Models of Octatonic and Whole-Tone Interaction: George Crumb and His Predecessors
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4. Non-Heptatonic Modes
Tagg: Everyday Tonality II — 4. Non‐heptatonic modes 151 4. Non‐heptatonic modes If modes containing seven different scale degrees are heptatonic, eight‐note modes are octatonic, six‐note modes hexatonic, those with five pentatonic, while four‐ and three‐note modes are tetratonic and tritonic. Now, even though the most popular pentatonic modes are sometimes called ‘gapped’ because they contain two scale steps larger than those of the ‘church’ modes of Chapter 3 —doh ré mi sol la and la doh ré mi sol, for example— they are no more incomplete FFBk04Modes2.fm. 2014-09-14,13:57 or empty than the octatonic start to example 70 can be considered cluttered or crowded.1 Ex. 70. Vigneault/Rochon (1973): Je chante pour (octatonic opening phrase) The point is that the most widespread convention for numbering scale degrees (in Europe, the Arab world, India, Java, China, etc.) is, as we’ve seen, heptatonic. So, when expressions like ‘thirdless hexatonic’ occur in this chapter it does not imply that the mode is in any sense deficient: it’s just a matter of using a quasi‐global con‐ vention to designate a particular trait of the mode. Tritonic and tetratonic Tritonic and tetratonic tunes are common in many parts of the world, not least in traditional music from Micronesia and Poly‐ nesia, as well as among the Māori, the Inuit, the Saami and Native Americans of the great plains.2 Tetratonic modes are also found in Christian psalm and response chanting (ex. 71), while the sound of children chanting tritonic taunts can still be heard in playgrounds in many parts of the world (ex. -
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. -
Theory Placement Examination (D
SAMPLE Placement Exam for Incoming Grads Name: _______________________ (Compiled by David Bashwiner, University of New Mexico, June 2013) WRITTEN EXAM I. Scales Notate the following scales using accidentals but no key signatures. Write the scale in both ascending and descending forms only if they differ. B major F melodic minor C-sharp harmonic minor II. Key Signatures Notate the following key signatures on both staves. E-flat minor F-sharp major Sample Graduate Theory Placement Examination (D. Bashwiner, UNM, 2013) III. Intervals Identify the specific interval between the given pitches (e.g., m2, M2, d5, P5, A5). Interval: ________ ________ ________ ________ ________ Note: the sharp is on the A, not the G. Interval: ________ ________ ________ ________ ________ IV. Rhythm and Meter Write the following rhythmic series first in 3/4 and then in 6/8. You may have to break larger durations into smaller ones connected by ties. Make sure to use beams and ties to clarify the meter (i.e. divide six-eight bars into two, and divide three-four bars into three). 2 Sample Graduate Theory Placement Examination (D. Bashwiner, UNM, 2013) V. Triads and Seventh Chords A. For each of the following sonorities, indicate the root of the chord, its quality, and its figured bass (being sure to include any necessary accidentals in the figures). For quality of chord use the following abbreviations: M=major, m=minor, d=diminished, A=augmented, MM=major-major (major triad with a major seventh), Mm=major-minor, mm=minor-minor, dm=diminished-minor (half-diminished), dd=fully diminished. Root: Quality: Figured Bass: B. -
The Group-Theoretic Description of Musical Pitch Systems
The Group-theoretic Description of Musical Pitch Systems Marcus Pearce [email protected] 1 Introduction Balzano (1980, 1982, 1986a,b) addresses the question of finding an appropriate level for describing the resources of a pitch system (e.g., the 12-fold or some microtonal division of the octave). His motivation for doing so is twofold: “On the one hand, I am interested as a psychologist who is not overly impressed with the progress we have made since Helmholtz in understanding music perception. On the other hand, I am interested as a computer musician who is trying to find ways of using our pow- erful computational tools to extend the musical [domain] of pitch . ” (Balzano, 1986b, p. 297) Since the resources of a pitch system ultimately depend on the pairwise relations between pitches, the question is one of how to conceive of pitch intervals. In contrast to the prevailing approach which de- scribes intervals in terms of frequency ratios, Balzano presents a description of pitch sets as mathematical groups and describes how the resources of any pitch system may be assessed using this description. Thus he is concerned with presenting an algebraic description of pitch systems as a competitive alternative to the existing acoustic description. In these notes, I shall first give a brief description of the ratio based approach (§2) followed by an equally brief exposition of some necessary concepts from the theory of groups (§3). The following three sections concern the description of the 12-fold division of the octave as a group: §4 presents the nature of the group C12; §5 describes three perceptually relevant properties of pitch-sets in C12; and §6 describes three musically relevant isomorphic representations of C12. -
Diatonic-Collection Disruption in the Melodic Material of Alban Berg‟S Op
