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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. -
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). -
Smufl Standard Music Font Layout
SMuFL Standard Music Font Layout Version 0.7 (2013-11-27) Copyright © 2013 Steinberg Media Technologies GmbH Acknowledgements This document reproduces glyphs from the Bravura font, copyright © Steinberg Media Technologies GmbH. Bravura is released under the SIL Open Font License and can be downloaded from http://www.smufl.org/fonts This document also reproduces some glyphs from the Unicode 6.2 code chart for the Musical Symbols range (http://www.unicode.org/charts/PDF/U1D100.pdf). These glyphs are the copyright of their respective copyright holders, listed on the Unicode Consortium web site here: http://www.unicode.org/charts/fonts.html 2 Version history Version 0.1 (2013-01-31) § Initial version. Version 0.2 (2013-02-08) § Added Tick barline. § Changed names of time signature, tuplet and figured bass digit glyphs to ensure that they are unique. § Add upside-down and reversed G, F and C clefs for canzicrans and inverted canons. § Added Time signature + and Time signature fraction slash glyphs. § Added Black diamond notehead, White diamond notehead, Half-filled diamond notehead, Black circled notehead, White circled notehead glyphs. § Added 256th and 512th note glyphs. § All symbols shown on combining stems now also exist as separate symbols. § Added reversed sharp, natural, double flat and inverted flat and double flat glyphs for canzicrans and inverted canons. § Added trill wiggle segment, glissando wiggle segment and arpeggiato wiggle segment glyphs. § Added string Half-harmonic, Overpressure down bow and Overpressure up bow glyphs. § Added Breath mark glyph. § Added angled beater pictograms for xylophone, timpani and yarn beaters. § Added alternative glyph for Half-open, per Weinberg. -
Standard Music Font Layout
SMuFL Standard Music Font Layout Version 0.5 (2013-07-12) Copyright © 2013 Steinberg Media Technologies GmbH Acknowledgements This document reproduces glyphs from the Bravura font, copyright © Steinberg Media Technologies GmbH. Bravura is released under the SIL Open Font License and can be downloaded from http://www.smufl.org/fonts This document also reproduces glyphs from the Sagittal font, copyright © George Secor and David Keenan. Sagittal is released under the SIL Open Font License and can be downloaded from http://sagittal.org This document also currently reproduces some glyphs from the Unicode 6.2 code chart for the Musical Symbols range (http://www.unicode.org/charts/PDF/U1D100.pdf). These glyphs are the copyright of their respective copyright holders, listed on the Unicode Consortium web site here: http://www.unicode.org/charts/fonts.html 2 Version history Version 0.1 (2013-01-31) § Initial version. Version 0.2 (2013-02-08) § Added Tick barline (U+E036). § Changed names of time signature, tuplet and figured bass digit glyphs to ensure that they are unique. § Add upside-down and reversed G, F and C clefs for canzicrans and inverted canons (U+E074–U+E078). § Added Time signature + (U+E08C) and Time signature fraction slash (U+E08D) glyphs. § Added Black diamond notehead (U+E0BC), White diamond notehead (U+E0BD), Half-filled diamond notehead (U+E0BE), Black circled notehead (U+E0BF), White circled notehead (U+E0C0) glyphs. § Added 256th and 512th note glyphs (U+E110–U+E113). § All symbols shown on combining stems now also exist as separate symbols. § Added reversed sharp, natural, double flat and inverted flat and double flat glyphs (U+E172–U+E176) for canzicrans and inverted canons. -
Pietro Aaron on Musica Plana: a Translation and Commentary on Book I of the Libri Tres De Institutione Harmonica (1516)
Pietro Aaron on musica plana: A Translation and Commentary on Book I of the Libri tres de institutione harmonica (1516) Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Matthew Joseph Bester, B.A., M.A. Graduate Program in Music The Ohio State University 2013 Dissertation Committee: Graeme M. Boone, Advisor Charles Atkinson Burdette Green Copyright by Matthew Joseph Bester 2013 Abstract Historians of music theory long have recognized the importance of the sixteenth- century Florentine theorist Pietro Aaron for his influential vernacular treatises on practical matters concerning polyphony, most notably his Toscanello in musica (Venice, 1523) and his Trattato della natura et cognitione de tutti gli tuoni di canto figurato (Venice, 1525). Less often discussed is Aaron’s treatment of plainsong, the most complete statement of which occurs in the opening book of his first published treatise, the Libri tres de institutione harmonica (Bologna, 1516). The present dissertation aims to assess and contextualize Aaron’s perspective on the subject with a translation and commentary on the first book of the De institutione harmonica. The extensive commentary endeavors to situate Aaron’s treatment of plainsong more concretely within the history of music theory, with particular focus on some of the most prominent treatises that were circulating in the decades prior to the publication of the De institutione harmonica. This includes works by such well-known theorists as Marchetto da Padova, Johannes Tinctoris, and Franchinus Gaffurius, but equally significant are certain lesser-known practical works on the topic of plainsong from around the turn of the century, some of which are in the vernacular Italian, including Bonaventura da Brescia’s Breviloquium musicale (1497), the anonymous Compendium musices (1499), and the anonymous Quaestiones et solutiones (c.1500). -
