A Study of Vocal Compositions for Two and Three Voices
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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. -
Shifting Exercises with Double Stops to Test Intonation
VERY ROUGH AND PRELIMINARY DRAFT!!! Shifting Exercises with Double Stops to Test Intonation These exercises were inspired by lessons I had from 1968 to 1970 with David Smiley of the San Francisco Symphony. I don’t have the book he used, but I believe it was one those written by Dounis on the scientific or artist's technique of violin playing. The exercises were difficult and frustrating, and involved shifting and double stops. Smiley also emphasized routine testing notes against other strings, and I also found some of his tasks frustrating because I couldn’t hear intervals that apparently seemed so familiar to a professional musician. When I found myself giving violin lessons in 2011, I had a mathematical understanding of why it was so difficult to hear certain musical intervals, and decided not to focus on them in my teaching. By then I had also developed some exercises to develop my own intonation. These exercises focus entirely on what is called the just scale. Pianos use the equal tempered scale, which is the predominate choice of intonation in orchestras and symphonies (I NEED VERIFICATION THAT THIS IS TRUE). It takes many years and many types of exercises and activities to become a good violinist. But I contend that everyone should start by mastering the following double stops in “just” intonation: 1. Practice the intervals shown above for all possible pairs of strings on your violin or viola. Learn the first two first, then add one interval at a time. They get harder to hear as you go down the list for reasons having to do with the fractions: 1/2, 2/3, 3/4, 3/5, 4/5, 5/6. -
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. -
Vocal Similarity Predicts the Relative Attraction of Musical Chords
Vocal similarity predicts the relative attraction of musical chords Daniel L. Bowlinga,1, Dale Purvesb,1, and Kamraan Z. Gillc aDepartment of Cognitive Biology, University of Vienna, Vienna 1090, Austria; bDuke Institute for Brain Sciences, Duke University, Durham, NC 27708; and cDepartment of Pathology, CBLPath, Rye Brook, NY 10573 Contributed by D. Purves, November 21, 2017 (sent for review July 27, 2017; reviewed by Aniruddh D. Patel and Laurel J. Trainor) Musical chords are combinations of two or more tones played to conspecific vocalization. Accordingly, we here ask whether the together. While many different chords are used in music, some are consonance of tone combinations in music can be rationalized on heard as more attractive (consonant) than others. We have pre- this basis: that is, whether our attraction to specific chords is pre- viously suggested that, for reasons of biological advantage, human dicted by their relative similarity to human vocalization. tonal preferences can be understood in terms of the spectral Answering this question requires perceptual data that docu- similarity of tone combinations to harmonic human vocalizations. ment the relative consonance of chords. Previous evaluations have Using the chromatic scale, we tested this theory further by focused on the two-tone combinations (“dyads”) that define the assessing the perceived consonance of all possible dyads, triads, chromatic scale, a set of 12 tones over an octave used in much and tetrads within a single octave. Our results show that the music worldwide (Table S1). Studies of dyadic consonance have consonance of chords is predicted by their relative similarity to been repeated many times over the last century and, despite some voiced speech sounds. -
Musical Techniques
Musical Techniques Musical Techniques Frequencies and Harmony Dominique Paret Serge Sibony First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2017 The rights of Dominique Paret and Serge Sibony to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2016960997 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-058-4 Contents Preface ........................................... xiii Introduction ........................................ xv Part 1. Laying the Foundations ............................ 1 Introduction to Part 1 .................................. 3 Chapter 1. Sounds, Creation and Generation of Notes ................................... 5 1.1. Physical and physiological notions of a sound .................. 5 1.1.1. Auditory apparatus ............................... 5 1.1.2. Physical concepts of a sound .......................... 7 1.1.3. -
Part-Writing Rules.FH9
