Synthesising the Geometry of Lichfield Cathedral with Music and Finding the Ark of the Covenant
“Don’t know much trigonometry. Don’t know much about algebra. Don’t know what a slide rule is for.” Sam Cooke
Introduction
In this article, we move on from the musical and spiritual principles of unison and look in more detail at the octave and how its spiritual principles are built into the gothic architecture of Lichfield Cathedral in the UK. We reference future articles in our exploration of how this applies to other world religions. We explore how this corresponds to the patriarch Abraham and the geometry of the cube. We then bring you back full circle using the Paleo-Hebrew alphabet of Moses in the architectural design of Lichfield Cathedral and how it represents, at the same time, a “House of God” as a place of worship and figuratively a model in stone of the structure of the human soul and how this is also a “House of God”.
It would be helpful to read our first article in this series on the musical interval of unison.
Spiritual Meanings of the Octave
In future articles, we will look at how the octave corresponds to spiritual concepts in some of the major world religions. These will appear as links in this article as these specific articles are written and include:
Later we will also visit the following:
- Christian mysticism
- Hinduism
- Shinto
- Zoroastrianism
The Octave is the Building Block of Melody
“There’s a blaze of light in every word. It doesn’t matter which you heard. The holy or the broken Hallelujah.” Leonard Cohen
When considering spiritual parallels with music, it is helpful to look at basic music theory. Earlier, we looked at different types of spiritual transformations. In this way, moving from one octave to another can be seen as a spiritual journey taken through life, each of us with our own melody, in the symphony of the human race.
The octave is a fundamental building block that shapes the structure and expression of melodies. The interval of the octave is the distance between two notes with the same pitch class, where the frequency of the higher note is twice that of the lower note.
The use of octaves in melody can create meaning and evoke emotions through several key principles:
Unity and familiarity
The octave interval is universally recognized as harmonious and consonant. As the fundamental interval of the harmonic series, it establishes a sense of unity and familiarity in a melody. The repetition of a note at different octaves can create a sense of cohesion and continuity within a musical piece. It is the unity between unisons. In the same way, there seem to be nodal points in our spiritual quests, where we reach a stage of enlightenment, only to find another path and gate in front of us asking us to look deeper.
Range and register
The octave provides a framework for organizing the range and register of a melody. Composers and performers can manipulate the octave in which a melody is played to convey different emotional qualities or to create contrast. For example, a melody played in a higher octave may be perceived as more energetic or uplifting, while the same melody in a lower octave might evoke a sense of depth or solemnity. In the same way, there seem to be various modes in our mystical life: knowing God, knowing ourselves, knowing those we love, knowing our enemies, knowing nature, and knowing time … there is no end of keys, modes and registers that colour our universe.
Doubling and harmonization
In many musical styles, octaves are used to double melodic lines or create harmony, reinforcing the melody and giving it a more robust sound. This technique can also be used to create a sense of depth or richness in the texture of a musical piece. We will look at doubling in reference to Abraham, the geometry of the cube and aspects of Lichfield Cathedral.
Variation and development
The octave can be used as a tool for developing and varying a melody. Composers may transpose a melodic phrase up or down an octave to create contrast or to explore different registers. This technique can give a melody new expressive qualities while still maintaining its recognizable identity. In the same way, as we grow spiritually, we gain a deeper understanding of the universal truths and principles that seem to cut across all aspects of our experience. In our pursuit of loving God, we recognise that the quality of our personal relationships very much tells us about the quality of our inner life.
Establishing tonality
The use of octaves in a melody can help establish the tonality or key of a piece, as the root note of a scale is often emphasized through octave leaps or repetitions. The tonic note, which is the first note of the scale, serves as the central pitch around which the melody is organized, providing a sense of stability and resolution. Spiritual growth and retardation often seem to follow a cycle, like a “wheel of fortune” a perception of difficulty, a dark night of the soul, a feeling of ascent and then an appreciation in wisdom of the balance and harmony of all our experiences.
Overall, the octave plays a crucial role in shaping the meaning and expression of melodies in music in the same way we experience our spiritual lives. Both provide a sense of unity, establish tonality, and offer opportunities for variation and development. The octave allows composers and performers to create a rich tapestry of emotional and sonic experiences within a melody. This is mirrored in life.
