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
beatpic7a
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)

beatpic1a

Now with frequencies 5 and 10
beatpic2a

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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NY Teacher Offers Advice to VAM-Loving Reader

howardat58:

In response to the last point – Maybe it does, nowadays.

Originally posted on Diane Ravitch's blog:

Regular readers may have noticed a flurry–one might say–a deluge of comments by a reader who signs as “Virginiasgp.” SGP stands for “student growth percentiles.” He believes with a religious fervor in student growth measures for evaluating teachers. He also says that he has worked in the U.S. Navy on a submarine. Another reader who signs as “NY Teacher” offered Virginiasgp some ideas about the deficiencies of test scores for teacher evaluation:

VAsgp
Apparently you think it’s a great idea to run public schools like the Navy runs its nuclear submarine fleet. Well thanks for the inspiration man. You really are a hoot-n-a-half on this. Shear genius. Now let’s take your fantabulous idea and put it to work for the Navy.
Don’t worry, I am highly qualified to help the US Navy mainly because I have zero experience with nuclear submarines. At least we’re square on the experience piece. Well…

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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:

beat harmonic scale
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)

beat tone semitone

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.

beat pitch comparison

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.

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

beat spinet 1704A 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.

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A change from math – my garden in Puerto Rico

Ten years ago I came to Puerto Rico, to Mayaguez on the west coast, to a house with the usual vast expanse of nicely trimmed grass. Now I have the controlled jungle I always wanted. More time spent cutting things back or down than spent trying to get things to grow.
annual leaf
One leaf like this each year. A less successful part of the jungle.
flamboyan
My flamboyan tree. Best year ever for flowers.
flamboyan from below
And here from underneath.
monster plant
The monster plant. From 1 foot to 6 feet in 8 years, and prickly.
ginger1
Ginger. Later the flower grows baby plants and more flowers.
baby apple bananas
Small apple bananas. Guineo manzano. Up to 300 on one stalk.
name unknown by me
I don’t know the name of this one, but it is 6 feet high and growing.
not weed
Definitely not weed.
voyager palm 2
Voyager palm, it flowered three times this year.
fancy bromelia
A fancy bromelia.weird fern tree
This is a fern, tree sized, with its annual one foot high flower.

There’s more, but I have to go out and cut some strays.

I am quite sure that there is math growing here, waiting to be spotted.

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Gross misuse of + and – and x and the one that’s not on my keyboard

Arithmetic is the art of processing numbers.
We have ADD, SUBTRACT, MULTPLY and DIVIDE
In ordinary language these words are verbs which have a direct object and an indirect object.

“Add the OIL to the EGG YOLKS one drop at a time”.
“To find the net return subtract the COSTS from the GROSS INCOME”.

In math things have got confused.
We can say “add 3 to 4″or we can say “add 3 and 4″.
We can say “multiply 3 by 4″ or we can say “multiply 3 and 4″.
At least we don’t have that choice with subtract or divide.

The direct + indirect form actually means something with the words used,
but when I see “add 3 and 4″ my little brain says “add to what?”.

There are perfectly good ways of saying “add, or multiply, 3 and 4″ which do not force meanings and usages onto words that never asked for them.
“Find the sum of 3 and 4″ and “Find the product of 3 and 4″ are using the correct mathematical words, which have moved on from “add” and “multiply”, and incorporate the two commutative laws.

If we were to view operations with numbers as actions, so that an operation such as “add” has a number attached to it, eg “add 7″, then meaningful arithmetical statements can be made, like

“start with 3 and then add 5 and then add 8 and then subtract 4 and then add 1″

which with the introduction of the symbols “+” and “-“, used as in the statement above allows the symbolic expression 3+5+8-4+1 to have a completely unambiguous meaning. It uses the “evaluate from left to right” convention of algebra, and does not rely on any notion of “binary operation” or “properties of operations”.

If we want to view “+” as a binary operation, with two inputs then, yes, we can ascribe meaning to “3+4″, but not in horrors such as the following (found in the CCSSM document):

To add 2 + 6 + 4, the second two numbers can be added to make a ten,
so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

If + is a binary operation, which are the two inputs for the first occurrence of + and which are the inputs for the second occurrence of + ?
The combination of symbols 2 + 6 + 4 has NO MEANING in the world of binary operations.

