Music as Meaningful Vibrations

by the Editor

Contents List:

Hearing, Noise, and Music
Psycho-Physics
Musical Terms
The Monocord
Mersenne's Laws
Mathematics
Major and Minor
Equal Temperament
Harmonics
Octave Numbers as Powers of Two
Extra-musical Interpretation

Return to:

Supplementary "Lectures"
"Campus"
Temple Library

See also:

Mind and Consciousness
Consciousness, Laws, and Influences
Vibration
Personal Library, Sacred Geometry
The Beginnings of Music and Religion
Math and Music Relating Science to Arts?(Internet)

Hearing, Noise, and Music

We normally take it for granted that we hear with our ears, and are content to let our ears "get on with it". But, like Seeing, hearing is a much more mysterious function than is generally imagined. The ear is only the sensor portion of a highly complex vibration-processing system, most of which operates below the level of consciousness before the results of its labours rise into consciousness as sound. Furthermore, our much-vaunted sciences are unable to explain convincingly why noise can be so infuriating or why music can "play" upon our emotions and by turns make us feel happy or sad, lively or restful.

We may define "noise" as any unwelcome sound. This can include music if one is not in the mood for it, so we need to differentiate music from "random" noise.

For most of my life, I have regretted that my early schooling did not include any formal instruction in music. Although I have always enjoyed listening to various kinds of music, I felt that my inability to appreciate the "finer points" constituted a kind of "deafness" that cut me off from the "light" that musical knowledge could cast on my relationship with the Universe. It was only when a comfortable retirement enabled me to devote some effort to self-education that I was able to start exploring what has proved to be a fascinating "field". Notice how, even in this short paragraph, I have felt obliged to use inverted commas on no fewer that four occasions suggesting that music is a subtly subjective subject that does not easily succumb to prosaic description.

I have learned many unexpected things in the last few years, and shall endeavour to share with you, in language as plain as I can muster, some of those aspects that I have found most helpful in "attuning" myself with the greater Cosmos in which I am a minuscule sub-system.

Psycho-Physics

In general, what "enters" the ear and is ultimately heard as sound is some kind of vibration to which the ear is sensitive. A vibration is a to-and-fro motion of some sort. Vibrations may be regular (i.e. repeated at regular intervals) or irregular i.e. random or meaningless. The rate at which regular vibrations are repeated is known as their frequency, and is measured as so many cycles (occurrences) in each second. The unit of frequency in common use is the Hertz (after Heinrich Hertz, 1857-94, German physicist), usually abbreviated to Hz, meaning 1 cycle per second. Other common units are KHz (1000 Hz) and MHz (1 million Hz).

The period of a regular vibration is the length of time required for the completion of one cycle. Thus the period of a note is inversely proprotional to its frequency.

Humans interpret as sound all vibrations within a range from about 16Hz to about 20 KHz, but it is known that other creatures can "hear" frequencies outside the normal range of the human ear. Frequencies below the human hearing range are subsonic; frequencies above the human hearing range are ultrasonic. Please note the subjective nature of these definitions, which mean little or nothing to anyone who is totally deaf. A vibration becomes sound only after it has been interpreted as such by the human psyche.

Sound propagation is the process by which sound vibrations are transmitted through the air or some other medium.

A source of sound can be any structure or system which produces vibrations within the sonic range of frequencies. Sources of "musical" sounds include the vocal cords in a bird's larynx, a stretched string as in a violin, a column of air as in an organ pipe, a diaphragm as in a drum.

The larger the to-and-fro motion (amplitude) of the vibration, the louder the sound.

Musical Terms

The basic "unit" of music is the note, that is, a sound which is recognised as having a definite pitch. Pitch is determined by frequency. The higher the frequency of a note, the higher its pitch, and vice versa right up to the point at which it becomes ultrasonic in one direction or subsonic in the other direction.

Just as one swallow doesn't make a summer, so one note doesn't make music. A melody is a series of notes of different pitches.

The difference in pitch between two notes is called an interval. Melodic intervals can vary in size. The smallest interval which is readily distinguishable by the "ear" constitutes a step. (The "ear" is here used as shorthand for whatever faculty in the human being recognises the difference in pitch between two notes.)

If all the notes of a melody are collected together and arranged in step order, then if all duplicate notes are deleted, what is left is a scale. Thus a scale may be regarded as a set of notes from which melodies can be composed.

One scale familiar to almost everybody is the so-called diatonic scale of Western music, which consists of seven notes which are designated by the letters A through G.

