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Ratio

Relation or proportion of one measure to another which can be linear, geometric or volumetric. It can be one sound to another or one volume to another or just about any one class of things, forces or vibrations. Can also be expressed in parts. [See Proportion, Reciprocating Proportionality]

A relationship, or interval, expressing the vibrations per second, or cycles, of the two tones concerned, generally in the lowest possible (integer) terms; simultaneously a representative of a tone and an implicit relationship to a "keynote" - or unity.

Euclid's definition: a mutual relation of two magnitudes of the same kind to one another in respect of quantity. [from Partch 1974, Genesis of a Music, 2nd ed., Da Capo Press, New York, p. 73]

The ratio expressed as 1:2 says the first quantity is half the second quantity which is twice the first quantity. Music intervals are generally expressed as ratios.

Simple ratio
1:2 (1 is to 2)

Ratio of two ratios
1:2::2:4 (1 is to 2 as 2 is to 4)

Ramsay
Euler, while treating of music, shows that there are just three mathematical primes, namely 2, 3, and 5, employed in the production of the musical notes - the first, in ratio of 2 to 1, producing Octaves; the second, in the ratio of 3 to 1, producing Fifths; and the third, in the ratio of 5 to 1, producing Thirds. [Scientific Basis and Build of Music, page 8]

"The numbers which express the motions of these twenty-five quantities have among themselves nineteen different ratios, or rates of meeting; and when these ratios are represented by the oscillations of twenty-five pendulums, at the number of 64 for the highest one, they will all have finished their periods, and meet at one for a new series. This is an illustration, in the low silence of pendulum-oscillations, of what constitutes the System of musical vibration in the much higher region of vibrating strings and other elastic bodies, and determines the number of undeveloped sounds which form the harmonious halo of one sound, more or less faintly heard, or altogether eluding our dull mortal ears; and which determines the number of sounds which, when developed, constitute the System of musical sounds." [Scientific Basis and Build of Music, page 16]

Different writers have put forth different views of what constitute a musical vibration, but their various views do not make any difference in the ratios which the notes of this sound-host bear to each other. Whether the vibrations be counted as single or double vibrations, the ratios of their relative motions are the same. Nevertheless, a musical vibration is an interesting thing in itself, and ought to be correctly defined.
A string when vibrating musically is passing and re-passing the central line of its rest or equilibrium with a certain range of excursion. Some writers have defined a vibration to be the passage of the string from one extreme of its excursion to the other, while some have preferred to define it as the passage of the string from the one extreme of its excursion to the other and back again. D. C. Ramsay has been led in his researches to define a vibration as the movement of the string from its central line of rest to the extreme of its excursion on one side, and back to the central line of rest; and from the central line of rest to the extreme of its excursion on the other side, and back again to the "right line," as he calls it, as a second vibration. His reasoning on this will be seen in what follows. (See Fig. 3, Plate IV.) [Scientific Basis and Build of Music, page 21]

"If we take a pendulum which goes from side to side 60 times in a minute, and another which goes from side to side 120 times in a minute, these two pendulums while oscillating will come to their first position 30 times during the minute. Now, if an oscillation is to be considered a natural operation, like the revolution of a wheel, or that of a planet in its orbit, which is completed when it returns to the place where the revolution began, then the pendulum's oscillation is not completed till it returns to the place of starting; and thus defined the oscillations of these two pendulums in the minute are not 60 and 120, but 30 and 60; 30 is the unit of measure in this case - 30 is the 1, and 60 is the 2; and this would establish the ratio of 1 to 2 in these two pendulums. And what is true in the ratio of 1 to 2 is true also of every other ratio, in this respect. This is a natural basis to work on, and defines the oscillation of a pendulum to be its excursion from extreme to extreme and back." [Scientific Basis and Build of Music, page 25]

"The system of musical sounds is derived from the laws of motion and a particular election of numbers which give the greatest variety of simple ratios.
There are three primary and pregnant ratios which produce the chords and scales. The first is the ratio of 1:2, producing Octaves, and nothing else; the second is the ratio of 2:3, producing Fifths; the third is the ratio of 4:5, producing Thirds." [Scientific Basis and Build of Music, page 26]

