Mathematics and Music David Wright Professor Department of Mathematics Washington University Saint Louis, MO, USA and Associate Director Ambassadors of Harmony Saint Charles, MO, USA Field of Dreams Conference Washington University Saint Louis, Missouri, November 4-5, 2016 Wright Mathematics and Music Wright Mathematics and Music Pythagoras (c.570 - c.495 BC) was interested in the relationships between harmonious tones. Wright Mathematics and Music J. S. Bach (1685-1750) was interested in the mathematical problem of tuning keyboards. Wright Mathematics and Music BEGINNING MATH CONCEPTS sets, equivalence relations functions, graphs integers, rational numbers, real numbers modular arithmetic trigonometry BEGINNING MUSIC CONCEPTS tempo, rhythm scales, key signatures melody, form pitch, intervals, tuning tone, timbre Wright Mathematics and Music 18 CHAPTER 2. HORIZONTAL STRUCTURE named according to that proportion. Thus, if the whole note has a certain duration in beats, then the half note has half that duration, the quarter note has one fourth that duration, etc. In the situation where a whole note gets four beats, then a half note gets two beats and the sixty-fourth note represents one sixteenth of a beat. We will use the (non-standard) term durational note to mean a note distinguished by its duration,suchashalfnoteorquarternote,independent of its associated pitch. Observe that these designations fornotestacitly employ the concept of equivalence class. Here we are declaring two notes to be equivalent if they have the same duration, so that “durational note” refers to the equivalence class of all notes having a given duration (e.g., “half note” designated to the equivalence class of all half notes, regardless of their pitch). This is to be distinguished from octave equivalence, discussed in Chapter 1, whose equivalence classes are called “note classes”. The pitch of a note is dictated by the vertical position of its notehead on the staff.Thedurationofthenoteisdictatedbyseveraldetails which we will discuss individually. They are: 1. whether the interior of the notehead is filled 2. the presence or absence of a note stem, and, if present, the number of flags on the stem or the number of beams attached to the stem 3. the number of dots following the note, if any 4. the tuplet designation of the note, if any Noteheads, Stems, Flags, and Beams. The whole note and half note Temporal notation 1 are written with an unfilled notehead. For n 2the 2n -th note is written 1 ≥ with a filled notehead. All 2n -th notes except the whole note (i.e., the case n =0)possessanotestem,whicheitherextendsupwardfromtheright side of the notehead or downward from the left side of the notehead.Forn 3, ≥ the 1 -th note’s stem is given n 2flags.Thusaneighthnote(n =3)has Music’s2n temporal notation− is based on powers of two. We divide onetime flag, intervals a sixteenth in note half. (n =4)hastwoflags,etc. whole¯ note half˘ note quarterˇ note eighthˇ note - sixteenthˇ note thirty secondˇ note sixty fourthˇ note . / 0 These are names for equivalence classes of notes. Wright Mathematics and Music Dots Music notation’s method of extending the duration of a note is by adding dots. If the note has duration d, then 3 1 · has duration 2 d = (1 + 2 )d 7 1 1 ·· has duration 4 d = (1 + 2 + 4 )d 15 1 1 1 ··· has duration 8 d = (1 + 2 + 4 + 8 )d This hearkens to the geometric series in mathematics: 1 1 1 1 1 + 2 + 4 + 8 + 16 + ··· = 2 Wright Mathematics and Music Meter Score Score Score[Title] [Title] [Title] Meter is given by a pair m where m represents the number of n [Composer] [Composer] [Composer] beats in a measure of time, n = 2r dictates the temporal note which receives one beat. & 23 ! &! 42 !! &!!86 ! !! ! !! ! !! ! !! ! ! ! ! ! ! When counting time the human brain is most comfortable with small primes,9 reflected in9 the fact that most9 time signatures involve 2 and 3. & ! ! & !! !!& !!! !!! !!! !!! !!! !! !! ! ! Audio Example 1 Audio Example 2 18 18 18 Counting in fives and sevens is less common. & ! ! & !! !!& !!! !!! !!! !!! !!! !! !! ! ! Audio Example 3 Audio Example 3 27 27 27 Wright Mathematics and Music & ! & !! & !!! !!! !!! !! ! CHAPTER 2. HORIZONTAL STRUCTURE 25 (horizontal translation) of the sequence of pitches or the rhythmic pattern. Here is a familiar example which illustrates rhythmic translation: G 2R Get(ˇ outˇ the(ˇ way,˘ oldˇ Dan Tuck-er! Get(ˇ outˇ the(ˇ way,˘ old Dan Tuck-er! 26 CHAPTERˇ (ˇ `ˇ 2. HORIZONTALˇ STRUCTUREˇ (ˇ `ˇ This2 Get type(ˇ outˇ ofthe transposition(ˇ way,˘ oldˇ Dan exemplifiedˇ Tuck-er! You’re abovetooˇ islate calledtochromaticcome to transposi-sup-per. Gtion. (ˇ ˇ` ˇ `ˇ (ˇ ˇ ˇ (ˇ ˇ` Melodic transformationNoteA that variant the form rhythm of transposition, of the first two called barsdiatonic is repeated transposition, twice, whileoccurs the sequencewhen a diatonic of pitches melody varies. is moved up or down by the same number of dia- tonicAn scale example tones, of producing melodic (as a wellmelody as rhythmic) having the translation same genera is fol shape,und in butthe 2 Many songsspiritualwithfeature chromaticWhen The intervals melodic Saints not Go perfectly Marching transpositions, preserved In, due to the analogousdiffering inter- to vals between adjacent diatonic notes. This occurs on the German Carol O geometricTannenbaum transformations (O Christmas Tree).