1. Newell, W. G. and H. A. Simon, "The Mind's Eye in Chess," in Visual Information Processing, ed. W.G. Chase (New York: Academic Press, 1973).

2. von Hippel, P. T., "The efficiency of improvisational thinking: Protocol analyses of student harmonizations," Industrial Affiliates Meeting, Center for Computer Research in Music and Acoustics (Stanford University, May 15, 1997).

3. Schoenberg, Arnold, Theory of harmony, trans. Roy Carter (London: Faber and Faber, 1911 / 1978), 4.

4. Prout, Ebenezer, Harmony: Its Theory and Practice, 32nd ed. (Boston: Boston Music, 1903), 37, emphasis in original.

5. Rimsky-Korsakov, Nikolay, Practical Manual of Harmony, trans. Joseph Achron, 12th ed. (New York: C. Fischer, 1886 / 1930), 17.

6. Tchaikovsky, Peter Ilich, Guide to the Practical Study of Harmony, German trans. P. Juon; English trans. Emil Krall and Jame Liebling (Canoga Park, CA: Summit, 1871 / 1970), 27.

7. Piston, Walter, Harmony, 3rd ed. (New York: W. W. Norton., 1962), 45.

8. Tweedy, Donald, Manual of Harmonic Technic Based on the Practice of J. S. Bach (New York: C. H. Ditson, 1928), 72.

9. Kostka, Stefan and Dorothy Payne, Tonal Harmony (New York : A. A. Knopf, 1984), 115.

10. Rameau, Jean Philippe, Treatise on Harmony, trans. Philip Gossett (New York: Dover, 1722 / 1971), 313-314.

11. Tweedy, 72.

12. Riemann, Hugo, Harmony Simplified, or the Theory of the Tonal Functions of Chords (London: Augener, 1893 / 1895), 12-14.

13. Mitchell, William J., Elementary Harmony (New York: Prentice-Hall, 1939), 45.

14. Weidig, Adolf, Harmonic Material and Its Uses (Chicago: Clayton F. Summy, 1923), 75.

15.  Schoenberg, 4.

16. Ibid., 61.

17. Weidig, 76.

18. Prout, 37.

19. e.g., Piston, 45; Des Marais, Paul, Harmony: A Workbook in Fundamentals (New York: W. W. Norton, 1962), 43; Warburton, Annie O., Harmony for Schools and Colleges (New York: Longmans, Green, 1938), 145.

20. Huron, David, "Chordal Tone Doubling and the Enhancement of Key Perception," Psychomusicology 12, no. 1 (1993): 73�83.

21. Alchin, Carolyn, Applied Harmony, part I, ed. Vincent Jones (Los Angeles: L. R. Jones, 1930), 42; Aldwell, Edward and Carl Schachter, Harmony and Voice Leading (San Diego: Harcourt Brace Jovanovich, 1989), 92; Ottman, Robert, Advanced Harmony: Theory and Practice (Englewood Cliffs, NJ: Prentice-Hall, 1961), 299; Tweedy, 72; and Weidig, 75.

22. Kostka and Payne, 115.

23. e.g., Kostka and Payne, 118.

24. e.g., Mitchell, 45.

25. Robinson, Raymond C., Progressive Harmony (Boston: B. Humphries, 1942), 73.

26. Alchin.

27. Aldwell & Schachter.

28. Benjamin, Thomas, Michael Horvit, and Robert Nelson, Techniques and Materials of Tonal Music (Boston: Houghton Mifflin, 1975).

29. Bussler, Ludwig, Elementary Harmony, trans. Theodore Baker, 2nd ed. (New York: G. Schirmer, 1891).

30. Chadwick, G. W., Harmony, a Course of Study (Boston: B.F. Wood music, 1897).

31. Christ, William; Richard DeLone, Vernon Kliewer, Lewis Rowell, and William Thomson, Materials and Structure of Music, 2nd ed. (Englewood Cliffs, N.J.: Prentice-Hall, 1972).

