Drawing chords in perspective

Drawing Chords

in Perspective

Harmonic sets in math, art

and music

H.S.M. Coxeter, Projective Geometry 2nd ed.

“6. Still working in the Euclidean plane, draw a line segment OC, take G two thirds of the way along it, and E two-fifths of the way from G to C.

(For instance, make the distance in centimeters OG = 10, GE = 2, EC = 3.)

If the segment OC represents a stretched string, tuned to the note C, the same string stopped at E or G will play the other notes of the major triad. By drawing a suitable triangle, verify experimentally that H(OE,CG).

(Such phenomena explain our use of the word harmonic.)”

Cremona, Elements of Projective Geometry, Oxford University Press, 1913

Matthews, Projective Geometry, Longmans, Green and co., 1914

Veblen and Young, Projective Geometry Volume 2, 1918

Young, Projective Geometry, Carus Mathematical Monographs, MAA, 1930

Baer, Linear Algebra and Projective Geometry, Academic Press, 1952

Seidenberg, Lectures in Projective Geometry, D Van Nostrand and Co, 1962

Pedoe, Introduction to Projective Geometry, Macmillan, 1963

Fishback, Projective and Euclidean Geometry 2ed, John Wiley & Sons, 1969

Bennett, Affine and Projective Geometry, John Wiley & Sons, 1995

Kadison and Kromann, Projective Geometry and Modern Algebra, Birkhauser, 1996

Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998

Casse, Projective Geometry, an Introduction, Oxford University Press, 2006

Linnaeus Wayland Dowling, Projective Geometry, McGraw-Hill Book Co, Inc., 1917 (Forgotten Books)

Notes.- The idea of four harmonic points, or harmonic division, was known to the early Greek geometers, but who first invented it is not definitely known. Apollonius of Perga (247 BC) mentions it is his book on conic sections.

The harmonic property of a complete quadrangle is contained in the Collections of Pappus (300AD). It was made the foundation for Von Staudt’s Geometric der Lage, 1847.

Three cords consisting of the same substance and having the same size and tension, and whose lengths are in harmonic progression, will vibrate in harmony when struck in unison. The name harmonic is probably due to that fact.

Lewis Goupy, Brook Taylor. 1720

Harmonics in music

Harmonic sets in mathematics

Drawing in perspective

Ceva’s Theorem

Menelaeus’ Theorem

Harmonic sets defined by ratios


Harmonic Set: H(AC,BD)




𝐴𝐵𝐴𝐷𝐵𝐶𝐷𝐶

=−1

The harmonic mean of two numbers

𝑎𝑏

h−𝑎h −𝑏

=−1𝑎𝑏

=𝑎− hh−𝑏

h𝑎 −𝑎𝑏=𝑎𝑏− h𝑏 h (𝑎+𝑏 )=2𝑎𝑏

h=2𝑎𝑏𝑎+𝑏

Harmonic set from a circle and its tangent



Harmonic set from two circles and their tangents


A = circumcenterB = centroidC = 9-point circle centerD = orthocenter

Harmonic set on Euler’s line

Harmonic sets in projective geometry…

What is projective geometry?

Abraham Bosse, 1665

Leon Battista Alberti, 1435 A real projective plane is an extension of the real Euclidean plane, extended by strategically adding points and a line at infinity.

A non-Euclidean geometry which developed out of the mathematics of perspective drawing.

A study of geometric properties that are invariant under projections.

Euclidean Geometry Projective Geometry

Constructions with compass and straightedge

Constructions with just a straightedge

Parallel Postulate: Given a line and a point not on the line, there is just one line through the point parallel to the line

There are no parallel lines: Any two lines are incident with a unique point.

A study of properties invariant under rigid motions, like length, angle, area

A study of properties invariant under projections, like…

Euclidean to Projective

Definition: A harmonic set is a set of four distinct points along a line in the projective plane such that and are the intersections of pairs of opposite sides of a quadrilateral and and are intersections of the diagonals of the quadrilateral with line .

Harmonic sets defined by quadrilaterals

Definition: A perspectivity through is a one-to-one mapping from points along line to those along line , with not incident with or , given by the following:

a point on line is mapped to .

Definition: A projectivity is a composition of perspectivities.

Theorem : A projectivity preserves harmonic sets.

Proof : It suffices to show that a perspectivity preserves harmonic sets. First, show .

Proof (cont.): Draw line . It intersects at .

Proof (cont.): Notice, forms a quadrilateral, with opposite sides intersecting at and and a diagonal intersecting at . Since and is unique, lies on line .

