Ratios and harmonics

new page - under development

In the whole of creation we find number and measure, and the question is how we can make the link with the wonderful complexity we find in creation in the four kingdoms of nature on Earth.

This page builds on the topic page Book of Ten Pages on 'numbers' and adds the next step of 'ratios' between numbers. This page also covers waves, standing waves and harmonics that follow from these ratios.

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A ratio is a mathematical term that expresses the relationship between two numbers and is typically written using whole numbers separated by a colon (e.g., 3:1) or as a fraction.

Ratios can be used to compare parts of a whole or to find equivalent relationships.

Ratios also define the relationships between the frequencies and wavelengths of waves, creating harmonics in a harmonic series. In vibrating systems like musical strings, the lowest resonant frequency is the fundamental frequency, and higher harmonics are integer multiples of this fundamental frequency. The integer ratios such as 1:2:3:4 dictate that each harmonic wave has a wavelength that is an integer fraction of the fundamental wavelength, leading to the formation of specific resonant patterns known as standing waves and a rich, complex sound.

Ratios of numbers give rise to waves and harmonics

simple integer ratios, like 2:1 (an octave) or 3:2 (a perfect fifth), are perceived by the brain as harmonious, whereas more complex, non-integer ratios sound dissonant
Simple whole-number ratios between frequencies create the harmonics found in a fundamental frequency. A fundamental frequency, which is the lowest frequency in a complex sound, produces its overtones or harmonics at frequencies that are integer multiples of itself (e.g., 1f, 2f, 3f, where f is the fundamental frequency).

The mathematical relationship defined by simple number ratios is the basis for how waves produce the pleasing and structured sounds we call musical harmony.

1. Fundamental Frequency: when a musical instrument produces a note, it creates a primary sound wave, the fundamental frequency (f).

2. Overtones: simultaneously, the string or air column vibrates in shorter segments, creating higher frequency overtones called harmonics.

3. Integer Ratios: the frequencies of these harmonics are always integer multiples of the fundamental, meaning they form a whole-number ratio.

For example a note played on a guitar will have harmonics with frequencies in ratios like 1:2:3:4:5, etc.

4. Perceived Harmony: when these simple whole-number ratios occur together, the combination of frequencies sounds pleasant and "consonant".

5. Dissonance: conversely, more complex or non-integer frequency ratios, such as a 7:5 ratio, result in a "dissonant" or less pleasing sound.

Harmonics are the frequencies produced by standing waves in a vibrating object, such as a guitar string or a column of air in a musical instrument.

A standing wave is a wave pattern that looks like it is "standing still" rather than moving through a medium. It is created when two identical waves (of the same frequency and amplitude) travel in opposite directions and interfere with each other, resulting in regions of fixed high and low displacement called antinodes and nodes. Nodes are points that stay completely still (zero displacement) due to destructive interference, antinodes: are points that oscillate with maximum amplitude due to constructive interference.

Unlike traveling waves, standing waves do not carry energy from one place to another but instead, energy oscillates locally.

Standing waves are the physical manifestations of the harmonic frequencies allowed by a system. Each standing wave that fits within the boundary conditions of a system is a harmonic, with the fundamental (first harmonic) being the simplest and lowest-frequency standing wave, and subsequent harmonics being integer multiples of this frequency.

Examples of standing waves in bounded systems: a string fixed at both ends, like on a guitar: when you pluck the string, the vibration reflects at both ends and produces standing waves that determine the musical pitch. Similar for air in tubes like organ pipes, or microwaves in an oven cavity.

Aspects

mathematics

ratios
- golden ratio (a+b is to a as a is to b)
- Fibonacci sequence (each element is the sum of the two elements that precede it)
- spiral (and golden spiral, etc…)
- see also: Formative forces#On Number, Form, Geometry (spirals, fibonacci, golden ratio)
two dimensions
- polygons: triangle, rectangle, hexagon, etc
three dimensions
- polyhedra: three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices
  - Platonic solids, Archimedean solids
- see also: Symbols#Note 1 - Platonic solids

in nature: wavefronts of air vibrations

sound and music is based on waves of air vibrations (on a string or air tube)
music: 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth)
speech