Important Rules for 4-Part Progressions In general, some theorists (including Ottman and myself) try to spend most of our time telling you what to do rather than what not to do. If you internalize all of our little procedures then you should be able to churn out progressions quickly and easily without really worrying about making mistakes. However, there are times when you really have to be familiar with the rules. Ottman sprinkles many of these around the later chapters of the text, and he tries to summarize everything you need to know in his Appendix A. Ive produced my own summary because I have a few slightly different ideas of what needs emphasis or de-emphasis. Please compare what I say here to what Ottman says. Making each triad Doubling the root. For triads in root position, try to cover all three chord tones in your upper voices. That means you will end up doubling the root. all three tones present doubled root Doubling any other tone in a root-position Ottman (p. 95) teaches that sometimes at triad will happen occasionally, but it is cadences you will end up tripling the root (and considered less good. I will sometimes subtract leaving out the fifth.) Use this for cadences a point for these bad doublings. only. (If you do it in the middle of a progression it will probably cause bad parallels afterwards.) LESS GOOD doubled doubled 3rd 5th tripled root IVI ©2004 Dave Smey. Reproduction and classroom use freely permitted. Using proper spacing (open position) Adjacent upper voices are not allowed to be more than an octave apart. -
11 – Music Temperament and Pitch
11 – Music temperament and pitch Music sounds Every music instrument, including the voice, uses a more or less well defined sequence of music sounds, the notes, to produce music. The notes have particular frequencies that are called „pitch“ by musicians. The frequencies or pitches of the notes stand in a particular relation to each other. There are and were different ways to build a system of music sounds in different geographical regions during different historical periods. We will consider only the currently dominating system (used in classical and pop music) that was originated in ancient Greece and further developed in Europe. Pitch vs interval Only a small part of the people with normal hearing are able to percieve the absolute frequency of a sound. These people, who are said to have the „absolute pitch“, can tell what key has been hit on a piano without looking at it. The human ear and brain is much more sensitive to the relations between the frequencies of two or more sounds, the music intervals. The intervals carry emotional content that makes music an art. For instance, the major third sounds joyful and affirmative whereas the minor third sounds sad. The system of frequency relations between the sounds used in music production is called music temperament . 1 Music intervals and overtone series Perfect music intervals (that cannot be fully achieved practically, see below) are based on the overtone series studied before in this course. A typical music sound (except of a pure sinusiodal one) consists of a fundamental frequency f1 and its overtones: = = fn nf 1, n 3,2,1 ,.. -
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. -
On the Notation and Performance Practice of Extended Just Intonation
On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation by Marc Sabat 1. Introduction: Two Different E’s Like the metric system, the modern tempered tuning which divides an octave into 12 equal but irrational proportions was a product of a time obsessed with industrial standardization and mass production. In Schönberg’s words: a reduction of natural relations to manageable ones. Its ubiquity in Western musical thinking, epitomized by the pianos which were once present in every home, and transferred by default to fixed-pitch percussion, modern organs and synthesizers, belies its own history as well as everyday musical experience. As a young musician, I studied composition, piano and violin. Early on, I began to learn about musical intervals, the sound of two tones in relation to each other. Without any technical intervention other than a pitch-pipe, I learned to tune my open strings to the notes G - D - A - E by playing two notes at once, listening carefully to eliminate beating between overtone-unisons and seeking a stable, resonant sound-pattern called a “perfect fifth”. At the time, I did not know or need to know that this consonance was the result of a simple mathematical relationship, that the lower string was vibrating twice for every three vibrations of the upper one. However, when I began to learn about placing my fingers on the strings to tune other pitches, the difficulties began. To find the lower E which lies one whole step above the D string, I needed to place my first finger down. -
Polytonal Non-Octave Complexes DMA Document Presented In