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. -
Tuning Continua and Keyboard Layouts
October 4, 2007 8:34 Journal of Mathematics and Music TuningContinua-RevisedVersion Journal of Mathematics and Music Vol. 00, No. 00, March 2007, 1–15 Tuning Continua and Keyboard Layouts ANDREW MILNE∗ , WILLIAM SETHARES , and JAMES PLAMONDON † ‡ § Department of Music, P.O. Box 35, FIN-40014, University of Jyvaskyl¨ a,¨ Finland. Email: [email protected] † 1415 Engineering Drive, Dept Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706 USA. ‡ Phone: 608-262-5669. Email: [email protected] CEO, Thumtronics Inc., 6911 Thistle Hill Way, Austin, TX 78754 USA. Phone: 512-363-7094. Email: § [email protected] (received April 2007) Previous work has demonstrated the existence of keyboard layouts capable of maintaining consistent fingerings across a parameterised family of tunings. This paper describes the general principles underlying layouts that are invariant in both transposition and tuning. Straightforward computational methods for determining appropriate bases for a regular temperament are given in terms of a row-reduced matrix for the temperament-mapping. A concrete description of the range over which consistent fingering can be maintained is described by the valid tuning range. Measures of the resulting keyboard layouts allowdirect comparison of the ease with which various chordal and scalic patterns can be fingered as a function of the keyboard geometry. A number of concrete examples illustrate the generality of the methods and their applicability to a wide variety of commas and temperaments, tuning continua, and keyboard layouts. Keywords: Temperament; Comma; Layout; Button-Lattice; Swathe; Isotone 2000 Mathematics Subject Classification: 15A03; 15A04 1 Introduction Some alternative keyboard designs have the property that any given interval is fingered the same in all keys. -
Encyclopedia of the History of Arabic Science. Volume 2, Mathematics
Encyclopedia of the History of Arabic Science Encyclopedia of the History of Arabic Science Volume 2 Edited by ROSHDI RASHED in collaboration with RÉGIS MORELON LONDON AND NEW YORK First published in 1996 by Routledge 11 New Fetter Lane, London EC4P 4EE 29 West 35th Street, New York, NY 10001 This edition published in the Taylor & Francis e-Library, 2009. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. Structure and editorial matter © 1996 Routledge The chapters © 1996 Routledge All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloguing-in-Publication Data A catalogue record for this book is available on request. ISBN 0-203-40360-6 Master e-book ISBN ISBN 0-203-71184-X (Adobe ebook Reader Format) ISBN 0-415-12411-5 (Print Edition) 3 volume set ISBN 0-415-02063-8 Contents VOLUME 1 Contents v Preface ix 1 General survey of Arabic astronomy Régis Morelon 1 2 Eastern Arabic astronomy between the eighth and the eleventh centuries 20 Régis Morelon 3 Arabic planetary theories after the eleventh century AD 58 George Saliba 4 Astronomy and -
On the Bimodular Approximation and Equal Temperaments
On the Bimodular Approximation and Equal Temperaments Martin Gough DRAFT June 8 2014 Abstract The bimodular approximation, which has been known for over 300 years as an accurate means of computing the relative sizes of small (sub-semitone) musical intervals, is shown to provide a remarkably good approximation to the sizes of larger intervals, up to and beyond the size of the octave. If just intervals are approximated by their bimodular approximants (rational numbers defined as frequency difference divided by frequency sum) the ratios between those intervals are also rational, and under certain simply stated conditions can be shown to coincide with the integer ratios which equal temperament imposes on the same intervals. This observation provides a simple explanation for the observed accuracy of certain equal divisions of the octave including 12edo, as well as non-octave equal temperaments such as the fifth-based temperaments of Wendy Carlos. Graphical presentations of the theory provide further insights. The errors in the bimodular approximation can be expressed as bimodular commas, among which are many commas featuring in established temperament theory. Introduction Just musical intervals are characterised by small-integer ratios between frequencies. Equal temperaments, by contrast, are characterised by small-integer ratios between intervals. Since interval is the logarithm of frequency ratio, it follows that an equal temperament which accurately represents just intervals embodies a set of rational approximations to the logarithms (to some suitable base) of simple rational numbers. This paper explores the connection between these rational logarithmic approximations and those obtained from a long-established rational approximation to the logarithm function – the bimodular approximation. -
Sagittal a Microtonal Notation System