standing waves underlying formation

Chladni patterns are visualizations of two-dimensional standing waves on a vibrating surface.
- When a plate is vibrated at a specific resonant frequency, it forms standing waves with stationary points (nodes) and points of maximum vibration (antinodes). Light particles like sand move away from the vibrating antinodes and collect at the stable nodes and result in the complex two-dimensional patterns of these standing waves. Sand highlights locations of zero displacement on the vibrating surface. The patterns can thus be seen as a visualization of sound.
- Hints of future etheric force technology#Note 2 - Cymatics, Chladni patters and Visual audio
- in water as a medium, eg the work of Masaru Emoto, Alexander Lauterwasser and others
on a spherical rotating sphere:
- Earth#Note 3 - Rossby planetary waves and their mathematical modelling
- (and see also Planetary seals)

various

Pythagorean teachings - see also Schema FMC00.680
- the image of the lyre with its strings, the harmony is not in the strings itself but in what emerges from them, a spirituality no longer bound to physicality (from 1901-11-23-GA087 on the 'Phaido' (by Plato, dialogue with Socrates) as basis to imagine the development of a Pythagorean)

Illustrations

Schema FMC00.680: shows,

on the left, Raphael's fresco painting 'The School of Athens' with a detail of the tablet held up before Pythagoras (ca. 570 to ca. 490 BC).

The tablet shows the tetractys below (see Schema FMC00.556 and variants), and above the U-bow representation of musical harmonic ratios.

In the middle, in colour, a legend with the greek and modern terms for what we know as the octave, fourth and fifth in music.

On the right is an example of the corresponding illustration from a 14th century book by Jean des Murs (ca 1291-1344) who revisited Boethius (ca 480-524) work in Latin on ancient Greek knowledge and ideas about music (including Pythagorean numerical ratios)

This example shows the lineage tradition of the U-bow representation shown on Raphael's work for showing ratios.

Many such illustrations can be found in the centuries that followed, when also other visualizations were used, ao with concentric circles and arc-based structured such as Robert Fludd's monochord and others.

shows, on the left, Raphael's fresco painting 'The School of Athens' with a detail of the tablet held up before Pythagoras (ca. 570 to ca. 490 BC). The tablet shows the tetractys below (see Schema FMC00.556 and variants), and above the U-bow representation of musical harmonic ratios. In the middle, in colour, a legend with the greek and modern terms for what we know as the octave, fourth and fifth in music. On the right is an example of the corresponding illustration from a 14th century book by Jean des Murs (ca 1291-1344) who revisited Boethius (ca 480-524) work in Latin on ancient Greek knowledge and ideas about music (including Pythagorean numerical ratios) This example shows the lineage tradition of the U-bow representation shown on Raphael's work for showing ratios. Many such illustrations can be found in the centuries that followed, when also other visualizations were used, ao with concentric circles and arc-based structured such as Robert Fludd's monochord and others.

Lecture coverage and references

attributed to Novalis

Novalis is often quoted along the lines of

all visible forms in the world are expressions of congealed movements, vibrations that have come to rest, or solidified sounds

or

all visible forms are frozen motions — vibrations stilled into shape, the music of the world made stone

however no original source could be found for this quote

1797-1798 - Novalis

from: collection 'Blüthenstaub', written 1797–1798 and published in 1798 (in the 'Athenäum' journal)

Nature is an aeolian harp, a musical instrument whose tones are the re-echo of higher strings within us.

An Aeolian harp is a string instrument sounded only by the wind, a favorite metaphor for the self and for nature’s mysterious resonance with human spirit. For Novalis, nature itself plays us — our souls vibrate in sympathy with the “higher strings” (a divine or transcendental source).

Other authors also used this metaphor (Herder, Emerson? .. tbc)

from 'Fragmente' (Fragments) written in the same period, around 1797–1798, and published posthumously in 1802.

That which the external world perceives as quite motionless has the appearance of being quite at rest. But, because it is inwardly in motion, it remains ever the same. Everything beautiful is a self-illuminated, perfect individual

Discussion

Note 1 - Mineral matter and the etheric formative forces

This is a draft note, WIP, as a narrative to concatenate various related aspects underlying the way how mineral matter can be understood in relation to the etheric formative forces. As background, see also Force substance representation and Spectrum of elements and ethers.

Rudolf Steiner uses the example of Chladni figures to show how light particles (such as sand) are brought into movement by a vibrating wavefront, and move and cluster into certain locations, the nodes of the standing wave. There is zero displacement (due to destructive interference), and energy oscillates locally.

