Counting to the Music of the Spheres THE QUADRIVIAL COSMOS

X X X X X X X X X X

Daniel Newsome dlnuzum@mifami.org

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 dlnuzum@mifami.org

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.