The Doubling Geometry of the Cube and the Musical Octave
“These are the days of Elijah, Declaring the word of the Lord. And these are the days of Your servant Moses.” Paul Wilbur
There is a mathematical relationship between the expanding geometry of the cube and the increasing frequencies of the notes of the musical octave. Specifically, the frequencies of the notes of the musical octave are related to the lengths of the sides of the cube.
To understand this relationship, let’s start with the concept of a “musical octave.” In Western music, an octave is defined as the interval between two notes where the higher note has twice the frequency of the lower note. For example, if we play a note with a frequency of 440 Hz (A4), the next note in the octave (one octave higher) will have a frequency of 880 Hz (A5).
Now, let’s consider the expanding geometry of the cube. If we start with a cube that has sides of length 1, and we double the length of each side to create a larger cube, the volume of the larger cube will be 8 times the volume of the smaller cube. This is because the volume of a cube is proportional to the cube of its side length. So, if we double the length of each side, we are effectively multiplying the volume by 2 x 2 x 2 = 8.
Similarly, if we double the length of the sides of the original cube again, we will create an even larger cube with sides of length 4. The volume of this cube will be 8 times the volume of the previous cube (with sides of length 2), which means it will be 64 times the volume of the original cube (with sides of length 1).
Now, let’s relate this to the frequencies of the notes of the musical octave. If we start with a note with a frequency of f, the next note in the octave (one octave higher) will have a frequency of 2f. If we continue to double the frequency of each note, we will eventually reach a note with a frequency of 2^n * f, where n is the number of doublings required to reach that note.
The relationship between the increasing frequencies of the notes and the expanding geometry of the cube becomes apparent when we realize that the ratio of the frequency of each note to the frequency of the previous note is always 2. This is analogous to the ratio of the length of the sides of the larger cube to the length of the sides of the smaller cube, which is also always 2.
Abraham, the Principle of Doubling and the Cube
“Look at the kindness of Abraham, the friend of God, He chose the path of love and was the father of many prophets. His love for the Creator made him reach the highest rank, Even the angels acknowledged his perfection and praised him.” Rumi
We have referenced Abraham in many of our earlier articles. In reference to the concept of the musical octave, Abraham may also be linked with the concept of doubling.
His name in Hebrew Gematria is 248, which is also the number of positive Mitzvahs in Judaism (there also being 365 negative Mitzvahs). In the Torah, there are considered 248 “organs” in the body, and there are 248 words in the twice-daily prayer, the Shema.
There are obvious Torah and New Testament references to this concept of doubling:
Genesis 17:3–6 — Then Abram fell on his face, and God talked with him, saying: “As for Me, behold, My covenant is with you, and you shall be a father of many nations. No longer shall your name be called Abram, but your name shall be Abraham; for I have made you a father of many nations. I will make you exceedingly fruitful; and I will make nations of you, and kings shall come from you.
Galatians 3:29 (NKJV) — And if you are Christ’s, then you are Abraham’s seed, and heirs according to the promise.
From the geometric perspective, the series 2, 4, and 8 are the numbers of vertices of the developing cube into 3-dimensional space from 1-dimension. We touched on this in one of our earliest articles.
The Geometry of the Tetrahedron and the Semitone (the harmonic series)
“Abraham, the patriarch, entered into partnership with the Holy One, blessed be He, in order to bring down the divine presence upon earth.” The Zohar
There is a mathematical relationship between the geometry of the tetrahedron and the semitone, but it is a bit more complex than the relationship between the expanding cube and the octave.
The relationship between the tetrahedron and the semitone is based on the concept of the “harmonic series,” which is a sequence of frequencies that are related to each other by integer multiples. The harmonic series is a fundamental concept in music theory, and it is the basis for many of the patterns and relationships that we hear in music.
To understand the relationship between the tetrahedron and the semitone, we need to start with the harmonic series. If we start with a base frequency of f, the harmonic series is a sequence of frequencies that are integer multiples of f:
f, 2f, 3f, 4f, 5f, …
Now, if we take the ratio of each frequency in the harmonic series to the previous frequency, we get a sequence of ratios:
2/1, 3/2, 4/3, 5/4, …
These ratios are known as the “harmonic ratios,” and they play an important role in the mathematical relationships that underlie music.
