Geometry of Music

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society".

Geometry is the study of shapes.  Musical information can often be represented naturally by shapes, allowing insights to be gained from geometric techniques.   One indication of the close connection between music and geometry comes from the fact that Euclid, whose Elements of Geometry (300 B.C.E)  is the founding document of modern geometry, also wrote a comprehensive treatise on the mathematics of musical pitches called Theory of Intervals.  The great 18^th century mathematician Leonhard Euler also developed geometric tools for music analysis.

Symmetry is one of the most powerful ideas in geometry.  No less so in the geometry of music, where symmetries abound.  Geometric techniques can be applied to musical scales, chords, and melodic lines.   Due to the concept of octave equivalence, the twelve pitches of the equally tempered chromatic scale are inherently cyclic in nature.  Thus the geometric theory of cyclic groups plays a major role in the mathematical description of scales and chords. 

Similarly, geometry can play a role in the analysis of musical rhythm, particular in musical forms based upon a repeating rhythmic motif.  In twentieth century atonal music, geometric ideas have been proposed as unifying theoretic structures to fill the role once played by tonal harmonic concepts. 

Symmetries in the Twelve Pitches of the Equally Tempered Scale

Two fundamental principles of modern musical analysis are octave equivalence and equal temperament.  Octave equivalence refers to the perception, believed to be universal in developed music cultures, that two pitches separated by an octave are members of the same pitch class.  Equal temperament refers to the system of musical intonation by which the twelve chromatic half steps within the octave represent uniform frequency scaling:  given a pitch with frequency f, the pitch one half step above has frequency 2^1/12 f.    In the equally tempered scale, enharmonically spelled notes such as C^# and D^b represent the same pitch. 

The twelve pitch classes are inherently cyclic.  This is represented in the left view of Figure 1.  Notice that this is identical to an analog clock face, with the traditional “12” replaced by “0”.   The diatonic scale (discussed in the Scales entry of this volume) is represented by the vertices of the inscribed polygon in the center view of Figure 1.  This arrangement of the seven diatonic pitches is the most even spacing possible for seven pitches in the twelve tone octave.  The evident symmetry about the 2—8 axis puts the complicated diatonic sequence of half steps and whole steps into a simpler conceptual framework. The figure illustrates that the Dorian Mode (which begins and ends on the second diatonic scale degree, given here as “D” or “2”) is unique among the diatonic modes in that it follows the same sequence of intervals both ascending and descending.

The six pairs of diametrically opposite pitch classes in the clock representation are separated by the interval of a tritone, so named because it contains three whole steps. In tonal music, the tritone is considered the most dissonant-sounding interval.  If the three odd numbered pitch class pairs on the clock face are reflected diametrically, the result is the circle of fifths shown in the right view of Figure 1.  The circle of fifths is familiar to music students as a mnemonic device for learning the musical key signatures:  the number of sharps increases by one (or alternatively, the number of flats decreases by one) at each step in the clockwise direction, while the number of flats increases (or sharps increase) at each step in the counterclockwise direction.   The circle of fifths is used extensively as an analytical tool for twentieth century music in the work of the American composer and music theorist Howard Hanson.

Figure 1 Top: The twelve pitches of the equally tempered chromatic scale arranged on a circle.  Center: The vertices of the inscribed polygon represent the pitches of the diatonic scale.  The diatonic arrangement is the most evenly spaced distribution of seven vertices in a twelve-sided figure.  Note the symmetry inherent in the Dorian mode, which begins and ends on pitch 2 (D).  Bottom: Diametric reflection of the odd numbered pitches results in the circle of fifths.

Tonnetz:  Representing Harmonic Structure in Two Dimensions

Beginning with the musical writings of the 18^th century mathematician Leonhard Euler, and continuing at least through the work of the influential music theorist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann) in the 19^th century, the representation of harmonic concepts in a two dimensional array called a Tonnetz (Tonal Network) has guided the understanding of tonal harmony.  In the tonnetz shown in Figure 2, the rows are simply the entries of the circle of fifths, while the columns are the twelve diatonic pitch classes arranged chromatically (by half steps).  The result is that the diagonals are made up of pitch classes separated by minor thirds (in the southeast direction) and major thirds (northeast).  In this arrangement, the sonorities of tonal harmony can be represented by polygonal groupings of the adjacent symbols:  triangles for major and minor triads, parallelograms for major and minor seventh chords, and similar structures for diminished, augmented and dominant seventh chords.  The musical theory of modulation (changing from one tonal center to another in the course of a musical composition) is aided by the geometric perspective of a tonnetz.  Tonal networks such as the one shown here are precursors of the contemporary musical theory of pitch class spaces.

