Development of Musical Scales in Europe
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Tonality in My Molto Adagio
Estonian Academy of Music and Theatre Jorge Gómez Rodríguez Constructing Tonality in Molto Adagio A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy (Music) Supervisor: Prof. Mart Humal Tallinn 2015 Abstract The focus of this paper is the analysis of the work for string ensemble entitled And Silence Eternal - Molto Adagio (Molto Adagio for short), written by the author of this thesis. Its originality lies in the use of a harmonic approach which is referred to as System of Harmonic Classes in this thesis. Owing to the relatively unconventional employment of otherwise traditional harmonic resources in this system, a preliminary theoretical framework needs to be put in place in order to construct an alternative tonal approach that may describe the pitch structure of the Molto Adagio. This will be the aim of Part 1. For this purpose, analytical notions such as symmetric pair, primary dominant factor, harmonic domain, and harmonic area will be introduced before the tonal character of the piece (or the extent to which Molto Adagio may be said to be tonal) can be assessed. A detailed description of how the composer envisions and constructs the system will follow, with special consideration to the place the major diatonic scale occupies in relation to it. Part 2 will engage in an analysis of the piece following the analytical directives gathered in Part 1. The concept of modular tonality, which incorporates the notions of harmonic area and harmonic classes, will be put forward as a heuristic term to account for the use of a linear analytical approach. -
Nora-Louise Müller the Bohlen-Pierce Clarinet An
Nora-Louise Müller The Bohlen-Pierce Clarinet An Introduction to Acoustics and Playing Technique The Bohlen-Pierce scale was discovered in the 1970s and 1980s by Heinz Bohlen and John R. Pierce respectively. Due to a lack of instruments which were able to play the scale, hardly any compositions in Bohlen-Pierce could be found in the past. Just a few composers who work in electronic music used the scale – until the Canadian woodwind maker Stephen Fox created a Bohlen-Pierce clarinet, instigated by Georg Hajdu, professor of multimedia composition at Hochschule für Musik und Theater Hamburg. Hence the number of Bohlen- Pierce compositions using the new instrument is increasing constantly. This article gives a short introduction to the characteristics of the Bohlen-Pierce scale and an overview about Bohlen-Pierce clarinets and their playing technique. The Bohlen-Pierce scale Unlike the scales of most tone systems, it is not the octave that forms the repeating frame of the Bohlen-Pierce scale, but the perfect twelfth (octave plus fifth), dividing it into 13 steps, according to various mathematical considerations. The result is an alternative harmonic system that opens new possibilities to contemporary and future music. Acoustically speaking, the octave's frequency ratio 1:2 is replaced by the ratio 1:3 in the Bohlen-Pierce scale, making the perfect twelfth an analogy to the octave. This interval is defined as the point of reference to which the scale aligns. The perfect twelfth, or as Pierce named it, the tritave (due to the 1:3 ratio) is achieved with 13 tone steps. -
The Chromatic Scale
Getting to Know Your Guitar: The Chromatic Scale Your Guitar is designed as a chromatic instrument – that is, guitar frets represent chromatic “semi- tones” or “half-steps” up and down the guitar fretboard. This enables you to play scales and chords in any key, and handle pretty much any music that comes from the musical traditions of the Western world. In this sense, the chromatic scale is more foundational than it is useful as a soloing tool. Put another way, almost all of the music you will ever play will be made interesting not by the use of the chromatic scale, but by the absence of many of the notes of the chromatic scale! All keys, chords, scales, melodies and harmonies, could be seen simply the chromatic scale minus some notes. For all the examples that follow, play up and down (both ascending and descending) the fretboard. Here is what you need to know in order to understand The Chromatic Scale: 1) The musical alphabet contains 7 letters: A, B, C, D, E, F, G. The notes that are represented by those 7 letters we will call the “Natural” notes 2) There are other notes in-between the 7 natural notes that we’ll call “Accidental” notes. They are formed by taking one of the natural notes and either raising its pitch up, or lowering its pitch down. When we raise a note up, we call it “Sharp” and use the symbol “#” after the note name. So, if you see D#, say “D sharp”. When we lower the note, we call it “Flat” and use the symbol “b” after the note. -
In Search of the Perfect Musical Scale
