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The Ubiquitous Octave

Since piano keys are laid out in a linear fashion, as opposed to the equivalent mechanisms on most other instruments(such as the valves of a trumpet or the strings on a guitar, where there doesn't seem to be any "organization" to their location), perhaps that is what made the observation that there is no readily apparent need for the cyclic nature of our musical scales all the more evidentnow, while it hadn't occurred to me before.  Indeed, one would think that each key could have its own unique designation, A through Z and beyond, rather than repeating the familiar A–B–C–D–E–F–G pattern.  Perhaps now that we plan to create our own keyboard layout from scratch, it will be a good time to try this!!In addition to this, we are free, of course, to set the spacing between tones to encompass any span desired.  For example, one might start with A = 50 Hz and proceed upward using a 25 Hz spacing as follows: B = 75, C = 100, D = 125, E = 150, F = 200, G = 225, H = 250, I = 275, J = 300, etc.

If we did so, you would notice that the human ear identifies something special between the 50 Hz and 100 Hz, and the 100 Hz and 200 Hz tones of the above example.  While it naturally detects the difference in pitch between any two notes of dissimilar frequency, we have just discoverd that when the notes are in a 2:1 ratio (which we call an "octave"), it interprets this as being something unique.  In this case the ear tends to identify the two tones as being the same note, rather than as two different notes, even though one of them is discerned to be at a higher pitch.  Compare the third set of tones in this example with the first two sets.   In every set each note is played separately twice, then played together twice.  Do you see how the tones in the last set blend together such that it sounds like a single note??  This unique characteristic was recognized, and taken advantage of, even among the early Greeks of 500 BC.

As for the scientific or mathematical basis behind why this happens, the reasons are a bit complex.   One explanation is that it is caused by a reinforcing (or additive) effect produced by a high number of overlapping harmonics within the first few terms of the Fourier expansion for the tones being combined.  These harmonics are not as prevalent in other interval ratios.  We'll discuss these further in later sections, but for now the important thing to note is that, whatever the cause, almost all musical systems make use of this phenomenon.  In fact, I have seen it claimed (although I have no proof of the veracity of this statement) that all music systems use octaves to partition their available tonal ranges.

So it turns out that there is, after all, a very "sound" reason for dividing the keyboard up into repeating groups of notes!!  Middle A = 440 Hz seems to have been the chosen starting point, with sub-octave divisions occurring at 27.5 Hz, 55 Hz, 110 Hz, and 220 Hz, while the octaves occur at 880 Hz, 1760 Hz, and 3520 Hz.  Since each of these tones sounds the same (except for the pitch being higher in each successive case), they are all given the same name, i.e. they are all known as the note "A".

But this is not to say that only the "A" tone has octaves – for that is not the case.   In fact, each and every key on the piano belongs to its own family of related harmonics.  If we thus group all the notes that sound alike under an identical name, we will have intuitively divided the keyboard up using a repetitive pattern such as: A, B, C, D, E...A, B, C, D, E...A, B, C, D, E...etc.  Then if one considers any two notes with the same name, the higher tone will always be some multiple of two times the frequency of the lower one.

I have inserted dots between the patterns given above to signify that we still haven't decided how many notes should occur between each octave yet.  It will take a bit more work before we are in a position to do that.

Dividing Things Fairly

At this point, we have several choices when it comes to assigning frequencies to our notes, and two obvious choices if we decide to divide things equally.  We could choose an arbitrary number of intervals to divide each octave into, and calculate the spacing that is then required between each consecutive note.  Or we could choose some desired spacing that would fit evenly, and assign enough notes to fill in between the end-tones.

For example, if one chose intervals of 20 Hz, the following assignments could be made.  This would result in creating 11 intervals per octave, while maintaining a 2:1 octave ratio between the first and last tones:

A = 220             E = 300             I = 380

B = 240             F = 320             J = 400

C = 260             G = 340             K = 420

D = 280             H = 360             A = 440

The next octave would then be calculated as below, so that a 2:1 ratio is maintained between each note in this octave and the note having the same name in the previous octave:

