The parallel equally spaced lines
and the desired length HI, of ribbon, wood, anything non-elastic.
The parallel equally spaced lines
and the desired length HI, of ribbon, wood, anything non-elastic.
The mystery is solved…….
Here is the line, with 0 and 1 marked. /You chose it already !
Here is a numbered line, any size, equally spaced, at intervals of one unit.
It only has to start from zero.
Now construct the line from point 5 to the “fraction” line at point 1, and a parallel line from point 1 on the numbered line.
The point of intersection of the parallel line and the “fraction” line is then 1/5 of the distance from 0 to 1 on the “fraction” line.
1/5, 2/5, 3/5, 4/5 and 1 are equally spaced on the fraction line.
L cannot be moved in the static picture.
Who needs LCM ?
First, three views of LCM with no comments :
Now we can do fraction addition without LCM. It just needs the use of the distributive law, and the result shows the way in which the divisors combine.
And now using 3/4
But the best one is via multiplication ……
Now for multiplication and division.
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:
Part three will look at beat frequencies and reasons for preferring the harmonic system, though not for keyboard instruments.
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
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)
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:
Next post: The equal tempered scale and some more math.
There may be other solutions. a = b = 3, c = 21. ( )
Mary’s mother brought a pizza
For her little kiddies, two.
“Johnny, you can have threequarters.
Mary, just a half will do.”.
Then the kiddies started eating.
Soon Mary grabbed her final piece.
“That’s mine” screamed Johnny, then the fighting
Broke the tranquil mealtime peace.
How much pizza then was eaten?
How much pizza on the floor?
Mother swore and left the building.
“I should have ordered just one more”.
So one day recently I was bored, and then the following rushed onto the page:
Half of a big pizza is equal to two small pizzas – rewrite this in as precise way as possible.
Is this 3 hours or 1/4 of a pizza?
How many hours equals 1/4 of a pizza?
Apologies to those for whom a clock face is a historical artifact.
How do I know it’s a pizza and not a cake?
Does half a day include half the night?
There are four feet in our yard, mine and my sister’s.
Would you prefer 1/2 of a round pizza or 1/2 of a square pizza?
Are ratios numbers or fractions (or neither) ?
Fractions are parts of the same whole. OK, I’ll have 5 quarters of that pizza (5/4 is a fraction, isn’t it?)
You cut, I choose !
This year fractions are parts of a whole. Next year fractions will be numbers. I guess the other party won the election.
2/3 and 4/6 are equivalent fractions. Equivalent to what ?
The word “fraction” has the same root as “fracture”. So something got broken. I think it was my faith in common sense.
The Common Core test question asked – How long is 3.25 hours. This could be 3 hours and 25 minutes or 3 hours and 15 minutes. I guess it depends on the grade level.
Back to the heavy stuff next time !
This was found on “talking math with your kids” as an example of the “strange” stuff that kids bring home and cause mystification in their parents.
“The whole is 8. One part is 8. What is the other part ?”.
Just what exactly is this supposed to mean?
That the whole always consists of two parts?
Since when did numbers have parts?
What is the definition of “part”?
Even if we are talking about 8 things, then the natural AND logical answer is “There isn’t another part”.
If I want to see ways of creating 8, using adding, then what is wrong with
8=1+7 8=2+6 8=3+5 … 8=7+1 and 8=8+0 for completeness’ sake.
To call zero a part of 8 is going to lay the groundwork for a feeling that math hasn’t got a lot to do with real life, which is a crying shame. This feeling can arise at any stage, we should give reality a chance at this level. To conclude “What a stupid question!”.
I did the sums, no hesitation.
But then it asked for explanation.
“I know it’s right”, I wrote down fast,
“I understood from first to last!”.
“I’m going to be a mathematician,
“Not a fingernail technician!”.