Polytonal Non-Octave Complexes DMA Document Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Musical Arts in the Graduate School of The Ohio State University By Luis Javier Obregón, B.A., M.M. Graduate Program in Music The Ohio State University 2012 DMA Document Committee: Dr. Marc Ainger, Advisor Dr. David Clampitt Dr. Donald Harris Dr. Robert C. Holub Copyright by Luis Javier Obregón 2012 ABSTRACT Through an exploration of dissonance by means of phenomenologically organized scalar complexes, this document describes an alternative to the octave-based scalar system that has prevailed over the past centuries. These complexes are derived from an adaptation of Ching Fang’s sixty-step division of the octave into systems that use the perfect fourth or the perfect fifth as their interval of periodicity. In this manner scalar complexes are created that span five octaves with the use of the fourth, and seven octaves, with the use of the fifth, and can contain thirty-six or more register dependant pitches. In this document I will explore the origins and methodology for deriving these complexes and, through the analysis of my own musical compositions, I will explain the compositional approaches and techniques that I have developed over the past four years using these complexes, which I have termed “Polytonal Non-Octave Complexes”. ii Dedicated to Ion, Balam and my parents iii AKNOWLEDGEMENTS First and foremost I would like to thank my mother for her help and support she has given me not only completing this document, but throughout my entire musical career. -
MTO 12.3: Duffin, Just Intonation in Renaissance Theory and Practice
Volume 12, Number 3, October 2006 Copyright © 2006 Society for Music Theory Ross W. Duffin ABSTRACT: Just intonation has a reputation as a chimerical, theoretical system that simply cannot work in practice. This is based on the assessment of most modern authorities and supported by misgivings expressed during the Renaissance when the practice was supposedly at its height. Looming large among such misgivings are tuning puzzles printed by the 16th-century mathematician, Giovanni Battista Benedetti. However, Renaissance music theorists are so unanimous in advocating the simple acoustical ratios of Just intonation that it seems clear that some reconciliation must have occurred between the theory and practice of it. This article explores the basic theory of Just intonation as well as problematic passages used to deny its practicability, and proposes solutions that attempt to satisfy both the theory and the ear. Ultimately, a resource is offered to help modern performers approach this valuable art. Received June 2006 Introduction | Theoretical Background | Benedetti's Puzzles | Problematic Passages | Is Just Tuning Possible? A New Approach | Problem Spots in the Exercises | Rehearsal Usage | Conclusion Introduction The idea . that one can understand the ratios of musical consonances without experiencing them with the senses is wrong. Nor can one know the theory of music without being versed in its practice. [1] So begins the first of two letters sent by the mathematician Giovanni Battista Benedetti to the composer Cipriano de Rore in 1563. Subsequently publishing the letters in 1585,(1) Benedetti was attempting to demonstrate that adhering to principles of Just intonation, as championed most famously by Gioseffo Zarlino,(2) would, in certain cases, cause the pitch of the ensemble to migrate. -
A New Keyboard for the Bohlen-Pierce Scale
A New Keyboard for the Bohlen-Pierce Scale Antonio B. Nassar Science Department The Harvard-Westlake School 3700 Coldwater Canyon N. Hollywood, CA 91604 And Department of Math and Sciences UCLA Extension 10995 Le Conte Avenue Los Angeles, CA 90024 Abstract: The study of harmonic scales of musical instruments is discussed in all introductory physics texts devoted to the science of sound. In this paper, we present a new piano keyboard to make the so-called Bohlen-Pierce scale more functional and pleasing for composition and performance. PACS: 01.55.+b; 01.90.+g 1. Introduction 1,2 The basis of harmonic scales of musical instruments is the following: If two tones with frequencies f1 and f2 are played together, the result is pleasant to the human ear if the ratio f1:f2 is equal to m:n, with m and n two integers. This discovery is attributed to the ancient Chinese and Greeks. A series of bells has been discovered in the tomb of the Count of Chin (around 200 B.C.). They were found to be tuned quite precisely to the harmonic scale. On the other hand, the Greeks, with their abundance of string instruments, discovered that dividing a string into equal parts or chopping of one-third of the string resulted in pleasant musical intervals: An octave (2:1) and a perfect fifth (3:2), respectively. The Pythagoreans asked themselves whether an integral number of octaves could be constructed from the fifth alone by repeated application of the simple frequency ratio 3:2. In mathematical notation, they asked for a solution to3 n 3 m 2 2 in positive integers n and m.