A = 2 x 220 = 440             E = 2 x 300 = 600             I = 2 x 380 = 760

B = 2 x 240 = 480             F = 2 x 320 = 640             J = 2 x 400 = 800

C = 2 x 260 = 520             G = 2 x 340 = 680             K = 2 x 420 = 840

D = 2 x 280 = 560             H = 2 x 360 = 720             A = 2 x 440 = 880

But notice that the interval between successive tones is now 40 instead of 20!!  Why did the spacing of the intervals change??  It is because the octave tones themselves are further apart.  In looking at the list of frequencies given for "A" in the previous section, it is immediately obvious now that the fourth and fifth instances are spaced much further apart (440 - 220 = 220) than the first two (55 - 27.5 = 27.5), but not as much as the last two (3520 - 1760 = 1760).  As it happens, they follow an exponential curve expressed by the equation:

An = A02n

For the non-mathematicians among us, don't worry –– there won't be a test.  This is algebraic shorthand used to indicate that the first octave (n = 1) is twice the frequency of the initial tone, just as expected.  But the next octave (n = 2) is four times the initial value because 22 = 4.  And seven octaves up, the final tone will be 27 = 128 times the frequency of the original!!

It certainly doesn't sound like there is a bigger jump between pitches in the higher octaves than in the lower ones!!  Evidently the human auditory system perceives exponential changes in pitch linearly, just as it does variations in loudness.  In the latter case, a sound that is twice as loud as another one doesn't seem to us as if it is twice as loud, and it takes a sound 22 = 4 times louder than that one to seem twice as loud as it does.  Thus differences in volume which follow an exponential curve are converted to a linear scale.  This is a good thing too, as this is what gives us the ability to hear sounds over such an enormous dynamic range.

Indeed, a little research into the field of biology will verify that the same thing applies to our ability to discern frequency intervals.  The higher the pitch, the further apart two tones will have to be in order for the spacing between them to sound equivalent to two tones of a lower pitch.

This brings up a problem with our scheme above.  Given the tendency of the human ear to interpret frequency changes exponentially, it may no longer make sense to divide each octave into equally spaced intervals (which are linear), because that results in a choppy transition between octaves, as shown in the figure.  Rather, it now appears that perhaps, like the octaves themselves, we need to also assign each tone exponentially, so that the intervals gradually increase between each consecutive note, from one end of the keyboard to the other, resulting in a smooth exponential curve overall.

To do this, we consider that if the octave is based on a whole integer power of two, then it would seem to make sense that its subdivisions should simply be fractional exponents of two.  In other words, for a twelve-step format, each of the twelve sequential tones should have intervals between them that are the twelfth root of two:

ΔA = 21/12

Or for an eight-step system:

ΔA = 21/8

In general, then, any interval – whether between two adjacent tones, or perhaps between the endnotes of what is known as a third or a fifth...or in fact the spacing between any two (not necessarily consecutive) notes on the keyboard – can be calculated by:

Ap = A02(p/q)

where "p" is the number of note spacings between A0 and Ap, and "q " is the number of tones per octave in the system.  Looks familiar, doesn't it??  It is just a more generalized form of our prior equation.

Using this new information, the following rounded values – which match exactly the rounded values of those that are actually used on the traditional keyboard familiar to modern western musicians – will be obtained by consecutively assuming p = 1, 2, 3, etc., with q = 12 and an initial tone of A0 = 110 Hz:

A0 = 110.00 = A           A6 = 155.56 = D#          A12 = 220.00 = A

A1 = 116.54 = A#         A7 = 164.81 = E           A13 = 233.08 = A#

A2 = 123.47 = B           A8 = 174.61 = F            A14 = 246.94 = B

A3 = 130.81 = C           A9 = 185.00 = F#          A15 = 261.63 = CA

4 = 138.59 = C#        A10 = 196.00 = G           A16 = 277.18 = C#

A5 = 146.83 = D         A11 = 207.65 = G#          A17 = 293.66 = D

    A48 = 1760.00              A60 = 3520.00

The reader will notice that each tone in the second octave (starting with A12) is twice the frequency of that bearing the same name in the first octave, each interval is slightly larger than the preceding one (i.e. A9 - A8 > A2 - A1), and octaves of the first note are represented precisely.   This configuration of tones, by the way, is called "equal temperament", and we arrived at it in one day rather than taking 2500 years as our ancestors did!!

This page was last updated on  February 12, 2023

Always remember to "Think Green" because good planets are hard to find!!  


 

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