Proof (cont.): Now that we’ve established that is a line, we can identify another quadrilateral, . Notice, the opposite sides intersect at and , and the diagonals intersect at and . So we have

Proof (cont.): Using , we show

Proof (cont.): Thus if , then Hence perspectivities preserve harmonic sets, and therefore projectivities preserve harmonic sets.

A projectivity maps lines to lines.

A projective collineation maps planes to planes, such that incidence is preserved.

Definition: A perspective collineation through is a one-to-one mapping from points and lines in a plane to points and lines in another plane (or the same plane) mapping collinear points to collinear points such that and its image, are collinear with . This mapping preserves incidence.

Definition: A projective collineation is a composition of perspective collineations.

Collineations in perspective drawingConsider translation from the red square to the blue square in perspective.

Translation in perspective is a perspective collineation.

Collineations in perspective drawing

English Boy Using Reflection in Mirror in Foyer of Grand Hotel to Fix His Tie Photographic Print by Alfred Eisenstaedt

Reflection in perspective is a perspective collineation.


180˚ rotation in perspective is a perspective collineation.


45˚ rotation in perspective is NOT a perspective collineation.


45˚ rotation in perspective is a projective collineation, the composition of two reflections.


then H(BD, AC) ?

Harmonic sets and perspective drawingIf such that

If such that

then H(BD, AC) ?

Harmonic sets and perspective drawing

If such that


then ?

If such that


then ?

then ?

If such that


Harmonic sets and the harmonic sequence

, where is the point at infinity

𝑯 (𝑩𝑹 ,𝑬𝑸)

𝑯 (𝑩𝑱 ,𝑴𝑬 )

𝑯 (𝑩𝑬 , 𝑱𝑹)

15

14

13

12

16

11

Harmonic sets, the harmonic sequence and the harmonic mean of two numbers

1/3 1/2 11/4 1/3 1/21/5 1/4 1/31/6 1/5 1/41/(n+2) 1/(n+1) 1/n

h (𝑎 ,𝑏)=2𝑎𝑏𝑎+𝑏

1/3 1/2 12/4 2/3 2/23/5 3/4 3/34/6 4/5 4/4n/(n+2) n/(n+1) n/n

Harmonic sets and its relationship withharmonics in music

Definition: The frequency of a sound wave is the number of cycles per second, measured in Hertz.

Definition: The pitch of a sound is the perception of frequency.

Harmonic sets and its relationship with…harmonics in music

Definition: A harmonic of a sound wave is an integer multiple of the fundamental frequency of the sound wave.

Harmonic sets and its relationship with…harmonics in music

Definition: Overtones are frequencies higher than the fundamental frequency.

Many musical instruments are created to have harmonic overtones.

The human voice can create overtones.

http://www.youtube.com/watch?v=DY1pcEtHI_w

1 55/55 1/2 55/110 1/3 55/1651 110/110 2/3 110/165 2/4 110/2201 165/165 3/4 165/220 3/5 165/2751 220/220 4/5 220/275 4/6 220/3301 275/275 5/6 275/330 5/7 275/385n/(n) … n/(n+1) … n/(n+2) …

Harmonic sets, the harmonic sequence, the harmonic mean of two numbers, and harmonics in music

These intervals, octave = 2:1, perfect fifth = 3:2, etc. is tuned to just intonation.Pianos, guitars and ukuleles however, are tuned to 12-tone equal temperament.

The octaves go up exponentially: 100Hz 200Hz 400Hz 800Hz 1600HzDivide adjacent notes evenly on a logarithmic scale. For ex.: A4 = 440 B4♭ = 440*21/12 = 466.16, B4 = 440*22/12 = 493.88, C4 = 440*2-9/12 = 261.63

Harmonic sets and its relationship withharmonics in music

Just intervals1 1/2 1/3 1:1/3 = 3:12 octaves 1:1/2 = 2:11 octave 1:2:31 2/3 2/4 1:1/2 = 2:11 octave 1:2/3 = 3:2Perfect fifth 2:3:41 3/4 3/5 1:3/5 = 5:3Major sixth 1:3/4 = 4:3Perfect fourth 3:4:51 4/5 4/6 1:2/3 = 3:2Perfect fifth 1:4/5 = 5:4Major third 4:5:61 5/6 5/7 1:5/7 = 7:5Subminor fifth 1:5/6 = 6:5Minor third 5:6:71 7/8 6/8 1:3:4 = 4:3Perfect fourth 1:7/8 = 8:7Supermajor second 6:7:81 8/9 7/9 1:7/9 = 9:7Supermajor third 1:8/9 = 9:8Major second 7:8:9

Harmonic sets and an android app by Stephen Brown

Education

Drawing chords in perspective