DOCSLIB.ORG

Mathematics and Music

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematics and Music 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.
Load more

Recommended publications
  • Shepard, 1982
    Psychological Review VOLUME 89 NUMBER 4 JULY 1 9 8 2 Geometrical Approximations to the Structure of Musical Pitch Roger N. Shepard Stanford University ' Rectilinear scales of pitch can account for the similarity of tones close together in frequency but not for the heightened relations at special intervals, such as the octave or perfect fifth, that arise when the tones are interpreted musically. In- creasingly adequate a c c o u n t s of musical pitch are provided by increasingly gen- eralized, geometrically regular helical structures: a simple helix, a double helix, and a double helix wound around a torus in four dimensions or around a higher order helical cylinder in five dimensions. A two-dimensional "melodic map" o f these double-helical structures provides for optimally compact representations of musical scales and melodies. A two-dimensional "harmonic map," obtained by an affine transformation of the melodic map, provides for optimally compact representations of chords and harmonic relations; moreover, it is isomorphic to the toroidal structure that Krumhansl and Kessler (1982) show to represent the • psychological relations among musical keys. A piece of music, just as any other acous- the musical experience. Because the ear is tic stimulus, can be physically described in responsive to frequencies up to 20 kHz or terms of two time-varying pressure waves, more, at a sampling rate of two pressure one incident at each ear. This level of anal- values per cycle per ear, the physical spec- ysis has, however, little correspondence to ification of a half-hour symphony requires well in excess of a hundred million numbers.
    [Show full text]
  • Docunint Mori I
    DOCUNINT MORI I. 00.175 695 SE 020 603 *OTROS Schaaf, Willias L., Ed. TITLE Reprint Series: Mathematics and Music. RS-8. INSTITUTION Stanford Univ., Calif. School Mathematics Study Group. SKINS AGENCY National Science Foundation, Washington, D.C. DATE 67 28p.: For related documents, see SE 028 676-640 EDRS PRICE RF01/PCO2 Plus Postage. DESCRIPTORS Curriculum: Enrichment: *Fine Arts: *Instruction: Mathesatics Education: *Music: *Rustier Concepts: Secondary Education: *Secondary School Mathematics: Supplementary Reading Materials IDENTIFIERS *$chool Mathematics Study Group ABSTRACT This is one in a series of SBSG supplesentary and enrichment pamphlets for high school students. This series makes available expository articles which appeared in a variety of athematical periodicals. Topics covered include: (1) the two most original creations of the human spirit: (2) mathematics of music: (3) numbers and the music of the east and west: and (4) Sebastian and the Wolf. (BP) *********************************************************************** Reproductions snpplied by EDRS are the best that cam be made from the original document. *********************************************************************** "PERMISSION TO REPRODUCE THIS U S DE PE* /NE NT Of MATERIAL HAS BEEN GRANTED SY EOUCATION WELFAIE NATIONAL INSTITUTE OF IEDUCAT1ON THISLX)( 'API NT HAS141 IN WI P410 Oti(10 4 MA, Yv AS WI t1,4- 0 4 tiONI TH4 Pi 40SON 4)44 fwftAN,IA t.(1N 041.G114. AT,sp, .1 po,sos 1)1 1 ev MW OP4NIONS STA 'Fp LX) NE.IT NI SSI144,t Y RIPWI. TO THE EDUCATIONAL RESOURCES SI NT (7$$ M A NAIhj N',1,11)11E 01 INFORMATION CENTER (ERIC)." E.),,f T P(1,1140N ,1 V 0 1967 by The Board cf Trustees of theLeland Stanford AU rights reeerved Junior University Prieted ia the UnitedStates of Anverka Financial support for tbe SchoolMatbernatks Study provided by the Group has been Nasional ScienceFoundation.
    [Show full text]
  • Physics 101 Physics
    PhysicsPhysics 101101 LectureLecture 2121 DopplerDoppler EffectEffect LoudnessLoudness HumanHuman HearingHearing InterferenceInterference ofof SoundSound WavesWaves ReflectionReflection && RefractionRefraction ofof SoundSound Quiz: Monday Oct. 18; Chaps. 16,17,18(as covered in class),19 CR/NC Deadline Oct. 19 1/32 DemoDemo:: DopplerDoppler ShiftShift Hear frequency as higher when whistle is moving towards you and hear it as lower when moving away from you. Summary: Source & observer approaching, fobs>fs Source & observer separating, fobs<fs Higher Shorter Wavelength Frequency Lower Frequency Longer Wavelength 13-Oct-10 2/32 LoudnessLoudness && AmplitudeAmplitude Loudness depends on amplitude of pressure and density variations in sound waves. 13-Oct-10 3/32 deciBelsdeciBels (dB) (dB) Loudness of sound depends on the amplitude of pressure variation in the sound wave. Loudness is measured in deciBels (dB), which is a logarithmic scale (since our perception of loudness varies logarithmically). From the threshold of hearing (0 dB) to the threshold of pain (120 dB), the pressure amplitude is a million times higher. At the threshold of pain (120 db), the pressure variation is still only about 10 Pascals, which is one ten thousandth of atmospheric pressure. 4/32 5/32 Human Hearing 6/32 7/32 HearingHearing LossLoss The hair cells that line the cochlea are a delicate and vulnerable part of the ear. Repeated or sustained exposure to loud noise destroys the neurons in this region. Once destroyed, the hair cells are not replaced, and the sound frequencies interpreted by them are no longer heard. Hair cells that respond to high frequency sound are very vulnerable to destruction, and loss of these neurons typically produces difficulty understanding human voices.
