There are other interesting features about this set of frequencies. Three times the frequency of C4 is almost equal to the frequency of G5! Five times the frequency of C4 is close to the frequency of E6! Or to put in simple terms, the ratio of frequencies of a G note to that of C note, in the same octave is 3:2, and that of notes E to C is 5:4. From here it will mostly be a reverse working to find out the answer to the question why 12 notes. Since it is a reverse working I am also going to start with my conclusion on this subject.
Our brain stores every detail in patterns and every rule that has been defined in the history owes to this biology of our brain. When I read all these articles the first thing that occurred to my mind was THE MATRIX movie. It seems like everything is mechanical and a code has already been written for the same and I want to get myself unplugged and find out some new form of music in arithmetic progression.
I am sure most of us have attended “paatu class” during some time of our schooling. Esp for singing during navarathri festival or some bhajanais. Of course some of us end up taking it as a career. I am no exception to this paatu class and singing for sundal during navarathri festival, but there was always one question which I never had answer; at least a scientific one until now.
WHY DO WE SING SA----PA----SA at the beginning of every class?
It is as simple as this. As mentioned above Pa will be of the ratio 3:2 in comparison to the Sa. Hence therefore making it exactly the middle note of the scale and hence stabilizing the vocal tone.
When you pluck a string on a guitar, it vibrates back and forth. This causes mechanical energy to travel through the air, in waves. The number of times per second these waves hit our ear is called the ‘frequency’. This is measured in Hertz (Hz). The more waves per second the higher the pitch. For instance, the A note below middle C is at 220 Hz. Middle C is at about 262 Hz.
Do we know what frequency means? The rate of repetition of cycles of periodic quantity such as sound wave is called frequency.
Now, to understand the concept of why 12 notes, again as I did earlier I am taking C Major Scale as example.
The above picture shows the sine waveform of C (red) & F# (green). Just on seeing the same we can say that these 2 sine waves doesn't match. The 2 wave pattern start together and in a given time (t) C’s wave pattern red is 8 lambda and in the same time Green wave is 11. And I do not like these numbers 11 & 8 as a pair for a simple reason I cannot simplify this ratio more. Yeah I know I am using a new term lambda without explaining what it is. I think this time instead of going tangent from the subject again I will reserve certain topics for next part of the intervals topic and for now lambda is just like meters is a measurement used to measure waveforms. One trough to the next trough is considered 1 lambda. Now let us see how C & G sine waves look like:
These patterns if carefully observed we see the red and the green trough merge in equal intervals.
2 lambda of red = 3 lambda of green. So this proves our case of 3:2 between C & G (SA & PA) in the old method of pictogram.
Is this is the secret for creating pleasing sounding note combinations: Frequencies that match up at regular intervals.
Now let’s look at the ratios of the notes in the C Major key in relation to C:
C – 1
D – 9/8
E – 5/4
F – 4/3
G – 3/2
A – 5/3
B – 17/9
To tell you the truth, these are approximate ratios. Remember when I said the ratio of E to C is about 5/4ths? The actual ratio is not 1.25 (5/4ths) but 1.2599. Why isn’t this ratio perfect? That’s a good question. When the 12-note ‘western-style’ scale was created, they wanted not only the ratios to be in tune, but they also wanted the notes to go up in equal sized jumps. Since they couldn’t have both at the same time, they settled on a compromise. Here are the actual frequencies for the notes in the C Major
Key:
Note | Perfect Ratio to C | Actual Ratio to C | Ratio off by | Frequency in Hz |
Middle C | 261.6 | |||
D | 9/8 or 1.125 | 1.1224 | 0.0026 | 293.7 |
E | 5/4 or 1.25 | 1.2599 | 0.0099 | 329.6 |
F | 4/3 or 1.333… | 1.3348 | 0.0015 | 349.2 |
G | 3/2 or 1.5 | 1.4983 | 0.0017 | 392.0 |
A | 5/3 or 1.666… | 1.6818 | 0.0152 | 440.0 |
B | 17/9 or 1.888… | 1.8877 | 0.0003 | 493.9 |
You can see that the ratios are not perfect, but pretty close. The biggest difference is in the C to A ratio. If the ratio was perfect, the frequency of the A above middle C would be 436.04 Hz, which is off from 'equal temperament' by about 3.96 Hz.
Now let’s look at a
chord, to find out why its notes sound good together. Here are the
frequencies of the notes in the C Major chord (starting at middle C):
C – 261.6
Hz
E – 329.6 Hz
G – 392.0 Hz
The ratio of E to C
is about 5/4ths. This means that every 5th wave of the E
matches up with every 4th wave of the C. The ratio of G to E
is about 5/4ths as well. The ratio of G to C is about 3/2. Since
every note’s frequency matches up well with every other note’s
frequencies (at regular intervals) they all sound good together!