Now, let’s consider the tetrahedron. If we start with a tetrahedron with edges of length 1 and we inscribe a sphere inside the tetrahedron (the red sphere in the diagram below), the radius of the sphere will be sqrt(2)/2.
This is because the sphere is tangent to each of the four faces of the tetrahedron, and each face is an equilateral triangle with edges of length 1. The height of each equilateral triangle is sqrt(3)/2.
So the distance from the centre of the tetrahedron to the midpoint of each edge is sqrt(2)/2.
Now, if we take the distance from the centre of the tetrahedron to one of its vertices, we get a distance of sqrt(6)/3.
This distance is related to the radius of the inscribed sphere by the harmonic ratio of 3/2:
(sqrt(6)/3) / (sqrt(2)/2) = sqrt(6/2) / 3 * sqrt(2) = 3/2
This harmonic ratio of 3/2 is also the ratio between the frequency of a note and the frequency of its perfect fifth. In Western music theory, the perfect fifth is a musical interval that is formed by the ratio of 3/2 between two notes.
This geometric link fits well with our earlier speculation on the mathematics of the angels seen by Ezekial in his vision of the heavens, assuming that the angels responsible for producing the perfect fifth and by extension, the semitones are the seraphim. We will explore this more when we review the tetrahedron in higher dimensions.
So, while the relationship between the tetrahedron and the semitone is not as direct as the relationship between the expanding cube and the octave, there is a mathematical connection between the harmonic ratios in music and the geometric ratios in the tetrahedron. This connection helps to explain some of the fundamental relationships that underlie the structure and harmony of music. So now, let’s look at the derivation of the semitone from the perfect fifth by different methods of tuning.
Deriving the Semitone from the Perfect Fifth (Pythagorean, Equal Temperament & Just Intonation Tuning)
“ … a vision softly creeping. Left its seeds while I was sleeping. And the vision that was planted in my brain. Still remains.” Paul Simon
So the semitone is the smallest interval in Western music, and it can be derived from the perfect fifth in a variety of ways. A perfect fifth is an interval that spans seven semitones or half-steps. Here are a few methods to derive the semitone from the perfect fifth in music theory:
Pythagorean tuning
In Pythagorean tuning, a perfect fifth is derived from a 3:2 frequency ratio. To derive the semitone, you can stack up twelve perfect fifths and then divide the result by a power of two (which represents octaves) until it falls within the range of a single octave.
Mathematically, this is represented as (3/2)¹² / 2⁷, which approximates a 2:1 ratio (an octave). The difference between the perfect fifth and the perfect fourth, which has a frequency ratio of 4:3, will give you a semitone.
The Pythagorean comma (or ditonic comma) comes from this frequency ratio not being precisely unity ((3/2)¹² / ²⁷ = 1.01364 )). Some speculate that the easterly-westerly kink of approximately 1.5 degrees in the longitudinal axis of Lichfield Cathedral may be an architectural reference to this musical adjustment. From an architectural engineering sound acoustics perspective, this may be an attempt to “tune” the building to dampen any differences between the Pythagorean and Just Intonation tuning systems that were prominent at the time of the building of the Cathedral.
Equal temperament
In 12-tone equal temperament, the octave is divided into 12 equal semitones. Each semitone is derived by taking the 12th root of 2 (approximately 1.05946).
To derive the semitone from the perfect fifth, you can calculate the frequency ratio of the perfect fifth in equal temperament, which is the 12th root of 2 raised to the power of 7 (approximately 1.4983).
Then you can calculate the difference in semitones between the perfect fifth and its corresponding perfect fourth (which is five semitones lower) by dividing the perfect fifth’s ratio by the 12th root of 2 raised to the power of 5 (approximately 1.3348). This difference will give you the semitone ratio in equal temperament.
Just intonation
In just intonation, the perfect fifth is derived from the 3:2 frequency ratio, similar to Pythagorean tuning. However, other intervals are based on different small whole-number ratios. For example, the major third has a 5:4 frequency ratio.
To derive a semitone from the perfect fifth in just intonation, you can calculate the difference between the perfect fifth (3:2) and another interval with a different frequency ratio, such as the just major third (5:4). In this case, you would calculate (3/2) / (5/4), which simplifies to 6:5, representing a whole tone.
