An 833 Cents Scale

An experiment on harmony

Latest update: December 9,2012

 

Introduction

Not every music theorist agrees that combination tones play a major role when it comes to defining harmony. Anyway, some do. We do not want to start this discussion here again; for the sake of this experiment, let's simply assume for the moment that the latter ones are right.

We start the experiment with a small exercise: We take an octave, built from two pure sinewave tones with the frequency relation 2:1 (1200 cents) and observe that there appear combination tones. The closest one has a frequency of three times the base frequency and thus forms a perfect fifth (3:2, or 702 cents) with the higher tone of our octave. If we repeat the exercise, this time starting with a pure sinewave fifth, we obtain a summation tone that forms a major sixth (5:3, or 884 cents) with the higher of the two interval tones. Continuing the game by each time making the new interval (between upper interval tone and summation tone) the base for the next step, the following table develops:

 Base Interval

New Interval [Ratio]

New Interval [Cents]

 2 : 1

3 : 2

701.96

3 : 2

5 : 3

884.36

5 : 3

8 : 5

813.69

8 : 5

13 : 8

840.53

13 : 8

21 : 13

830.25

21 : 13

34 : 21

834.17

34 : 21

55 : 34

832.68

55 : 34

89 : 55

833.25

89 : 55

144 : 89

833.03

 144 : 89

233 :144

833.11

It is quite obvious that the new found intervals converge to a value close to 833 cents. That means nothing else than that for instance for an interval of 144:89 (833.11 cents) both the summation and the difference tone appear in again 833 cents distance from this interval:

55 : 89 : 144 : 233
(833 cents + 833 cents + 833 cents),

thus creating something like a harmonic series. This implies that an 833 cents chord might sound special, despite representing a gross dissonance in traditional opinion. Reason enough to investigate this further. (It is by the way unimportant which interval we choose as a starting point for the above exercise; the result is always 833 cent.)

 

833 cents, an interval with unique properties

The exact value that the 833 cents interval converges to can easily be calculated. If we call the two members of the base interval A and B and the summation tone C, as follows:

A : B : C,

then they have to satisfy the following conditions, with x as the sought after ratio:

x = C/B = B/A, and C = A + B.

That yields an equation

x = 1/x + 1

with the solutions

x1 = 1/2 + (5/4)1/2 = 1.618034
x2 = 1/2 - (5/4)1/2 = - 0.618034

From the view point of music theory, both results are identical:

x1 = 1/|x2| = 1.618034 = Z.

Hence, we need only x1, and for convenience, we call it Z in the following text. Note that Z has funny, though trivial, mathematical properties:

1/Z = 0.618034, Z = 1.618034, Z2 = 2.618034.

The desired more exact value in cents for our unique interval is

1200 cents * lg Z / lg 2 = 833.09 cents.

 

(At this point it would have dawned to the author that he had hit upon the Fibonacci Sequence and the Golden Ratio or Golden Section, if he hadn't been living in total ignorance regarding the Fibonacci Sequence at all, and if he had known more about the Golden Ratio than just its name. Shame on him, but is was not until he found a note relating to the Fibonacci series on the tuning list [tuning@onelist.com] that he realized that he had just added another example to the long list of phenomena that follow this pattern. Anyway, for the following considerations regarding the musical properties of the 833 cents interval this fact is of secondary importance only.)

 

An 833 cents scale

If we intend to create a quasi harmonic scale within the frame of the 833 cents chord we will have to take into account that ordinary sounds are not sinewaves; so there will be harmonics. Let's consider five stacked 833 cent chords, numbered -II, -I, 0, I, II, and consisting of the tones I, J, K, L, M, N, covering a range of 4165 cent:

 Tone

 I

J

K

L

M

N

 Cents

 -1666

-833

0

833

1666

2499

 Ratio

 0.382

0.618

1

1.618

2.618

4.236

The harmonics and "subharmonics" up to the fourth order that fall into this range are listed in the following table. The restriction to four orders is a willful act to avoid too small intervals. (In each field: top: harmonic respectively "subharmonic"; center: actual cents; bottom: cents related to the respective 833 cents frame interval.)

