Intervals and Transposition
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Tonalandextratonal Functions of Theaugmented
TONAL AND EXTRATONAL FUNCTIONS OF THE AUGMENTED TRIAD IN THE HARMONIC STRUCTURE OF WEBERN'S 'DEHMEL SONGS' By ROBERT GARTH PRESTON B. Mus., Brandon University, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (School of Music, Music Theory) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1989 © Garth Preston, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) ii ABSTRACT: TONAL AND EXTRATONAL FUNCTIONS OF THE AUGMENTED TRIAD IN THE HARMONIC STRUCTURE OF WEBERN'S 'DEHMEL SONGS' The composing of the 'Dehmel Songs' marks a pivotal juncture both in Webern's oeuvre and in the history of music in general. The years that saw the birth of this cycle of five songs, 1906-8, comprise what is generally regarded as a period of transition, in the work of Schoenberg, Webern and Berg, from a 'late tonal' style of composition to an early 'atonal' style. In this study I approach the 'Dehmel Songs' from the perspective that its harmonic structure as a whole can be rendered intelligible in a theoretical way by combining a simple pitch-class-set analysis, which essentially involves graphing the pattern of recurrence of the 'augmented triad' as a motivic harmonic entity—a pattern which is in fact serial in nature-through the course of the unfolding harmonic progression, with a tonal interpretation that uses that pattern as a referential pitch-class skeleton. -
The 17-Tone Puzzle — and the Neo-Medieval Key That Unlocks It
The 17-tone Puzzle — And the Neo-medieval Key That Unlocks It by George Secor A Grave Misunderstanding The 17 division of the octave has to be one of the most misunderstood alternative tuning systems available to the microtonal experimenter. In comparison with divisions such as 19, 22, and 31, it has two major advantages: not only are its fifths better in tune, but it is also more manageable, considering its very reasonable number of tones per octave. A third advantage becomes apparent immediately upon hearing diatonic melodies played in it, one note at a time: 17 is wonderful for melody, outshining both the twelve-tone equal temperament (12-ET) and the Pythagorean tuning in this respect. The most serious problem becomes apparent when we discover that diatonic harmony in this system sounds highly dissonant, considerably more so than is the case with either 12-ET or the Pythagorean tuning, on which we were hoping to improve. Without any further thought, most experimenters thus consign the 17-tone system to the discard pile, confident in the knowledge that there are, after all, much better alternatives available. My own thinking about 17 started in exactly this way. In 1976, having been a microtonal experimenter for thirteen years, I went on record, dismissing 17-ET in only a couple of sentences: The 17-tone equal temperament is of questionable harmonic utility. If you try it, I doubt you’ll stay with it for long.1 Since that time I have become aware of some things which have caused me to change my opinion completely. -
An Adaptive Tuning System for MIDI Pianos
David Løberg Code Groven.Max: School of Music Western Michigan University Kalamazoo, MI 49008 USA An Adaptive Tuning [email protected] System for MIDI Pianos Groven.Max is a real-time program for mapping a renstemningsautomat, an electronic interface be- performance on a standard keyboard instrument to tween the manual and the pipes with a kind of arti- a nonstandard dynamic tuning system. It was origi- ficial intelligence that automatically adjusts the nally conceived for use with acoustic MIDI pianos, tuning dynamically during performance. This fea- but it is applicable to any tunable instrument that ture overcomes the historic limitation of the stan- accepts MIDI input. Written as a patch in the MIDI dard piano keyboard by allowing free modulation programming environment Max (available from while still preserving