You can then derive the semitone by finding a ratio that, when combined with the whole tone, approximates the perfect fourth (4:3). In just intonation, one possible semitone ratio is 16:15, which, when combined with the whole tone (6:5), approximates the perfect fourth.
Subtraction method
Another simple way to derive the semitone from the perfect fifth is to subtract the number of semitones in a perfect fourth (5) from the number of semitones in a perfect fifth (7). The difference, 2, represents a whole tone. Since a whole tone is composed of two semitones, you can divide this interval by 2 to obtain a single semitone.
Keep in mind that these methods are based on different tuning systems, and the precise size of a semitone may vary slightly between these systems.
The Gematria of 134
“Come and see: the world was created by the Torah, as it is written, ‘In the beginning (Beresheet) God created,’ and it is also written, ‘The Lord created me as the beginning (Reishit) of His way’ (Proverbs 8:22). What is ‘me’? It is the Torah.” The Zohar
Just as the cube has a value of 248, the tetrahedron has a value of 134.
As you now know, in Hebrew, each letter has a numerical value, and the Gematria is the sum of the numerical values of the letters in a word or a name. Here are some Hebrew words and names from the Bible that have a Gematria value of 134:
- גְּבַרְיָה (Gevarya): This is a Hebrew name meaning “God is my strength.” It appears in the Bible in the book of 1 Chronicles 25:4, referring to a Levite musician.
- יְדוּתוּן (Yedutun): This is another Hebrew name, meaning “praiser.” It also appears in the book of 1 Chronicles 25:1.
- מַלְכִּיאֵל (Malkiel): This Hebrew name means “God is my king.” It appears in the Bible in the book of Numbers 26:45, referring to a descendant of Asher (whose name means “happy”).
- יְעֻשָּׂה (Ye’usha): This is a Hebrew name meaning “God saves.” It appears in the Bible in the book of Nehemiah 3:19, referring to one of the builders of the wall of Jerusalem.
- יְחִידָה (Yechida): This is a Hebrew word meaning “unity” or “unique.” Although not directly mentioned in the Bible, it is derived from the word יָחִיד (Yachid), which means “only” or “solitary,” and appears in the Bible in Psalms 22:21 and Psalms 35:17.
The Hierarchy of Angels & the Ark of The Covenant
“The shape of the Heavens is determined by the needs of the Earth.”
Intuition suggests the octahedron, as the dual of the cube (the tetrahedron is self-dual), plays a role in there being eight octaves. In our mathematical theory of angels, the octahedron would correspond to the Cherubim. Let’s review the hierarchy of angels according to Thomas Aquinas.
Thomas Aquinas, a prominent 13th-century theologian and philosopher, was heavily influenced by the works of Aristotle and Christian theology. In his seminal work, “Summa Theologica,” Aquinas discussed the nature and hierarchy of angels. He drew upon earlier works, especially the writings of Pseudo-Dionysius the Areopagite, who first proposed a celestial hierarchy.
According to Aquinas, angels are organized into a hierarchy of nine orders or choirs, which are further grouped into three spheres or triads. Here is the first triad:
First Sphere (closest to God)
- Seraphim
The highest order of angels, the Seraphim, are characterized by their intense love and closeness to God. They perpetually worship and praise God, and their primary function is to contemplate and radiate divine love.
2. Cherubim
The Cherubim are associated with divine wisdom and knowledge. They are often depicted as guardians of sacred spaces and bearers of the throne of God. Cherubim contemplate and reflect on the divine wisdom and knowledge of God.
3. Thrones
The Thrones symbolize divine justice and authority. They serve as channels for God’s will and maintain the cosmic order, ensuring that divine justice prevails.
There is no precise correspondence between these angels and the Kabbalah Tree of Life, but we could imagine that the elements connecting the Sephirot, represented by the 22 Hebrew letters of the alphabet, are actually angels or the combined actions of the angels, how they relate and how they transmit the voice of God. In this way:
- The Seraphim relate to the element connecting Sephirah Keter (Crown) with Chockmah (Wisdom)
- The Cherubim relate to the element connecting Chockmah to Binah (Understanding)
- The Thrones relate to lower elements in the Tree of Life
- The Orphanim, like waves interfering with the voices of the other angels, help reinforce the light that is communicated
So, with the octaves as the expanding cube, the seraphim responsible for the perfect fifth/semitone and the octahedron possibly corresponding to the eight octaves, we have a nice geometric parallel with the shape of the Ark of the Covenant. In summary, then, by correlating the geometric correspondences of music with Lichfield Cathedral, we can make a suggestion that this Cathedral geometrically symbolises the Ark of the Covenant.