"notes which are produced by the two primes, 3 and 5. As the quadrant contains all the angles which give the different proportions in form, so does the ratio of 1:2, or the area of an octave, contain all the different notes in music. The ratio of 1:2 corresponds to unity, and, like the square and the circle in form, admits of no varieties. Half the length of a string gives an octave when the string is homogeneous and uniform; if the one half has more gravity than the other, the center of gravity of the whole string gives the octave. The ratio of 1:2 rests on the center of gravity. [Scientific Basis and Build of Music, page 27]

"lower effect than the fifth; the seventh, B, has a higher effect than the sixth; but the eighth, C, has a lower effect than the seventh. If the effects of notes or chords depended wholly on the mathematical primes by which they are measured and located, or the ratios inherent in them, then the effects of the tonic, subdominant, and dominant chords would have been alike, for these chords are measured by exactly the same primes, and have exactly the same ratios. It is the position of the tonic chord which gives it its importance and not any special primes by which it is produced, nor any special ratios inherent in it. Notes by the power of 2 have a pure unmixed and invariable character. Notes by the first, second, and third powers of 3 have different degrees of centrifugal force; and the character of the notes produced by the first power of 5 depends on the character of the notes from which they are derived. The final character of notes and chords is determined by the amount of force which they have acquired from the way in which they have been derived, and from their position in the system. And no matter where these notes may be afterwards placed, like chemical elements, they never lose their original forces and tendencies. What Tyndal says of the inorganic chemical elements of the brain is true of the inorganic notes of music, "They are all dead as grains of shot." It is the organic state which gives the notes and chords their gravities and (levity|levities, and these two tendencies, the one upward and the other downward, constitute the vital principle of music. It is true that the mathematical operation is required to give birth and life to music, and that the mathematical system gives the knowledge of causes down to the law of gravitation, yet the artistic effects are fully realised from the tempered system deriving its organic harmony from this vital principle of music. The centrifugal tendencies of the notes of the subdominant, are too strong to be at all disturbed by the system being tempered. The enormous power of these chords corrects the effect which might otherwise arise from tempering, as the enormous power of the sun corrects the perturbations of the planets." [Scientific Basis and Build of Music, page 29]

"The ratio of 2:3 twelve times, in Fifths, is so near the ratio of 1:2 seven times, in Octaves, as to allow this cycle of the mathematical scales to be closed without losing any of its vitality. The reason why there are thirteen instead " [Scientific Basis and Build of Music, page 29]

Getting Fifths as we ascend toward the number twelve they are in themselves the same, but with regard to their relationships they are quite different. Before and up to the twelfth fifth no scale has all the notes at the same distance above the first scale of the series. But after twelve, the thirteenth scale for example, B#, supposing the scale to be marked by sharps only, is a comma and a very small ratio above C; Cx is the same distance above D of the first scale; Dx the same above E; E# is the same distance above F; Fx the same distance above G; Gx the same distance above A; and Ax the same distance above B. So the scale of B# is just the scale of C over again at the distance of twelve-fifths, only it is a comma and the apotome minor higher; and each series of twelve-fifths is this distance higher than the preceding one. [Scientific Basis and Build of Music, page 30]

With twelve divisions in the Octave, each note is adapted to serve in any capacity, and does serve in every capacity by turns. It is quite clear that this cannot be said of the mathematically perfect notes. And this is where it is seen that what is perfect in mathematical ratios becomes imperfect in the Musical System. Indeed, the mathematical intonation does not give a boundary within which to constitute a System at all, but goes off into never-ending cycles.
In music, Nature begins by producing the Diatonic Octave of seven notes, derived by the mathematical ratios2; [Scientific Basis and Build of Music, page 34]

she is found to have produced the Chromatic scale of twelve semitones, derived from her own vital operations; so that there are no anomalies. It is a degradation of the mathematical primes to apply them to the getting of the semitones of the chromatic scale, as even Euler himself mistakenly does. The mathematical ratios lead the way in getting the notes of the diatonic scale, and that is all that is required of them. The true praise to the ratios is that they have constituted an organic structure with form and life-powers adapted for self-development. It would be little credit to the mother if the child required to be all its life-long pinned to her apron-strings. As the bird when developed so far leaves the shell, and is afterwards fully developed in new conditions; so the System of music when developed so far leaves the law of ratios, its mathematical shell, and is afterwards fully developed by other laws. Music has an inspirational as well as a mathematical basis, and when mathematicians do not recognize this they reckon without their host. [Scientific Basis and Build of Music, page 35]