In Search of the Perfect Musical Scale J. N. Hooker Carnegie Mellon University, Pittsburgh, USA [email protected] May 2017 Abstract We analyze results of a search for alternative musical scales that share the main advantages of classical scales: pitch frequencies that bear simple ratios to each other, and multiple keys based on an un- derlying chromatic scale with tempered tuning. The search is based on combinatorics and a constraint programming model that assigns frequency ratios to intervals. We find that certain 11-note scales on a 19-note chromatic stand out as superior to all others. These scales enjoy harmonic and structural possibilities that go significantly beyond what is available in classical scales and therefore provide a possible medium for innovative musical composition. 1 Introduction The classical major and minor scales of Western music have two attractive characteristics: pitch frequencies that bear simple ratios to each other, and multiple keys based on an underlying chromatic scale with tempered tuning. Simple ratios allow for rich and intelligible harmonies, while multiple keys greatly expand possibilities for complex musical structure. While these tra- ditional scales have provided the basis for a fabulous outpouring of musical creativity over several centuries, one might ask whether they provide the natural or inevitable framework for music. Perhaps there are alternative scales with the same favorable characteristics|simple ratios and multiple keys|that could unleash even greater creativity. This paper summarizes the results of a recent study [8] that undertook a systematic search for musically appealing alternative scales. The search 1 restricts itself to diatonic scales, whose adjacent notes are separated by a whole tone or semitone. -
Lissajous Figures, Musical Intervals, and the Just Diatonic Scale," the Physics Teacher 56 424 (October 2018)
Michael J. Ruiz, "Lissajous Figures, Musical Intervals, and the Just Diatonic Scale," The Physics Teacher 56 424 (October 2018). Lissajous Figures, Musical Intervals, and the Just Diatonic Scale Michael J. Ruiz University of North Carolina at Asheville, Asheville, NC The frequency ratios for the just diatonic scale are obtained by identifying musical intervals corresponding to Lissajous figures. The demonstration integrates the engineering physics of Lissajous patterns with the recognition of musical intervals through simple ear training. A free HTML5 app has been developed for this class activity and the program can be run online or downloaded to the desktop. A video1 is provided illustrating the use of the software. Background Discussion of musical scales can be found in an early issue (1934) of The American Physics Teacher2 and a recent issue (2015) of its successor journal, the American Journal of Physics.3 Both references include discussion of the equal-tempered tuning we use today as well as tunings based on the perfect integer ratios described in this paper. These tunings are listed in Table 1, where the first note in the major scale has been set to 240 Hz. For years I have used 240 Hz for the first degree of the major major since all the frequencies with integer ratios work out easily as whole-number frequencies without a calculator. Reference 2 lists the relative frequencies as whole numbers where 24 is assigned to Do (the note C), a procedure that was made by the legendary Helmholtz as far back as 1895 in his masterpiece On the Sensations of Tone as a Physiological Basis for the Theory of Music.4 Table I. -
Theory IV – Study Guide Dr. Amy Dunker Clarke College Dubuque, IA 52001 Classical Serialism Arnold Schoenb
Theory IV – Study Guide Dr. Amy Dunker Clarke College Dubuque, IA 52001 www.amydunker.com Classical Serialism Arnold Schoenberg composed the first twelve-tone piece in the summer of 1921 (Suite, Op. 25 (completed in 1923). Schoenberg had developed a method of composing with twelve tones that are related only with one another. He saw twelve-tone or serial composition as the natural extension of chromaticism on the tonal system. Anton Webern and Alban Berg: Schoenberg’s two pupils who composed in the twelve- tone method. Tone Row (also called , Row, Set, Basic Set, Series): an arrangement of the twelve pitches of the chromatic scale so that no notes repeat (except immediately after it is heard and trills/tremolos) until all pitches of the row have sounded in order. Dodecaphonic Scale: Twelve tone scale Four Forms of the Tone Row: Prime: The original set (do not confuse this with the terms use in Non-Serial Atonality) Retrograde: The original set in reverse order (i.e. backwards) Inversion: The mirror inversion of the original set Retrograde Inversion: The inversion in reverse order Abbreviations: P=Prime R=Retrograde I=Inversion RI=Retrograde Inversion *In addition, each of the four basic forms has twelve transpositions Order Numbers: numbers assigned to the row which indicate each notes intervallic distance from the first note of the row. The first note of the row is assigned the number zero (0). Twelve-Tone Matrix (“Magic Square”): a method of determining all 48 possible versions of the tone row. To construct a Twelve-Tone Matrix do the following: 1.) Fill in the Prime or Original row across the top ( from left to right) using the row’s order numbers. -
The Consecutive-Semitone Constraint on Scalar Structure: a Link Between Impressionism and Jazz1
