On this page, I will explain some rules which are applicable when we calculate with intervals and their frequencies.
Intervals are fractions
An interval ranges from a lower tone to a higher one. The fraction of the interval is calculated by dividing the frequency of the higher tone by the frequency of the lower tone, for instance
E = 330 Hz
A = 440 Hz
440/330 = 4/3. This is a fourth. The interval of the fourth is always 4/3: in the fourth, the higher tone is precisely 4/3 times as fast as the lower tone.
What counts here are only the relative values, not the absolute ones. Whether I start the fourth with the E (E-A) or with the C (C-F) is immaterial; the relative frequency ratio is always 4/3. In other words, intervals are always relative.
The exponential progression of the frequencies
When we compare the intervals with each other, there is a crucial feature: the progression of the frequencies is exponential.
Fig. 1: The progression of the frequencies is exponential
In Fig. 1, you can see the frequencies of various A tones, from the great A (capital letter) to the treble a” (small letter with two inverted commas). On the piano keyboard, it looks as if the distances between all four As are the same, but if we look at the frequencies, the distances become increasingly bigger. In other words: the frequencies increase faster than the intervals. In mathematical terms, the intervals progress in a linear fashion whereas the frequencies progress exponentially. This has some consequences for calculations with intervals.
Adding intervals
When we add two intervals, then this is a multiplication with regard to frequencies. Thus the tone of the (great) A has the frequency of 110 Hz. When we move up an octave, then the (small) a has twice that frequency, namely 220 Hz. The distance between 110 Hz and 220 Hz is 110 Hz. Yet this 110 Hz is only an octave if we start with the capital A. If we move up another octave from the small a, we must not add the 110 Hz of the lower octave (which would result in 330 Hz), but have to add 220 Hz, thus getting from 220 Hz to 440 Hz.
Our spontaneous idea that an octave corresponds to a value in Hz is incorrect. An octave means that the lower frequency is multiplied by 2 (with 2 because the octave always doubles). The addition of the intervals becomes a multiplication. At first sight, this change in arithmetic operations may be confusing, but once we’re aware of it, the matter is not so difficult. We must therefore remember:
the addition (of intervals) becomes a multiplication (of frequencies)
Subtracting intervals
Unsurprisingly, subtractions work in perfect analogy. Subtractions become divisions.
Example
We are looking for the distance between a major third and the fifth above it. Our knowledge of music tells us that the distance between the two intervals is a minor third. Can we also calculate this?
In this comparison, we subtract the major third from the fifth. But instead of subtracting, we divide:
Fifth = 3/2
Major third = 5/4
Fifth – major third → 3/2 : 5/4 = 3×4 / 2×5 = 12/10 = 6/5
As we know, 6/5 is the minor third. This method always works, for any interval:
To check, we can add the two thirds again and the result – of course with a multiplication – is:
5/4 x 6/5 = 30 / 20 = 3/2
The major and minor thirds again result in the fifth (3/2) in this way.
The advantage: we are able to reduce!
The shift from addition and subtraction to multiplication and division has the advantage that with fractions we are able to reduce, as shown in the above examples, we are able to reduce.
This has a direct impact on the resonances: whenever we are able to reduce, the numbers in the fractions become smaller – and small numerators and denominators in the fractions are an advantage for resonance. This also explains why we prefer not to have any prime numbers higher than 5 in the intervals. Non-prime numbers such as 6, 8, 9, 10, etc. can be reduced, which is why we can find a major second (9/8) in common musical scales, but no intervals with the fractions of 7/4 or 8/7.
Relating scale tones to each other
When we compare two tones within a musical scale in order to decide whether they resonate with each other, we always relate them to their shared fundamental tone. This is essentially connected with the character of musical scales (and chords). All tones of a musical scale are based on this one fundamental tone (tonality).
Since the intervals are always relative, the crucial factor is not the absolute frequency of a tone, but its relation to the fundamental tone. When we set the
fundamental tone = 1,
all scale tones are described as relations of their frequency to the frequency of the fundamental tone.
Distance between two scale tones
What, for example, is the relationship between the fourth and the fifth? The fourth is 4/3 above the fundamental tone, and the fifth is 3/2. If we want to compare these two tones, we can calculate the distance between them. The distance is a subtraction, and with frequencies, a subtraction is a division. We divide the higher tone by the lower one, and the result is: 3/2 : 4/3 = 9/8.
9/8 is a major second; the distance between the fourth and the fifth is a major second.
Resonance between the fundamental tone and two scale tones
Whenever we look at two scale tones, they always have the fundamental tone in their foundation. The fundamental tone determines the tonality, and the tonality places the musical scale on an absolute basis.
But how do the three tones mix – fundamental tone, fourth and fifth?
For resonance to be generated, all three tones must stand on a shared base, or mathematically speaking, they require something like a shared counting time for all three frequencies, i.e. for the fundamental tone (1), the fourth (4/3) and the fifth (3/2).
For this purpose, we look for the smallest denominator which tallies for all three figures. In the example, this is the denominator 6:
Fundamental tone 1 = 6/6
Fourth 4/3 = 8/6
Fifth 3/2 = 9/6
This common denominator is always the lcm, the least common multiple of all the individual denominators involved.
A further example
Fundamental tone 1 = 15/15
Minor third 6/5 = 18/15
Major sixth 5/3 = 25/15
The lcm of 1, 5 and 3 is 15.
What does this common denominator of several tones mean?
I propose the hypothesis that resonance occurs more easily, the smaller this common denominator is. For our consideration of resonance, the following applies on the basis of this hypothesis:
Tones which have a low common denominator mix easily.
The higher the common denominator, the smaller the internal resonance of the tones.
The consequences of these conclusions are dealt with in the text about specific musical scales and chords.
This is where you will find an overview of the texts about the theory of the three worlds.