just-tuned intervals in www.cycling74.com), the adaptive tuning logic is all keys. modeled after a system developed by Norwegian Keyboard tunings are compromises arising from composer Eivind Groven as part of a series of just the intersection of multiple—sometimes oppos- intonation keyboard instruments begun in the ing—influences: acoustic ideals, harmonic flexibil- 1930s (Groven 1968). The patch was first used as ity, and physical constraints (to name but three). part of the Groven Piano, a digital network of Ya- Using a standard twelve-key piano keyboard, the maha Disklavier pianos, which premiered in Oslo, historical problem has been that any fixed tuning Norway, as part of the Groven Centennial in 2001 in just intonation (i.e., with acoustically pure tri- (see Figure 1). The present version of Groven.Max ads) will be limited to essentially one key. -
The Lost Harmonic Law of the Bible
The Lost Harmonic Law of the Bible Jay Kappraff New Jersey Institute of Technology Newark, NJ 07102 Email: [email protected] Abstract The ethnomusicologist Ernest McClain has shown that metaphors based on the musical scale appear throughout the great sacred and philosophical works of the ancient world. This paper will present an introduction to McClain’s harmonic system and how it sheds light on the Old Testament. 1. Introduction Forty years ago the ethnomusicologist Ernest McClain began to study musical metaphors that appeared in the great sacred and philosophical works of the ancient world. These included the Rg Veda, the dialogues of Plato, and most recently, the Old and New Testaments. I have described his harmonic system and referred to many of his papers and books in my book, Beyond Measure (World Scientific; 2001). Apart from its value in providing new meaning to ancient texts, McClain’s harmonic analysis provides valuable insight into musical theory and mathematics both ancient and modern. 2. Musical Fundamentals Figure 1. Tone circle as a Single-wheeled Chariot of the Sun (Rg Veda) Figure 2. The piano has 88 keys spanning seven octaves and twelve musical fifths. The chromatic musical scale has twelve tones, or semitone intervals, which may be pictured on the face of a clock or along the zodiac referred to in the Rg Veda as the “Single-wheeled Chariot of the Sun.” shown in Fig. 1, with the fundamental tone placed atop the tone circle and associated in ancient sacred texts with “Deity.” The tones are denoted by the first seven letters of the alphabet augmented and diminished by and sharps ( ) and flats (b). -
Introduction to GNU Octave
Introduction to GNU Octave Hubert Selhofer, revised by Marcel Oliver updated to current Octave version by Thomas L. Scofield 2008/08/16 line 1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 8 6 4 2 -8 -6 0 -4 -2 -2 0 -4 2 4 -6 6 8 -8 Contents 1 Basics 2 1.1 What is Octave? ........................... 2 1.2 Help! . 2 1.3 Input conventions . 3 1.4 Variables and standard operations . 3 2 Vector and matrix operations 4 2.1 Vectors . 4 2.2 Matrices . 4 1 2.3 Basic matrix arithmetic . 5 2.4 Element-wise operations . 5 2.5 Indexing and slicing . 6 2.6 Solving linear systems of equations . 7 2.7 Inverses, decompositions, eigenvalues . 7 2.8 Testing for zero elements . 8 3 Control structures 8 3.1 Functions . 8 3.2 Global variables . 9 3.3 Loops . 9 3.4 Branching . 9 3.5 Functions of functions . 10 3.6 Efficiency considerations . 10 3.7 Input and output . 11 4 Graphics 11 4.1 2D graphics . 11 4.2 3D graphics: . 12 4.3 Commands for 2D and 3D graphics . 13 5 Exercises 13 5.1 Linear algebra . 13 5.2 Timing . 14 5.3 Stability functions of BDF-integrators . 14 5.4 3D plot . 15 5.5 Hilbert matrix . 15 5.6 Least square fit of a straight line . 16 5.7 Trapezoidal rule . 16 1 Basics 1.1 What is Octave? Octave is an interactive programming language specifically suited for vectoriz- able numerical calculations. -
MTO 20.2: Wild, Vicentino's 31-Tone Compositional Theory