[When we press all the
white keys in an octave the sound that we hear can be described as
just noise. However when we play chords like added 9th, 11th,
13th (i.e. C-E-G-B-D-F-A) it sounds awesome. I am sure most
of you would have and yeah as you have rightly thought the reason is
wavelength of these notes and their ratio from the tonic C]So it all comes down to number of times a note strikes... that is number of troughs and the crests created... and the mathematical fluency between them.
But then comes a big question; there are just 7 notes that I have mentioned here with this ratio but aren’t we looking to find out how 12 notes? Yeah, exactly that’s when I got hit with the concept of Consonance and Dissonances and that is precisely the topic as I have told above I am going to deal with in my next blog. I know I have promised to write on black and white keys in our piano but this a supplement to the current blog. So for this article let us find out the secret behind the 12 notes.
Keeping in mind the rule of octave i.e. the middle C to the C above it the frequency will exactly be double.
The previous list shows only the 7 notes in the C Major key, not all 12 notes in the octave. Each note in the 12 note scale goes up an equal amount, that is, an equal amount exponentially speaking.
And here starts the mathematical part of the reverse working. The moment I say Math, I have got nothing much to do in this part irrespective of whether it is simple or complex. So obviously it was my friend “Dr.Mathematics” Sriram who explained it in his article physics of music and I understood the same from that article (so here starts the cut copy paste part). Since we are following the rule of octave and fitting a G.P. into it, it is evident that the common ratio should be something of the form 21/n, where n is number of notes. But, I want 3f/2 to be a note (to be more specific the middle note based on the sine wave form above and f is the frequency of the tonic note which is C in our case) which means I should choose “n” such that
3/2 = 2a/n
where “a” would represent the number of terms after which 3/2 would occur in the G.P. Calling a/n as x, we have
3/2 = 2x
If you had not understood how the above equation came up, no worries! Because even I did not at first. Only on doing the following steps I realized the logic behind it. Every note is assumed to be in GP, therefore if “a” is the 1st note then a+1 will be the next until we reach a + (n-1) = n. So which would mean notes will be 21/n, 22/n… 2n/n. And this is precisely the GP we have been mentioning all along. So as per this GP if I require the ratio of the 7th note then the formula will be 27/12. And we are doing the reverse working to find out “n”
Taking logarithms on both sides, we get x = log(3/2)/log2 which is approximately, 0.5849 (OMG I am re learning how to use the Log tables, probably using it for the first time after my 12th exam. So it is solid 8 years). Obviously x can never be a rational number, so we can never have integer solutions for n and a. So we choose integers n and a, which will yield a fairly good approximation for x. It so happens that choosing n=12 and a=7 gives x=0 .5833 or 2x=1.4982, which is indeed a very good approximation to 3/2. If a similar exercise is carried out for the note 5f/4, then choosing n=12 and a=4 yields 2x=1.2599, which is also quite close to 5/4. Thus, with n=12 we get fairly decent approximations of 3f/2 and 5f/4. This is the reason why there are twelve notes to an octave!
Take up the case of our own Carnatic music. The music which is taught for the beginners, Sarali Varisai, consists of just seven notes. To the beginner the other Ri’s and Ga’s and such are essentially non-existent. When you consider this, the number of notes is 7 and the placement of Pa is the fourth note after Sa. So for this case, n=7 and a=4, which gives x=0.5714 or 2x=1.4859! Using a Geometric Progression in your system of notes just enables your system to have something called equal temperament! You can even do without it!!
Now I am thinking of creating music system with Arithmetic progression in place of Geometric Progression. The hurdle I will be facing is, I need to make people listen to the AP music system for at least a century to remove the GP swarams that got etched in our gene and make the brain to command wow! Really good! on the other system. The music system with AP should have 48 swarams for a simple reason, as a composer I will have lot of permutations and combinations of swarams that I can use and thereby I can keep creating different kinds of music at least for my life time. But for that I should live 100 years first.
On intervals part, concepts of consonances, dissonances, principle of super position, constructive interference, destructive interference (this particular physics theory is the most important part of sound proofing), Fourier law, equal temperament, etc. are very important.
So apparently it has taken about some 22 pages for me to explain 1/10th of the things I want to. And respecting the complaints from all my friends that a 16 page article is too difficult to read, I am going to continue the remaining in my next parts. And people this time it is just a 4 page article.
