**"...the
aim and final reason...of all music...should be none else but the
Glory of God and the recreation of the mind." (J.S.Bach)**

**1
- Preamble **

Judged simply as a
number, 1.05946309... is hardly likely to stir the emotions. Yet
it lies at the heart of that which now - in one form or another -
possesses this peculiar quality in abundance. In an age in which * naturalistic
speculation* in high places is presented as

It is the purpose of this page to draw attention to some of Western music's objective characteristics so that the reality of its close links with our Creator, Jesus Christ, and with the Act of Creation, may become clear.

**2
- The Piano Keyboard**

Our formal musical education may well have involved a keyboard of 6 or so octaves, as represented here:

A cursory study of this layout reveals a number of simple facts, viz

the repeating pattern of 12 keys - 7 white, and 5 black - is termed an

octavethe

sounded when any key is depressed is said to possess a certainnote- this related to the principalpitchof the string(s) associated with that particular keyfrequency of vibration

these frequencies

at each octavedouble

the ratio of the frequencies of adjacent notes is uniform, and hence equal to the

- or 1.05946309...twelfth root of 2

this indivisible step in pitch is termed a

semitone

each complete octave therefore comprises 12 semitones, or 6 tones

within each octave, the 7 white notes are named using the alphabetic sequence A - G

These facts are assimilated in the following diagram where bottom

has been taken as theC- having an assumed principal frequency ofreference noteunits:f

Assuming all notes to have been numbered sequentially from the point of reference (bottom C), a frequency may be assigned to each note that begins a subsequent octave. At the same points, the current counts are shown - for the semitones (12 per octave);for the tones, or semitone-pairs (6 per octave); and for the notes which participate in the diatonic scale of C major (in this case, all the white notes, totalling 7 per octave). Clearly, these relationships hold good throughout the keyboard and are completely independent of the choice of starting position.

Readers of earlier pages may observe that there are some familiar numbers represented here.

Highlighted in yellow, at the sixth octave, we find the conjunction {64, 73, 37, 43}, and are reminded particularly of the following 'creation geometries':

Observe that,

(a) in what must be considered a

typicalrepresentation of this cube, one or more faces of precisely 37 of the 64 component unit cubes are visible; and that the product of what is 'visually apparent' by what is 'actual' is 37 x 64, or 2368 - the Creator's number(b) 37 x 73 - another product of related figures - generates 2701, or Genesis 1:1

The only non-geometrical item, 43, is highest prime factor of 86 - value of 'elohim' (i.e. God), the 3rd word of Genesis 1:1 - representing a further reference to the Creator

Highlighted in yellow, at the third octave, we find the conjunction {8, 37, 19}, and are reminded of the following:

Observe that,

(a) these two primes, 19 and 37, are the factors of 703 - 37th triangular number and sum of words 6 and 7 of Genesis 1:1

(b) the product (8 x 37) = 296 = 7th word of Genesis 1:1 and factor of both the Lord's Name and Title, viz 888 and 1480; it is also distinguished by the fact that it represents the difference between the cubes of 8 (= 512) and 6 (= 216)

Further details of these conjuctions may be found here.

Highlighted in green, at the fourth octave, we find the conjunction of three squares, 16, 25 and 49 - the two latter marking the position of the centroid counter of the Genesis 1:1 triangle, 2701; further, 25-as-rhombus and 49-as-rhombus are generators of the hexagon/hexagram pairs 19/37 and 37/73, respectively

Highlighted in green, at the second octave, we find 7 and 13 - the factors of 91, sole trifigurate companion of 37

Further details of these conjuctions may be found here.