The Consecutive-Semitone Constraint on Scalar Structure: A Link Between Impressionism and Jazz1 Dmitri Tymoczko The diatonic scale, considered as a subset of the twelve chromatic pitch classes, possesses some remarkable mathematical properties. It is, for example, a "deep scale," containing each of the six diatonic intervals a unique number of times; it represents a "maximally even" division of the octave into seven nearly-equal parts; it is capable of participating in a "maximally smooth" cycle of transpositions that differ only by the shift of a single pitch by a single semitone; and it has "Myhill's property," in the sense that every distinct two-note diatonic interval (e.g., a third) comes in exactly two distinct chromatic varieties (e.g., major and minor). Many theorists have used these properties to describe and even explain the role of the diatonic scale in traditional tonal music.2 Tonal music, however, is not exclusively diatonic, and the two nondiatonic minor scales possess none of the properties mentioned above. Thus, to the extent that we emphasize the mathematical uniqueness of the diatonic scale, we must downplay the musical significance of the other scales, for example by treating the melodic and harmonic minor scales merely as modifications of the natural minor. The difficulty is compounded when we consider the music of the late-nineteenth and twentieth centuries, in which composers expanded their musical vocabularies to include new scales (for instance, the whole-tone and the octatonic) which again shared few of the diatonic scale's interesting characteristics. This suggests that many of the features *I would like to thank David Lewin, John Thow, and Robert Wason for their assistance in preparing this article. -
Music Perception 13
Music Perception 13 LEARNING OBJECTIVES ISLE EXERCISES 13.1 Kurdish Music Example 13.1 Explain how frequency is related to pitch, chroma, and the octave. 13.2 Javanese Gamelan Music Example 13.2 Summarize the basic neuroscience of music, including how training and experience can affect the representation of music in the brain. 13.3 Ancient Greek Music 13.4 35,000-Year-Old Flute 13.3 Discuss how learning and culture affect music perception. distribute 13.5 Is This Music? 13.6 The Octave and or Tone Similarity INTRODUCTION 13.7 Pentatonic Music 13.8 Meter and Beat Wherever you travel, you will find music. It may sound very different from the music you are accustomed to hearing, but you will recognize it instantly as music. In Kurdistan, we 13.9 Bolero Clip find a unique culture of music featuring such instrumentspost, as the tanbur (a fretted string 13.10 Attack and Decay instrument), the qernête (a double-reed wind instrument), and the şimşal (a flutelike 13.11 Examples of Melody instrument) (Figure 13.1). Although most of you may never have heard of these instru- ments and may never have heard Kurdish music before, you would instantly recognize 13.12 Types of Scales them as musical instruments, and you might even like Kurdish music (see ISLE 13.1 for 13.13 Gestalt Principles an example of Kurdish music). Review copy, © Aurora Photos/Alamy 13.14 Gestalt Principle: Proximity: Bach’s Partita No. 3 in E major not 13.15 A Shave and a Haircut 13.16 Cross-Modal Matchings as a Simulation of Synesthesia Do 13.17. -
The Processing of Pitch and Scale: an ERP Study of Musicians Trained Outside of the Western Musical System
Empirical Musicology Review Vol. 1, No. 4, 2006 The Processing of Pitch and Scale: An ERP Study of Musicians Trained Outside of the Western Musical System LAURA BISCHOFF RENNINGER [1] Shepherd University MICHAEL P. WILSON University of Illinois EMANUEL DONCHIN University of South Florida ABSTRACT: The current study extends the efforts of Bischoff Renninger, Granot and Donchin (2003) to non-Western musical systems and focuses specifically on Event- Related Potential (ERP) responses to scalar deviations within the Javanese pélog scale by groups of musicians trained within the Western and Javanese systems. The principal aim is to ascertain whether results found in previous experiments may be obtained cross- culturally. Participants include five subjects trained in the Western system only (control group) and five subjects trained in both the Western and Javanese systems (experimental group). Tasks include identifying scalar deviations within the Western diatonic scale, identifying scalar deviations within the Javanese pèlog scale, and identifying target letters in a visual control task. ERP and overt behavioral responses are recorded. Results show significant differences between group responses to diatonic and Javanese scale conditions. Interesting results also arise in terms of how tones and intervals in the different scales are conceptualized by subjects in both groups. Control subjects especially tend to assimilate pélog scale intervals to intervals in the Western equal- tempered scale. Submitted 2006 June 21; accepted 2006 June 26. KEYWORDS: music, scale, ERP, event-related potential, tuning, interval, enculturation INTRODUCTION IN human culture, music is as far-reaching a phenomenon as language. And, like language, learning another culture’s music can be both complicated and challenging. -