˚ in mathematics.Note that the first¨ bracketed Here sequence are be- two types: low is shifted downward by one diatonic scale tone in the second bracketed Oh, when the Saints go march-ing in, Oh, when the Saints (1) diatonicsequence.G 2R ˇ ˇ ˇ ¯ ˇ ˇ ˇ ˇ ¯ ˇ ˇ ˇ ˇ ˘ where the melodic sequence F-A-B!-C appears three times consecutively. 3 Transposition.4 When a repeating pattern is being represented melodically, itG is2 possibleO Christ- tomas alsotree, applyO(ˇ Christ- a` vertical(ˇ mas)ˇ tree,`ˇ shiftYour(ˇ or col-transposition,(ˇ or(ˇ isˇ un- analogouschang-ˇ ing to replacingˇ the` graph(ˇ )ˇ of`ˇ y = f(x) by that of y = f(x)+c. Suchˇ a shiftˇ may repeat a melodic excerpt, transposing each note upward or downward by a fixed chromatic interval, as in the first sixteen bars of George and Ira 2 Whenˇ fromˇ all trees`ˇ theˇ col-(ˇ ors(ˇ go,`ˇ Youˇ stillˇ are green`ˇ a-ˇ midst(ˇ the snow. Gershwin’sG ? Strike Up(ˇ The Band, in which the second(ˇ eight measures(ˇ repeatˇ the melody!- of- the first, transposed"#- up by$ the!- interval- of a"# four- th. $ (2) chromaticRetrogression. Yet another form of transformation in music is retrogres- sion, which is analogous to the mathematical notion of horizontal reflection. Such a reflection is exemplified; when we replace the graph= of y = f(x) with that of y = f(x), reflecting the graph around the y-axis. In music, 22R Let the drums− roll out! Let the trum- pet call! While the G“retrogression”` (ˇ 6 )ˇ meansˇ ˇ “inverting˘ `˘ the order` (ˇ 6 )ˇ of notes”,ˇ 2ˇ so˘ that the`˘ resulting` (ˇ 6 )ˇ sequence forms a reflection of the initial˚ one. In this excerpt from; Raindrops Keep Falling On; My Head, note the symmetry of the melody around the point22peo- designatedple shout! by Strike: up the band!¯ `˘ Hear`ˇ the cym-ˇ balsˇ ring!˘ `˘ Call-`ˇ ing G ˇ ˝^ˇ ˘ ˘ ∧ (ˇ ˇ (ˇ )ˇ )ˇ ; ; - ˚ - 2 one 2andˇ all!˘ `˘ To` theÏmar-Ï˝^tialˇ swingĽ˘Ľ ˘ ÉÉStrike upˇ the band!¯ ˘ G 2 ˇ 44 Rain- ˇ drops)ˇ ˇkeepÌ fall- ingĘ on myˇÆ head, ˇ they 2This exampleG is givenˇ in [4],- `ˇ as areˇ the excerpts`ˇ ˇ from`ˇ Oˇ- Tannenbaumˇ - ` (ˇ and Raindrops Keep FallingAudio On My Example Head, which appear 3 a little Audio later in this Example discussion. 3 Form. The sequence of larger sections of% music into which music may be organized is sometimes calledWrightform. TheMathematics number of measures and Music in a section is often a power of 2. For example, ragtime compositions typically consist of three or four sections, each section having 16 measures; sometimes one or more of these sections is repeated once. These sections are distinguishable by the listener by virtue of different rhythmic and melodic character. If a composition consisted of three sections, we might denote the form by: ABC. If the first two sections were repeated, the form would be AA BB C. Scott Patterns of m on n in music The modular integers, mod n, are the elements of the set Zn = {[0], [1], [2],..., [n − 1]} Modular arithmetic relates to music in several ways. Here is one. Composers sometimes create ingenious musical passages by imposing a pattern of m notes or beats against a pattern of n such, where gcd (m, n) = 1. This technique exploits (perhaps unknowingly by the composer) the fact that [m] is a generator in Zn (and vice versa). Wright Mathematics and Music CHAPTER 8. PROPERTIES OF INTEGERS 91 the big band song In the Mood. Here m =3andn =4.Thesong’s“hook” lies in the repetition of the rhythmic figure comprising four eighth notes in swing time, shown below. 3 3 One way this can occur is by cycling= m pitches through a repeated rhythmic patternˇ ˇ ˇ of ˇn notes.ˇ This(ˇ ˇ is(ˇ exemplified in the main melodic line of the big band song In the Mood. Here m = 3 and n = 4. The song’s “hook” lies in the repetition of The melody repeats the sequence of three pitches C ,E! ,A! ,throughthe the rhythmic figure comprising four eighth notes4 4 in4 swing time. above rhythmic pattern as follows: 1 2 3 4 1 2 3—fl4flfl— 1—flfl2fl— 3 4 6 22 1 2 3 1 2 3 1 2 3 1 2 3 G 2 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˘ ˇ Audioˇ Exampleˇ ˇ Note that both patterns end their cycle on the twelfth eighth notes and not before.