C      G      D      A      E       B     F#     Db    Ab    Eb    Bb         F         C

     Eb    Bb    F      C      G      D      A       E       B     F#      Db     Ab

B       F#    Db   Ab    Eb    Bb     F      C       G      D      A         E          B

      D      A      E       B      F#    Db   Ab    Eb    Bb     F        C          G  

 Bb      F      C       G      D      A       E     B     F#    Db    Ab      Eb        Bb

      Db    Ab    Eb     Bb     F       C     G      D     A       E        B        F#      

Figure 2  The first six rows of a Tonnetz (or Tone Network).  The pitch classes of the circle of fifths are arranged horizontally.  The vertical alignment of the pitch classes is chromatic.  Diagonals in the southeast direction progress by intervals of the minor third.  Northeast diagonals progress by major thirds.  All tonal sonorities are given in this representation by polygons containing adjacent pitches.  For example, major triads are given by triangles with vertex at top and minor triads are given by triangles with a vertex at the bottom, as shown above for the C major (C E G) and A minor (A C E) triads. 

Rhythmic Symmetry

Like the twelve pitch classes, the metrical organization of music in time is also highly cyclic, allowing similar geometric techniques to be applied to rhythm.  The left view of Figure 3 shows the eighth-note subdivisions of a 4/4 measure.  The vertices of the inscribed polygon represent the rhythmic placement within the measure of the hand-clap rhythm from the iconic 1956 Elvis Presley recording of “Hound Dog”.  This complicated rhythm has a simple symmetric structure when viewed geometrically.  Similarly, the center view in Figure 3 shows the clave rhythm familiar to listeners of Afro-Cuban music, with its line of symmetry.  The left view of Figure 3 shows a characteristic bossa nova rhythm (which can be heard on the cowbell in Quincy Jones’s “Soul Bossa Nova”) and its line of symmetry.

Figure 3  Eighth-note subdivisions of rhythmic units arranged around a circle.  Top: The vertices of the inscribed polygon represent the hand-clap rhythm heard in the Elvis Presley recording of “Hound Dog”.  Center: The vertices of the inscribed polygon represent the well-known clave rhythm heard in Afro-Cuban music.  Bottom: The bossa nova cowbell rhythm heard in Quincy Jones’s “Soul Bossa Nova”.

FURTHER READINGS:

Archibald, R.C. “Mathematicians and Music.” American Mathematical Monthly, (v. 31/1, 1924).

Buena Vista Social Club. The Buena Vista Social Club, Nonesuch, 1997

Demaine, E. D., Gomez-Martin, F., Meijer, H., Rappaport, D., Taslakian, P., Toussaint, G. T., Winograd, T., and Wood, D. R. “The Distance Geometry of Music,” Computational Geometry: Theory and Applications, (v. 42/5, 2009).

Hanson, Howard.  Harmonic Materials of Modern Music: Resources of the Tempered Scale.        New York:  Appleton, 1960

Johnson, Timothy. Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals, Lanham, Maryland:  2008

Jones, Quincy. The Reel Quincy Jones, Hip-O Records, 1999

Nolan, Catherine. “Music Theory and Mathematics,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen.  Cambridge:  Cambridge University Press, 2002

Presley, Elvis.  Elvis 75, RCA/Legacy, 2010

Composing

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society".

Throughout the history of western music, composers have utilized mathematical techniques in creating musical works.  From Pythagoras, Plato, and Ptolemy in ancient Greece to the sixth century music theorist Boethius, music was thought to be a corollary of arithmetic.  With the widespread development of written musical notation beginning in the Renaissance Age, compositional craft became more and more highly developed.  Especially prized were compositions intertwined with mathematical patterns.  The eighteenth century composer and theorist Jean-Philippe Rameau was unequivocal in his views on the connection between mathematics and music in his 1722 Treatise on Harmony: “Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics.”  With fugal composition techniques in the high baroque period, mathematical techniques in musical composition reached a high point. The classical and romantic eras, characterized by a movement away from polyphonic music, produced less obvious mathematically oriented composition technique.  In the twentieth century however, mathematical formalisms were fundamental as replacements for the outdated tonal structures of the romantic era.