Michael Schnitzius Diatonic-Collection Disruption in the Melodic Material of Alban Berg‟s Op. 5, no. 2 The pre-serial Expressionist music of the early twentieth century composed by Arnold Schoenberg and his pupils, most notably Alban Berg and Anton Webern, has famously provoked many music-analytical dilemmas that have, themselves, spawned a wide array of new analytical approaches over the last hundred years. Schoenberg‟s own published contributions to the analytical understanding of this cryptic musical style are often vague, at best, and tend to describe musical effects without clearly explaining the means used to create them. His concept of “the emancipation of the dissonance” has become a well known musical idea, and, as Schoenberg describes the pre-serial music of his school, “a style based on [the premise of „the emancipation of the dissonance‟] treats dissonances like consonances and renounces a tonal center.”1 The free treatment of dissonance and the renunciation of a tonal center are musical effects that are simple to observe in the pre-serial music of Schoenberg, Berg, and Webern, and yet the specific means employed in this repertoire for avoiding the establishment of a perceived tonal center are difficult to describe. Both Allen Forte‟s “Pitch-Class Set Theory” and the more recent approach of Joseph Straus‟s “Atonal Voice Leading” provide excellently specific means of describing the relationships of segmented musical ideas with one another. However, the question remains: why are these segmented ideas the types of musical ideas that the composer wanted to use, and what role do they play in renouncing a tonal center? Furthermore, how does the renunciation of a tonal center contribute to the positive construction of the musical language, if at all? 1 Arnold Schoenberg, “Composition with Twelve Tones” (delivered as a lecture at the University of California at Las Angeles, March 26, 1941), in Style and Idea, ed. -
Unified Music Theories for General Equal-Temperament Systems
Unified Music Theories for General Equal-Temperament Systems Brandon Tingyeh Wu Research Assistant, Research Center for Information Technology Innovation, Academia Sinica, Taipei, Taiwan ABSTRACT Why are white and black piano keys in an octave arranged as they are today? This article examines the relations between abstract algebra and key signature, scales, degrees, and keyboard configurations in general equal-temperament systems. Without confining the study to the twelve-tone equal-temperament (12-TET) system, we propose a set of basic axioms based on musical observations. The axioms may lead to scales that are reasonable both mathematically and musically in any equal- temperament system. We reexamine the mathematical understandings and interpretations of ideas in classical music theory, such as the circle of fifths, enharmonic equivalent, degrees such as the dominant and the subdominant, and the leading tone, and endow them with meaning outside of the 12-TET system. In the process of deriving scales, we create various kinds of sequences to describe facts in music theory, and we name these sequences systematically and unambiguously with the aim to facilitate future research. - 1 - 1. INTRODUCTION Keyboard configuration and combinatorics The concept of key signatures is based on keyboard-like instruments, such as the piano. If all twelve keys in an octave were white, accidentals and key signatures would be meaningless. Therefore, the arrangement of black and white keys is of crucial importance, and keyboard configuration directly affects scales, degrees, key signatures, and even music theory. To debate the key configuration of the twelve- tone equal-temperament (12-TET) system is of little value because the piano keyboard arrangement is considered the foundation of almost all classical music theories. -
AP Music Theory Course Description Audio Files ”
MusIc Theory Course Description e ffective Fall 2 0 1 2 AP Course Descriptions are updated regularly. Please visit AP Central® (apcentral.collegeboard.org) to determine whether a more recent Course Description PDF is available. The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of more than 5,900 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success — including the SAT® and the Advanced Placement Program®. The organization also serves the education community through research and advocacy on behalf of students, educators, and schools. For further information, visit www.collegeboard.org. AP Equity and Access Policy The College Board strongly encourages educators to make equitable access a guiding principle for their AP programs by giving all willing and academically prepared students the opportunity to participate in AP. We encourage the elimination of barriers that restrict access to AP for students from ethnic, racial, and socioeconomic groups that have been traditionally underserved. Schools should make every effort to ensure their AP classes reflect the diversity of their student population. The College Board also believes that all students should have access to academically challenging course work before they enroll in AP classes, which can prepare them for AP success. -
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. -
Pitch-Class Set Theory: an Overture
Chapter One Pitch-Class Set Theory: An Overture A Tale of Two Continents In the late afternoon of October 24, 1999, about one hundred people were gathered in a large rehearsal room of the Rotterdam Conservatory. They were listening to a discussion between representatives of nine European countries about the teaching of music theory and music analysis. It was the third day of the Fourth European Music Analysis Conference.1 Most participants in the conference (which included a number of music theorists from Canada and the United States) had been looking forward to this session: meetings about the various analytical traditions and pedagogical practices in Europe were rare, and a broad survey of teaching methods was lacking. Most felt a need for information from beyond their country’s borders. This need was reinforced by the mobility of music students and the resulting hodgepodge of nationalities at renowned conservatories and music schools. Moreover, the European systems of higher education were on the threshold of a harmoni- zation operation. Earlier that year, on June 19, the governments of 29 coun- tries had ratifi ed the “Bologna Declaration,” a document that envisaged a unifi ed European area for higher education. Its enforcement added to the urgency of the meeting in Rotterdam. However, this meeting would not be remembered for the unusually broad rep- resentation of nationalities or for its political timeliness. What would be remem- bered was an incident which took place shortly after the audience had been invited to join in the discussion. Somebody had raised a question about classroom analysis of twentieth-century music, a recurring topic among music theory teach- ers: whereas the music of the eighteenth and nineteenth centuries lent itself to general analytical methodologies, the extremely diverse repertoire of the twen- tieth century seemed only to invite ad hoc approaches; how could the analysis of 1. -