The twelve semitones being the practical fulfillment of the ratios when the life-force of the notes is considered, the great masters had the ratios in their most workable form invested in their key-boards, and this along with the musical ear was the sure word of prophecy to them.1 In their great works they have thus been enabled to develop the science of music, and to express it in the language of its art. [Scientific Basis and Build of Music, page 36]

When the major scale has been generated, with its three chords, the subdominant, tonic, and dominant, by the primary mathematical ratios, it consists of forms and orders which in themselves are adapted to give outgrowth to other forms and orders by the law of duality and other laws. All the elements, orders, combinations, and progressions in music are the products of natural laws. The law of Ratio gives quantities, form, and organic structure. The law of Duality gives symmetry, producing the minor mode in response to the major in all that belongs to it. The laws of Permutations and Combinations give orders and rhythms to the elements. The law of Affinity gives continuity; continuity gives unity; and unity gives the sweetness of harmony. The law of Position gives the notes and chords their specific levities and gravities; and these two tendencies, the one upward and the other downward, constitute the vital principle of music. This is the spiritual constitution of music which the Peter Bell mathematicians have failed to discern: [Scientific Basis and Build of Music, page 37]

If the effects of notes and chords had depended entirely on their mathematical ratios, then the effect of the subdominant, tonic, and dominant would have been alike; for these three chords have exactly the same ratios. It is the law of position which gives the tonic chord its importance, and not any special ratios embodied in its structure. The ratio of 2 to 1 has a pure, unmixed, invariable character, always realized in the interval of the octave. The notes produced from 1 by the first, second, and third powers of 3 have different degrees of centrifugal force. The character of the notes produced by the first power of 5 depends on the character of the notes from which they are derived, namely, 1, 3, and 9. The final character of the notes and chords derived by the same ratios is determined by the amount of force which they have acquired from the way in which they have been derived, and from their position in the system; and no matter where these notes may afterwards be [Scientific Basis and Build of Music, page 37]

The mathematical scales, if followed out regardless of other laws which rule in music, would read like a chapter in Astronomy. They would lead us on like the cycles of the moon, for example. In 19 years we have 235 moons; but the moon by that time is an hour and a-half fast. In 16 such cycles, or about 300 years, the moon is about a day fast; this, of course, is speaking roughly. This is the way seemingly through all the astronomical realm of creation. And had we only the mathematical ratios used in generating the notes of the scale as the sole law of music, we should be led off in the same way. And were we to follow up into the inaudible region of vibrations, we should possibly find ourselves where light, and heat, and chemical elective motions and electric currents are playing their unheard harmonies; or into the seemingly still region of solid substances, where an almost infinite tremor of vibrations is balancing the ultimate elements of the world. Music in this case would seem like some passing meteor coming in from among the silent oscillations of the planetary bodies of the solar system, and flashing past with its charming sound effects, and leaving us again to pass into the higher silence of those subtle vibrations to which we have referred, having no infolding upon itself, no systematic limit, no horizon. But music is not such a passing thing. Between the high silence of these intense vibrations, and the low silence of oscillating pendulums and revolving planets, God has constituted an audible sphere of vibrations, in which is placed a definite limit of systematic sounds; seven octaves are carried like a measuring line round twelve fifths; and motion and rest unite in placing a horizon for the musical world, and music comes [Scientific Basis and Build of Music, page 39]

The specific levity of notes increases in proportion to the number of times the ratios are multiplied in order to produce them, going upward by sharps; and their specific gravity increase in proportion to the number of times the ratios are divided in order to produce them, going downward by flats. The knowledge of this is attained when everything is in its perfect order. It is the discovery of the Law of Duality in music which shows the method of applying the ascending and the descending ratios so as to exhibit that perfect order of Nature. [Scientific Basis and Build of Music, page 43]

The individual character of any note, and the comparative degree of contrast between any two notes in the system, depends on at least three different causes. The first is the genetic relation of the two notes. If the one note has 2 vibrations and the other 3, or the one 4 and the other 5, or the one 5 and the other 8, because of this, and because of the excess of the vibration of the one over the other, "a third sound" or "grave harmonic" being awakened between them, the different ratios have different degrees of complexity, and, in a general way, the greater the complexity the greater the [Scientific Basis and Build of Music, page 60]

contrast. In the fifth, the ratio being 2:3, the excess of 3 above 2 is 1; this 1 bears a simple relation to both the notes which awaken it. The grave harmonic in this case gives the octave below the lower of the two sounds; 1 is an octave below 2. This is the simplest relation "a third sound" can have to the two which awaken it, and that is why the fifth has the smallest possible degree of contrast. The octave, the fifth, and the fourth may be reckoned as simple ratios; the major and minor thirds and their inversions as moderately complex; the second, which has the ratio of 9:10, and the major fourth F to B and its inversion, are very complex. [Scientific Basis and Build of Music, page 61]