The Renaissance Canon

During the Renaissance Age, as musical craft was becoming highly prized, mathematical devices were developed to a considerable degree by Northern European composers. In the canons of Johannes Ockeghem a single melodic voice provides the basis by which one or more additional voices are composed according to various mathematical transformations of the original: mirror reflection of musical intervals (inversion), time translation, mirror reflection in time (retrograde), or a non-unit time scaling (mensuration canon).  Composers of this period understood the word canon to mean a rule by which secondary voices could be derived from a given melody, in contrast to our modern usage of the word meaning a simple duplication with later onset time, as in the nursery rhyme round “Row, Row, Row Your Boat”.

Bach: The Canon Master

J.S. Bach was undoubtedly a master of canonic composition.  Bach’s canons challenged performers to solve puzzles he set before them.  Examples abound in A Musical Offering (BWV 1079), a collection of pieces written for King Frederick the Great of Prussia in 1747.  At that time, Frederick was the employer of Bach’s son C.P.E. Bach.   The first of two Canon a 2 (canon for two voices) from Musical Offering appears to be written for a single voice, but with two different clef symbols: one at the beginning of the first measure, and one at the end of the last.  The canon, then, is to be performed with one voice reading from beginning to end, the other voice reading from end to end.  In this small piece, Bach provides an example of retrograde or cancrizan (crab) canon.  The puzzle in the second Canon a 2 is even more cleverly concealed:  a single line with two clef signs in the first measure, one upside down.  The cryptic instruction Quaerendo invenietis "Seek and ye shall find” is inscribed at the top of the manuscript. After deducing that the second, inverted clef sign indicates that the second voice of the canon is to proceed in inversion, the performer is left to “seek” the appropriate time translation at which the second voice should begin.

Other examples of Bach’s masterful canonic treatment are BWV 1074: Kanon zu vier Stimmen, which with its numerous key signatures, clefs, and repeat signs can be played from any viewing angle, and BWV 1072: Trias Harmonica, a single line of manuscript that when fully realized yields an eight-voice canon.

A late example of the technical canonic craft is the Der Spiegel-Duett (mirror duet) for two violins, attributed to Mozart, which maintains the retrograde inversion transformation for its entire 63 measures.  An example of table canon, the sheet of music for this piece is placed on a table between two violinists.  The players read the music from opposite sides:  one right side up from beginning to end, and the other upside down from end to beginning.  The musical artistry in this mechanistic device lies in the harmonic consistency that is maintained throughout the work.

A General Representation of Mathematical Transformations in Composition

A musical composition can be represented as a sequence of points from the module M over the cyclic groups of integers Z_p:

M = Z_p1 ×  Z_p2  ×  Z_p3 ×  Z_p4

with the coordinates representing (respectively) onset time, pitch, duration, and loudness.  For example, the twelve notes of the chromatic scale would be represented in the second coordinate by Z₁₂.   In this schematic, a point  (x₁, x₂, x₃, x₄)  in a musical motif  would be represented as (x₁+a, x₂, x₃, x₄+β) in a later repetition of that motif at a different volume level.  Similarly, a point in the transposed motif would be (x₁, x₂+a, x₃, x₄).  Retrograde is represented as (μ-x₁, x₂, x₃, x₄).  Inversion takes the form (x₁, 2a-x₂, x₃, x₄).  Mensuration, as in the canons of Ockegham, is written (x₁, x₂, a×x₃, x₄).  Transformations of this form were used extensively in the Renaissance and Baroque eras and, as we will see, played a fundamental role in post-tonal era of the twentieth-century.