32. DeLone, Richard, Music: Patterns and Style (Reading, MA: Addison-Wesley, 1971).

33. Foote, Arthur and Walter Spalding, Modern Harmony in its Theory and Practice (New York: Arthur P. Schmidt, 1936).

34. Gow, George, The Structure of Music (New York: G. Schirmer, 1895).

35. Jones.

36. Kostka and Payne.

37. Kraft, Leo, Gradus: An Integrated Approach to Harmony, Counterpoint, and Analysis (New York: W. W. Norton, 1976).

38. Levarie, Siegmund, Fundamentals of Harmony (New York, Ronald Press, 1954).

39. Macpherson, S., Practical harmony (London: Williams, 1907).

40. Des Marais, Paul, Harmony: A Workbook in Fundamentals (New York: W. W. Norton, 1962).

41. Mitchell.

42. Morris, Reginald, Foundations of Practical Harmony and Counterpoint (London: Macmillan, 1943).

43. Ottman, Robert, Elementary Harmony: Theory and Practice (Englewood Cliffs, NJ: Prentice-Hall, 1961).

44. Ottman, Advanced Harmony.

45. Owen, Harold, Music Theory Resource Book (New York: Oxford University Press, 2000).

46. Piston.

47. Prout.

48. Ratner, Leonard G., Harmony: Structure and Style (New York: McGraw-Hill, 1962).  

49. Riemann.

50. Rimsky-Korsakov.  

51. Robinson.  

52. Schoenberg.

53. Serly, Tibor, A Second Look at Harmony (New York: S. French, 1964).

54. Shinn, Frederick G., A Method of Teaching Harmony, Based Upon Systematic Ear-Training (London: Augener, 1904).

55. Siegmeister, Elie, Harmony and Melody Workbook, vol. 1 (Belmont, CA: Wadsworth, 1965).

56. Stainer, John, Harmony (New York: Novello, Ewer, 1891).

57. Tapper, Thomas, First Year Harmony (Boston: Arthur P. Schmidt, 1908).

58. Tchaikovsky.

59. Tischler, Hans, Practical Harmony (Boston: Allyn and Bacon, 1964).

60. Trotter, T. H. Yorke, Constructive Harmony (London: Bosworth, 1911).

61. Tweedy.

62. Warburton.

63. Weidig.

64. York, Francis L., A Practical Introduction to Composition; Harmony Simplified (Boston: O. Ditson, 1909).

65. Bullis, Carleton, Harmonic Forms (Cleveland: The Clifton Press, 1933).

66. Murphy, Howard and Edwin Stringham, Creative Harmony and Musicianship (New York: Prentice-Hall, 1951).

67. Huron, David, and Peter Sellmer, "Critical Bands and the Spelling of Vertical Sonorities," Music Perception 10, no.2 (1992): 129-149.

68. We would like to thank David Huron for this suggestion.

69. These effects can also be observed in randomly-generated triads. (See �4.2.) Among chorale-style random triads whose largest space is between the bass and tenor, 54% double the bass, whereas only 45% do so when the largest space is not between the two lowest parts. Quartet-style random triads have fewer spacing constraints and do not exhibit this tendency. Among triads whose largest space is between the cello and viola, 44% double the bass, compared to 48% when this is not the case.

70. McHose, Allen, The Contrapuntal Harmonic Technique (New York: F. S. Crofts, 1947).

71. CCARH, The MuseData Database (Stanford, CA: Center for Computer Assisted Research in the Humanities, 2001); available from http://www.musedata.org.

72. Huron, David, The Humdrum Toolkit: Software for Music Researchers (v. 2.2b) [Three computer disks and 16-page installation guide] (Stanford, CA: Center for Computer Assisted Research in the Humanities, 1993).

73. Temperley, David, "An Algorithm for Harmonic Analysis," Music Perception 15, vol. 1 (1997): 31-68; Temperley, David, and Daniel Sleator, "Modeling Meter and Harmony: A Preference-Rule Approach," Computer Music Journal 23, vol. 1 (1999): 10�27.

74. Additional details about the process of analyzing Humdrum data with Melisma can be found in Aarden, Bret, "An Empirical Study of Chord-Tone Doubling in Common Era Music" (Master's Thesis, The Ohio State University, 2001).

75. Further communication with the author suggested that increasing Melisma's penalties against modulation may have resulted in a greater number of usable analyses. For those analyses that were retained, however, the settings we used produced more conservative estimates of how early the first modulations occurred.

76. Burns, Lori, Bach's Modal Chorales (Stuyvesant: Pendragon, 1995).

77. For example, Komar, Arthur J., Theory of Suspensions: A Study of Metrical and Pitch Relations in Tonal Music (Princeton, N.J.: Princeton University Press, 1971).

78. Apel, Willi, The New Harvard Dictionary of Music, 2nd edition, Don Randel, ed. (Cambridge, MA: Belknap Press of Harvard University Press, 1986).

79. Kirk, R. E., Experimental design: Procedures for the behavioral sciences (Pacific Grove, CA: Brooks Cole, 1995).

80. Long, J.S., Regression Models for Categorical and Limited Dependent Variables (Thousand Oaks, CA: Sage, 1997).

81. Breslow, N.E., "Covariance Adjustment of Relative-Risk Estimates in Matched Studies," Biometrics 38 (1982): 661-672.

82. We thank Althea Warren for this suggestion.

83. Krumhansl, C.L., and E. J. Kessler, "Tracing the Dynamic Changes in Perceived Tonal Organization in a Spatial Representation of Musical Keys," Psychological Review 89 (1982): 334-368.

84. Some readers may notice that the probabilities in this example are higher than the probabilities in the linked graphs. This is because the probability that a triad is composed depends on the triad it is paired with. In the example, the triad in the right is pretty obviously random; it is badly spaced and it doubles the leading tone, which is very unusual among composed triads. The unusual features of the right triad increase the probability that the left triad is composed.

End of footnotes