Expanding tonal space/planar extensions

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This is Part II of a small series of articles discussing the model of tonal space.
On this subpage, we explore what happens beyond the represented boundaries of tonal space (from Part I) as we extend the plane in different directions.

Extending tonal space upward (mode axis)

If we go up the vertical axis from Mode 16 of the overtone scale to the 6th octave, we pass Modes 17 through 31 to reach Mode 32.

The Mode 32 Horizon Chart

Fig.1: The Horizon Chart illustrating overtone scales up to Mode 32

There is an exponential growth in the number of intervals per row as we step up the logarithmic mode axis in octaves. Each additional octave in mode direction doubles the number of intervals per row.

The actual top row (Mode 32) is equivalent to 32afdo (Fig.1).

The Genesis scale

When we try to visualize Harry Partch's 43 note Genesis scale by mapping it into tonal space, we even have to resort to a Mode 81 overtone scale (Fig.2).

Fig.2: Mapping Harry Partch's Genesis scale into tonal space

It is worth noting that the Genesis scale covers all intervals of tonal space from Mode 1 to Mode 11 without a gap (with the exception of intervals containing prime factors of 13 or greater, which was a design choice). The marker for these 11-limit intervals (Fig.2) is a small plus sign (+). The ♦-markers indicate Partch’s multiple-number ratios beyond the 11-limit, x-markers indicate unimplemented intervals.

Horizontal extension of tonal space (interval axis)

Horizontal tiling

Since many overtone scales are defined as octave repeating, identical tiles placed along the horizontal interval axis can illustrate the use of cross-octave intervals or the possibility of rendering intervals in the next higher (blue frame) or lower (green frame) octave (Fig.3).

Fig-3 Extending 0847g tiles.png
Fig.3: Three copies of a Mode 16 Horizon Chart along the interval axis

Extending tonal space to the right

As we know, interval spacing actually gets narrower in the next higher octave of an overtone scale. For a more natural appearance of adjacent octaves, we can shift the two corresponding frames (e.g. orange and blue) vertically by one octave (Fig.4). This even leaves room for the more dense intervals of the 6th octave in the upper right corner.

Fig-4 Extending 450 1-32 2oct.png
Fig.4: Two octaves view of tonal space

The slanted fine blue lines connect intervals that share a common numerator, since Harry Partch also known as utonalities (for example starting at harmonic h6: [math]( \frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3}, \frac{6}{2})[/math] ).

Fig.5: 11-limit tonality diamond mapped to tonal space

To the editor's surprise, Partch's 11-limit tonality diamond can be mapped seamlessly to tonal space right at the boundary between two octaves (Fig.5).

Extending tonal space downward

By extending tonal space downward, we enter unfamiliar territory.

Remember that the frequency ratio r of the m-th element of an overtone scale is...

[math] r=\frac{n+m}{n}[/math] , where
  • r is a rational frequency ratio
  • n is the mode of an overtone scale, typically an integer
  • m addresses (indexes, counts) the elements of each overtone scale in horizontal direction from the tonic (left, starting at 0) to the right.

So far we have normalized the tonic of any mode n of an overtone scale to 1.0 by dividing n by itself, such as
[math] r=\frac{3}{3}[/math] = 1.0 for Mode 3 and [math] r=\frac{2}{2}[/math] = 1.0 for Mode 2 and so on. The result of these operations is that in tonal space all modes refer to a common tonic of 0 cents.

Obviously, we run out of integers, when we try to address modes of overtone scales smaller than 1. Mode 0 is undefined, and negative mode numbers make no sense. A possible solution to this problem is to represent the octave numbers in exponential form such as

23 for the fourth octave1) (Mode 8)
22 for the third octave (Mode 4)
21 for the second octave (Mode 2)
20 for the first octave (Mode 1)

1 Note that by convention, the octave number is 1 greater than the exponent

The next logical steps for the area of tonal space below Mode 1 are

2-1 for a fractional Mode 1/2 = 0.5 2)
2-2 for a fractional Mode 1/4 = 0.25

2 The exponent [math]x[/math] for an arbitrary given fractional mode can be calculated as [math]x=\frac{1}{ln(2)}\cdot ln(n)[/math]
So we extend the plane of tonal space downward by fractional modes < 1 and enter the field of tritaves, pentaves and doubled octaves (also known as tetraves).

Fig.6: Extending tonal space downward

Tritave

In a Mode 2-1 (=0.5) overtone scale the first interval above the fundamental is the tritave (3/1, Fig.6), which substitutes the octave (2/1), the first interval in the Mode 1 overtone scale. Incrementing m in steps of 1, we find only odd harmonics (e.g. 3, 5, 7, 9, 11) and no octaves in this fractional mode of the overtone scale.

[math] r=\frac{0.5+1}{0.5}=\frac{3}{1}[/math]
[math] r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 1902[/math] ¢

Pentave

In a Mode 2-2 (=0.25) overtone scale the first interval above the fundamental is the pentave (5/1), which substitutes the octave (2/1), the first interval in the Mode 1 overtone scale. This fractional mode of the overtone scale contains no octaves. It contains only odd harmonics with a spacing of 4 (e.g. 5, 9, 13, 17, 21, 25).

[math] r=\frac{0.25+1}{0.25}=\frac{5}{1}[/math]
[math] r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} \approx 2786[/math] ¢

Double octave

In a Mode(1/3) overtone scale the first interval above the fundamental is the double octave (4/1), sometimes called the tetrave. It substitutes the octave (2/1), the first interval in the Mode 1 overtone scale. This fractional mode of the overtone scale contains every second octave. Octaves with 1200 ¢, 3600 ¢,… are not present in this mode. It contains harmonics with a spacing of 3 (e.g. 4, 7, 10, 13, 16).

[math] r=\frac{\frac{1}{3}+1}{\frac{1}{3}}=\frac{4}{1}[/math]
[math] r_{cents}= ln(r)\cdot\frac{1200}{ln(2)} = 2400[/math] ¢

To find the exponent x for the mode number in exponential form we solve [math]2^x\,=\frac{1}{3}[/math] for x:

[math]x = \frac{ln(\frac{1}{3})}{ln(2)} \approx -1.585[/math]

Find out more about tonal space…

Part I: Expanding tonal space

Part III: Projections

See also...

Otonal sequence (OS)