Sagittal A Microtonal Notation System by George D. Secor and David C. Keenan Introduction George Secor began development of the Sagittal (pronounced “SAJ-i-tl”) notation system in August 2001. In January 2002 he presented what he had developed to the Yahoo group tuning and offered to consider suggestions for improvements. At that stage the system could notate the equal temperaments with 17, 19, 22, 29, 31, 41, and 72 divisions per octave. Little did he know that he had provided a unifying symbolic principle which would ultimately be developed into a system capable of notating almost any conceivable microtonal tuning. In the early stages of the discussion, one idea that surfaced repeatedly was that a set of symbols indicating prime-number commatic alterations to tones in a Pythagorean series might be used to notate both rational intervals (e.g. just intonation) and equal divisions of the octave. Similar ideas had previously been proposed by Hermann Helmholtz, Alexander Ellis, Carl Eitz, Paul Rapoport, Daniel Wolf, Joe Monzo, and others1, but these had never been implemented in a performance notation to such an extent as was being discussed. As a first step, Gene Ward Smith suggested that it would be desirable that there be no more than one comma symbol per prime number, and a selection of 19-limit commas was tentatively identified to define the semantics of the notation. David Keenan enthusiastically observed that these commas came in an assortment of sizes that made the objective theoretically feasible, but he also realized that devising a set of distinct and recognizable symbols suitable for a performance notation was not going to be a simple matter. -
La Gamme De Pythagore Est Fondée Sur Deux Intervalles Géné- Rateurs : La Quinte Et L’Octave Et Ils Sont Incommensurables
Jean-Louis MIGEOT Membre de l’Académie Classe Technologie et Société DES CHIFFRES ET DES NOTES QUAND LA SCIENCE PARLE À LA MUSIQUE ACADÉMIE ROYe ALE des Sciences, des Lettres et des Beaux-Arts DE BELGIQUE DES CHIFFRES ET DES NOTES COLLECTION TRANSVERSALES 1. Jean-Louis MIGEOT, Des chiffres et des notes. Quand la science parle à la musique, 2015. 2. Hugues BERSINI, Quand l’informatique réinvente la sociologie ! (à paraître) JEAN-LOUIS MIGEOT Membre de l’Académie, Classe Technologie et Société DES CHIFFRES ET DES NOTES QUAND LA SCIENCE PARLE À LA MUSIQUE Académie royale de Belgique Crédits Rue Ducale, 1 © Jean-Louis Migeot, pour le texte et les figures 1000 Bruxelles, Belgique (sauf mentions contraires) [email protected] www.academie-editions.be Suivi et couverture : Loredana Buscemi et Grégory Van Aelbrouck, Transversales Académie royale de Belgique Classe Technologie et Société Volume 1 Impression : nº 2108 IMP Printing s.a., 1083 Ganshoren © 2015, Académie royale de Belgique ISBN 978-2-8031-0508-3 Dépôt légal : 2015/0092/25 Ce livre est dédié… À ceux qui savent tout ce que je leur dois. À ceux qui ont tort de douter de ce qu’ils m’ont apporté. À ceux qui ne savent pas encore que je ne suis rien sans eux. À ceux à qui il est trop tard pour exprimer ma reconnaissance. Préface « Sans la musique, la vie serait une erreur », dit Nietzsche… et qui lui donnerait tort ? Des berceuses psalmodiées par nos parents dès notre naissance aux orgues majestueuses qui scelleront peut-être notre passage sur terre, des musiques endiablées sur lesquelles nous dansons aux accords subtils des plus belles sonates écoutées dans le recueillement d’une salle de concert, des bribes de mélodie entendues à la radio au chant de notre voisine sous la douche, de l’intensité d’un récital à l’inattentive perception du décor sonore d’un restaurant, chaque instant de notre vie, du plus solennel au plus anodin, est baigné de musique. -
Cell Phones Silent Clickers on Physics 1240 Lecture 14
This is PHYS 1240 - Sound and Music Lecture 12 Professor Patricia Rankin TA: Tyler McMaken Cell Phones silent Clickers on Physics 1240 Lecture 14 Today: Scales, Tutorial Next time: Review for midterm physicscourses.colorado.edu/phys1240 Canvas Site: assignments, administration, grades Homework – HW7 Not due till Wed March 11th 5pm Homelabs – Hlab4 Not due till March 16th HW 6 review open-open pipe closed-open pipe 3. (avg score: 30%) What would happen to the frequency of the 푛 = 1: second mode (the next member of the harmonic series after the fundamental) of an open-open pipe 푛 = 2: if a cap was placed on one end? 푛 = 3: A) It would increase by a factor of 2 B) It would decrease by a factor of 2 푛 = 4: C) It would decrease by a factor of 3 D) It would stay the same E) It would change by some other 푛 = 5: factor 푣푠 푣 푓 = 푛 ∙ 푓 = 푛 ∙ 푠 푛 2퐿 푛 4퐿 Review • Consonance: tones have whole number frequency ratios • Dissonance: harsh sound when 2 tones (or upper harmonics) produce beats within the same critical band • Harmonic series → Pythagorean intervals perfect perfect major minor major minor octave fifth fourth third third second second (2/1) (3/2) (4/3) (5/4) (6/5) (9/8) (16/15) ………… Questions: 1) Why does a piano have 12 notes in each octave? 2) How do we tune those 12 notes (how do we decide what frequencies to assign to each note)? Pythagoras of Samos • 500s BCE • Founded school of numerology • Music of the spheres • Pythagorean Hypothesis: Consonant musical intervals are related to low integer ratios of frequencies Clicker 14.1 Two monochords are plucked to produce sound.