See A. N. Whitehead in “Introduction to Mathematics” 1911.
here are the relevant pages:
whitehead numbers 1
whitehead numbers 2a
whitehead numbers 2b
whitehead numbers 3a
whitehead numbers 3b

And here are two more delights from the CCSSM document
subtract 10 – 8
add 3/10 + 4/100 = 34/100

In addition I would happily replace the term “algebraic thinking” in grades 1-5 by”muddled thinking”.

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A Common Core Absurdity: Teaching Health

howardat58:

You gotta read this, it’s priceless !

Originally posted on Diane Ravitch's blog:

A principal sent this account of a teacher’s experience to me:

“Common Core Training for ENCORE Teachers

“(ENCORE = subjects like Health, Physical Education, Art, Music, Technology, Home & Careers)

“The ENCORE subjects were assigned a period to meet with a Common Core Specialist. We were told to bring a sample lesson or activity that we use in class.

“I presented a project that I use at the end of my Violence Prevention Unit. This project allows students to research and bring in any article that interests them about Bullying. The article could represent facts about bullying, prevention tips, victim accounts or any other related material. The article could be from a magazine, a newspaper or an on-line source. Students then are asked to answer 5 questions based on the article they chose.

1. Summarize the Article

2. Personal Reaction to The Article

3. Victim’s Reaction – if you were…

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More bad language in math

howardat58:

I have to put this one up at least twice a year, so here it is again.

Originally posted on Saving school math:

Here is another horror which I found recently:

The distributive law of addition: a(b + c) = ab + ac (OK, it’s a definition)

distributive property really

The current school math explanation:
You take the a and distribute it to the b to get ab
and then you distribute the a to the c to get ac
and then you add them together to get ab + ac

I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is…

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Vertex of a parabola – language in math again

Here are some definitions of the vertex of a parabola.

One is complete garbage, one is correct  though put rather chattily.

The rest are not definitions, though very popular (this is just a selection). But they are true statements

Mathwarehouse: The vertex of a parabola is the highest or lowest point, also known as the maximum or minimum of a
parabola.
Mathopenref: A parabola is the shape defined by a quadratic equation. The vertex is the peak in the curve as shown on
the right. The peak will be pointing either downwards or upwards depending on the sign of the x2 term.
Virtualnerd: Each quadratic equation has either a maximum or minimum, but did you that this point has a special name?
In a quadratic equation, this point is called the vertex!
Mathwords: Vertex of a Parabola: The point at which a parabola makes its sharpest turn.
Purplemath: The point on this axis which is exactly midway between the focus and the directrix is the “vertex”; the vertex is the point where the parabola changes direction.
Wikibooks: One important point on the parabola itself is called the vertex, which is the point which has the smallest distance between both the focus and the directrix. Parabolas are symmetric, and their lines of symmetry pass through the vertex.
Hotmath: The vertex of a parabola is the point where the parabola crosses its axis of symmetry

Scoring is 10 points for finding the garbage definition and 5 points for the correctish definition !!!! Go for it!

When I studied parabolas, back in 1958 or so (!) the parabola had an apex. So I checked the meaning of vertex, and found that the word was frequently misused.

Here is a good account: https://en.wikipedia.org/wiki/Vertex_(curve)

Basically a vertex of a curve is a point where the curvature is a maximum or a minimum (in non math terms, most or least curved).

Here are two fourth degree polynomials, one has three vertices and the other has five. The maximum curvature points are indicated. The minimum curvature points are at the origin for the first curve, and at the points of inflexion for the second curve (curvature = zero)

01Gquartic201Gquartic0

A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis.

For a circle, which has constant curvature, every point is a vertex.

The center of curvature at a (nice) point on a curve is the center of the closest matching circle at that point. This circle will usually lie “outside” the curve on one side of the point, and “inside” the curve on the other side. Look carefully at the picture. It is called the osculating or kissing circle (from the Latin).

The center of curvature can be estimated by taking two point close to the point of interest, finding the tangents at these points, and then the lines at right angles to them and through the points. the center of curvature is roughly at the point of intersection of these two lines

01center of curvature

The diagram below shows this estimate, for the blue parabola, at the vertex.

02center of curvature

Finally (this has gone on further than expected!) I found this delightful gif.

01Lissajous-Curve+OsculatingCircle+3vectors_animated

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