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17 November

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Scales


The scale used extensively in the West has 13 notes from octave to octave and 12 intervals. In order for a scale to 'work' there should be:
* a minimum of dissonance when different notes across the whole range of pitches are sounded together
* an effective mapping of the harmonics of low notes onto higher notes, and an effective mapping onto the harmonics of higher notes
* the possibility of key modulation which does not result in further frequency mismatches.
The scale which has been widely adopted to fulfil these criteria is based on mathematics, such that the ratio of the frequency of any note to the frequency of the note a semitone above is constant. This is particularly useful in the extent to which it allows key modulation. However, this 'equitempered' scale is a compromise solution, because the frequency ratios of all intervals except the octave differ slightly from the 'perfect' intervals that the human ear really expects to hear.
The 'exact' interval of a fifth, for instance, is found by multiplying the frequency of the fundamental by 3/2, the fourth is found by multiplying the fundamental frequency by 4/3, and the major third interval is found by multiplying the fundamental frequency by 5/4. (Other intervals involve slightly less obvious fractions).
The problem with a scale built on fractional values like this, however, is that the increments from note to note are not constant (eg 5/4 - 4/3 does not equal 4/3 - 3/2) which creates difficulties when the required key for a piece is different to that of the fundamental from which the scale is constructed. For example, if we move up an octave from C by adding a fifth, and then adding a fourth, then the resulting high C will have a different frequency to that arrived at if our key is F, and we try to arrive at the same high C by adding a major third and then a minor third to that fundamental F. So in the equitempered scale all semitone increments have been 'tempered' such that they are always a little flat, or a little sharp.
The constant value on which this scale is based is 1.0594630915, such that if we call this value S, then the semitone above a fundamental note is found by multiplying the frequency of the fundamental by S to the power of 1.
The second above the fundamental is found by multiplying it's frequency by S to the power of 2, and so on until the octave above the fundamental is found by multiplying it's frequency by S to the power of 12. The value of the constant S is the 12th root of 2, since in order to find the twelve equal divisions between two notes an octave apart, where the frequency of the octave is twice that of the fundamental, the 12th root of 2 is the value we are looking for.
Other divisions of the octave have been proposed, such as a 19-step octave, and a 53-step octave. The maths for these 'works' although these 'scales' may be harder to use effectively. The maths for the 53-division scale is particularly elegant in fact, and closer to a 'perfect' musical scale than the 13-note scale which we currently use. (In that case the constant value for each successive interval is found by using the 53rd root of 2, ie 1.013164143).
The 'exact' scale, built on the 'perfect' intervals that the ear expects to hear, has much to recommend it if one key is kept to. This scale, however, has fifteen intervals and fourteen notes, since in the first octave there are all the notes of the equitempered scale (at slightly different frequencies) but there is also both a 'major whole tone' and a 'minor whole tone', and both an 'augmented fourth' and a 'diminished fifth'. (In the second octave there is both an 'augmented octave' and a 'dimished ninth' and also both an'augmented eleventh' and a 'diminished twelfth'). Thus successive octaves above the fundamental differ from each other in the way that they are put together. Furthermore, however, when we look at the extent to which the frequencies of harmonics of exact-scale notes match notes higher up in the exact scale, we see that we can list the intervals octave, fifth, fourth, major third, major sixth, minor third, minor sixth in terms of increasing dissonance, so that in the case of the minor sixth, if we look at all harmonics up to the twelfth, only one 'matches'.
The mathematical elegance of the 53-division scale should make it a more appropriate scale for dealing both with key modulation, and a preoccupation with harmonics.
The 53-interval scale uses eneven 'chunks' of these 53rd-of-an-octave division to create the notes of the diatonic scale. The size of each incremental chunk is as follows:
C->D: 9
D->E: 8
E->F: 5
F->G: 9
G->A: 8
A->B: 9
B->C: 5
and with the semitones:
C->D: 4+5=9
D->E: 4+4=8
E->F: 5
F->G: 4+5=9
G->A: 4+4=8
A->B: 4+5=9
B->C: 5

The Physiological Effects Of Sound


The accepted view of researchers into the physiological effects of sound is that 'no non-auditory [ie physiological] effects are noted until the loudness exceeds approximately 120Db. 120Db is VERY loud, in fact it's at the limit of what can be heard before physical damage is caused to the ear. However, research has been carried out into the extent to which vibration at frequencies within the audible range can be transmitted through the body. Different parts of the body have differing optimum resonance frequencies. Some of these are: (The first one is conjecture, the rest are documented).
The eyeball: 5Hz. (Low frequencies such as this are known as 'infrasound').
The jaw: 6-8Hz.
The chest, nose and throat cavities: somewhere in the region of 10-75Hz.
The whole skull: 200Hz.
The front and the back of the skull: 800Hz, where the front and back parts vibrate in opposite phase.
The front, left side, back and right sides of the skull: 1600Hz. At this frequency each of the four sides vibrates independently of the others. The exact frequencies for skull resonance vary from individual to individual due to variations in skull size, however; in fact all values given here can only be approximate averages.
The bones of the middle ear (the 'ossicular' system) resonate at 2000Hz.
The air within the middle ear resonates at 2500Hz.
The resonant frequency of the outer ear is 3150Hz. (Sounds in the surrounding frequency range, from about 3000Hz to 3500Hz are amplified several times by this effect).
There is some evidence to suggest that the middle ear when exposed to ultra-sound (ie sound above the threshold of audible frequencies) creates subharmonics within the audible range.

 

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