Mathematical Structure in Atonal Music

At the turn of the twentieth century, as musical composition moved further from traditional concepts of tonality, music theorists and composers looked for new organizing principles on which atonal music could be structured.  Older formalisms emphasized tonal center, repetition and development of themes.  In a conscious movement away from such notions, the groundbreaking composer Arnold Schoenberg turned to the idea of serialism, in which a given permutation of the twelve chromatic pitches constitutes the basis for a composition.  The new organizing principle called for the twelve pitches of this tone row to be used, singly or as chords at the discretion of the composer, always in the order specified by the row.  When the notes of the row have been used, the process repeats from the beginning of the row.  Composers like Webern, Boulez, and Stockhausen consciously used geometric transformations of onset time, pitch, duration, and loudness as mechanisms for applying the tone row in compositions.  In the latter half of the twentieth century, set theoretic methods on pitch class sets dominated the theoretical discussion.  Predicated on the notions of octave equivalence and the equally tempered scale, Howard Hanson and Allen Forte developed mathematical analysis tools that brought a sense of theoretical cohesion to seemingly intractable modern compositions. 

Another mathematical approach to composition without tonality is known as Aleatoric Music, or chance music.  This technique encompasses a wide range of spontaneous influences in both composition and performance.  One notable exploration of aleatoric music can be seen in the stochastic compositions of Iannis Xenakis from the 1950s.  Xenakis’s stochastic composition technique, in which musical scores are produced by following various probability models, was realized in the orchestral works “Metastasis” and “Pithoprakta” which were subsequently performed as ballet music in a work by George Ballanchine.

FURTHER READINGS:

Beran, Jan. Statistics in Musicology. Boca Raton: Chapman & Hall/CRC, 2003.

Forte, Allen.  The Structure of Atonal Music.  New Haven:  Yale University Press, 1973

Grout, Donald Jay. A History of Western Music. New York: Norton, 1980.

Xenakis, Iannis. Formalized Music: Thought and Mathematics in Composition.   Hillsdale, NY: Pendragon Press, 1992

Harmonics

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society"

Harmonics are components of a musical sound with well-defined frequency relationships to one another. For a pitch of frequency f, typically measured in units of cycles per second or Hertz (Hz), the nth harmonic has frequency n × f.   In this context the frequency f  is referred to as the fundamental frequency.  Harmonics are closely related to overtones (or equivalently partials), which are defined to be secondary pitches that audibly resonate when a fundamental pitch sounds.  The number and strength of these secondary pitches are responsible for the distinct timbres perceived in different instruments or voices.  The Overtone Series in music (also called the Harmonic Series at the risk of confusion with the infinite sum of the same name) refers to the sequence of ascending harmonics with frequencies f, 2f, 3f, 4f….  With only a few exceptions, the pitches of the lower harmonics match well with the frequencies of twelve pitches of the equally tempered scale.   Further along the overtone series, the pitch spacing becomes very small --- smaller than the traditional half step --- and these upper harmonics, if heard, would sound distinctly out of tune.  With the discovery of the overtone series by Jean-Philippe Rameau in the eighteenth century, the notion of musical consonance as the exclusive natural and rational sonic phenomenon --- pursued by mathematicians from Pythagorus to Euler --- began to fade.  There is a close physical relationship between the harmonic frequencies and the length of the vibrating medium.  This is exploited in the performance practices of musical instruments.  

Vibrating Media and the Overtone Series

For vibrating strings (e.g. violins and guitars) and open vibrating air columns (the western concert flute and some organ pipes) the words harmonic, partial, and overtone are essentially synonymous, with a slight difference in the enumeration:  the  fundamental pitch (frequency f ) is referred to as the 1^st harmonic.  The first overtone (frequency 2f ) refers to the 2^nd harmonic, etc. In stopped air columns (the clarinet and some organ pipes), the overtone series omits certain harmonic frequencies.  For vibrating membranes (percussion instruments), overtones may exist at non-harmonic frequencies. 

It is therefore a slight abuse of terminology to refer, as is commonly done, to the sequence of harmonics as the overtone series. Physically, the overtone series is seen by observing the motion of a vibrating string of length L and natural frequency f.  If forced to vibrate at frequencies n × f (for n = 2,3,…) , n-1 stationary points (nodes) appear along the string, at intervals of L/n.  In effect, the string moves as n strings of length L/n joined end to end.  String performers utilize this fact by lightly stopping the string at lengths L/2, L/3, etc to produce flute-like harmonic tones (sometimes called flageolet tones).

