4 … G 3/2 … B-flat 7/4 … C’ 2
or even better
C 4/4 … E 5/4 … G 6/4 … B-flat 7/4 … C’ 8/4

There do seem to be some holes in this, when comparing with the piano.
Where are D, F, A and B ?
We could go up in eighths, which does produce good values for D and B:
C 8/8 … D 9/8 … E 10/8 … G 12/8 … B-flat 14/8 … B 15/8 … C’ 8/4
but F at 11/8 and A at 13/8 don’t look good.
So we look at G, the first of the “new” notes, and think of it as a new home note. Then we can figure out
the B and the D in relation to the G.

The three notes C, E and G form a major triad (the chord of C major), and have the ratios 1, 5/4, 3/2
So to get the ratios of B and D to G just divide all three by the G to C ratio:
(3/2)/(3/2), (15/8)/(3/2) and (9/4)/(3/2) (using 2 times 9/8 as the ratio for D ‘)
and get 1, 5/4 and 3/2, which shows that G B D’ forms a major triad, just like C E G

We still have a problem with F and A, so let us try to set up F A C’ as a major triad by using some simple fraction manipulation.
F to C’ is to be the same ratio interval as C to G, so using the letter F to stand for its ratio to C we want
C’/F = G/C = 3/2
but C’ stands for the ratio of C’ to C, which is 2
So 2/F = 3/2, and solving we get F = 4/3

You can now do the calculations for A, which ends up as A = 5/3, so our full scale, including B-flat, is
C 1 … D 9/8 … E 5/4 … F 4/3 … G 3/2 … A 5/3 … B-flat 7/4 … B 15/8 … C’ 2

Observe that these frequency ratios determine the notes for the scales of C and F, and the scale of G without the F-sharp.
More ratios of interest can be found, as we can see that the intervals CD, DE, FG, GA, AB are whole tones, and EF and BC’ are half or semitones.

Up to the time when Johann Sebastian Bach intruduced the equal tempered scale together with a set of pieces written in each of the possible keys the methods of tuning keyboard instruments (harpsichord, clavichord, spinet) were based on some variant of the harmonic approach.

Here is a link for some heavier stuff:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Next post: The equal tempered scale and some more math.