Volume 20, Number 2, June 2014 Copyright © 2014 Society for Music Theory Genus, Species and Mode in Vicentino’s 31-tone Compositional Theory Jonathan Wild NOTE: The examples for the (text-only) PDF version of this item are available online at: http://www.mtosmt.org/issues/mto.14.20.2/mto.14.20.2.wild.php KEYWORDS: Vicentino, enharmonicism, chromaticism, sixteenth century, tuning, genus, species, mode ABSTRACT: This article explores the pitch structures developed by Nicola Vicentino in his 1555 treatise L’Antica musica ridotta alla moderna prattica . I examine the rationale for his background gamut of 31 pitch classes, and document the relationships among his accounts of the genera, species, and modes, and between his and earlier accounts. Specially recorded and retuned audio examples illustrate some of the surviving enharmonic and chromatic musical passages. Received February 2014 Table of Contents Introduction [1] Tuning [4] The Archicembalo [8] Genus [10] Enharmonic division of the whole tone [13] Species [15] Mode [28] Composing in the genera [32] Conclusion [35] Introduction [1] In his treatise of 1555, L’Antica musica ridotta alla moderna prattica (henceforth L’Antica musica ), the theorist and composer Nicola Vicentino describes a tuning system comprising thirty-one tones to the octave, and presents several excerpts from compositions intended to be sung in that tuning. (1) The rich compositional theory he develops in the treatise, in concert with the few surviving musical passages, offers a tantalizing glimpse of an alternative pathway for musical development, one whose radically augmented pitch materials make possible a vast range of novel melodic gestures and harmonic successions. -
The Devil's Interval by Jerry Tachoir
Sound Enhanced Hear the music example in the Members Only section of the PAS Web site at www.pas.org The Devil’s Interval BY JERRY TACHOIR he natural progression from consonance to dissonance and ii7 chords. In other words, Dm7 to G7 can now be A-flat m7 to resolution helps make music interesting and satisfying. G7, and both can resolve to either a C or a G-flat. Using the TMusic would be extremely bland without the use of disso- other dominant chord, D-flat (with the basic ii7 to V7 of A-flat nance. Imagine a world of parallel thirds and sixths and no dis- m7 to D-flat 7), we can substitute the other relative ii7 chord, sonance/resolution. creating the progression Dm7 to D-flat 7 which, again, can re- The prime interval requiring resolution is the tritone—an solve to either a C or a G-flat. augmented 4th or diminished 5th. Known in the early church Here are all the possibilities (Note: enharmonic spellings as the “Devil’s interval,” tritones were actually prohibited in of- were used to simplify the spelling of some chords—e.g., B in- ficial church music. Imagine Bach’s struggle to take music stead of C-flat): through its normal progression of tonic, subdominant, domi- nant, and back to tonic without the use of this interval. Dm7 G7 C Dm7 G7 Gb The tritone is the characteristic interval of all dominant bw chords, created by the “guide tones,” or the 3rd and 7th. The 4 ˙ ˙ w ˙ ˙ tritone interval can be resolved in two types of contrary motion: &4˙ ˙ w ˙ ˙ bbw one in which both notes move in by half steps, and one in which ˙ ˙ w ˙ ˙ b w both notes move out by half steps. -
PEDAL ORGAN Feet Pipes
Henry Willis III 1927, 2005 H&H PEDAL ORGAN Feet Pipes Open Bass 16 32 Open Diapason * 16 32 Bordun 16 32 Lieblich Bordun (from Swell) 16 - Principal * 8 32 Flute * 8 32 Fifteenth 4 32 Mixture * (19.22.26.29) I V 128 Ophicleide (from Tuba Minor) 16 12 Trombone * 16 32 CHOIR ORGAN Quintaton 16 61 Violoncello * (old bass) 8 61 Orchestral Flute 8 61 Dulciana 8 61 Unda Maris (bass from Dulciana) 8 49 Concert Flute 4 61 Nazard * 2 2/3 61 Harmonic Piccolo 2 61 Tierce * 1 3/5 61 Corno di Bassetto 8 61 Cor Anglais 8 61 Tremolo Tuba Minor 8 61 Tuba Magna * (horizontal, unenclosed) 8 61 GREAT ORGAN Double Open