A second cause of difference in degree of contrast between two notes and other two notes in which the ratios are the same lies in this - whether the two notes belong to one chord or to different chords. Two notes in the subdominant chord have a different contrast from two in the dominant chord which have the same ratio. [Scientific Basis and Build of Music, page 61]

F is soft, grand, and solemn;
C is melodious and soft;
A is interesting and soft.

C is melodious and soft;
G is melodious and vigorous;
E is interesting and melodious.

G is melodious and vigorous;
D is interesting and vigorous;
B is light, airy, and vigorous.


Although the system is composed of only three ratios, which in themselves moreover, are of a very fixed character, yet mobility and variety are chief features among the notes of the system. Great changes are effected by small means. By lowering the second of the major D one comma, the ratio of 80:81, [Scientific Basis and Build of Music, page 61]

Music, and mathematics have nothing more to do with it. Already the Law of Position has guided the genesis upward in the major; and while mathematical primes were generating the chords one after another in precisely the same way and form, like peas in a pod, the Law of Position was arranging them one over the other, and so appointing them in their relative position each its own peculiar musical effect bright and brighter. And when the major had been thus evolved and arranged by ratios and position, another law, the Law of Duality, gave the mathematical operation its downward direction in the minor; and while the primes which measured the upward fifths of the major also measure the downward fifths of the minor, the Law of Position is placing them in their relative position, and appointing each its own peculiar effect grave and graver. [Scientific Basis and Build of Music, page 68]

If it be asked why no more primes than 2, 3, and 5 are admitted into musical ratios, one reason is that consonances whose vibrations are in ratios whose terms involve 7, 11, 13, etc., would be less simple and harmonious than those whose terms involve the lesser primes only. Another reason is this - as perfect fifths and other intervals resulting from the number 3 make the schism of a comma with perfect thirds and other intervals resulting from the number 5, so intervals resulting from the numbers 7, 11, 13, etc., would make other schisms with both those kinds of intervals. [Scientific Basis and Build of Music, page 75]

The root of the subdominant is F, in the key of C major; and the top of the dominant is D. The difference between these two notes at the top and bottom of the chord-scale, is the quantity which two octaves is more than three fifths; it is the ratio of 27 to 30, a comma less than the minor third whose ratio is 5 to 6. [Scientific Basis and Build of Music, page 76]

the excess of the vibrations of the one note over the other makes one or more sounds which are called "grave harmonics;" e.g., in the interval of the fifth, in the ratio of 2:3, the excess of 3 over 2 is 1, so the grave harmonic is an octave below the lowest of the two notes, that is, the ratio of 1:2. This reinforces the lowest note, 2, and gives it a solid effect. In this way the octave is incorporated into the fifth, and unity with variety is combined with the law of continuity at the very threshold of harmony. In 32 of the 42 intervals the grave harmonics are notes which belong to the natural scale. In the 10 remaining intervals which have not the exact number of vibrations found anywhere in the natural scale, 6 of them are from the number 7, thus - 7, 7, 7, 21, 21, 35; the remaining 4 are from 11, 13, 13, and 19. [Scientific Basis and Build of Music, page 77]

See Also


Figure 8.11 - Four Fundamental Phases of a Wave
Figure 8.9 - Four Fundamental Motions of a Pendulum
Figure 6.17 - Areas and Volumes - Relations and Proportions
Figure 6.19 - Sphere to Cube - Relations and Proportions
Figure 14.10 - Proportionate Tonal Relations dictate Contraction or Expansion
Fraction
Fundamental
Interval
Keynote
Laws of Music
Laws of Ratios
Reciprocating Proportionality
Table 2 - Controlling Modes and Proportions
Table of Plate Harmonics and Intervals
Universal Ratios
6.8 - Proportionate and Relative Geometries
9.12 - Velocity of Sound and its Propagation Rate are Proportional
12.00 - Reciprocating Proportionality
3.13 - Reciprocals and Proportions of Motions and Substance
13.15 - Principle of Proportion

Created by Dale Pond. Last Modification: Saturday November 21, 2020 05:05:20 MST by Dale Pond.