Now consider what we call atoms, see The nature of atoms.

Mineral science considers that space is empty and atoms are the most solid entities in existence that whirl around in this empty space. That is not the case, as the view is based on illusion (see also Cosmic fractal).

For spiritual science, atoms are like blown up bubbles filled with ahrimanic substance, like bubbles of air seen in soda-water where there is no water. Where they are space is hollow; but impact occurs because when pushing against hollowness, an effect is produced. (1917-08-07-GA176, see The nature of atoms#1 - Bubbles in the ether).

Now the consecutive planetary stages of evolution caused the creation of the Spectrum of elements and ethers and what we currently call (and physics in mineral science differentiates as) substance and forces.

These formative forces gave matter inner form and organization according to the laws of (sound) waves .. just as with Chladni figures one can evoke the figures from the tone color. (1907-03-04-GA096).

Imagine you have not a sound wave but different spiritual wavefronts, that 'come down from' or better 'operate from' the higher spirit world, lower spirit world, or astral world. They superimpose, and so one can imagine phenomena related to what we know as standing waves in music or in twodimensional Chladni patterns.

These influences work on whatever is there as medium or substance, it can be warmth, or air, or water.

Imagine there were not yet any mineral substance as we know in the mineral kingdom today, that everything was contained in an ocean of fluidic substance with airy warmth, a mixture of the other three elements, so to say. Then one can imagine wavefronts working fluidic mixture so that the nodes of the standing waves cause a clustering of what we currently call the atom. Only, instead of a tiny piece of solid matter that we think is the atom, now imagine two spiritual wavefronts that we call etheric formative forces superimposing then creating matter (as the node of standing waves) and by miracle also causing the creation of form in matter.

In fact, various ancient myths describe this process of creation of our current world as the formation through gradual densification or solidification of a fluidic or oceanic primordial world or cosmos through vibrating and resonant spiritual forces. To help develop an imaginative image in our mind, this is beautifully visualized through the correspondence between sound and form of sand on a plate or water surfaces: form as solidified standing sound waves.

Some reference quotes - taken from Hints of future etheric force technology#Note 2 - Cymatics, Chladni patters and Visual audio:

On Earth, after the separation of Sun, the parts of the Earth which had become fluid ... this fluid-earthly water contained all those substances which exist separately today as metals, minerals, etc ... and these fluids were permeated by sphere-harmonies, tones which streamed into the Earth from the universe in every possible harmony. The result of this action of sound in the water-element was a very important one: tone created forms in the water. Similar to Chladni figures, the instreaming music (of the spheres) gave rise to most manifold forms and figures, and the substances dissolved in the water (which were themselves watery) listened to the cosmic music and arranged themselves in conformity to it. (1908-03-16-GA102)
The Earth became more dense, moved into a watery state (thicker than our ocean water) that contained everything which is firm and mineral today. "With the liquefying process a new element emerges. To the extent to which water appears, the music of the spheres, the cosmic sounds, are emanating from the cosmos and out of the Earth. ... The material substances separate out from a larger mass of undifferentiated matter. Under the influence of the music of the spheres, the Earth’s substances begin to 'dance'. This is the differentiation of substances into purely organic substances, for example, into protein. This is how organic matter, the protoplasm, came about under the influence of the music of the spheres, like today’s Chladni sound figures." (1908-03-17-GA102)
the inner organs of the human being (not the muscles or the bones) are formed out of the formative forces (that are cointained in the air - see also fine breathing), as can be grasped with the cognitive faculty of inspiration. "What has been formed out of the air has something of the nature of music about it, just as the musical element is at the basis of the Chladni sound figures." (1924-01-07-GA316)

To be continued (WIP)

With 'music of the spheres' is referenced what is 'heard' when exploring the lower spirit world with the cognitive faculty of inspiration.

Now things are more complex as the orchestra of forces of creation should be seen as the 'superposition' of 'wavefronts' from different planes or worlds of consciousness. Rudolf Steiner describes this as the forces from the zodiac and from the planets, or also as cosmic syllables and vowels.

This working down from the higher worlds to the wonderful creation that what we know as the four kingdoms of nature on Earth, is also called Homer’s Golden Chain. More on this on: Overview Golden Chain

Things to be added/integrated in the above storyline: