Introducing the mathematics of
QUIDDITY SEQUENCES - Optimal Non-convergence
This above diagram is drawn from research on topological triangulations representing certain Cluster Algebras in current mathematical research.
An optimal model for a quiddity sequence, drawn from frieze patterns, with respect to their relation to cluster algebra is explored.
A frieze pattern which grows and shrinks is understood to represent a finite pattern.
The study focuses on exploring growth which varies between arithmetic and exponential sequences, with infinite pattern growths in rubber geometry typologies.
By focusing on the limitations imposed by patterns formed when working with triangulations (where each unit has a maximum of 3 vertices), where the model has no superfluous points, a constrained environment is used to find architectures where non-convergence of form is optimised. By combining the patterns of growth – represented by the “diamond rule” (where ab – 1 = cd), a formal constraint is used to generate finding a loft form within the rubber geometries – where the curved-line-of-thought process for the mathematician is used as a a defined NURBS (non-uniform rational B-spline) curve, containing an inherent new geometric quality for an architectural form.
MYSTICAL WOMEN: The Choirs of Angels, by Hildegard of Bingen
Sep17
Scivias, Book One, Vision Six
Then I saw in the secret places in the heights of Heaven two armies of heavenly spirits who shone with great brightness. Those in one of the armies had on their breasts wings, with forms like human forms in front of them, on which human features showed as if in clear water. Those in the second army also had wings on their breasts, which displayed forms like human forms, in which the image of the Son of Man shone as if in a mirror. And I could see no other form either in these or in the others. These armies were arrayed in the shape of a crown around five other armies. Those in the first of these five armies seemed as if they had human forms that shone with great splendor from the shoulders down. Those in the second shone with such great brightness that I could not look at them. Those in the third had the appearance of white marble and heads like human heads, over which torches were burning, and from the shoulders down they were surrounded by an iron-gray cloud. Those in the fourth had forms like human forms and feet like human feet, and wore helmets on their heads, and marble tunics. And those in the fifth had nothing human in their appearance, and shone red like the dawn. And I saw no other form in them.
But these armies were also arrayed like a crown around two others. Those in the first of these other armies seemed to be full of eyes and wings, and in each eye appeared a mirror and in each mirror a human form, and they raised their wings to a celestial height. And those in the second burned like fire, and had many wings, in which they showed as if in a mirror all the Church ranks arrayed in order. And I saw no other shape either in these or in the others. And all these armies were singing with marvelous voices all kinds of music about the wonders that God works in blessed souls, and by this God was magnificently glorified.
1 God wonderfully formed and ordered His creation
Almighty and Ineffable God, Who was before all ages and had no beginning and will not cease to be when all ages are ended, marvelously by His will created every creature and marvelously by His will set it in its place. How? He destined some creatures to stay on the earth, but others to inhabit the celestial regions. He also set in place the blessed angels, both for human salvation and for the honor of His name. How? By assigning some to help humans in their need, and others to manifest to people the judgments of His secrets.
Therefore you see in the secret places in the heights of Heaven two armies of heavenly spirits who shine with great brightness; thus, as is shown to you in the height of secret places that the bodily eye cannot penetrate but the inner sight can see, these two armies indicate that the human body and soul should serve God, since they are going to have the brightness of eternal blessedness with the citizens of Heaven.
2 On the appearance of the angels and its meaning
And those in one of the armies have on their breasts wings, with forms like human forms in front of them, on which human features show as if in clear water. These are the angels, who spread the desires in the depths of their minds like wings; not that they have wings like birds, but that in their desires they are quick to accomplish God’s will, the way a person’s thoughts speed swiftly; and by their forms they display in themselves the beauty of reason, by which God closely examines human deeds; for as a servant who hears his master’s words carries them out according to his will, so the angels pay attention to God’s will for humans and show Him human actions in themselves.
3 On the appearance of the archangels and its meaning
And so those in the second army also have wings on their breasts, which display forms like human forms, in which the image of the Son of Man shines as if in a mirror. These are the archangels, who contemplate God’s will in the desires of their intellect and display in themselves the beauty of reason; they magnify the Incarnate Word of God in the purest way, because, knowing God’s secret decrees, they have often prefigured the mysteries of the Incarnation of the Son of God. And you can see no other form either in these or in the others; for in both the angels and the archangels there are many secret mysteries that the human intellect, weighed down by the mortal body, cannot understand. But these armies are arrayed in the shape of a crown around five other armies. This shows that the human body and soul must, by virtue of their strength, contain the five human senses, purify them by the five wounds of My Son, and lead them to the righteousness of governance from within.
4 On the appearance of the Virtues and its meaning
And so those in the first of these five armies seem as if they have human forms that shine with great splendor from the shoulders down. These are the Virtues, which spring up in the hearts of believers and in ardent charity build in them a lofty tower, which is their works; so that in their reason they show the deeds of the elect, and in their strength bring them to a good end with a great glory of blessedness. How? The elect, whose inner understanding is clear, cast away all their wickedness of evil, being enlightened by these Virtues in the enlightenment of My will, and fight vigorously against the snares of the Devil; and these Virtues unceasingly show to Me their Creator these struggles against the Devil’s throng. For people have within themselves struggles of confession and of denial. How? Because this one confessed Me, and that one denies Me. And in this struggle the question is: Is there a God or not? And the answer comes from the Holy Spirit Who dwells in the person: God is, and created you, and also redeemed you.
But as long as this question and answer are in a person, the power of God will not be absent from him, for this question and answer carries with it penitence. But when this question is not in a person, neither is the answer of the Holy Spirit, for such a person drives out God’s gift from himself and, without the question that leads to penitence, throws himself upon death. And the Virtues display to God the battles of these wars, for they are the seal that shows God the intention that worships or denies Him.
Scivias I.6: The Choirs of Angels. From the Rupertsberg manuscript, fol. 38r.
credits: Neue Vocalsolisten (SSATB) Sylvie Lacroix (flute)
SALT New Music Festival & Symposium Voices without Borders, Victoria, Canada 23 July, 2015
Attractive Privacies of Breathing Borders, in one Solitary Universe.
(for solo violin with chamber ensemble & electronics)
What does all this mean? In the diagram above you can see different hierarchies of instrumental relationships evolving. For me, it uncovers a new way of trying to understand the psychological means of "relating" to various instrumental forces with an awakened sense of trying to intuitively communicate. Through guided diagram sketches as above, [and in the attached PDF of Initial sketch phase diagrams], the piece attempts to "evolve on its own", through an intense opportunity to "hear" different relations of sounds. The composer records/scripts various readings of these relationships, working together with musicians (one session) to intuitively build the score around these discovered fragments, using the diagrams as a guide to the overall form and detailed compositional structure.
Attractive Privacies of Breathing Borders, in one Solitary Universe.
for SSATB and flute
Such topological triangulations give rise to rich numerical structures called frieze patterns. The study focuses on exploring growth which varies between arithmetic and exponential friezes, where borders can be bent, like in the underlying topological rubber geometry. By focusing on the limitations imposed by patterns formed when working with triangulations (where each unit has a maximum of 3 vertices), where the model has no superfluous points, a constrained environment is used to find compositional architectures where non-convergence of form is optimised.
After exploring architectural forms, using these constraints, I became interested in the metaphysical nature of these 'borders of non-convergence'. It seemed a natural evolution to try to explore what these borders meant with sound. As sound is most malleable witht he voice
