This document discusses harmonic sets in projective geometry and their relationship to harmonics in music. It defines harmonic sets as sets of points on a line that divide the line harmonically. Harmonic sets are preserved under projectivities and collineations. They relate to the harmonic sequence and harmonic mean. In music, harmonic sets correspond to just musical intervals formed by ratios of small whole numbers, like octaves, perfect fifths and major thirds. An android app was created by Stephen Brown to demonstrate harmonic sets.

2. 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.)”

3. 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

4. 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.

5. Harmonic sets in mathematics
Drawing in perspective
Lewis Goupy, Brook Taylor. 1720
Harmonics in music

6. Harmonic sets defined by ratios
Ceva’s Theorem
Menelaeus’ Theorem

7. Harmonic sets defined by ratios

8. Harmonic sets defined by ratios
Harmonic Set: H(AC,BD)

9. Harmonic sets defined by ratios
Harmonic Set: H(AC,BD)

11. Harmonic set from a circle and its tangent
Harmonic Set: H(AC,BD)

12. Harmonic set from two circles and their tangents
Harmonic Set: H(AC,BD)

13. Harmonic set on Euler’s line
A = circumcenter
B = centroid
C = 9-point circle center
D = orthocenter
Harmonic Set: H(AC,BD)

14. Harmonic sets in projective geometry…
What is projective geometry?
A non-Euclidean geometry which developed
out of the mathematics of perspective drawing.
A study of geometric properties that are
invariant under projections.
Abraham Bosse, 1665
A real projective plane is an Leon Battista Alberti, 1435
extension of the real Euclidean plane, extended
by strategically adding points and a line at infinity.

15. Euclidean Geometry Projective Geometry
Constructions with compass and Constructions with just a straightedge
straightedge
Parallel Postulate: Given a line and a There are no parallel lines: Any two
point not on the line, there is just one lines are incident with a unique point.
line through the point parallel to the line
A study of properties invariant under A study of properties invariant under
rigid motions, like length, angle, area projections, like…

16. Euclidean to Projective

17. Harmonic sets defined by quadrilaterals

19. Definition: A projectivity is a composition of perspectivities.

20. Theorem : A projectivity preserves harmonic sets.

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

28. Collineations in perspective drawing
Consider translation from the red square to the blue square in
perspective.

29. Collineations in perspective drawing
Translation in perspective is a perspective collineation.

30. Collineations in perspective drawing
Reflection in perspective is a perspective collineation.
English Boy Using Reflection in Mirror in Foyer of Grand Hotel
to Fix His Tie Photographic Print by Alfred Eisenstaedt

31. Collineations in perspective drawing
180˚ rotation in perspective is a perspective collineation.

32. Collineations in perspective drawing
45˚ rotation in perspective is NOT a perspective collineation.

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

34. Harmonic sets and perspective drawing
then H(BD, AC) ?

35. Harmonic sets and perspective drawing
then H(BD, AC) ?

36. Harmonic sets and perspective drawing

37. Harmonic sets and perspective drawing

38. Harmonic sets and perspective drawing

39. Harmonic sets and the harmonic sequence

43. Harmonic sets, the harmonic sequence
and the harmonic mean of two numbers
1/3 1/2 1 1/3 1/2 1
1/4 1/3 1/2 2/4 2/3 2/2
1/5 1/4 1/3 3/5 3/4 3/3
1/6 1/5 1/4 4/6 4/5 4/4
1/(n+2) 1/(n+1) 1/n n/(n+2) n/(n+1) n/n

44. Harmonic sets and its relationship with
harmonics 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.

45. 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.

46. 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.

47. Harmonic sets, the harmonic sequence, the harmonic
mean of two numbers, and harmonics in music
1 55/55 1/2 55/110 1/3 55/165
1 110/110 2/3 110/165 2/4 110/220
1 165/165 3/4 165/220 3/5 165/275
1 220/220 4/5 220/275 4/6 220/330
1 275/275 5/6 275/330 5/7 275/385
n/(n) … n/(n+1) … n/(n+2) …

49. Harmonic sets and its relationship with
harmonics in music
Just intervals
1 1/2 1/3 1:1/3 = 3:1 1:1/2 = 2:1 1:2:3
2 octaves 1 octave
1 2/3 2/4 1:1/2 = 2:1 1:2/3 = 3:2 2:3:4
1 octave Perfect fifth
1 3/4 3/5 1:3/5 = 5:3 1:3/4 = 4:3 3:4:5
Major sixth Perfect fourth
1 4/5 4/6 1:2/3 = 3:2 1:4/5 = 5:4 4:5:6
Perfect fifth Major third
1 5/6 5/7 1:5/7 = 7:5 1:5/6 = 6:5 5:6:7
Subminor fifth Minor third
1 7/8 6/8 1:3:4 = 4:3 1:7/8 = 8:7 6:7:8
Perfect fourth Supermajor second
1 8/9 7/9 1:7/9 = 9:7 1:8/9 = 9:8 7:8:9
Supermajor third Major second

50. Harmonic sets and an android app
by Stephen Brown