Color poster with text and diagrams.This study explored an overlap between combinatorics and music.  The goal was to show chord progressions that are common to a specific collection of music, composer, or era.University of Wisconsin--Eau Claire Office of Research and Sponsored Programs

Kiefer, Peter

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Minds@University of Wisconsin

Combinatorics of Chord ProgressionsPeter KieferFaculty Mentor: Manda RiehlUniversity of Wisconsin-Eau Claire1. Music Background•One of the first people to make a connection between mathematics and music was Pythagoras ofSamos, nearly 2500 years ago. Since then, many mathematicians have explored areas of overlapbetween mathematics and music. In this project, we explored an overlap between combinatoricsand music. Our goal is to show chord progressions that are common to a specific collection ofmusic, composer, or era.• To understand chord progressions, however, we must first understand the musical scale and itsrespective chords. A musical scale consists of seven notes and repeats at the octave, e.g., theC Major Scale is C-D-E-F-G-A-B-(C)-. . .. The notes of a musical scale are numbered by theirsteps from the root of the scale. In a C Major Scale, the root is C, the second D, the third E,and so on to the seventh B, at which point the scale starts over again one octave higher.•A chord is three or more notes played simultaneously. In our work, we consider seven chordsdefined on a musical scale. The C Major scale, for example, consists of the I chord (C-E-G),ii (D-F-A), iii (E-G-B), IV (F-A-C), V (G-B-D), vi (A-C-E), and vii◦ (B-D-F). An upper caseRoman numeral indicates a major chord where as a lower case Roman numeral indicates aminor chord.•A chord progression, then, is the movement from one chord to another in a musical scale. Forexample, I-IV-V-I and I-vi-IV-V are two very common chord progressions used in pop music.2. Mathematics and Music• For example, suppose a piece of music has the following probabilities changing from one chordto another,•We would represent this situation mathematically with the following Initial State matrix:I IV VI 0 .60 .40IV .20 0 .80V .70 .30 0•Using Markov Chain Analysis, we find that the steady state is( I IV V.322 .305 .373... ... ...)which tells us that we expect 32.2% of chord changes to progress to I chords, 30.5% to IVchords, and 37.3% to V chords.3. Chord Analysis - Initial StateMatrices• Each entry of the matrices below represents the probability that the chord in the columnimmediately follows the chord in the row. For example, in the first row of the Bach Minormatrix, the .41 means that there is a 41% chance that any I chord is followed by a V chord.Note that each row sums to 1, or 100%.Bach Minor I ii iii IV V vi vii◦I 0 .18 .01 .20 .41 .09 .12ii .01 0 .03 0 .89 0 .07iii .06 .06 0 .25 .19 .31 .13IV .22 .14 0 0 .48 0 .15V .80 0 .02 .06 0 .10 .02vi .03 .54 .03 .14 .19 0 .08vii◦.81 0 .01 .03 .15 0 0Bach Major I ii iii IV V vi vii◦I 0 .15 .01 .28 .41 .09 .06ii .01 0 0 0 .71 .01 .25iii .03 .03 0 .52 .06 .32 .03IV .22 .13 0 0 .39 .02 .23V .82 .01 0 .07 0 .09 0vi .15 .29 .05 .11 .32 0 .09vii◦.91 0 .01 .02 .04 .03 0Mozart Minor I ii iii IV V vi vii◦I 0 .08 0 .07 .68 .06 .11ii .37 0 0 0 .46 0 .17iii 0 0 0 1 0 0 0IV .42 .10 0 0 .39 0 .09V .82 0 0 .05 0 .07 .05vi .14 .51 0 .16 .05 0 .14vii◦.76 .01 0 0 .23 0 0Mozart Major I ii iii IV V vi vii◦I 0 .13 0 .15 .62 .05 .05ii .49 0 .01 0 .40 .01 .09iii .67 0 0 0 0 .33 0IV .64 .14 0 0 .15 0 .07V .94 0 0 .01 0 .04 .01vi .11 .51 0 .14 .20 0 .04vii◦.82 0 .01 .01 .16 0 0Palestrina I ii iii IV V vi vii◦I 0 .15 .13 .28 .14 .22 .08ii .08 0 .15 .13 .28 .14 .22iii .22 .08 0 .15 .13 .28 .14IV .14 .22 .08 0 .15 .13 .28V .28 .14 .22 .08 0 .15 .13vi .13 .28 .14 .22 .08 0 .15vii◦.15 .13 .28 .14 .22 .08 0Beethoven I ii iii IV V vi vii◦I 0 .10 .01 .13 .52 .02 .22ii .06 0 .02 0 .87 0 .05iii 0 0 0 0 .67 .33 0IV .33 .03 .07 0 .40 .03 .13V .56 .22 .01 .04 0 .07 .11vi .06 .44 0 .06 .11 0 .33vii◦.80 0 0 .03 .17 0 04. Markov Chain Analysis - Steady StateMatrices(Bach Minor I ii iii IV V vi vii◦.333 .109 .015 .010 .306 .063 .075... ... ... ... ... ... ...) (Bach Major I ii iii IV V vi vii◦.344 .089 .016 .131 .273 .067 .080... ... ... ... ... ... ...)