Diapason 16 61 Open Diapason No. 1 * (old bass) 8 61 Open Diapason No. 2 * (old bass) 8 61 Claribel Flute * (bass from Stopped Diapason) 8 61 Stopped Diapason * (wood) 8 61 Principal * 4 61 Chimney Flute * 4 61 Fifteenth * 2 61 Full Mixture * (15.19.22.26) IV 244 Sharp Mixture * (26.29.33) III 183 Trumpet * 8 61 SWELL ORGAN Lieblich Bordun 16 61 Geigen Diapason 8 61 Rohr Flute 8 61 Echo Viole 8 61 Voix Célestes (tenor c) 8 61 Geigen Principal 4 61 Flûte Triangulaire 4 61 Flageolet 2 61 Sesquialtera (12.17) II 122 Mixture (15.19.22) III 183 Oboe 8 61 Waldhorn 16 61 Trumpet 8 61 Clarion 4 61 Tremolo New stops are indicated by *. Couplers I Choir to Pedal XII Choir to Great II Choir Octave to Pedal XIII Choir Octave to Great III Great to Pedal XIV Choir Sub Octave to Great IV Swell to Pedal XV Swell to Great V Swell Octave to Pedal XVI Swell Octave to Great XVII Swell Sub Octave to Great VI Choir Octave VII Choir Sub Octave XVIII Swell Octave -
The Unexpected Number Theory and Algebra of Musical Tuning Systems Or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72
The Unexpected Number Theory and Algebra of Musical Tuning Systems or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72 Matthew Hawthorn \Music is the pleasure the human soul experiences from counting without being aware that it is counting." -Gottfried Wilhelm von Leibniz (1646-1716) \All musicians are subconsciously mathematicians." -Thelonius Monk (1917-1982) 1 Physics In order to have music, we must have sound. In order to have sound, we must have something vibrating. Wherever there is something virbrating, there is the wave equation, be it in 1, 2, or more dimensions. The solutions to the wave equation for any given object (string, reed, metal bar, drumhead, vocal cords, etc.) with given boundary conditions can be expressed as a superposition of discrete partials, modes of vibration of which there are generally infinitely many, each with a characteristic frequency. The partials and their frequencies can be found as eigenvectors, resp. eigenvalues of the Laplace operator acting on the space of displacement functions on the object. Taken together, these frequen- cies comprise the spectrum of the object, and their relative intensities determine what in musical terms we call timbre. Something very nice occurs when our object is roughly one-dimensional (e.g. a string): the partial frequencies become harmonic. This is where, aptly, the better part of harmony traditionally takes place. For a spectrum to be harmonic means that it is comprised of a fundamental frequency, say f, and all whole number multiples of that frequency: f; 2f; 3f; 4f; : : : It is here also that number theory slips in the back door. -
Generalized Interval System and Its Applications
Generalized Interval System and Its Applications Minseon Song May 17, 2014 Abstract Transformational theory is a modern branch of music theory developed by David Lewin. This theory focuses on the transformation of musical objects rather than the objects them- selves to find meaningful patterns in both tonal and atonal music. A generalized interval system is an integral part of transformational theory. It takes the concept of an interval, most commonly used with pitches, and through the application of group theory, generalizes beyond pitches. In this paper we examine generalized interval systems, beginning with the definition, then exploring the ways they can be transformed, and finally explaining com- monly used musical transformation techniques with ideas from group theory. We then apply the the tools given to both tonal and atonal music. A basic understanding of group theory and post tonal music theory will be useful in fully understanding this paper. Contents 1 Introduction 2 2 A Crash Course in Music Theory 2 3 Introduction to the Generalized Interval System 8 4 Transforming GISs 11 5 Developmental Techniques in GIS 13 5.1 Transpositions . 14 5.2 Interval Preserving Functions . 16 5.3 Inversion Functions . 18 5.4 Interval Reversing Functions . 23 6 Rhythmic GIS 24 7 Application of GIS 28 7.1 Analysis of Atonal Music . 28 7.1.1 Luigi Dallapiccola: Quaderno Musicale di Annalibera, No. 3 . 29 7.1.2 Karlheinz Stockhausen: Kreuzspiel, Part 1 . 34 7.2 Analysis of Tonal Music: Der Spiegel Duet . 38 8 Conclusion 41 A Just Intonation 44 1 1 Introduction David Lewin(1933 - 2003) is an American music theorist. -