    [Show full text]
  • Sine Waves and Simple Acoustic Phenomena in Experimental Music - with Special Reference to the Work of La Monte Young and Alvin Lucier
    Sine Waves and Simple Acoustic Phenomena in Experimental Music - with Special Reference to the Work of La Monte Young and Alvin Lucier Peter John Blamey Doctor of Philosophy University of Western Sydney 2008 Acknowledgements I would like to thank my principal supervisor Dr Chris Fleming for his generosity, guidance, good humour and invaluable assistance in researching and writing this thesis (and also for his willingness to participate in productive digressions on just about any subject). I would also like to thank the other members of my supervisory panel - Dr Caleb Kelly and Professor Julian Knowles - for all of their encouragement and advice. Statement of Authentication The work presented in this thesis is, to the best of my knowledge and belief, original except as acknowledged in the text. I hereby declare that I have not submitted this material, either in full or in part, for a degree at this or any other institution. .......................................................... (Signature) Table of Contents Abstract..................................................................................................................iii Introduction: Simple sounds, simple shapes, complex notions.............................1 Signs of sines....................................................................................................................4 Acoustics, aesthetics, and transduction........................................................................6 The acoustic and the auditory......................................................................................10
    [Show full text]
  • A Biological Rationale for Musical Consonance Daniel L
    PERSPECTIVE PERSPECTIVE A biological rationale for musical consonance Daniel L. Bowlinga,1 and Dale Purvesb,1 aDepartment of Cognitive Biology, University of Vienna, 1090 Vienna, Austria; and bDuke Institute for Brain Sciences, Duke University, Durham, NC 27708 Edited by Solomon H. Snyder, Johns Hopkins University School of Medicine, Baltimore, MD, and approved June 25, 2015 (received for review March 25, 2015) The basis of musical consonance has been debated for centuries without resolution. Three interpretations have been considered: (i) that consonance derives from the mathematical simplicity of small integer ratios; (ii) that consonance derives from the physical absence of interference between harmonic spectra; and (iii) that consonance derives from the advantages of recognizing biological vocalization and human vocalization in particular. Whereas the mathematical and physical explanations are at odds with the evidence that has now accumu- lated, biology provides a plausible explanation for this central issue in music and audition. consonance | biology | music | audition | vocalization Why we humans hear some tone combina- perfect fifth (3:2), and the perfect fourth revolution in the 17th century, which in- tions as relatively attractive (consonance) (4:3), ratios that all had spiritual and cos- troduced a physical understanding of musi- and others as less attractive (dissonance) has mological significance in Pythagorean phi- cal tones. The science of sound attracted been debated for over 2,000 years (1–4). losophy (9, 10). many scholars of that era, including Vincenzo These perceptual differences form the basis The mathematical range of Pythagorean and Galileo Galilei, Renee Descartes, and of melody when tones are played sequen- consonance was extended in the Renaissance later Daniel Bernoulli and Leonard Euler.
    [Show full text]
  • A Study to Explore the Effects of Sound Vibrations on Consciousness
    International Journal of Social Work and Human Services Practice Horizon Research Publishing Vol.6. No.3 July, 2018, pp. 75-88 A Study to Explore the Effects of Sound Vibrations on Consciousness Meera Raghu Independent Researcher, New Zealand Abstract Sound is a form of energy produced by cause happiness, joy, courage or calmness, dissonant vibrations caused by movement of particles. Sound can intervals can cause tension, anger, fear or sadness, thereby travel through solids (such as metal, wood, membranes), affecting the emotional aspect of consciousness. liquids (water) and gases (air). The sound vibrations that reach our ear are produced by the movement of particles in Keywords Consciousness, Sound, Vibration, Music, the air surrounding the source of sound. The movement or Emotion, Chladni, Pattern, Energy vibration of particles produces waves of sound. Sound waves are longitudinal and travel in the direction of propagation of vibrations. The pitch of sound is related directly to its frequency, which is given by the number of Introduction vibrations or cycles per second. The higher the pitch of sound, the higher is its frequency, and the lower the pitch, Sound is everywhere. There is perpetual movement and the lower is its frequency. Human ear can hear sounds of action in the world around us, and this produces a variety of frequencies ranging from 20 – 20,000 cycles per second (or sounds, such as those coming from Nature, from animals, Hertz – Hz). Sound waves can be visually seen and studied those generated by humans in the form of speech or music, using ‘Chladni’ plates, which was devised and those that are generated by vehicles, machines, gadgets that experimented by Ernst Chladni, a famous physicist with a are used for comfort, leisure and convenience.