The Bohlen-Pierce Scale: Continuing Research
The Bohlen-Pierce Scale: Continuing Research Elaine Walker - Final Project in Advanced Acoustics, NYU, Spring 2001 The purpose of this project was to conduct a study on the perception of scales and chord progressions in the Bohlen-Pierce Scale. The version of the Bohlen-Pierce Scale I used in these experiments is an equal-tempered scale based on the 13th root of 3, which gives 13 equal steps spanning the “tritave” (an octave and a fifth). This scale has been shown to have perceptible properties such as consonant and dissonant chords, and a rich structure that is comparable in complexity to the 12 tone equal tempered scale, but completely different in detail. This study included a set of listening examples and data sheets for the subjects to fill out. For the test subjects, I primarily chose trained musicians, so that they would understand the terminology like “major”, “minor” or “cadence” that were prevalent in the questions relating to the listening examples. In general, a much different set of questions and experiments would need to be written for test subjects with no musical training. Figure 1: Bohlen in 1973, tinkering with the transcription of a Christmas carol into his "13-Step Scale" (BP) The Bohlen-Pierce Scale was designed by Heinz Bohlen, a microwave electronics and communications engineer, in the early 1970's in just tuning. He discovered its suitability to be turned into an equally tempered scale with a minimal defect just days after inventing the tuning. About six years later, another microwave electronics and communications engineer, John R. Pierce ("Father of the Communication Satellite" and creator of the "Pierce (electron) gun") discovered the same scale again, also proposing that it be used in the equal tempered form. -
The Death and Resurrection of Function
THE DEATH AND RESURRECTION OF FUNCTION A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By John Gabriel Miller, B.A., M.C.M., M.A. ***** The Ohio State University 2008 Doctoral Examination Committee: Approved by Dr. Gregory Proctor, Advisor Dr. Graeme Boone ________________________ Dr. Lora Gingerich Dobos Advisor Graduate Program in Music Copyright by John Gabriel Miller 2008 ABSTRACT Function is one of those words that everyone understands, yet everyone understands a little differently. Although the impact and pervasiveness of function in tonal theory today is undeniable, a single, unambiguous definition of the term has yet to be agreed upon. So many theorists—Daniel Harrison, Joel Lester, Eytan Agmon, Charles Smith, William Caplin, and Gregory Proctor, to name a few—have so many different nuanced understandings of function that it is nearly impossible for conversations on the subject to be completely understood by all parties. This is because function comprises at least four distinct aspects, which, when all called by the same name, function , create ambiguity, confusion, and contradiction. Part I of the dissertation first illuminates this ambiguity in the term function by giving a historical basis for four different aspects of function, three of which are traced to Riemann, and one of which is traced all the way back to Rameau. A solution to the problem of ambiguity is then proposed: the elimination of the term function . In place of function , four new terms—behavior , kinship , province , and quality —are invoked, each uniquely corresponding to one of the four aspects of function identified. -
Scales, Modes, and Chord/Cluster Concepts for 20Th-Century Techniques Exam
Scales, Modes, and Chord/Cluster Concepts for 20th-Century Techniques Exam Traditional Modes -The easiest way to remember these is to know the modes in order of their alphabetical starting pitch: A, B, C, D, E, F, G -To build one of these modes on a different starting pitch, just follow its exact whole-step, half-step model Mode Structure Whole-step/half-step Model Aeolian (A) A B C D E F G A half steps are between 2-3 and 5-6, (just like "natural" minor) Locrian (B) B C D E F G A B half steps are between 1-2 and 4-5 Ionian (C) C D E F G A B C half steps are between 4-5 and 7-8, (just like a "major" scale) Dorian (D) D E F G A B C D half steps are between 2-3 and 6-7 Phrygian (E) E F G A B C D E half steps are between 1-2 and 5-6 Lydian (F) F G A B C D E F half steps are between 4-5 and 7-8 Mixolydian (G) G A B C D E F G half steps are between 3-4 and 6-7 Modern Modes and Scales - To build one of these modes on a different starting pitch, just follow its exact whole-step, half-step model Mode/Scale Structure Whole-step/half-step Model Lydian Minor (F)-- F G A B C Db Eb F half steps are between 4-5 and 5-6 used in jazz F G A B C D E F (compare to Lydian): Whole-Tone Scale C D E F# G# A# B# (=C) All whole steps (only 7 notes in an octave) Octatonic Scale (in C D Eb F Gb Ab A B C The most common ones alternate whole-step/half-step or half- jazz, this is called a C Db Eb E F# G A Bb C step/whole-step to create 9 notes in an octave "Diminished" Scale) (in jazz, these are the two options for a "diminished scale"; in classical music there are