 Stacked chord #

 -II

 -I

 0

 I

 II

 Harmonics of tone K

K/2
-1200
466

 -

 -

 2K
1200
367

3K / 4K
1902/2400
236/734

 Harmonics of tone L

 L/4 / L/3
-1567/-1069
99 / 597

L/2
-367
466

-

 -

 2L
2033
367

 Harmonics of tone J

-

-

2J
367
367

3J / 4J
1069 / 1567
236 / 734

-

 Harmonics of tone M

-

M/4 / M/3
-734 / -236
99 /597

M/2
466
466

-

 -

 Harmonics of tone I

 -

2I
-466
367

3I / 4I
236 / 734
236/734

 -

 -

 Harmonics of tone N

 -

 -

N/4 / N/3
99 / 597
99 / 597

N/2
1299
466

 -

It is quite obvious that there is a number of repeatedly appearing tones, making suitable candidates for steps in the intended scale. The result is listed in the following table.

 Step

Ratio (dec.)

Ratio (cents)

Diff. to previous step (cents)

 0

 1.0000

0

 -

 1

 1.0590

 99.27

 99.27

 2

 1.1459

 235.77

 136.50

 3

 1.2361

 366.91

 131.14

 4

 1.3090

 466.18

 99.27

 5

 1.4120

 597.32

 131.14

 6

 1.5279

 733.82

 136.50

 7

 1.6180

 833.09

 99.27

This is a rather short scale, but that does not mean a restriction. In accordance with its design the scale repeats above and below. It contains a complex network of harmonic relations. A few examples, related to the base tone: Step 10 turns out to be identical with the octave (1200 cents) to the base tone, at the same time featuring the Golden Ratio to step 3. Step 16 represents the perfect twelfth (1902 cents) on the base tone, a perfect fifth to step 10, and is also twice the Golden Ratio to step 2. Step 20 is the double octave (2400 cents) of the base tone, a perfect fourth to step 10 and twice the Golden Ratio to step 6; and so on.

More specifically the single intervals have the following properties:

- 2 steps: A variety of whole tones in transition to minor thirds. Those from step 1 to step 3 and from step 4 to step 6 are within fractions of a cent identical with the septimal or subminor third (7/6), and those from step 2 to step 4 and from step 3 to step 5 are, again within fractions of a cent, identical with the septimal second (8/7).

- 3 steps: A mixture of uncertain thirds, right between minor and major.

- 4 steps: Perfect (or Pythagorean) fourths from step 1 to 5 and from step 2 to 6, as to be expected from the 2-step intervals. All others are near fourths.

- 5 steps: A variety of tritones!

- 6 steps: Two perfect fifths from step 3 to step 9 and from step 5 to step 11, two meantone fifths from step 2 to 8 and from step 6 to 12; the other four "larger than life" fifths.

- 7 steps: All "Golden Section" intervals, by design.

- 8 steps: Steps 0 to 8, 3 to 11, 6 to 14 and 7 to 15 form four intervals that are, within a fraction of a cent, identical with the septimal major sixth (12/7), and steps 1 to 9 and 5 to 13 represent, with a similar small deviation, the harmonic seventh (7/4). The remaining two are about 5 cents smaller than the latter ones.

- 9 steps: A variety of major sevenths

- 10 steps: Most of them perfect or near perfect octaves, but beware of steps 5 to 15 and 6 to 16; they fall 32 cents short!

It goes without saying that temperation would be contrary to the intention of the scale; to achieve the intended effect all intervals have to be tuned fairly accurately. However, an equal division of the octave into 36 steps provides a suitable approximation:

 Step (just)

Cents (just)

Step (36/octave)

Cents (36/octave)

 0

0

0

0

 1

99.27

3

100.00

 2

235.77

7

233.33

 3

366.91

11

366.67

 4

466.18

14

466.67

 5

597.32

18

600.00

 6

733.82

22

733.33

 7

833.09

25

833.33

All right, here is a scale, available for experimenting to everybody who dares. And here is the example of somebody who did, and here is a second one.

Heinz Bohlen


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