An octave is the interval between two notes, the upper note of which has twice the frequency of the lower. In a diatonic scale, the eighth note which finishes the octave has the same name as the starting note because it also starts a new, higher, octave. It may be helpful to think of individual notes as the posts supporting a long fence. Every eighth post simultaneously marks the end of one octave of seven intervals and the beginning of another.

An octave may begin on any note of the scale, and a melody composed using the notes of that octave is said to be in the key of the starting note. For example, if the octave uses C as the starting (key) note, then the melody is said to be in the key of C. The keyboard in the illustration shows three complete octaves, each starting and finishing with C.

An alternative nomenclature is the sol-fa system (do, re, mi, fa, so, la, ti, do) as popularised in the film, The Sound of Music. It uses a "moveable" 'do', i.e. the keynote of whatever scale is used becomes 'do' in the sol-fa system.

Two or more notes played or sung at the same time may, if the combination is agreeable to the "ear", be considered harmonious; the notes are then said to be consonant. If their combination is not pleasing to the "ear", the notes are said to be dissonant because their combination is disagreeable or discordant. A discord is any jarring combination of sounds. Here, as in melody, the "ear" determines the rules.

A chord is the effect produced by sounding two or more notes simultaneously.

The Monocord

Much may be learned about the physics underlying the production of musical notes with the aid of the monochord. The following description is taken from The Science of Sound by Sir James Jeans, 1877-1946, English physicist and mathematician.

"A wire, with one end A fastened rigidly to a solid framework of wood, passes over a fixed bridge B and a movable bridge C, after which it passes over a freely turning wheel D, its other end supporting a weight W. The weight keeps the wire in a state of tension which we can make as large as we please by altering the weight. Only the piece BC of the string is set into vibration, and as the bridge C can be moved backwards and forwards, this can be made of any length we please. It can be set in vibration in a variety of ways by striking it, as in the piano; by stroking it with a bow, as in the violin; by plucking it, as in the harp; possibly even by blowing over it as in the Aeolian harp, or as the wind makes the telegraph wires whistle on a cold windy day.

"On setting the string vibrating in any of these ways, we hear a musical note of definite pitch. While this is still sounding, let us press with our hand on the weight W. We shall find that the note rises in pitch, and the harder we press on the weight, the greater the rise will be. The pressure of our hand has of course increased the tension in the string, so we learn that increasing the tension of the string raises the pitch of the note it emits. This is how the violinist and the piano tuner tune their strings and wires; when one of these is too low in pitch, they screw up the tuning-key. By varying W, we shall find that the frequency is proportional to the square root of the tension in the string.

"By changing the effective length of the string without affecting the tension, we find that the pitch rises as we shorten the string. If we halve the length, the pitch rises by exactly an octave, showing that the period of vibration has also been halved. By experimenting with the bridge C in a range of positions, we shall find that the period is exactly proportional to the length of the string, so that the frequency of vibration varies inversely as the length of the string. In the violin, the same string is made to give out different notes by altering its effective length by touching it with the finger.

"We may experiment in the same way on the effect of changing the thickness or material of the wire.

Mersenne's Laws

"The knowledge gained from all these experiments can be summed up in the following laws, which were first formulated by the French theologian, natural philosopher, and mathematician, Marin Mersenne, 1588-1648, in Harmonie Universelle published in 1636:

I. When a string and its tension remain unaltered, but the length is varied, the period of vibration is proportional to the length. (The "Law of Pythagoras".)
II. When a string and its length remain unaltered, but the tension is varied, the frequency of vibration is proportional to the square root of the tension.
III. For different strings of the same length and tension, the period of vibration is proportional to the square root of the weight of the string.

"The operation of all these laws is illustrated in the ordinary pianoforte. The piano-maker could obtain any range of frequencies he wanted by using strings of different lengths but similar structure, the material and tension being the same in all. But the 7.25 octaves range of the modern piano contains notes whose frequencies range from 27 to 4096 Hz. If the piano-maker relied on the law of Pythagoras alone, his longest string would have to be more than 150 times the length of the shortest, so that either the former would be inconveniently long, or the latter inconveniently short. He accordingly avails himself of the two other laws of Mersenne. He avoids undue length of his bass (low-frequency) strings by increasing their weight usually by twisting thinner copper wire spirally round them. He avoids inconvenient shortness of his treble (high-frequency) strings by increasing their tension. This had to be done with caution in the old wooden-framed piano, since the combined tension of more than 200 stretched strings imposes a great strain on a wooden structure. The modern steel frame can, however, support a total tension of about 30 tons with safety, so that piano-wires can now be screwed up to tensions which were formerly quite impracticable."