The author accepts this requires some imagination. However, in later articles, we will draw the same conclusion in different ways.
Finding Music in Lichfield Cathedral, the Ratio of 2:1 & 3:2
Unison
If you have not read our article on the spiritual principles of unison and an overview of other musical intervals, you may wish to visit the below article first:
How Music Unfolds in the Geometry of Lichfield Cathedral & How it Becomes a House of God & a Model of the Human Soul
“There’s nothing you can do that can’t be done. Nothing you can sing that can’t be sung.” Lennon & McCartney
The Hebrew alphabet letter Aleph (the letter also has the value 1) corresponds to the Infinite Light of God and describes the infinite potential of His presence in all things.
The foundations of Lichfield Cathedral can be seen as attempting to tell us this, also as it stands halfway up the side of a hill, as Moses did when addressing the Hebrews (and also Jesus when giving his Sermon on the Mount, and as Mohammed did when receiving monthly revelation from Allah via the angel Gibril in the cave of Mount Hira).
In all cases, they remained grounded by their wives (and various speculated female characters for Jesus). Lichfield Cathedral is grounded by St Mary’s Church, which sits at the lowest point of the hill in the city centre’s Market Square. These female elements parallel the “Shekinah”.
For example, we looked at how musical scales and the structural dynamics of sound can be found in the solar system:
Taken from the tip of the Central Spire how the concept of doubling using the golden spiral / Fibonacci sequence corresponds to the Jewish doctrine of the structure of the soul.
How this also draws out the Paleo-Hebrew (Moses) alphabet letter for Beth (“House of God”). The letter Beth also has the value 2. Therefore the ratio/relation of Aleph to Beth (the Infinite Light to the soul) is 2:1, the same as the octave. The Hebrew letter with the value 3 is Gimmel, and so the perfect fifth is the ratio of Gimmel to Beth. And in the tuning of the semitone, the requirement of the perfect fourth (4:3), we are introduced to the Hebrew letter Dalet. This completes the Hebrew letters lined inside the bays of the Lady Chapel of Lichfield Cathedral.
You can get a taste of the spiritual meanings of the Hebrew letters in the following article:
You can find more about the principles of the soul in this earlier article:
The Geometric Nave of Lichfield Cathedral
“We know where we’re from. We’re leaving Babylon. We’re going to our Father’s land.” Bob Marley
In the next article, we will begin to explore the vertical elevations of Lichfield Cathedral and re-synthesise the many universal spiritual principles we have learned about in earlier articles.
Practical Applications: The Octave in World Religions & Culture
Listen to your favourite song with someone you love and tell them why you love the song so much.
Listen to any piece of music, presumably one close to your heart, and focus your attention on just one instrument or element in it, such as one voice. Try to make this instrument the centre of your experience in the music. Listen to the same piece of music and focus on a different instrument and see how this changes your experience.
Are there any links to music in your own culture? Consider differences to the Western structure and tuning of music as explored here, with your own culture.
Look for the ratio of 2:1 in a sacred building near you. Any sacred building of any faith will do. Look for combinations of items, expansions of space, and tesselations of shapes such as tiles.
Look to see if these shapes and spaces form a series of similar musical series, such as 7/8 (as in a diatonic scale) or 11/12 (as in the semitones).
Here are a couple of well-known secular melodies built around the major scale from a Western and Eastern perspective.
A Western Melody
An Eastern Melody
Dr Nick Stafford
Eye of Heaven — Lichfield Cathedral, a Theory of Everything
Sometimes too hot the eye of heaven shines — William Shakespeare, Sonnet 18
Footnotes
This article is purely speculative, and the author claims no truth or originality in it.
Photograph of Geometric Nave by Diliff copyright 2014 use under Creative Commons.
Likely vertical elevation ratios:
Lady Chapel = 3:1
Quire = ? 4:3 or 3:2
Quire aisle = 2:1
North transept = unison
South transept = unison
Nave aisles = 2:1
Nave = 4 segments, 2:1:1
West face from inside = 1:1
Artificial intelligence was used to assist in the research for this article.
