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This chapter was originally part of The Fine Art of Combining Harmonics chapter. As the topic became larger, I had to split the information. The link above will take you back to the section from which this part was cut.

Tuning Theory Examples

There are only two kinds of scales derived from the harmonic series:

1. Radial, and
2. Cyclic.

The third would be a strange mixture of the two. Note that the second is more like a spiral, and despite this fact it's still called "cyclic". I'll explain why when we'll get there. From another point of view, cyclic scales are just a special case of radial systems.

But let's leave things as they are, and for the joy of setting labels let's continue. Every one of the above can further have two forms, depending on the side of the harmonic series from which its family of notes is derived:

a. Undertones, and
b. Overtones.

And of course the mixture of the two, which brings us back to the same oneness.

My presentation takes into accound both under- and overtones, at the same time. The creation of scales based on the two series is presented simultaneously. Thus every step will occur mirrored, for the convenience of catching two rabits in the same net, at the same time.

Let's have a look at how a radial tuning system is created. This will be the foundation for understanding the cyclic system, and the difference between the two.

So far we saw how the interval between the second and third harmonics started the process of scale creation. Let's continue expanding the system.

Searching for further sonic material, we skip the fourth harmonic, because is just a halving/duplication of the second, which is in turn a halving/duplication of the first.

The fifth harmonic comes with a new sound, and the whole process described so far gets repeted. Here it is briefly:

These new notes represent the interval between the fourth and fifth harmonics, having as starting point the fundamental. From looking at the picture, 6/4 is exactly 3/2. 6 is the double of 3, 4 is the double of 2. THIS IS THE REASON FOR SIMPLIFYING - that is, for multiplying and dividing by 2 (also 4, 8...)

Remember that an interval is the distance between 2 harmonics. In this case, the distance is between harmonics 4 and 7, having as starting point the fundamental.

every new interval from the 8th double/half will fill an empty space between existing intervals

Another way of looking at it is this:

notice how the interval between first and second harmonics is the same as between the 8th and 16th harmonics - in both directions (which is the same interval as between the 2nd and the 4th, the same as between the 4th and the 8th).

also, the interval between second and third harmonics is the same as between the 8th and the 12th (which is the same interval as between the 4th and the 6th).

the interval between 4th and 7th harmonics is the same as between the 8th and 14th.


8th harmonic turns into fundamental
9th harmonic 9/1 (distance from fundamental to 9), turns into dist. from 8 to 9
10/1 becomes 10/8 which is 5/4

THE COMPLETE SYSTEM

together with the intervals from which it was derived

next image presents the order in which intervals appear. this has great significance for the perceived quality of intervals: first created will generally sound more "consonant". the reason for this can ve explained with the image of the string oscillating in equal segments.

There is still another way of looking at intervals smaller than 2

CYCLIC TUNING

now there's no space to see how far the 27th harmonic stretches. making the number line smaller...

giving up the rainbow colors to make a better point. this is just temporarely.

shrinking even more the number line.

JUST FOR CLARITY AND ERGONOMICS:
representing only harmonics 2 & 3
using 2 separate number lines
using simple (harmonic) numbers instead of ratios.
left and right are opposite, even if numbers don't reflect it.

the result: "cyclic" tuning system (needs to be colored)

the right part of the above set up, the overtones of 12 "cyclic" 3/2's are the starting point for messing around with the 12 notes we all know from the piano, guitar, and basically all modern instruments.

This chapter was originally part of The Fine Art of Combining Harmonics chapter. As the topic became larger, I had to split the information. The link below will take you back to the section from which this part was cut.

<- back to The Fine Art of Combining Harmonics
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