From the overtone perspective, only lower harmonics are perceptible to the hearer of a fundamental pitch.  The first six harmonics are perceived by the modern hearer as in tune within the twelve pitches of the equally tempered scale, in which the octave (the distance between the first and second harmonic) is divided into 12 equal half step intervals.  The frequency difference between successive pitches in this twelve-tone system is given by f_n+1= 2^1/12 f_n.  The 2^nd, 4^th, 8^th, harmonics, at octaves above the fundamental, sound perfectly in tune..  Upper harmonics can sound significantly out of tune however.  The 7^th harmonic sounds uncomfortably flat compared to its nearest corresponding equal temperament pitch.  The 11^th harmonic has a frequency almost equidistant between adjacent notes of the equally tempered scale, causing it to sound very out of tune --- likewise for the 13^th and 14^th harmonics.  These considerations are significant for period instrument brass performers, whose instruments, like the so-called natural trumpet, are nothing more than long tubes without the length-changing system of valves of modern trumpets.  Performers play tunes on these instruments by producing overtones, typically between the 3^rd and 16^th in the series.  While skillful performers can compensate for the most problematic overtones, composers in the baroque era typically avoided these notes or used their sonic character for special effect.  Modern composers have experimented with specially tuned pianos and electronic instruments to directly explore the sonorities of harmonics.

The first twenty-four harmonics are listed in the table with fundamental pitch taken as the A below middle C.  Harmonics with frequencies that differ significantly from the equally tempered scale are indicated in bold.

Other uses of the word Harmonic in Mathematics

In mathematics, the word harmonic appears in a number of contexts, all of which trace their origins to the overtone series and associated physical vibrations.  A harmonic progression is defined as the term by term reciprocal of an arithmetic progression.  For example, the arithmetic sequence a₁=1, a₂=2, a₃=3, … gives rise to the harmonic sequence h₁=1, h₂=1/2, h₃=1/3, ... . In this example, the arithmetic sequence gives the frequency multiples for the overtone series, and the harmonic sequence corresponds to the wavelengths of the respective overtones.  The harmonic mean is the reciprocal of the arithmetic mean of reciprocals.  For example the harmonic mean of two numbers x and y is defined as 2(1/x + 1/y)^–1.    The harmonic series in mathematics is the infinite sum 1+1/2+1/3+…, providing the canonical example of a series whose terms approach zero, but nevertheless the sum diverges. The harmonic oscillator is a differential equation whose solutions are sinusoidal functions that can be used to model musical sounds.  Harmonic analysis is the study of functions (or signals) by decomposition into fundamental component functions by means of the Fourier transform or other techniques. In the study of complex variables, harmonic functions are generalizations of the sinusoidal functions that model fundamental vibrations.

FURTHER READINGS:

Cohen, H.F. Quantifying Music:  The Science of Music at the First Stage of the Scientific Revolution, 1580—1650. Dordrecht: D. Reidel Publishing Company, 1984

Gouk, Penelope. “The Role of Harmonics in the Scientific Revolution,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen.  Cambridge:  Cambridge University Press, 2002

Johnston, Ben.  Suite for Microtonal Piano. Robert Miller, piano; New World Records, 80203-2.

Sundberg, Johan. The Science of Musical Sounds. San Diego: Academic Press, 1991

Scales

A version of the following will appear in the 2011 volume "Encyclopedia of Mathematics in Society"

Western music is based on a system of twelve pitches within each octave.  The interval between adjacent pitches in this twelve-tone system is called a half step or semitone.  Pitches separated by two successive semitones are said to be at the interval of a whole step, or a tone.  Based on a variety of theoretical underpinnings, the concept and sound of tones and semitones have evolved throughout the history of western music.  In modern music practice, a uniform division of the octave into twelve equally spaced pitches, known as Equal Temperament, holds sway.  Scales are arrangements of half and whole step intervals in the octave.  Denoting a half step as h and a whole step as w, the familiar diatonic major scale is defined by the sequence wwhwwwh.  The diatonic natural minor scale is whwwhww.  Twenty-four distinct diatonic scales result from beginning these patterns from each of the twelve pitches.  This suggests a set-theoretic description, by which each major scale can be represented as a transposition (in algebra this would be called a translation) of the set of pitches C-D-E-F-G-A-B-C.  In the twentieth century, such mathematical formalisms have led to the conceptualization of non-diatonic scales with special transposition properties.