The Lowdown on the Octatonic Scale by Frank Horvat
The Lowdown on the Octatonic Scale By Frank Horvat Within the last year, RCM Examinations has revised its Theory Syllabus. A number of new topics have been added to the course of study. In Grade One Rudiments, scale types such as the octatonic scale are now expected to be part of a student's knowledge base. An octatonic scale includes eight pitches per octave arranged in a pattern of alternating semitones and whole tones (for example: c c# d# e f# g a a# c). Octatonic scales can start on any pitch and do not require a key signature. Although the concept of the octatonic scale might be foreign to some, one does not have to look too far for musical examples. One of the first composers to use this scale was Rimsky-Korsakov who looked to "exotic" non-Western scales as a basis for some of his melodic ideas. It was Rimsky-Korsakov who would introduce it to the next generation of Russian composers including Skryabin & Stravinsky who used it in a variety of their compositions in the early 1900s. Bartok also used the octatonic scale in a number of his works including pieces within the Mikrokosmos. Look to Nos. 101 & 109 for clear examples of usage. Later in the 20th century, Messiaen would use it in works as a basis for his tone rows, as part of his philosophy of integral serialism. In order to help your students grasp this scale, here are a few activities that you can try in your studio: • Draw the scale on manuscript paper assigning a variety of starting notes • Have them sing or play it so they can hear the uniqueness of the set of pitches • Sight read pieces from excerpts of Bartok's Mikrokosmos • Suggest that they compose or improvise a short piece based on a melody constructed from the notes of an octatonic scale. -
Chromatic Sequences
CHAPTER 30 Melodic and Harmonic Symmetry Combine: Chromatic Sequences Distinctions Between Diatonic and Chromatic Sequences Sequences are paradoxical musical processes. At the surface level they provide rapid harmonic rhythm, yet at a deeper level they function to suspend tonal motion and prolong an underlying harmony. Tonal sequences move up and down the diatonic scale using scale degrees as stepping-stones. In this chapter, we will explore the consequences of transferring the sequential motions you learned in Chapter 17 from the asymmetry of the diatonic scale to the symmetrical tonal patterns of the nineteenth century. You will likely notice many similarities of behavior between these new chromatic sequences and the old diatonic ones, but there are differences as well. For instance, the stepping-stones for chromatic sequences are no longer the major and minor scales. Furthermore, the chord qualities of each individual harmony inside a chromatic sequence tend to exhibit more homogeneity. Whereas in the past you may have expected major, minor, and diminished chords to alternate inside a diatonic sequence, in the chromatic realm it is not uncommon for all the chords in a sequence to manifest the same quality. EXAMPLE 30.1 Comparison of Diatonic and Chromatic Forms of the D3 ("Pachelbel") Sequence A. 624 CHAPTER 30 MELODIC AND HARMONIC SYMMETRY COMBINE 625 B. Consider Example 30.1A, which contains the D3 ( -4/ +2)-or "descending 5-6"-sequence. The sequence is strongly goal directed (progressing to ii) and diatonic (its harmonies are diatonic to G major). Chord qualities and distances are not consistent, since they conform to the asymmetry of G major. -
Harmonic Dualism and the Origin of the Minor Triad
45 Harmonic Dualism and the Origin of the Minor Triad JOHN L. SNYDER "Harmonic Dualism" may be defined as a school of musical theoretical thought which holds that the minor triad has a natural origin different from that of the major triad, but of equal validity. Specifically, the term is associated with a group of nineteenth and early twentieth century theo rists, nearly all Germans, who believed that the minor harmony is constructed in a downward ("negative") fashion, while the major is an upward ("positive") construction. Al though some of these writers extended the dualistic approach to great lengths, applying it even to functional harmony, this article will be concerned primarily with the problem of the minor sonority, examining not only the premises and con clusions of the principal dualists, but also those of their followers and their critics. Prior to the rise of Romanticism, the minor triad had al ready been the subject of some theoretical speculation. Zarlino first considered the major and minor triads as en tities, finding them to be the result of harmonic and arith metic division of the fifth, respectively. Later, Giuseppe Tartini advanced his own novel explanation: he found that if one located a series of fundamentals such that a given pitch would appear successively as first, second, third •.• and sixth partial, the fundamentals of these harmonic series would outline a minor triad. l Tartini's discussion of combination tones is interesting in that, to put a root un der his minor triad outline, he proceeds as far as the 7:6 ratio, for the musical notation of which he had to invent a new accidental, the three-quarter-tone flat: IG.