Monthly Archives: October 2015

How Twelveness creates Harmony


12-ness is the product of 3 and 4, three being a prime number and 4 being the first square number, of two squared. One of twelve’s most profound manifestations is found in musical harmony because there is, to the human hearing of an octave, a natural sense of 12 semitones organised as 5 tones and two semitone intervals. The semitones can fall in different places within the octave of a given reference tone, commonly called do, to define the mode (as in C Major and D flat etc) and in different modes the semitones are always separated from each other by at least two tones or not more than three tones, when the octave is viewed as a tone circle. When ancient and modern tuning methods are considered*, these all reveal a unique role for 12 but then as the ratio 3 over 4 or 4 over 3 where the denominator times the numerator causes the notes to fall according to a 12 fold modulus because 3 times 4 equals 12. However, it is the relationship of the octave to the fifth (1:2 to 2:3) which enforces 12-ness in Harmony.

* the cycle of fifths, diatonic, and its embellishment, Just intonation.

Tuning by Fifths

When viewed numerically, the technique of tuning by fifths naturally projects a series of new tones, upwards or downwards, in a way that will exceed or decede the range of a single octave.

The rising tone interval of a fifth is 3 over 2 so that downwards will be the falling fourth of 4 over 3. But in practice the fifth must often be 3 over 4 to stay within the single octave model* just as the falling fourth must sometimes be 3 over 2, the octaval number 2 being applied to translate tones back into the octave.

* Two out of three applications of the fifth =9/4 to be reduced to
9/8, the whole tone difference between two successive fifths.
The reverse is inevitably true of successive fourths which
require 4/9 to be reset as 8/9, effectively making 4/3 into 2/3.

If an octave is populated with fifths opposed fourths, up to 13 semitones appear in symmetry about the starting point, between 12 tones, to obtain Pythagorean chromatism. The Pythagorean heptatonic requires three ascending intervals (3/2 x 3/4 x 3/4) and descending intervals (4/3 x 4/3 x 2/3) either side the reference tone. This means that octaval resetting is DEFINED within such tuning schemes by the modulus of 3 x 4 equal to mod(TWELVE). When 3 is divided by 4 there is a common unit of 1 over 12 dividing the turning circle and therefore a modulus of 12 is inherent but it is not quite clear that is really the real cause.

Just Intonation

The Modulo 12 effect continues to operate even when the interval ratios employ the prime number 5 and hence the tuning operator 5 over 4 (major third) is used, instead of the fifth and fourth, and from this usage new differential intervals, a Just tone and semitone, emerge, deviating very slightly from the same fundamental pattern of 12 semitones. From a modal perspective, this gives rise to 5 tones + 2 semitones. It is because the resulting tones differ little from the Pythagorean tones (by just 81/80), that the modulo 12 operates with Just tones, then approximating 2 ET units and semitones approximating 1 ET unit.

Equal Temperament

It is therefore no surprise that equal temperament has chosen the 12th root of 2, the octave, to create a perfect compromise between historical tuning methods, so that the ET units just mentioned are perfectly followed and hence the modulo 12 equal semitones are to be seen (geometrically), equal in the (logarithmic) tone circle used in Ernest G McClain’s work and hence in Harmonic Explorer’s display. The circle has the circumference of one octave, perceived logarithmically as from a log(do1) equal 1 to a log2(do2) equal to 1200 cents and each semitone 100 cents.

The 12 ET semitones were designed around the cycle of fifths and other tuning systems adapted from it such as Just intonation, which employs the number 5 to better approach equal divisions of the octave into five notes and two semitones.

Harmony within the Earliest Numbers

The unique nature of the fifth (3/2), employing the earliest two numbers, equates (as stated) to 3 over 4 in practice, when employed to suit the simplest interval of all, the octave (2/1). Ernst Levy, called the first six numbers (1:2:3:4:5:6) Senarius since they define the five primary intervals of octave: fifth: fourth: major third: minor third. These can be seen clearly seen in Harmonic Explorer’s (HE’s) tone circle for limit=60, where 5 first allows fifth and fourth to be accompanied by Just b and f.

The outer graticule of HE is twelve fold, indicating therefore equal temperament’s twelve semitone intervals. Parallel to this is the number of cents given to each tonal brick in the brick pile organised in tuning orders: horizontal being tuning as if* by fifths and upwards and right, tuning as if by major thirds. The fifth and fourth of G-D-A can be seen to be the first creation of tuning by fifths, first seen at all when the limit is 12, to yield Plato’s World Soul.

* I say “as if” because McClain realised that larger tonal matrices
had to have been the result of a purely arithmetical calculational process,
and that a number of alternative methods evolved in different regions of the ancient world.

The number of cents in A (=3/2) is 702, just 2 greater than seven ET semitones, and it is this which creates the modularity of the octave.

One can study modularity by dividing a circle into N equal portions, and HE has its tone circle displaying 12 divisions. Moving to A one can move a further fifth (3/2) then bringing the E produced back into the octave with 3/4 instead. The deviance from ET is then 1404 cents minus 1200 (the octave) so as to be E = 204 cents (see limit = 864, the Pythagorean heptatonic). Twelve intervals will return to D as 24 cents, a sharp excess called the Pythagorean comma, which has become audible, yet the twelvefold pattern of fifths is a dominant pattern, unbreakable exactly because of the proximity of the fifth to the ET semitone of 700 cents.

It is the proximity of the fifth (3/2) to seven twelfth roots of 2,
which makes the octave (1:2) home to
the modular Twelveness found within Harmony.

Once this pattern is established, between the two (numerically) simplest intervals, Just intonation is able to mitigate the accumulating comma inherent within successive fifths by clearing the accumulation of threes and arriving at new tones 10/9 and 16/15 from the Pythagorean tones, both of these intervals dividing by 3, twice or once, and giving us the familiar Just intonation found in Greek modal music. In the melodic view, a scale is a journey which must end in doubling or halving using seven intervals, five of which must be twice as large as the other two, i.e. 12 units. One must double or halve using 12 units, which is perhaps why J.G. Bennett sometimes gave the number twelve the systemic attribute of Autocracy.