Octave Equivalence

Fundamental to understanding musical scales is the concept of octave:  the musical interval between notes with frequencies that differ by a factor of two.   In western music notation, pitches separated by an octave are given the same note name.  The piano keyboard provides a visual representation of this phenomenon. Counting up the white keys from middle C as “1”, the 8th key in the sequence is again called C.  This eight-note distance explains the etymology of the word octave.  The perception and conceptualization of such pairs of pitches as higher or lower versions of the same essential pitch is called octave equivalence. Octave equivalence is thought to be common to all systematic musical cultures.  Evidence of octave equivalence is found in ancient Greek and Chinese music.  Recent psycho-acoustic research suggests a neurological basis for octave equivalence in auditory perception.

The mathematical explanation of octave equivalence comes from the fact that the sound of a musical pitch is a combination of periodic waveforms that can be modeled as sinusoidal functions of time. In the two periodic functions f (t)=sin(t) and g(t)=sin(2t), with frequencies 2p and p, every peak of the lower frequency function coincides with a peak of the high-frequency function. In sonic terms, this is the highest degree of consonance possible for two pitches of different frequencies.

History of Scales

As western music developed from the middle ages through the twentieth century, the central construct was the diatonic scale.   This arrangement spans an octave with seven distinct pitches arranged in a combination of five whole steps and two half steps.  It is important to note that the pattern of intervals (and not the absolute pitch of the starting note) was the only distinguishing feature of scales until the rise of tonal harmony in the seventeenth century. The pitch-specific examples given here are intended illustrate the interval patterns in terms familiar to the modern reader.

The diatonic scale traces its origins to the ancient Greek genus of the same name, referring to a particular tuning of the four-stringed lyre (tetrachord) consisting of two whole steps and one half step in descending succession.  An example of this can be constructed with the pitches A-G-F-E.   Concatenization of two diatonic tetrachords [A-G-F-{E]-D-C-B} produces the pitches of the diatonic scale (the piano white keys).  In medieval European musical practice, the distinct Church Modes (Lydian, Phrygian, etc.) developed from the diatonic scale by the assignment of a tonal anchor or final tone.  For example, the Dorian mode is characterized by the sequence of ascending half and whole steps in the diatonic scale whwwwhw, e.g. D-E-F-G-A-B-C-D, while the Phrygian mode is hwwwhww: E-F-G-A-B-C-D-E.  The diatonic major scale wwhwwwh  (C-D-E-F-G-A-B-C) came into widespread use in the seventeenth century.  The diatonic natural minor scale is whwwhww (A-B-C-D-E-F-G-A). 

Intervals, Ratios and Equal Temperament

The simplest musical interval is the octave.  The frequency ratio between pitches separated by an octave is 2:1.  The interval of a perfect fifth has frequency ratio 3:2.    Using these two ratios, pitches and corresponding intervals for the diatonic scale can be assigned according to the Pythagorean Tuning.  Simpler diatonic scales based on ratios of small integers are known as Just Tunings.  Western music in the modern era uses a symmetric assignment of intervals known as Equal Temperament.  In equal temperament, the twelve half steps that comprise the frequency doubling octave each have frequency ratio 2^1/12 ≈ 1.0595.   For these three tuning schemes, frequency ratios relative to the starting pitch and intervals between adjacent scale notes are illustrated and compared in the Table 1.

Modern Scales

In contrast to the idiosyncratic pattern of intervals that comprise the diatonic scales, the chromatic scale hhhhhhhhhhh is perfectly symmetric.  In particular, the set of pitches that form the chromatic scale is unchanged by transposition --- there is only one set of pitches with this intervallic pattern.  We could refer to this set of pitches as having order 1.  The elements of the pitch set forming a diatonic scale, which as we have seen generates twelve diatonic scales by transposition, has order 12.  This point of view suggests other scales of interest with respect to transposition.  The set of six pitches in a whole-tone scale wwwwww  (e.g. C-D-E-F^#-G^#-A^#-C) are unchanged by transposition by an even number of half steps.  A transposition by an odd number of half steps results in the whole tone scale containing the remaining six pitches (C^#-D^#-F-G-A-B- C^#).  Thus, the set of pitches in the whole-tone scale has order 2. Whole-tone scales are a characteristic feature in much of the music of Claude Debussy.  The twentieth-century composer and music theorist Olivier Messiaen codified a number of eight-tone “scales of limited transposition”.  Among these are the order 3 scales hwhwhwhw and whwhwhwh, which are called octatonic scales in the music of Stravinsky and sometimes referred to as diminished scales in jazz performance. It can be seen that transposition by one and two half steps produce new diminished scales, but transposition by 3 half steps leaves the original set of pitches unchanged. 

