A Treatise on the neddiH dlroW of Intervals
© 1992 Paul Cooijmans B., B.

1 Tones and Intervals
Twelve tones per octave over about ten octaves give more than a hundred tones.

If, as is common, we assume tones with exactly an octave between them are related, twelve essentially distinct tones remain:
C   C#   D   D#   E   F   F#   G   G#   A   A#   B

In terms of intervals between tones, simplification can be taken further still:
C' - E' - C''

The intervals C' - E' and E' - C'' for instance are different in size but related because they consist of essentially the same tones (note: C'' is the tone one octave above C'). When all intervals are examined in this sense, seven essentially distinct groups remain:
[1] Prime - Octave
[3] Perfect Fourth - Perfect Fifth
[5] Major Third - Minor Sixth
[6] Minor Third - Major Sixth
[7] Tritone
[9] Major Second - Minor Seventh
[12] Minor Second - Major Seventh

The numbers in square brackets are meant as "shorthand" for the seven groups, and based on the first occurrence of those intervals in the series of natural harmonics. E.g., a major third first occurs at the fifth harmonic.

2 Interval Combinations
In music, intervals are combined with other intervals. These combinations can be written using the aforesaid shorthand, e.g.:
[7, 1]
[9, 5, 3]
[6] or
[12, 9, 7, 6, 5, 3, 1]

127 combinations are possible, 2 of which are considered below.

[9, 7, 5, 1] - A set of tones containing this interval combination is:

The [9] is between the a and b, the [7] between the f and b, the [5] between the f and a, and the [1] occurs three times: between the f and itself, between the a and itself, and between the b and itself.

For the combination [5, 3, 1] this is different. No set of tones contains exactly these intervals.

When all 127 interval combinations are examined, it becomes apparent that only 31 thereof can occur in a set of tones; the remaining 96 are theoretical.

A fascinating discovery: there are 96 imaginary interval combinations - 96 chords that have never yet sounded!

Below is a schematic chart of the 31 combinations that may sound; in vertical notation, and the [1] has been left out in each combination for conciseness (except in the actual combination [1] itself).

And here the remaining combinations; again, [1] has been left out, so each set represents two interval combinations, one with and one without [1]. All the below combinations are imaginary, either with or without [1]. The combinations in the above chart are imaginary without [1] and real with [1].
 [7, 6, 5, 3] [9, 7, 5, 3] [9, 7, 6, 3] [9, 7, 6, 5] [12, 7, 5, 3] [12, 7, 6, 3] [12, 9, 5, 3] [12, 9, 7, 5] [12, 9, 7, 6] [7, 5, 3] [7, 6, 3] [7, 6, 5] [9, 5, 3] [9, 6, 5] [9, 7, 3] [9, 7, 6] [12, 6, 3] [12, 7, 5] [12, 7, 6] [12, 9, 3] [12, 9, 5] [12, 9, 7] [5, 3] [6,3] [6, 5] [7, 3] [7, 5] [9, 6] [9, 7] [12, 3] [12, 6] [12, 7]

3 Relations

[9, 3, 1] [9, 3, 1]

Above the intervallic content of two sets of tones is given. But that's not all that is going on; between sets of tones are intervals too:

[12, 7, 5]

When written as an interval combination, this is one of the "imaginary" ones, as one can see in chapter 2. The example [5, 3, 1], found to be imaginary in chapter 2, may in fact occur as a relation too:

Further research shows all 127 combinations may occur as relations. So the 127 fall apart into 31 that may occur either in a set of tones or as a relation between sets of tones, and 96 that can only be relations.

The 96 play their role in music in a hidden way; never heard as actual sound, they exist in the infinitely short time slice between two sets of tones, deciding how we perceive the next sound. 4 Refining the notation
Intervallic content may also be expressed like this:

 Chord Relation Chord 0 x [12] 4 x [12] 0 x [12] 1 x [9] 0 x [9] 1 x [9] 0 x [7] 3 x [7] 0 x [7] 0 x [6] 0 x [6] 0 x [6] 0 x [5] 2 x [5] 0 x [5] 2 x [3] 0 x [3] 2 x [3] 3 x [1] 0 x [1] 3 x [1]

A shorter form of this notation is obtained by giving only the occurrence frequencies of each interval type:
(0, 1, 0, 0, 0, 2, 3)
(4, 0, 3, 0, 2, 0, 0)
(0, 1, 0, 0, 0, 2, 3)

As occurrence frequencies are mostly in the 1-digit range, commas, zeros on the left side and parentheses may be left out for shortness, resulting in numbers:
100 023
4 030 200
100 023

In the case of chords, this number indicates the amount of dissonance of the chord, as demonstrated below:

5 The Hidden World
The most important level of meaning of this notation can be found by taking the occurrence frequencies of the interval types as coordinates in a grid. For example, consider the combination 12 - in full, (0, 0, 0, 0, 0, 1, 2). This means 2 times [1], 1 time [3], and absence of the other five types. As only two interval types actually occur, we need only draw two axes of the coordinate grid:

The point 12 (in the graph vertical notation is used with the [1] occurrence at the bottom) corresponds to the interval combination 12. Similarly, all interval combinations can be taken as points in the coordinate grid. Distinction between sets of tones and relations vanishes.
6 Surveying
For the relevance of the coordinate grid, see the following examples:

This musical example corresponds to a journey from 01 to 10 and back again. We now calculate the distance covered; according to Pythagoras, this is the root of the sum of squares of the coordinate differences. So in this case:
√(1+1) = √2

As this distance is covered twice, the example corresponds to a journey of 2 √2 interval units in the interval world. One more example, with calculation of distance shown step by step:

100 023 (chord)
4 030 200 (relation)
100 023 (chord)

Vertical notation, chord to relation:
0 4
1 0
0 3
0 0
0 2
2 0
3 0

Differences:
0 - 4 = -4
1 - 0 = 1
0 - 3 = -3
0 - 0 = 0
0 - 2 = -2
2 - 0 = 2
3 - 0 = 3

Squares of differences:
-42 = 16
12 = 1
-32 = 9
02 = 0
-22 = 4
22 = 4
32 = 9

Sum of squares of differences:
16 + 1 + 9 + 0 + 4 + 4 + 9 = 43

Square root thereof:
√ 43

Multiply by two because relation to second chord gives same result:
2 √43

Now a longer chord progression with distances given between each two chords:

And finally the distances between the twelve diatonic scales; as seen below, these correspond to their position in the "circle of fifths".

7 Conclusion
The examples in chapter 6 show that when occurrence frequencies of the 7 interval types in chords and relations are taken as coordinates in a 7-dimensional grid, the distances from chord to relation to chord in that grid correspond to the amount of discordance perceived in that chord progression.

End of Treatise

Supplement 1: Analysis of J.S. Bach's Prelude I
Supplement 2: Qoymans Intervallic Converter - a program that executes the calculations explained above