    [Show full text]
  • The Perceptual Representation of Timbre
    Chapter 2 The Perceptual Representation of Timbre Stephen McAdams Abstract Timbre is a complex auditory attribute that is extracted from a fused auditory event. Its perceptual representation has been explored as a multidimen- sional attribute whose different dimensions can be related to abstract spectral, tem- poral, and spectrotemporal properties of the audio signal, although previous knowledge of the sound source itself also plays a role. Perceptual dimensions can also be related to acoustic properties that directly carry information about the mechanical processes of a sound source, including its geometry (size, shape), its material composition, and the way it is set into vibration. Another conception of timbre is as a spectromorphology encompassing time-varying frequency and ampli- tude behaviors, as well as spectral and temporal modulations. In all musical sound sources, timbre covaries with fundamental frequency (pitch) and playing effort (loudness, dynamic level) and displays strong interactions with these parameters. Keywords Acoustic damping · Acoustic scale · Audio descriptors · Auditory event · Multidimensional scaling · Musical dynamics · Musical instrument · Pitch · Playing effort · Psychomechanics · Sound source geometry · Sounding object 2.1 Introduction Timbre may be considered as a complex auditory attribute, or as a set of attributes, of a perceptually fused sound event in addition to those of pitch, loudness, per- ceived duration, and spatial position. It can be derived from an event produced by a single sound source or from the perceptual blending of several sound sources. Timbre is a perceptual property, not a physical one. It depends very strongly on the acoustic properties of sound events, which in turn depend on the mechanical nature of vibrating objects and the transformation of the waves created as they propagate S.
    [Show full text]
  • The Sounds of Music: Science of Musical Scales∗ 1
    SERIES ARTICLE The Sounds of Music: Science of Musical Scales∗ 1. Human Perception of Sound Sushan Konar Both, human appreciation of music and musical genres tran- scend time and space. The universality of musical genres and associated musical scales is intimately linked to the physics of sound, and the special characteristics of human acoustic sensitivity. In this series of articles, we examine the science underlying the development of the heptatonic scale, one of the most prevalent scales of the modern musical genres, both western and Indian. Sushan Konar works on stellar compact objects. She Introduction also writes popular science articles and maintains a Fossil records indicate that the appreciation of music goes back weekly astrophysics-related blog called Monday Musings. to the dawn of human sentience, and some of the musical scales in use today could also be as ancient. This universality of musi- cal scales likely owes its existence to an amazing circularity (or periodicity) inherent in human sensitivity to sound frequencies. Most musical scales are specific to a particular genre of music, and there exists quite a number of them. However, the ‘hepta- 1 1 tonic’ scale happens to have a dominating presence in the world Having seven base notes. music scene today. It is interesting to see how this has more to do with the physics of sound and the physiology of human auditory perception than history. We shall devote this first article in the se- ries to understand the specialities of human response to acoustic frequencies. Human ear is a remarkable organ in many ways. The range of hearing spans three orders of magnitude in frequency, extending Keywords from ∼20 Hz to ∼20,000 Hz (Figure 1) even though the sensitivity String vibration, beat frequencies, consonance-dissonance, pitch, tone.
    [Show full text]
  • Mathematics and Music
    Mathematics and Music David Wright April 8, 2009 2 i About the Author David Wright is professor of mathematics at Washington University in St. Louis, where he currently serves as Chair of the Mathematics Department. He received his Ph.D. in Mathematics from Columbia University, New York City. A leading researcher in the fields of affine algebraic geometry and polynomial automorphisms, he has produced landmark publications in these areas and has been an invited speaker at numerous international mathematics conferences. He designed and teaches a university course in Mathematics and Music, the notes from which formed the beginnings of this book. As a musician, David is an arranger and composer of vocal music, where his work often integrates the close harmony style called barbershop harmony with jazz, blues, gospel, country, doo-wop, and contemporary a cappella. He is Associate Director of the St. Charles Ambassadors of Harmony, an award winning male chorus of 160 singers. He also serves as a musical consultant and arranger for numerous other vocal ensembles. He is active in the Bar- bershop Harmony Society and was inducted into its Hall of Fame in 2008. As arranger and music historian David has been featured in national radio and TV broadcasts at home and abroad, and has authored several articles on vocal harmony. ii Contents Introduction v 1 Basic Concepts 1 2 Horizontal Structure 17 3 Harmony and Related Numerology 31 4 Ratios and Musical Intervals 45 5 Logarithms and Musical Intervals 53 6 Chromatic Scales 61 7 Octave Identification 69 8 Properties of Integers 87 9 The Integers as Intervals 97 10 Timbre and Periodic Functions 105 11 Rational Numbers As Intervals 125 12 Rational Tuning 139 iii iv CONTENTS Introduction The author’s perspective.