Mathematics

I have long felt that mathematics is a faculty which unites the intellect with the emotions, and results in the ability to appreciate beauty in form. Nowhere is this better demonstrated than in considering the "musical" properties of a stretched string from a strictly rational point of view. The proportion, or ratio, between any two measures is the root of the word "rational". Thus musical experiment shows that qualitative "psychic" effects or "values" can be associated with quantitative physical ratios. This provides a rational foundation for metaphysics and mysticism.

In his experiments with the monochord, Pythagoras found that the four string lengths 6, 8, 9, and 12 between them provided all the possibilities of consonance which the Greeks recognised, namely 2:1, 3:2, and 4:3. This may explain why the early Greek lyre had four strings.

An extension of this principle of proportionality applies to the structure of the diatonic scale. This is the "natural" or "true" scale which was used in European music until the seventeenth century. In distinguishing between two notes, Pythagoras would have thought in terms of the ratio of string lengths required to produce them, and this Mersenne showed to be the same as the ratio of their respective periods. That is probably how the term "interval" came to be used. As the frequency of a note is inversely proportional to its period, the interval ratios relevant to period must be inverted when applied to frequency.

The table on the right (which is taken from The Book of Music published by Macdonald Educational Ltd) shows that the intervals of the scale are inter-related by whole-number ratios. It also makes clear why these ratios are called by names like "thirds" and "fifths", something which used to puzzle me because they are not the same as the thirds and fifths of arithmetic.

In musical terminology, an interval is the difference in pitch (i.e., frequency) between two notes. In Western music, intervals are named by the number of scale notes which they span. Thus a second is the interval between two adjacent notes on the keyboard; a third is the interval spanned by three consecutive notes; a fourth is the interval spanned by four consecutive notes; and so on, up to the seventh, which is the interval spanned by seven consecutive notes. The next largest interval is the complete octave.

When an interval is sounded by simultaneously playing its two end notes, the effect on the ear may be consonant or dissonant. "Perfect" consonances, which are found to have the simplest frequency ratios, are: the unison (1:1) i.e. two of the same note; octave (1:2), fifth (2:3) and fourth (3:4).

We see from the diagram that the interval between C and D is called a tone, and that between B and C at the upper end of the scale is called a semitone. The interval between E and F is also a semitone. Thus there are two places in the natural octave where an apparent difference in pitch between two notes is not reflected by a smooth progression in their frequency ratios.

The frequency ratios between successive notes are:

Interval RatioStep
Between C and D 8/9 Tone
Between D and E 9/10 Tone
Between E and F 15/16 Semitone
Between F and G 8/9 Tone
Between F and A 9/10 Tone
Between A and B 8/9 Tone
Between B and C 15/16 Semitone

We observe that the "fence-posts" are "staggered" and, particularly, that the intervals between E and F and between B and C are appreciably larger than the remaining five intervals. These two intervals are named semitones possibly because of the shorter length of string required to span the gap between the notes.

The table is expressed in terms of the intervals between successive notes. The period of a note, i.e. the reciprocal of its pitch or frequency, is proportional to the length of string required to produce the note. The "rational" structure of the octave reflects the position of bridge C on the monochord required to produce each note. It's as if the procession of notes "slows down" or "hiccups" in spanning the semitones.

Major and Minor

The terms major (greater) and minor (smaller) are used to distinguish between intervals that cover the same number of scale steps but which are of different sizes. In the case of C major which we have just been considering, the major third C-E covers three successive notes (C-D-E) and two tones (C-D and D-E). The minor third D-F also covers three scale notes (D-E-F) but only a tone (D-E) and a semitone (E-F).

The same terminology is extended to distinguish between the two main kinds of scale used in Western music. The diatonic scale is called the major scale because the interval between its first and third notes is a major third.

In a minor scale, the interval between its first and third notes is a minor third (tone and semitone). A minor scale has two forms, melodic and harmonic. In the melodic minor scale, the order of tones (T) and semitones (S) depends on whether it is ascending, when the order is TSTTTTS, or descending, when the order is TTSTTST. The harmonic minor scale is the same both ways, but it has one step of a tone and a semitone (T+S = A) which is known as an "augmented second". Its interval order is TSTTSAS.

Equal Temperament

The natural scale of eight notes was used in European music until the seventeenth century. Its intervals were in just intonation, i.e. tuned to the simple "natural" ratios we have been discussing.