FURTHER READINGS:

Grout, Donald Jay. A History of Western Music. New York: Norton, 1980.

Hanson, Howard. Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts, 1960.

Johnson, Timothy. Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals, Lanham, Maryland:  2008

Pope, Anthony. “Messiaen’s Musical Language: An Introduction.,” in The Messiaen Companion, ed. Peter Hill.  Portland: Amadeus Press, 1995

Sundberg, Johan. The Science of Musical Sounds. San Diego: Academic Press, 1991

Simple Linear Transformations of the Octave

A few weeks ago, a colleague in the music department sent me a page he had written describing some remarkable counting properties of the twelve pitches of the musical octave, with the circle of fourths and circle of fifths most remarkable.   Here is a generalization of his ideas.  There is some cool group theory on Z₁₂,  the set of integers 0,1,2,3,4,5,6,7,8,9,10,11 lurking here, but that will wait for a future post.  In the meantime, suppose we enumerate the twelve chromatic tones of a musical octave as in the following table:

C  C#  D  D#  E  F  F#  G  G#  A  A#  B  C
0   1   2   3   4   5   6    7   8   9   10   11   0

Notice that we will count modulo12  (mod 12) so that the C at the octave,which would have been numbered 12, becomes 0 instead.  Dave Benson refers to this as "clock arithmetic" in his comprehensive work Music: A Mathematical Offering.

The Chromatic Scale
We could generate the ascending chromatic scale with a very simple formula:

f(x)= x +1 (mod 12).

Notice that from any starting pitch x, every pitch of the chromatic scale can be expressed by n iterations of the function f.  That is,

 f (f (...f (x)))= x + n for n =1, 2, 3,....


Because the chromatic scales contains all twelve pitches, we can start with any choice of x  and eventually reach every tone in the octave. Another way to say this is that it takes twelve iterations of the generating function f to return to the starting pitch (modulo the octave). Notice the shift term “1” in f(x)is relatively prime to 12.

The Whole Tone Scales are generated by f (x)= x +2 (mod 12): 

0, 2, 4, 6, 8, 10, 0
and
1, 3, 5, 7, 9, 11, 1.

The shift term “2” in the generating function divides 12 evenly: 12÷2 =6.  Notice that six iterations of the generating function are required to return to the starting pitch, and that we need two distinct whole tone scales to reach all twelve tones.

The Diminished Seventh Arpeggios
are generated by f (x)= x +3 (mod 12):

0, 3, 6, 9, 0
1, 4, 7, 10, 1
and
2, 5, 8, 11, 2

The shift term “3” in the generating function divides 12 evenly: 12 ÷ 3 =4. Notice that four iterations of the generating function are required to return to the starting pitch, and that we need three distinct diminished seventh arpeggios to reach all twelve tones.

The Augmented Arpeggios are generated by f (x)= x +4 (mod 12):

0, 4, 8, 0
1, 5, 9, 1
2, 6, 10, 2
and
3, 7, 11, 3.

The shift term “4” in the generating function divides 12 evenly: 12 ÷ 4 =3. Notice that three iterations of the generating function are required to return to the starting pitch,and that we need four distinct augmented arpeggios to reach all twelve tones.

The Circle of Fourths is generated by f (x)= x +5 (mod 12): 

0, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7, 0.

The shift term “5” in the generating function is relatively prime with 12.
Notice that twelve iterations of the generating function are required to return
to the starting pitch, and that we need only one distinct circle of fourths to
reach all twelve tones.

The The Tritone Arpeggios are generated by f (x)= x +6 (mod 12):

0, 6, 0
1, 7, 1
 2, 8, 2
3, 9, 3
4, 10, 4
and
5, 11, 5.

The shift term “6” in the generating function divides 12 evenly: 12 ÷ 6 =2. Notice that two iterations of the generating function are required to return to the starting pitch,and that we need six distinct tritone arpeggios to reach all twelve tones.