    [Show full text]
  • Perspectives on Memory for Musical Timbre
    Perspectives on Memory for Musical Timbre Kai Kristof Siedenburg Music Technology Area Department of Music Research Schulich School of Music McGill University Montreal, Canada March 2016 A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Doctor of Philosophy. c 2016 Kai Siedenburg 2016/03/29 bong bing bang bung bäng für Eliza ii Contents Abstract/Résumé .................................. vi Acknowledgments ................................. x Contribution of authors .............................. xii List of Figures ................................... xvii List of Tables .................................... xx List of Acronyms .................................. xxi 1 Introduction 1 1.1 Ideas and questions ............................. 1 1.2 Methods ................................... 8 1.3 Thesis outline ................................ 9 I Background 13 2 Three conceptual distinctions for timbre 15 2.1 Introduction ................................. 15 2.2Soundeventvs.timbre........................... 16 2.3 Qualitative vs. source timbre ........................ 18 2.4 Timbre on different scales of detail .................... 19 2.5 Conclusion .................................. 21 3 A review of research on memory for timbre 23 3.1 Introduction ................................. 23 3.2 Basic concepts in memory ......................... 25 3.3 Key findings in auditory memory ..................... 28 Contents iii 3.3.1 Auditory sensory memory ..................... 29 3.3.2 Memory for noise .........................
    [Show full text]
  • Absolute Pitch
    Psychological Bulletin Copyright 1993 by the American Psychological Association, Inc. 1993. Vol. 113. No. 2. 345-361 0033-2909/93/53.00 Absolute Pitch Annie H. Takeuchi and Stewart H. Hulse Absolute pitch (AP) is the ability to identify a tone's pitch or to produce a tone at a particular pitch without the use of an external reference pitch. AP exists in varying degrees among people generally described as AP possessors. AP possessors vary not only in the accuracy with which they can identify pitches but also in their ability to produce pitches absolutely and in their ability to identify tones of various timbres and in various pitch registers. AP possessors' memory for pitches is mediated by verbal pitch names; they do not have superior memory for pitches per se. Although the etiology of AP is not yet completely understood, evidence points toward the early-learning theory. This theory states that AP can be learned by anyone during a limited period early in development, up to about age 6, after which a general developmental shift from perceiving individual features to perceiving relations among features makes AP difficult or impossible to acquire. Absolute pitch (AP) is the ability to identify the pitch of a work with units of a single tone, whereas composers without AP musical tone or to produce a musical tone at a given pitch must work with pitch relations that require a minimum unit of without the use of an external reference pitch. Most humans two tones. process musical pitch relatively rather than absolutely. They Psychologists have puzzled over several questions about AP process the melodic and harmonic relations among pitches in- for over 100 years.
    [Show full text]
  • The Psychological Representation of Musical Pitch in a Tonal Context
    COGNITIVE PSYCHOLOGY 11, 346-374 (1979) The Psychological Representation of Musical Pitch in a Tonal Context CAROL L. KRUMHANSL Stanford University In this series of experiments, evidence was found for a complex psychological representation of musical pitch. The results of a scaling study, in which subjects judged the similarities between pairs of tones presented in an explicitly tonal context, suggest that musical listeners extract a pattern of relationships among tones that is determined not only by pitch height and chroma, but also by member- ship in the major triad chord and the diatonic scale associated with the tonal system of the context. Multidimensional scaling of the similarity ratings gave a three-dimensional conical structure around which the tones were ordered ac- cording to pitch height. The major triad components formed a closely related cluster near the vertex of the cone; the remaining diatonic scale tones formed a less closely related subset farther from the vertex; and, the nondiatonic tones, still farther from the vertex, were widely dispersed. The results also suggest that, in the psychological representation, tones less closely related to the tonality are less stable than tones closely related to the tonality, and that the representation incor- porates the tendency for unstable tones to move toward the more stable tones in time, reflecting the dynamic character of musical tones. In the similarity ratings of the scaling study, tones less related to the tonality were judged more similar to tones more related to the tonality than the reverse temporal order. Furthermore, in a delayed recognition task memory performance for nondiatonic tones was less accurate than for diatonic tones, and nondiatonic tones were more often confused with diatonic tones than diatonic tones were confused with nondiatonic tones.
    [Show full text]