Chapter 29 Dictation Handout
NAME _______________________________________ Chapter 29 Dictation Exercises Section 1. Melodic Dictation You will hear melodies that are four measures in length. Both major and minor keys are possible. Each melody will contain altered tones. The key signature and time signature are provided to you on each blank staff. Click the Play button to hear each exercise. Before each melody begins, you will hear the notes of the entire scale played stepwise up and down, then you will hear the tempo counted off. Write each melody using the correct rhythm. 1. 2. 3. 4. 5. 6. page 1 Section 2. Interval Identification a. Thirds You will hear a variety of thirds. Listen to each interval and identify it as one of the listed qualities. A diminished third sounds identical to a major 2nd, and an augmented third sounds identical to a perfect 4th. 1. major 3rd minor 3rd diminished 3rd augmented 3rd 2. major 3rd minor 3rd diminished 3rd augmented 3rd 3. major 3rd minor 3rd diminished 3rd augmented 3rd 4. major 3rd minor 3rd diminished 3rd augmented 3rd 5. major 3rd minor 3rd diminished 3rd augmented 3rd 6. major 3rd minor 3rd diminished 3rd augmented 3rd 7. major 3rd minor 3rd diminished 3rd augmented 3rd 8. major 3rd minor 3rd diminished 3rd augmented 3rd 9. major 3rd minor 3rd diminished 3rd augmented 3rd 10. major 3rd minor 3rd diminished 3rd augmented 3rd b. Sixths You will hear a variety of sixths. Listen to each interval and identify it as one of the listed qualities. An augmented 6th sounds identical to a minor 7th. -
Frequency Ratios and the Perception of Tone Patterns
Psychonomic Bulletin & Review 1994, 1 (2), 191-201 Frequency ratios and the perception of tone patterns E. GLENN SCHELLENBERG University of Windsor, Windsor, Ontario, Canada and SANDRA E. TREHUB University of Toronto, Mississauga, Ontario, Canada We quantified the relative simplicity of frequency ratios and reanalyzed data from several studies on the perception of simultaneous and sequential tones. Simplicity offrequency ratios accounted for judgments of consonance and dissonance and for judgments of similarity across a wide range of tasks and listeners. It also accounted for the relative ease of discriminating tone patterns by musically experienced and inexperienced listeners. These findings confirm the generality ofpre vious suggestions of perceptual processing advantages for pairs of tones related by simple fre quency ratios. Since the time of Pythagoras, the relative simplicity of monics of a single complex tone. Currently, the degree the frequency relations between tones has been consid of perceived consonance is believed to result from both ered fundamental to consonance (pleasantness) and dis sensory and experiential factors. Whereas sensory con sonance (unpleasantness) in music. Most naturally OCCUf sonance is constant across musical styles and cultures, mu ring tones (e.g., the sounds of speech or music) are sical consonance presumably results from learning what complex, consisting of multiple pure-tone (sine wave) sounds pleasant in a particular musical style. components. Terhardt (1974, 1978, 1984) has suggested Helmholtz (1885/1954) proposed that the consonance that relations between different tones may be influenced of two simultaneous complex tones is a function of the by relations between components of a single complex tone. ratio between their fundamental frequencies-the simpler For single complex tones, ineluding those of speech and the ratio, the more harmonics the tones have in common.