The Circle of Fifths
is generated by f (x)= x +7 (mod 12):

0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, 0.

The shift term “7” in the generating function is relatively prime with 12.
Notice that twelve iterations of the generating function are required to return
to the starting pitch, and that we need only one distinct circle of fifths to
reach all twelve tones. Because x +7 = x - 5 (mod 12) we see that the
ascending circle of fifths is equivalent to the descending circle of fourths.

We could continue with generating functions f (x)= x +k
(mod 12), for k =8, 9, 10, 11. However,we know that because x +8 = x - 4 (mod 12), we would simply generate the descending augmented arpeggios.
Likewise x + 9=x - 3 (mod12) generates the descending diminished seventh arpeggios, x +10 = x - 2 (mod 12) generates the descending whole tone scale, and x +11 = x - 1 (mod 12) generates the descending chromatic scale.

Coffee sounds

I was at a dinner last week with physicists and mathematicians from the College, along with a visiting scientist who was on campus to give a lecture. During dinner, I suggested that it was possible to identify by ear the difference between pouring hot water and pouring cold water. As we ate our dessert, I did a small demonstration. I poured ice-cold lemonade into a china cup, then poured hot coffee into an identical cup. A few of the diners didn't notice a difference. Most of us did hear the two liquids differently, but no one could accurately describe what we had heard: "more tinkly", "more splashy", "higher", "lower" were some of the descriptive words we suggested.

This weekend I conducted a more careful experiment, making a sound recording of each.
I poured very hot water (fresh from the tea kettle) from my teapot into a mug:  pouring hot water
Then I poured chilled water (fresh from the fridge) from the same teapot into the same mug:  pouring cold water

Do you hear a difference?

My wife said she thinks the cold water sounds higher pitched.  I agreed (she has a very good ear), but I was unsure.  I loaded these recorded sound files into Audacity and computed the power spectra.

 In the following figure, the hot water spectrum is shown in the upper view, with the cold water spectrum in the lower.

The first few peaks (114 Hz, 175 Hz, 358 Hz, 444 Hz, 536 Hz, 650 Hz) are common to both spectra.  I think this must be due to the physical properties of the teapot spout, the mug, and my kitchen.  Then comes the difference:  the strongest peak for the hot water is 895 Hz, while the strongest peak for the cold water is lower, at 780 Hz.   Overall, the strongest frequency of the hot water is 15% higher.  In terms of musical pitch, the MIDI digital music encoding format has a neat way to identify a pitch, given its frequency.  The standard reference pitch at 440 Hz is the musical note "A" above middle C on the piano.  MIDI calls this pitch number 69.  Every other pitch number is determined by the formula

.

For us then, the hot water has a frequency component with pitch number 12.2 half-steps above A440, while the cold water has a strong frequency component 9.9 half-steps above  A440 ---  a difference of about one whole step.  This seems like a small difference to discern in two sequential experiments! And it is counter to our perception that the cold water had a higher pitch.  Maybe something else is going on.

I listened to the recordings a few more times, and realized that the last few seconds of each one are a lot different:


The last 2.5 seconds of
hot water.

The last 2.5 seconds of
cold water.

Notice that the end of the cold water sound much higher pitched than the end of the hot water.
Here are spectrum plots for the final 2.5 seconds  (upper view is hot water, lower view is cold water):

Notice that all the peaks up to about 1300Hz are the same in each case.  In the cold water spectrum, there are pronounced peaks at  about 2700 Hz, 4100 Hz and 5000 Hz.  There are none of these high frequency peaks in the hot water spectrum.

There is a difference at the beginning of the pour as well, with the cold water having a lower frequency sound.  The frequency difference  in the strongest peaks we saw in the full spectral decomposition is prominent in the first 2.5 seconds,  as illustrated by these final spectrum plots:

None of this explains why hot and cold pouring water have different sounds.  A quick Google search shows that this issue has been the subject of discussion boards from time to time.  The proposed physical causes are many and varied:  Temperature dependent changes in water viscosity, surface tension, density, and gas content.  There are suggestions that the mechanics of the pourer are different for hot and cold waters (like maybe we're naturally more careful with hot water).  Maybe the sound waves travel differently in the hot, swirling, steamy air inside the hot-water cup.  I'm eager to know the cause, but I'm not in a position to work it out now.  Maybe a later post!