(Mozart Minor I ii iii IV V vi vii◦.400 .066 0 .056 .345 .049 .084... ... ... ... ... ... ...) (Mozart Major I ii iii IV V vi vii◦.435 .086 .001 .073 .330 .037 .038... ... ... ... ... ... ...)(Palestrina I ii iii IV V vi vii◦.143 .143 .143 .143 .143 .143 .143... ... ... ... ... ... ...) (Beethoven I ii iii IV V vi vii◦.317 .120 .012 .060 .328 .034 .130... ... ... ... ... ... ...)• For example, the results for Bach Minor show that there is a 33.3% probability of progressingto the I chord from the previous chord, 10.9% to ii, 1.5% to iii, 1.0% to IV, 30.6% to V, 6.3%to vi, and 7.5% to vii◦.5. Mathematical Analysis• By analysing chord sequences in compositions by Bach, Mozart, Palestrina, and Beethoven, weused Markov Chains to obtain the probability of any chord progression in that composer’s workand show the most probable chord progression. Our analysis shows that that progressions to theI and V chords are most probable in all compositions (except Palestrina), whereas progressionsto the iii chord are the least probable.Noteworthy ResultsComposer Most Probable Chord(s) Least Probable Chord(s)Palestrina Equal Probability (14.3%) Equal Probability (14.3%)Bach I (33.9%) and V (29.0%) iii (1.5%)Mozart I (41.8%) and V (33.8%) iii (0%)Beethoven V (32.8%) and I (31.7%) iii (1.2%)•Note the time line of the composers analysed and the musical eras they lived in:Palestrina (1525 - 1594) - Renaissance (1400 - 1600)Bach (1685 - 1750) - Baroque (1600 - 1760)Mozart (1756 - 1791) - Classical (1750 - 1830)Beethoven (1770 - 1827) - Classical, Romantic (1815 - 1910)6. Music Analysis•Music from the Renaissance Period emphasized the self, and introduced the concept of chordprogressions. The Church, which emphasized the divine, disapproved of the Renaissance Periodbecause of its contrasting view. Palestrina, a composer of sacred music, composed at a levelthe Church approved of - a level prior to the Renaissance Period when chord progressions didnot exist. Our results on Palestrina’s work correlate with this musical history because there isan equal probability of progressing to any chord; i.e., it suggests that Palestrina did not followany chord progression model.•Our results for Bach and Mozart agree with the Baroque and Classical eras, respectively, becausethey show the dominance of the I and V chord and the lack of the iii chord. Beethoven’s PianoQuartet No. 1, which was composed in 1783, during the Classical Period, also conforms to thistrend.•Romantic music broke down formal structures, such as chord progression models, from theClassical Period. We would expect an analysis of chord progressions of Romantic compositionsto show no trend. In particular, we would expect the dominance of the I and V chord and thelack of the iii chord from the Classical Period to disappear almost entirely.•Our research can serve as a launching pad for chord and Markov Chain Analysis for any numberof compositions and composers. With enough data, it would be possible to show the changes inmusical eras, say from the Classical Period to the Romantic Era, and therefore defining featuresof each musical time period.References[1] Tymoczko, Dmitri. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York: Oxford UP,2011. Print.[2] Tymoczko, Dmitri. Initial State Matrices. (personal communication).Acknowledgments•Office of Research and Sponsored Programs, UW-Eau Claire• Professor James Walker, UW-Eau Claire• Professor Dmitri Tymoczko, Princeton University

A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice.

Initial State Matrices.

Combinatorics of Chord Progressions

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Combinatorics of Chord Progressions

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