**3
- Western Music: a concise background**

It was Pythagoras (ca 600 BC) who first realised that music, at its deepest level, is mathematical in nature. He saw that a musical scale derives from the mapping of certain proportions onto the indefinite continuum of musical pitch. Thus, as observed above, the eighth of a rising succession of notes in any major or minor scale will be found to have exactly twice the frequency of the first and, sounded together, they are said to be

in unison. The interval of pitch represented is called 'an eighth' or 'octave'. Intermediate intervals are similarly described, e.g. that between the first and fifth notes is called 'a fifth'. In an idealnatural scalethe frequency ratios arevulgar fractions, thus:

The 'Just Intonation' or 'Natural' ScaleThe notes of the scale are named and numbered in the usual way and, by setting C = do, the scale may be played entirely on the white notes of a keyboard. Irrespective of absolute pitch, or key, a succession of notes having these frequency ratios will be instantly recognised by the normal human ear as a

. The relationships, of course, extend above and below the selected octave - the same ratios relating to the corresponding octave notes. Any attempt to to play a major scale from some other point in this sequence must fail because the succession of intervals is then different. Indeed, to allow this scale to be played in any other key would require the availability of very many additional notes. Clearly, the limitations imposed by the natural scale do not permit harmony and modulation in the fullest sensemajor scale*. Hence, early music - so strange-sounding to the modern ear was largelymonodici.e. consisted of a single melody line; it recognised a system ofwhich provided the basic material of Western music well into the Middle Ages.modes

* However, it is interesting to observe that a group of choristers singing without instrumental accompaniment will tend to harmonise automatically using these natural intervals; similarly, instruments that can bend pitches enough to fine-tune them during a performance - and this includes most orchestral instruments - also tend to play "pure" intervals. But for many instruments that cannot be fine-tuned quickly - such as piano, organ and harp -tuningbecomes a big issue.It is helpful to present the foregoing data in the form of a 'Log Frequency/Time' graph, as follows:

Observe that the steps 'mi-fa' and 'ti-do' are approximately half those of the remaining 5 intervals. A logical development (which reputedly occurred in the late Sixteenth Century) was to halve each of these wider intervals so as to divide the octave into a

of 12chromatic scale, thus:semitonesAnd so our familiar keyboard was born!

A glance at the rightmost column, however, reveals that the division was not uniform; in other words, the frequency ratio between successive semitones varied. Thus, the moves to free music from its natural 'straightjacket' had also to include adjustments to the frequency ratios in this 'chromatic' scale so that one could play equally well in all keys.

Many so-called

tunings were devised to achieve this - the aim always being to minimise the inevitable clashes, whilst yet retaining the individual 'character' of each key. For example, in respect of Bach'smeantoneThe Well-Tempered Clavier- two sets of 24 Preludes and Fugues in all major and minor keys - it is highly probable that the form of tuning known as 'Werckmeister III' provided a satisfactory basis.

[Readers interested in the finer points of instrument tuning will findNigel Taylor's tuning pagea rich source of information]

Equal temperament- the ultimate, bland,of the 12 pitches of the octave - is essentially a Twentieth-Century phenomenonequal spacingand, in Western music, is now considered the- even for voice and those instruments capable of being played in just intonationstandard.Observe that in equal temperament, the onlypureintervals are the octaves. This form of tuning is well suited to music that changes key often, is very chromatic, or is harmonically complex. Nevertheless, because it lacks pure intervals, and because the individual character of each key is lost, many musicians find it unsatisfying.In the following diagram, a comparison is made between natural and equal temperament tunings - the numbers representing the frequency ratios between major scale notes.

Observe that whilst most of the intervals match reasonably well, there are problems with the 6th and 7th degrees of the scale, where errors approaching +1% are incurred.