When composers started to write music that moved from one key to another, the natural scale caused problems for players of keyboards and some other instruments which had to be pre-tuned unlike stringed instruments on which the performer was able to adjust the pitch of each note as he played. The "natural" semitone (15:16) did not divide a tone exactly, and was too large to be used to divide the octave into twelve equal parts. As a compromise between nature and art, the octave was divided into twelve equal semitones by giving the semitone a ratio of about 84:89, and the interval of a tone was made exactly equal to two semitones. In this way, the scale could start on any of the twelve semitones. On instruments such as pianos tuned by equal temperament, all intervals except the octave are slightly out of tune compared with the intervals of just intonation. A finely-tuned "ear" can tell that there is "something not quite right" about this stratagem, but it does make it possible for a pianist to play any scale in twelve keys. I guess that whether one likes it or not depends on one's temperament.

The added notes are represented by the five black keys in the octave. The interval between a black note and the white note on either side of it is a semitone. Each black note may have two names depending on whether the scale is played upwards (ascending) or downwards (descending) in pitch. For example, the black note between C and D may be called C sharp (ascending) or D flat (descending).

Harmonics

Later experiments with the monochord showed not only that a string vibrates as a whole, but also that various fractions of the string vibrate at their own, higher, frequencies. A plucked string can vibrate not only with the whole of its length but also in fractions of, e.g., 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, etc. of its length, and therefore produce harmonics "subsidiary" notes whose frequencies are whole-number multiples of the fundamental frequency of the string. In other words, the fundamental note has superimposed on it further notes 2, 3, 4, 5, 6, 7, etc. octaves higher in pitch but much less loud than the fundamental. These partials or overtones vary from one instrument to another because of differences in their structure. This enables the "ear" to recognise them by their characteristic timbres.

Octave Numbers as Powers of Two

We have seen that the last note of an octave has double the frequency of the first note of the octave. This suggests that we might explore the usefulness of applying binary theory to all sorts of vibrations.

If we assume that the first octave starts from silence, (i.e. 0 Hz), the frequency of the highest "note" of the first octave will be 1 Hz which is also the first note of the next higher octave. "Doubling up" in succession, we get 2 Hz, 4 Hz, 8 Hz, 16 Hz, etc. Expressing these as "powers" of 2, we get 20, 21, 22, 23, 24, etc. This gives us a convenient way of numbering octaves, as in the table below.

Octave NumberUpper Frequency (Hz)
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
15 32,768
20 1,048,576
30 1.0737*109
40 1.0995*1012
60 1.1529*1018
80 1.208910*1024

The notation "x*10y" means a number 'x' multiplied by 10 'y' times.

I have always been attracted to the idea that "Middle C" (the C nearest the middle of the piano keyboard) should in the "natural" order of things vibrate at exactly 256 Hz. However, the architects of "equal temperament" chose to standardise on A above Middle C tuned to 440 Hz. Working "backwards" from there, Middle C on your piano is normally tuned to vibrate at approximately 261 Hz.

Extra-musical Interpretation

We see from the table that the frequencies we interpret as "sound" are contained in octave numbers 4 through 15. Physics also tells us that what we commonly call "light" is contained within octaves 48 to 51.

There is a big gap between the group of octaves we interpret as sound and the group we interpret as light, and there is apparently unlimited scope beyond that. We (or at least those scientists who specialise in these matters) already know the octaves associated with radio, television, radar, X-rays, etc., but these still leave plenty of possible octaves for transmission of other kinds of "intelligence". Many other parts of the spectrum of octaves might, or might not, have some effect on human sensibilities. Just because we cannot assign a definite place in the spectrum to, for example, "mystical" faculties such as telepathy, clairvoyance, distant healing, etc., is no reason to assume that they are merely imaginary.

There is evidence that other creatures may be able to hear or see or otherwise respond to vibrations outside the ranges we interpret as sound or light. Perhaps we, too, have vibrational potentials of which we are not aware and which, consequently, we don't use. It might for a start be profitable for us to explore the possibility that what we call taste, smell, touch, and pain have their own distinctive frequency ranges. And who knows what other "senses" we may find if we make a diligent search for them within ourselves? So it seems that both physiology and harmonics, as well as psychology and mathematics, may have contributions to make to our knowledge of how we collectively relate to the Universe.

I leave you with a final thought. 1 vibration per second is the same as 60 vibrations per minute, 1440 vibrations per diem, 518,400 vibrations per annum, and so on for as many centuries, millennia, and ages as you care to take into account. Would it not be interesting to explore the various "cycles" found in individual lives and in collective experience as recorded in history, archaeology, etc., in search of any possible correlations with the vibrational structure of the natural octave? Did I hear someone whisper "astrology", or "music of the spheres"?