Counting to the Music of the Spheres · Diapason Duplex 1200 cents 4:3:2:1 x x x x x x x x x x 1 3...
Transcript of Counting to the Music of the Spheres · Diapason Duplex 1200 cents 4:3:2:1 x x x x x x x x x x 1 3...
Counting to the Music of the Spheres THE QUADRIVIAL COSMOS
X X X X X X X X X X
Daniel Newsome [email protected]
1 3 2 9 6 4 27 18 12 8
1 + 2 + 3 + 4 = 10
Counting to the Music of the Spheres THE QUADRIVIAL COSMOS
X X X X X X X X X X
Daniel Newsome [email protected]
1 3 2 9 6 4 27 18 12 8
1 + 2 + 3 + 4 = 10
Students in a class at the University of Bologna. Tomb of Johannes of Legano -Law Prof. – Bologna - ca.1350
Class at the University of Paris – 14th c. The “tonsure,” the clerical haircut.
Frontispiece from Gregorius Reisch’s Margarita Philosophica (Philosophical Pearl) - 1517
The Trivium:
Grammar
Rhetoric
Logic
The Quadrivium:
Arithmetic
Music
Geometry
Astronomy
The sleeping student seems to be a standard feature on many of the tombstones of famous teachers.
Detail from the Tomb of Mondino de Luizzi, a famous teacher of medicine and author of a very influential text on anatomy.
ca. 1318 - Bologna
Doctorates in … the Arts (the Liberal Arts and Philosophy) Law Medicine Theology
The Magister Teaching University of Paris – 14th c.
THE QUADRIVIUM, FOUR PATHS TO A THEORY OF STRUCTURE
ARITHMETIC or perhaps more accurately,
Number Theory The Art of Counting…
1 + 1 + 1 = 3 and 1
Greborius Reisch “Typus Arithmeticae” from the Margarita Philosophica - 1517
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3
4 9 8 27
Unity Oneness The One
“Apples and Oranges”
are comparable in the world of light.
1 Apple = 1 Orange
….. so long as you divide out the world of change.
(1 Apple) ÷ (world of change) = (1 Orange) ÷ (world of change)
1 = 1
or simply One-ness
1 + 1 + 1 = 3 or 3-ness … and Unity The Trinity as Unity
Square numbers……………...…..and pentagonal numbers. 1 4 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 5 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Greborius Reisch “Typus Arithmeticae” from the Margarita Philosophica - 1517
1 3 2
9 4 27 8
Greborius Reisch “Typus Arithmeticae” from the Margarita Philosophica - 1517
1 3 2
9 4 27 8
Greborius Reisch “Typus Arithmeticae” from the Margarita Philosophica - 1517
1 3 2
9 4 27 8
The Platonic “Lambda”
L
Uni-verse Uni-ty
See also Courbet’s painting “Origin of the World” from 1866. From F. Giorgi, De Harmonia Mundi, 1525
1 3 2
9 4 27 8
MUSIC
Musica instrumentalis
Musica mundana
Musica humana
x
x x x x x
x x x x
1 + 2 + 3 + 4 = 10 4 qualities 4 elements 4 humors 4 seasons 4 ages of life 4 winds 4 vices 4 virtues
Junkstore Guitar Monochord – Frets removed, bridge and nut raised
4:3:2:1 x
x x x x x
x x x x
1 + 2 + 3 + 4 = 10 Tetractys
4 qualities 4 elements 4 humors 4 seasons 4 ages of life 4 winds 4 vices 4 virtues
1 3 2
9 4 27 8
4:3:2:1 x
x x x x x
x x x x
1 + 2 + 3 + 4 = 10 Tetractys
4 qualities 4 elements 4 humors 4 seasons 4 ages of life 4 winds 4 vices 4 virtues
1 3 2
9 4 27 8
4:3:2:1 x
x x x x x
x x x x
1 + 2 + 3 + 4 = 10 Tetractys
4 qualities 4 elements 4 humors 4 seasons 4 ages of life 4 winds 4 vices 4 virtues
1 3 2
9 4 27 8
2:2 [1:1] Uni-son One sound Uni-ty Uni-verse
1:1 Uni-son One sound Uni-ty Uni-verse
1:1 2:1 3:2 4:3 3:1 4:1
x x x
x x x x x x x
1:1 = 2:2 = 3:3, etc.
0 cents
1 + 2 + 3 + 4 = 10 Tetractys
4:3:2:1
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
2:1 Octave P8 Double Diapason Duplex 1200 cents
This is exactly the same interval as an octave in our modern, equal-tempered system.
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
This is exactly the same interval as an octave in our modern, equal-tempered system.
2:1 Octave P8 Double Diapason Duplex 1200 cents
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
3:2 Perfect Fifth P5 Sesquialtera Diapente 702 cents
Our modern equal-tempered fifth is slightly flatter (700 cents) compared with this perfect fifth.
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
3:2 Perfect Fifth P5 Sesquialtera Diapente 702 cents
Our modern equal-tempered fifth is slightly flatter (700 cents) compared with this perfect fifth.
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
4:3 Perfect Fourth P4 Sesquitertia Diatessaron 498 cents
The modern fourth is a bit sharper than the perfect fourth
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
4:3 up 3:2 is an octave, 4:2 or 2:1
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
4:3 up 3:2 is an octave, 4:2 or 2:1
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
4:3 up 3:2 is an octave, 4:2 or 2:1 4/3 x 3/2= 12/6 or 2/1
4:3:2:1
Consonant Intervals
2:1 3:2 4:3
Superparticular
4:3:2:1 x
x x x x x
x x x x 1
3 2 9 4
27 8
9:8 – (204 cents) Tone (not functioning)
256:243 – (90 cents)
Semitone
9/8 x 9/8 x 256/243 = 4/3
Open string pitch. up 9:8 from opening pitch. up 9:8 from previous, 81:64 total from open string pitch. up 256:243 from previous, 4:3 total. up 9:8 from previous, 3:2 total. up 9:8 from previous, 27:16 total. up 9:8 from previous, 243:128 total. up 256:243 from previous, 2:1 total. An Octave.
Technically, this is Ptolemy’s Diatonic Ditonic
left - Newton’s Opticks, 1704 Colorized with lables and monochord added for clarity.
The Quadrivium:
Multitude: Arithmetic
& Music
Magnitude Geometry
& Astronomy
“Amazing Grace” sung in the Tuvan throat-singing style.