Expressed
mathematically, the audio frequencies associated with any
equally-tempered chromatic scale form a **geometric*** progression *- first term (

**4
- The Geometrical Analogues**

3.1 - Just Intonation - IThe principal frequency ratios of the natural scale may be expressed very simply and elegantly in geometric terms, as follow:

A geometrical expression of the natural scale ratiosThe basis of the analogy is the symmetrical hexagram (

Y) constructed from 12 identical equilateral triangular tiles. Central to this figure is the symmetrical hexagon (X) - comprising 6 of these elements. The large triangle (T), and the rhombus (R), which are each capable of generatingX/Y(by self-intersection/self-union, following one rotational step of 180 degrees, or two of 60 degrees, repectively) incorporate 9, and 8 of these elemental tiles, respectively.Linking these observations with the just intonation ratios provided above, we may write:

Y:X = 2:1 = the eighth, or octave

T:X = Y:R = 3:2 = 1.500 :1 = the fifth

R:X = Y:T = 4:3 = 1.333 :1 = the fourth

T:R = 9:8 = 1.125 :1 = the second

Clearly, we witness here a marked correspondence with the chief intervals of the natural scale.

3.2 - Just Intonation - IIBut such geometries may also appear in an alternative 'digitised' form - the elemental triangles now replaced by standard representations of

triangular numbers. Consider, for example,In this illustration, observe that the 12 elemental triangles of the hexagram (

Y) - each of 6 units - are set around a single 'centroid' counter. Clearly, the totals of units for this and the associated figures are as follows:Y = 12x6 + 1 = 73; X = 6x6 + 1 = 37; T = 9x6 + 1 = 55; R = 8x6 + 1 = 49

and their ratios:

Y:X = 73:37 = 1.973 :1 2:1 (error = - 1.35%)

T:X = 55:37 = 1.486 :1 3:2 (error = - 0.93%)

R:X = 49:37 = 1.324 :1 4:3 (error = - 0.70%)

T:R = 55:49 = 1.122 :1 9:8 (error = - 0.27%)

Clearly, these errors are already of the same order of size as those between the scales of equal temperament and just intonation; with the introduction of larger elemental triangles, they may be reduced - without limit.

3.3 - Equal TemperamentThe final analogue concerns the scale of equal temperament (ET) which, as already observed, has become the universal standard tuning procedure. However, before turning to that let us clarify any doubts there may be concerning the two alternative ways of generating of

X/Ypairs - symmetrical figures which have already featured large in this analysis. The matter is discussed here..A principal feature of ET is that the octave is divided into 12 equal, indivisible, steps of pitch - the semitones - these, the subjective interpretation of a geometric progression of frequencies, common ratio, (=1.05946309...). This is closely approximated by the rational number 196/185 (= 1.05945945...) which differs from the true semitone ratio by a mere 0.00034% (and therefore negligible when compared with the inherent errors of ET discussed earlier).

Now 196/185 is a fraction of some considerable interest, for observe:

Its numerator, 196 = 2^2 x 7^2 = 14^2 = 4 x 49 = 7 x 28 = 7 x 7th triangular number= (order of triangle generating 37-as-hexagram) x (that triangle)

= (No. of Words in Genesis 1:1) x (No. of Letters in the same verse)

Its denominator, 185 = 5 x 37= (order of rhombus generating 37-as-hexagram) x (that hexagram)

= [(sum of words 1 to 5) - (sum of words 6 to 7)] / 7

= [1998 - 703] / 7 = 1295 / 7 = 185

For convenience, the geometries referred to above are reproduced here:

At (a) and (b) we have the figures capable of generating the particular hexagram (c)

It is therefore abundantly clear that with the introduction of the scale of equal temperament in the Twentieh Century and its adoption as the international tuning standard came these powerful links with the elements of numerical geometry, and with the opening words of the Judeo-Christian Scriptures.

**5
- Concluding Remarks**

The more one
contemplates the phenomenon of music, the more remarkable it
appears. It not only enables man to express his many moods and to
pour forth his innermost thoughts and emotions in a natural way
but, as we have seen in this brief anaysis, it enables him
through reasoned argument to know his true roots. Is man God's **Special
Creation***,* as the Bible maintains, or is he
rather a * cosmological/biological accident*?
The answer now seems clear enough! Music reveals this
much-disputed truth. In itself, of course, music is

Vernon Jenkins MSc

2007-07-07

**email: ****vernon.jenkins@virgin.net**